Chapter 1
Introduction
The Nature of Derivatives
A derivative is an instrument whose value depends on the values of other more basic underlying variables
Examples of Derivatives
Swaps
Options
Forward Contracts
Futures Contracts
Derivatives Markets
Exchange Traded
standard products
trading floor or computer trading
virtually no credit risk
Over-the-Counter
non-standard products
telephone market
some credit risk
Ways Derivatives are Used
To hedge risks
To reflect a view on the future direction of the market
To lock in an arbitrage profit
To change the nature of a liability
To change the nature of an investment without incurring the costs of selling one portfolio and buying another
Forward Contracts
A forward contract is an agreement to buy or sell an asset at a certain time in the future for a certain price (the delivery price)
It can be contrasted with a spot contract which is an agreement to buy or sell immediately
How a Forward Contract Works
The contract is an over-the-counter (OTC) agreement between 2 companies
The delivery price is usually chosen so that the initial value of the contract is zero
No money changes hands when contract is first negotiated and it is settled at maturity
The Forward Price
The forward price for a contract is the delivery price that would be applicable to the contract if were negotiated today (., it is the delivery price that would make the contract worth exactly zero)
The forward price may be different for contracts of different maturities
远期价格
我们把使得远期合约价值为零的交割价格称为远期价格（Forward Price）。
这个远期价格显然是理论价格，它与远期合约在实际交易中形成的实际价格（即双方签约时所确定的交割价格）并一定相等。
一旦理论价格与实际价格不相等，就会出现套利（Arbitrage）机会。
远期价格与远期价值的区别
远期价格指的是远期合约中标的物的远期价格，它是跟标的物的现货价格紧密相联的.
远期价值则是指远期合约本身的价值，它是由远期实际价格与远期理论价格的差距决定的。
远期合约的由来和优缺点
远期合约是适应规避现货交易风险的需要而产生的。
远期合约是非标准化合约,灵活性较大.
缺点：效率较低,流动性较差,违约风险较高。
Terminology
The party that has agreed to buy has what is termed a long position
The party that has agreed to sell has what is termed a short position
Example (page 3)
On January 20, 1998 a trader enters into an agreement to buy £1 million in three months at an exchange rate of
This obligates the trader to pay $1,619,600 for £1 million on April 20, 1998
What are the possible outcomes?
Profit from a
Long Forward Position
profit
K
Price of Underlying
at Maturity, ST
Profit from a
Short Forward Position
ST
K
Futures Contracts
Agreement to buy or sell an asset for a certain price at a certain time
Similar to forward contract
Whereas a forward contract is traded OTC a futures contract is traded on an exchange
1. Gold: An Arbitrage Opportunity?
Suppose that:
The spot price of gold is US$300
The 1-year forward price of gold is US$340
The 1-year US$ interest rate is 5% per annum
Is there an arbitrage opportunity?
2. Gold: Another Arbitrage Opportunity?
Suppose that:
The spot price of gold is US$300
The 1-year forward price of gold is US$300
The 1-year US$ interest rate is 5% per annum
Is there an arbitrage opportunity?
The Forward Price of Gold
If the spot price of gold is S , the forward price for a contract deliverable in T years is F, then
F = S (1+r )T
where r is the 1-year (domestic currency) risk-free rate of interest.
In our examples, S=300, T=1, and r= so that
F = 300(1+) = 315
1. Oil: An Arbitrage Opportunity?
Suppose that:
The spot price of oil is US$19
The quoted 1-year futures price of oil is US$25
The 1-year US$ interest rate is 5% per annum
The storage costs of oil are 2% per annum
Is there an arbitrage opportunity?
2. Oil: Another Arbitrage Opportunity?
Suppose that:
The spot price of oil is US$19
The quoted 1-year futures price of oil is US$16
The 1-year US$ interest rate is 5% per annum
The storage costs of oil are 2% per annum
Is there an arbitrage opportunity?
Exchanges Trading Futures
Chicago Board of Trade
Chicago Mercantile Exchange
BM&F (Sao Paulo, Brazil)
LIFFE (London)
TIFFE (Tokyo)
and many more (see list at end of book)
Options
A call option is an option to buy a certain asset by a certain date for a certain price (the strike price)
A put is an option to sell a certain asset by a certain date for a certain price (the strike price)
Long Call on IBM (Figure , Page 5)
Profit from buying an IBM European call option: option price = $5, strike price = $100, option life = 2 months
30
20
10
0
-5
70
80
90
100
110
120
130
Profit ($)
Terminal
stock price ($)
Short Call on IBM (Figure , page 9)
Profit from writing an IBM European call option: option price = $5, strike price = $100, option life = 2 months
-30
-20
-10
0
5
70
80
90
100
110
120
130
Profit ($)
Terminal
stock price ($)
Long Put on Exxon (Figure , page 6)
Profit from buying an Exxon European put option: option price = $7, strike price = $70, option life = 3 mths
30
20
10
0
-7
70
60
50
40
80
90
100
Profit ($)
Terminal
stock price ($)
Short Put on Exxon (Figure , page 7)
Profit from writing an Exxon European put option: option price = $7, strike price = $70, option life = 3 mths
-30
-20
-10
7
0
70
60
50
40
80
90
100
Profit ($)
Terminal
stock price ($)
Payoffs from Options
What is the Option Position in Each Case?
X = Strike price, ST = Price of asset at maturity
Payoff
Payoff
ST
ST
X
X
Payoff
Payoff
ST
ST
X
X
Types of Traders
Hedgers
Speculators
Arbitrageurs
Some of the large trading losses in derivatives occurred because individuals who had a mandate to hedge risks switched to being speculators
Hedging Examples (pages 10)
A US company will pay £1 million for imports from Britain in 3 months and decides to hedge using a long position in a forward contract
An investor owns 500 IBM shares currently worth $102 per share. A two- month put with a strike price of $100 costs $4. The investor decides to hedge by buying 5 contracts
Speculation Example
An investor with $7,800 to invest feels that Exxon’s stock price will increase over the next 3 months. The current stock price is $78 and the price of a 3-month call option with a strike of 80 is $3
What are the alternative strategies?
Arbitrage Example (page 11)
A stock price is quoted as £100 in London and $172 in New York
The current exchange rate is
What is the arbitrage opportunity?
Exchanges Trading Options
Chicago Board Options Exchange
American Stock Exchange
Philadelphia Stock Exchange
Pacific Stock Exchange
European Options Exchange
Australian Options Market
and many more (see list at end of book)
Chapter 2
Futures Markets and
the Use of Futures for Hedging
Futures Contracts
Available on a wide range of underlyings
Exchange traded
Specifications need to be defined:
What can be delivered,
Where it can be delivered, &
When it can be delivered
Settled daily
Margins
A margin is cash or marketable securities deposited by an investor with his or her broker
The balance in the margin account is adjusted to reflect daily settlement
Margins minimize the possibility of a loss through a default on a contract
Example of a Futures Trade
An investor takes 2 long positions in 2 December gold futures contracts on June 3
contract size is 100 oz.
futures price is US$400
margin requirement is US$2,000/contract (US$4,000 in total)
maintenance margin is US$1,500/contract (US$3,000 in total)
A Possible Outcome
Table , Page 25
Daily
Cumulative
Margin
Futures
Gain
Gain
Account
Margin
Price
(Loss)
(Loss)
Balance
Call
Day
(US$)
(US$)
(US$)
(US$)
(US$)
4,000
3-Jun
(600)
(600)
3,400
0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
11-Jun
(420)
(1,340)
2,660
1,340
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
17-Jun
(1,140)
(2,600)
2,740
1,260
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
24-Jun
260
(1,540)
5,060
0
+
=
4,000
3,000
+
=
4,000
<
Other Key Points About Futures
They are settled daily
Closing out a futures position involves entering into an offsetting trade
Most contracts are closed out before maturity
Delivery
If a contract is not closed out before maturity, it usually settled by delivering the assets underlying the contract. When there are alternatives about what is delivered, where it is delivered, and when it is delivered, the party with the short position chooses.
A few contracts (for example, those on stock indices and Eurodollars) are settled in cash
Some Terminology
Open interest: the total number of contracts outstanding
equal to number of long positions or number of short positions
Settlement price: the price just before the final bell each day
used for the daily settlement process
Volume of trading: the number of trades in 1 day
Convergence of Futures to Spot (Figure , page 32)
Time
Time
(a)
(b)
Futures
Price
Futures
Price
Spot Price
Spot Price
Questions
When a new trade is completed what are the possible effects on the open interest?
Can the volume of trading in a day be greater than the open interest?
Regulation of Futures
Regulation is designed to protect the public interest
Regulators try to prevent questionable trading practices by either individuals on the floor of the exchange or outside groups
Accounting & Tax
If a contract is used for
Hedging: it is logical to recognize profits (losses) at the same time as on the item being hedged
Speculation: it is logical to recognize profits (losses) on a mark to market basis
Roughly speaking, this is what the treatment of futures in the many other countries attempts to achieve
Long & Short Hedges
A long futures hedge is appropriate when you know you will purchase an asset in the future & want to lock in the price
A short futures hedge is appropriate when you know you will sell an asset in the future & want to lock in the price
Basis Risk
Basis is the difference between spot & futures
Basis risk arises because of the uncertainty about the basis when the hedge is closed out
Long Hedge
Suppose that
F1 : Initial Futures Price
F2 : Final Futures Price
S2 : Final Asset Price
You hedge the future purchase of an asset by entering into a long futures contract
Cost of Asset=S2 -F2+F1 = F1 + Basis
Short Hedge
Suppose that
F1 : Initial Futures Price
F2 : Final Futures Price
S2 : Final Asset Price
You hedge the future sale of an asset by entering into a short futures contract
Price Realized=S2 -F2+F1 = F1 + Basis
Choice of Contract
Choose a delivery month that is as close as possible to, but later than, the end of the life of the hedge
When there is no futures contract on the asset being hedged, choose the contract whose futures price is most highly correlated with the asset price. There are then 2 components to basis:
F1+(S2*-F2)+(S2-S2*)
Optimal Hedge Ratio
Proportion of the exposure that should optimally be hedged is where
S : spot price,
F : futures price,
sS : standard deviation of DS ,
sF : standard deviation of DF
r: coefficient of correlation between DS & DF
Hedge Ratio
Short Hedge
Rolling The Hedge Forward
We can use a series of futures contracts to increase the life of a hedge
Each time we switch from 1 futures contract to another we incur a type of basis risk
期货合约与远期合约比较
标准化程度不同
交易场所不同
违约风险不同
价格确定方式不同
履约方式不同
合约双方关系不同
结算方式不同
期货市场的功能
转移价格风险的功能
价格发现功能
*
*
Chapter 3
Forward and Futures Prices
*
*
This Chapter Covers:
Relationships between forward/futures prices and the price of the underlying.
Forward price and futures price are very close to each other.
Distinguish between investment assets and consumption assets.
No arbitrage pricing methods
*
*
Compounding Frequency
The compounding frequency used for an interest rate is the unit of measurement
The difference between quarterly and annual compounding is analogous to the difference between miles and kilometers
*
*
Terminal Values
Compounded once per annum:
A(1+R)n
Compounded m times per annum:
A(1+R/m)mn
Compounded continuously:
*
*
Continuous Compounding
Compounded continuously is almost equal to compounded per day.
$100 grows to $100eRT when invested at a continuously compounded rate R for time T
$100 received at time T discounts to $100e-RT at time zero when the continuously compounded discount rate is R
*
*
Conversion Formulas
Define
Rc : continuously compounded rate
Rm: same rate with compounding m times per year
*
*
Conversion Formulas
When m=1
Rc=ln(St/St-1)
because: Rm=(St-St-1)/St-1=St/St-1-1
1+Rm=St/St-1
*
*
Short Selling
Short selling involves selling securities you do not own
Your broker borrows the securities from another client and sells them in the market in the usual way
*
*
Short Selling
(continued)
At some stage you must buy the securities back so they can be replaced in the account of the client
You must pay dividends & other benefits the owner of the securities receives
*
*
Assumptions
No transaction costs
Same tax rate
Borrow or lend at the same risk-free rate of interest
No arbitrage opportunities
*
*
Gold Example
For the gold example in chapter 1,
F0 = S0(1 + r )T
(assuming no storage costs)
If r is compounded continuously instead of annually
F0 = S0erT
*
*
Extension of the Gold Example
For any investment asset that provides no income and has no storage costs
F0 = S0erT
*
*
When an Investment Asset Provides a Known Dollar Income
F0 = (S0 – I )erT
where I is the present value of the income
*
*
When an Investment Asset Provides a Known Dividend Yield
F0 = S0 e(r–q )T
where q is the average dividend yield during the life of the contract
*
*
Valuing a Forward Contract
Suppose that
K is delivery price in a forward contract &
F0 is forward price that would apply to the contract today
The value of a long forward contract, ƒ, is ƒ = (F0 – K )e–rT
Similarly, the value of a short forward contract is
(K – F0 )e–rT
*
*
Forward vs Futures Prices
Forward and futures prices are usually assumed to be the same. When interest rates are uncertain they are, in theory, slightly different:
A strong positive correlation between interest rates and the asset price implies the futures price is slightly higher than the forward price
A strong negative correlation implies the reverse
*
*
Stock Index
Can be viewed as an investment asset paying a continuous dividend yield
The futures price & spot price relationship is therefore
F0 = S0 e(r–q )T
where q is the dividend yield on the portfolio represented by the index
*
*
Stock Index
(continued)
For the formula to be true it is important that the index represent an investment asset
In other words, changes in the index must correspond to changes in the value of a tradable portfolio
The Nikkei index viewed as a dollar number does not represent an investment asset
*
*
Index Arbitrage
When F0>S0e(r-q)T an arbitrageur buys the stocks underlying the index and sells futures
When F0<S0e(r-q)T an arbitrageur buys futures and shorts or sells the stocks underlying the index
*
*
Index Arbitrage
(continued)
Index arbitrage involves simultaneous trades in futures & many different stocks
Very often a computer is used to generate the trades
Occasionally (., on Black Monday) simultaneous trades are not possible and the theoretical no-arbitrage relationship between F0 and S0 may not hold
*
*
Hedging Using Index Futures
To hedge the risk in a portfolio the number of contracts that should be shorted is
βP/A
where P is the value of the portfolio, b is its beta, and A is the value of the assets underlying one futures contract
*
*
Changing Beta
What position in index futures is appropriate to change the beta of a portfolio from b to b*
(β*-β)P/A
negative result means short position;
Positive result means long position.
*
*
Futures and Forwards on Currencies
A foreign currency is analogous to a security providing a continuous dividend yield
The continuous dividend yield is the foreign risk-free interest rate
It follows that if rf is the foreign risk-free interest rate
F0=S0e(r-rf)T
*
*
Futures on Consumption Assets
F0 £ S0 e(r+u )T
where u is the storage cost per unit time as a percent of the asset value.
Alternatively,
F0 £ (S0+U )erT
where U is the present value of the storage costs.
*
*
The Cost of Carry (Page 73)
The cost of carry, c , is the storage cost plus the interest costs less the income earned
For an investment asset F0 = S0ecT
For a consumption asset F0 £ S0ecT
The convenience yield on the consumption asset, y , is defined so that F0 = S0 e(c–y )T
*
*
Futures Prices & Expected Future Spot Prices
Suppose k is the expected return required by investors on an asset, If the underlying asset has no income,
E(ST)=S0ekT
F0=S0erT
So, F0=E(ST)e(r-k)T
The same result will be got for the underlying asset paying fixed income or fixed rate income.
*
*
Futures Prices & Future Spot Prices (continued)
If the asset has
no systematic risk, then k = r and F0 is an unbiased estimate of ST
positive systematic risk, then
k > r and F0 < E (ST )
negative systematic risk, then
k < r and F0 > E (ST )
*
*
Relationship between F and S
F,S
FT=E(ST)
F0
S0, (S0-I),
or Se-qT
Time premium
Time
Risk premium
Chapter 4
Interest Rates and Duration
Types of Rates
Treasury rates
LIBOR rates
Repo rates
Zero Rates
A zero rate (or spot rate), for maturity T, is the rate of interest earned on an investment that provides a payoff only at time T
Example
Bond Pricing
To calculate the cash price of a bond we discount each cash flow at the appropriate zero rate
In our example, the theoretical price of a two-year bond providing a 6% coupon semiannually is
Bond Yield(到期收益率)
The bond yield is the discount rate that makes the present value of the cash flows on the bond equal to the market price of the bond
Suppose that the market price of the bond in our example equals its theoretical price of
The bond yield is given by solving
to get y= or %.
问题:请问投资者按市价购买该债券并持有到期,其实际收益率等于多少?
Par Yield(平价收益率)
The par yield for a certain maturity is the coupon rate that causes the bond price to equal its face value.
In our example we solve
Par Yield continued
In general if m is the number of coupon payments per year, P is the present value of $1 received at maturity and A is the present value of an annuity of $1 on each coupon date,then
问题
假设你是财政部国债司司长,你的目标是使国债发行实际收入尽量等于计划收入,你应如何确定国债的票面利率?
Determining Zero Rate
Sample Data
Bond
Time to
Annual
Bond
Principal
Maturity
Coupon
Price
(dollars)
(years)
(dollars)
(dollars)
100
0
100
0
100
0
100
8
100
12
100
10
The Bootstrap Method
An amount can be earned on during 3 months.
The 3-month rate is 4 times or % with quarterly compounding
This is % with continuous compounding
Similarly the 6 month and 1 year rates are % and % with continuous compounding
The Bootstrap Method continued
To calculate the year rate we solve
to get R = or %
Similarly the two-year rate is %
The Bootstrap Method continued
The cash flows of the sixth bond are:
3 months later $5
9 months later $5
years later $5
years later $5
years later $5
years later $105
The Bootstrap Method continued
Using interpolation Method(线性插值法)to find the other zero rates:
9 months %(=(%+%)/2)
%
%
So the present value of the first four cash flows is:
The Bootstrap Method cont.
The present value of the last two cash flows is:
=
Let the -year zero rate is R,using interpolation method we can find the -year zero rate:
.1081×2/3+R/3=.0721+R/3
So we have:
Solved by trial and error:
R=
Zero Curve Calculated from the Data
Forward Rates
The forward rate is the future zero rate implied by today’s term structure of interest rates
Calculation of Forward Rates
Zero Rate for
Forward Rate
an
n
-year Investment
for
n
th Year
Year (
n
)
(% per annum)
(% per annum)
1
2
3
4
5
Formula for Forward Rates
Suppose that the zero rates for time periods T1 and T2 are R1 and R2 with both rates continuously compounded.
The forward rate for the period between times T1 and T2 is
Instantaneous Forward Rate
The instantaneous forward rate(RF) for a maturity T is the forward rate that applies for a very short time period starting at T. It is
where R is the T-year rate
Because
RF=R2+(R2-R1)T1/(T2-T1)
Upward vs Downward Sloping
Yield Curve
For an upward sloping yield curve:
Fwd Rate > Zero Rate > Par Yield
For a downward sloping yield curve
Par Yield > Zero Rate > Fwd Rate
Forward Rate, Zero Rate and Par Yield
Interest Rate
Par Yield
Zero Rate
Forward Rate
T
Theories of the Term Structure
Expectations Theory: forward rates equal expected future zero rates
Market Segmentation: short, medium and long rates determined independently of each other
Liquidity Preference Theory: forward rates higher than expected future zero rates
远期利率协议
远期利率协议是空方承诺在未来的某个时刻（T时刻）将一定数额的名义本金（A）按约定的合同利率（rK）在一定的期限（T*-T）贷给多方的远期协议 。
远期利率协议的定价
多方（即借入名义本金的一方）的现金流为：
T时刻：A
T*时刻：
这些现金流的现值即为远期利率协议多头的价值。
远期利率协议的定价（2）
为此，我们要先将T*时刻的现金流用T*-T期限的远期利率贴现到T时刻，再贴现到现在时刻t。
Day Count Conventions
in the .
Treasury Bonds: Actual/Actual (in period)
Corporate Bonds:30/360
Money Market Instruments: Actual/360
Treasury Bond Price Quotes
in the
Cash price = Quoted price +
Accrued Interest
Treasury Bill Quote in the .
If Y is the cash price of a Treasury bill that has n days to maturity, the quoted price is:
It’s called discount rate(折扣率). It’s the annualized dollar return expressed as a percentage of the par value. It’s not the same as the rate of return investors really earned.
中长期国债期货的定价
中长期国债属附息票债券，属支付已知现金收益的证券，因此公式（）和（）适用于中长期国债期货的定价。只是由于其报价和交割制度的特殊性，使这些公式的运用较为复杂而已。
以下我们以美国芝加哥交易所的长期国债期货为例来说明其定价问题，其结论也适用于中期国债期货。
长期国债现货和期货的报价与现金价格的关系
长期国债期货的报价与现货一样，以美元和32分之一美元报出，所报价格是100美元面值债券的价格，由于合约规模为面值10万美元，因此90—25的报价意味着面值10万美元的报价是90，美元。
应该注意的是，报价（净价）与购买者所支付的现金价格（全价）是不同的。全价（即期货价格）与净价的关系为：
全价=净价+上一个付息日以来的累计利息
交割券与标准券的转换因子
芝加哥交易所规定，空头方可以选择期限长于15年且在15年内不可赎回的任何国债用于交割。由于各种债券息票率不同，期限也不同，因此芝加哥交易所规定交割的标准券为期限15年、息票率为8%的国债，其它券种均得按一定的比例折算成标准券。这个比例称为转换因子（Conversion Factor ）。
转换因子等于面值为100美元的各债券的现金流按8%的年利率（每半年计复利一次）贴现到交割月第一天的价值，再扣掉该债券累计利息后的余额。在计算转换因子时，债券的剩余期限只取3个月的整数倍，多余的月份舍掉。如果取整数后，债券的剩余期限为半年的倍数，就假定下一次付息是在6个月之后，否则就假定在3个月后付息。转换因子由交易所计算并公布。
空方收到的现金
算出转换因子后，我们就可算出空方交割100美元面值的债券应收到的现金：
空方收到的现金=期货净价交割债券的转换因子+交割债券的累计利息
确定交割最合算的债券
由于转换因子制度固有的缺陷和市场定价的差异决定了用何种国债交割对于双方而言是有差别的，而空方可选择用于交割的国债多达30种左右，因此空方应选择最合算的国债用于交割。
交割最合算债券就是购买交割券的成本与空方收到的现金之差最小的那个债券。
交割差距=债券净价+累计利息—[（期货净价转换因子）+累计利息]
=债券净价—（期货净价转换因子）
国债期货价格的确定
由于国债期货的空方拥有交割时间选择权和交割券种选择权，因此要精确地计算国债期货的理论价格也是较困难的。但是，如果我们假定交割最合算的国债和交割日期是已知的，那么我们可以通过以下四个步骤来确定国债期货价格：
根据交割最合算的国债的净价，算出该交割券的全价。
根据交割券的全价算出交割券期货的理论全价。
根据交割券期货的全价算出交割券期货的净价。
将交割券期货的净价除以转换因子即为标准券期货净价，也是标准券期货的全价。
Eurodollar Futures
The variable underlying the 3-month Eurodollar futures is the 3-month Eurodollar interest rate(LIBOR) applicable to a 90-day period beginning on the 3rd Wed. of the delivery month.
If Z is the quoted price of a 3-month Eurodollar futures contract, the value of one contract is 10,000[(100-Z)](注意期货与现货报价的不同).
A change of one basis point or in a Eurodollar futures quote corresponds to a contract price change of $25
Eurodollar Futures continued
A Eurodollar futures contract is settled in cash
When it expires (on the third Wednesday of the delivery month) Z is set equal to 100－R(the 90 day Eurodollar interest rate) and all contracts are closed out
Forward Rates and Eurodollar Futures
Eurodollar futures contracts last out to 10 years
For Eurodollar futures lasting beyond two years we cannot assume that the forward rate equals the futures rate
Forward Rates and Eurodollar Futures continued
Duration
Duration of a bond that provides cash flow c i at time t i is
where B is its price & y is its yield (continuously compounded)
This leads to
Duration Cont.
It’s because:
Duration Continued
When the yield y is expressed with annual compounding, we have Macaulay Duration(DM):
Duration cont.
Duration Cont.
Let
D’ is called modified duration.
We have:
Duration Cont.
Generally,when y is expressed with a compounding frequency of m times per year, we have:
Duration Matching
This involves hedging against interest rate risk by matching the durations of assets and liabilities
It provides protection against small parallel shifts in the zero curve
Convexity
The convexity of a bond is defined as（第四版）
问题1
有些公司并不能确切知道支付外币的确切日期，这样它就希望与银行签订一种在一段时期中都可交割的远期合同。公司希望拥有选择确切的交割日期的权力以匹配它的现金流。如果把你自己放在银行经理的位置上，你会如何对客户想要的这个产品进行定价?
答案
银行在定价时可假定客户会选择对银行最不利的交割日期。我们可以很容易证明，如果外币利率高于本币利率，则拥有远期外币多头的客户会选择最早的交割日期，而拥有远期外币空头的客户则会选择最迟的交割日期。相反，如果外币利率低于本币利率，则拥有远期外币多头的客户会选择最迟的交割日期，而拥有远期外币空头的客户则会选择最早的交割日期。只要在合约有效期中，外币利率和本币利率的高低次序不变，上述分析就没问题，银行可按这个原则定价。
但是当外币利率和本币利率较为接近时，两者的高低次序就有可能发生变化。因此，客户选择交割日期的权力就有特别的价值。银行应考虑这个价值。
如果合约签订后，客户不会选择最有利的交割日期，则银行可以另赚一笔。
问题2
一家银行为其客户提供了两种贷款选择，一是按年利率11%（一年计一次复利）贷出现金，一是按年利率2%（一年计一次复利）货出黄金。黄金贷款用黄金计算，并需用黄金归还本息。假设市场无风险连续复利年利率为%。储存成本为每年%（连续复利）。请问哪种贷款利率较低？
答案
将上述贷款利率转换成连续复利年利率，则正常贷款为%,黄金贷款为％。
假设银行按S元/盎司买了1盎司黄金，按％的黄金利率贷给客户1年，同时卖出盎司1年远期黄金，根据黄金的储存成本和市场的无风险利率，我们可以算出黄金的1年远期价格为元/盎司。也就是说银行1年后可以收到+=元现金。可见黄金贷款的连续复利收益率为%。显然黄金贷款利率高于正常贷款。
Chapter 5
Swaps
Nature of Swaps
A swap is an agreement to exchange cash flows at specified future times according to certain specified rules
互换与掉期的区别
互换和掉期在英文中都叫Swap，因此很多人误把它们混为一谈。实际上，两者有很大区别。
合约与交易的区别
掉期是外汇市场上的一种交易方法，是指对不同期限，但金额相等的同种外汇作两笔反方向的交易，它并没有实质的合约，更不是一种衍生工具。而互换则有实质的合约，是一种重要的衍生工具。
有无专门市场不同
掉期在外汇市场上进行，它本身没有专门的市场。互换则在专门的互换市场上交易。
比较优势理论与互换原理
比较优势（Comparative Advantage）理论是英国著名经济学家大卫李嘉图（David Ricardo）提出的。
李嘉图的比较优势理论不仅适用于国际贸易，而且适用于所有的经济活动。只要存在比较优势，双方就可通过适当的分工和交换使双方共同获利。人类进步史，实际上就是利用比较优势进行分工和交换的历史。
互换是比较优势理论在金融领域最生动的运用。根据比较优势理论，只要满足以下两种条件，就可进行互换：双方对对方的资产或负债均有需求；双方在两种资产或负债上存在比较优势。
The Comparative Advantage Argument
Company A wants to borrow floating
Company B wants to borrow fixed
Fixed
Floating
Company A
%
6-month LIBOR + %
Company B
%
6-month LIBOR + %
合作收益
不合作:(LIBOR+%)+%
=LIBOR+%
合作:10%+(LIBOR+1%)=LIBOR+11%
合作的总收益:%
假设双方平分合作收益,则A的筹资成本应为(LIBOR+%)%=LIBOR+%, B的筹资成本应为%%=%
The Swap
A
B
LIBOR
LIBOR+1%
%
10%
Criticism of the Comparative Advantage Argument
The % and % rates available to A and B in fixed rate markets are 5-year rates
The LIBOR+% and LIBOR+1% rates available in the floating rate market are six-month rates
B’s fixed rate depends on the spread above LIBOR it borrows at in the future
Although A faces no market risk, he does face default risk.
金融互换的功能
通过金融互换可在全球各市场之间进行套利，从而一方面降低筹资者的融资成本或提高投资者的资产收益，另一方面促进全球金融市场的一体化。
利用金融互换，可以管理资产负债组合中的利率风险和汇率风险。
金融互换为表外业务，可以逃避外汇管制、利率管制及税收限制。
金融互换的种类
金融互换虽然历史较短，但品种创新却日新月异。除了传统的货币互换和利率互换外，一大批新的金融互换品种不断涌现。
An Example of a “Plain Vanilla” Interest Rate Swap
On March 1,1999,an agreement by “Company B” to receive 6-month LIBOR & pay a fixed rate of 5% per annum every 6 months for 3 years on a notional principal of $100 million
Next slide illustrates cash flows
Cash Flows to Company B
---------Millions of Dollars---------
LIBOR
FLOATING
FIXED
Net
Date
Rate
Cash Flow
Cash Flow
Cash Flow
, 1999
%
Sept. 1, 1999
%
+
–
–
, 2000
%
+
–
–
Sept. 1, 2000
%
+
–
+
, 2001
%
+
–
+
Sept. 1, 2001
%
+
–
+
, 2002
%
+(+100)
–(-100)
+
Typical Uses of an
Interest Rate Swap
Converting a liability from
fixed rate to floating rate
floating rate to fixed rate
Converting an investment from
fixed rate to floating rate
floating rate to fixed rate
A and B Transform a Liability
A
B
LIBOR
5%
LIBOR+%
%
Financial Institution is Involved
A
.
B
LIBOR
LIBOR
%
%
%
LIBOR+%
A and B Transform an Asset
A
B
LIBOR
5%
%
%
Financial Institution is Involved
A
.
B
LIBOR
LIBOR
%
%
%
%
Valuation of an Interest Rate Swap
Interest rate swaps can be valued as the difference between the value of a fixed-rate bond & the value of a floating-rate bond
Alternatively, they can be valued as a portfolio of forward rate agreements (FRAs)
Valuation in Terms of Bonds
The fixed rate bond is valued in the usual way
The floating rate bond is valued by noting that it is worth par immediately after the next payment date
Valuation in Terms of FRAs
Each exchange of payments in an interest rate swap is an FRA
The FRAs can be valued on the assumption that today’s forward rates are realized
An Example of a Currency Swap
An agreement to pay 11% on a sterling principal of £10,000,000 & receive 8% on a US$ principal of $15,000,000 every year for 5 years
Exchange of Principal
In an interest rate swap, the principal is not exchanged
In a currency swap the principal is exchanged at the beginning & the end of the swap
The Cash Flows
Dollars Pounds
Years
$
------millions------
0
–
+
1
+
–
2
+
–
3
+
–
4
+
–
5
+
£
Typical Uses of a
Currency Swap
Conversion from a liability in one currency to a liability in another currency
Conversion from an investment in one currency to an investment in another currency
Comparative Advantage Arguments for Currency Swaps
Company A wants to borrow AUD
Company B wants to borrow USD
USD
AUD
Company A
%
%
Company B
%
%
Valuation of Currency Swaps
Like interest rate swaps, currency swaps can be valued either as the difference between 2 bonds or as a portfolio of forward contracts
Swaps & Forwards
A swap can be regarded as a convenient way of packaging forward contracts
The “plain vanilla” interest rate swap in our example consisted of 6 FRAs
The “fixed for fixed” currency swap in our example consisted of a cash transaction & 5 forward contracts
Swaps & Forwards
(continued)
The value of the swap is the sum of the values of the forward contracts underlying the swap
Swaps are normally “at the money” initially
This means that it costs NOTHING to enter into a swap
It does NOT mean that each forward contract underlying a swap is “at the money” initially
Credit Risk
A swap is worth zero to a company initially
At a future time its value is liable to be either positive or negative
The company has credit risk exposure only when its value is positive
其他种类的互换
交叉货币利率互换。交叉货币利率互换（Cross—Currency Interest Rate Swaps）是利率互换和货币互换的结合，它是以一种货币的固定利率交换另一种货币的浮动汇率。
增长型互换、减少型互换和滑道型互换。在标准的互换中，名义本金是不变的，而在这三种互换中，名义本金是可变的。其中增长型互换（Accreting Swaps）的名义本金在开始时较小，尔后随着时间的推移逐渐增大。减少型互换（Amortizing Swaps）则正好相反，其名义本金随时间的推移逐渐变小。近年来，互换市场又出现了一种特殊的减少型互换，即指数化本金互换（Indexed Principal Swaps），某名义本金的减少幅度取决于利率水平，利率越低，名义本金减少幅度越大。滑道型互换（Roller－Coaster Swaps）的名义本金则在互换期内时而增大，时而变小。
基点互换。在普通的利率互换中，互换一方是固定利率，另一方是浮动利率。而在基点互换（Basis Swaps）中，双方都是浮动利率，只是两种浮动利率的参照利率不同，如一方为LIBOR，另一方为基准利率。
可延长互换和可赎回互换。在标准的互换中，期限是固定的。而可延长互换（Extendable Swaps）的一方有权在一定限度内延长互换期限。可赎回互换（Puttable Swaps）的一方则有权提前中止互换。
零息互换。零息互换（Zero—Coupon Swaps）是指固定利息的多次支付流量被一次性的支付所取代，该一次性支付可以在互换期初也可在期未。
其他种类的互换（继续）
后期确定互换。在普通涉及到浮动利率的互换中，每次浮动利率都是在该计息期开始之前确定的。后期确定互换（Back—Set Swaps）的浮动利率则是在每次计息期结束之后确定的。
差额互换。差额互换（Differential Swaps）是对两种货币的浮动利率的现金流量进行交换，只是两种利息现金流量均按同种货币的相同名义本金计算。如互换一方按6月期美元的LIBOR对1000美元的名义本金支付利息，另一方按6月期德国马克的LIBOR减去%的浮动利率对1000万美元的名义本金支付以美元表示的利息。
远期互换。远期互换（Forward Swaps）是指互换生效日是在未来某一确定时间开始的互换。
互换期权。互换期权（Swaption）从本质上属于期权而不是互换，该期权的标的物为互换。例如，利率互换期权本质上是把固定利率交换为浮动利率，或把浮动利率交换为固定利率的权利。但许多机构在统计时都把互换期权列入互换的范围。
股票互换。股票互换（Equity Swaps）是以股票指数产生的红利和资本利得与固定利率或浮动利率交换。投资组合管理者可以用股票互换把债券投资转换成股票投资，反之亦然。
Chapter 6
Options Markets
Assets Underlying
Exchange-Traded Options
Stocks
Foreign Currency
Stock Indices
Futures
Specification of
Exchange-Traded Options
Expiration date
Strike price
European or American
Call or Put (option class)
Terminology
Moneyness :
At-the-money option(平价期权)
In-the-money option(实值期权)
Out-of-the-money option(虚值期权)
Terminology
(continued)
Option class(all options of the same type(call or put)
Option series(a particular contract that is traded,that is all the options of a given class with the same expiration date and strike price,.,The IBM 110 January calls are an option series.)
Intrinsic value
Time value
Dividends & Stock Splits
Suppose you own N options with a strike price of X :
No adjustments are made to the option terms for cash dividends
When there is an n-for-m stock split,
the strike price is reduced to mX/n
the no. of options is increased to nN/m
Stock dividends are handled in a manner similar to stock splits
Dividends & Stock Splits
(continued)
Consider a call option to buy 100 shares for $20/share
How should terms be adjusted:
for a 2-for-1 stock split?
for a 5% stock dividend?
Organization of Trading
Types of traders:
Market makers
Floor brokers
Alternative systems for limit orders
Order book officials(指令登记员)
Specialists
Margins
Margins are required when options are sold
When a naked option is written the margin is the greater of:
A total of 100% of the proceeds of the sale plus 20% of the underlying share price less the amount (if any) by which the option is out of the money
A total of 100% of the proceeds of the sale plus 10% of the underlying share price
For other trading strategies there are special rules
Warrants
Warrants are options that are issued (or written) by a corporation or a financial institution
The number of warrants outstanding is determined by the size of the original issue & changes only when they are exercised or when they expire
Warrants
(continued)
Warrants are traded in the same way as stocks
The issuer settles up with the holder when a warrant is exercised
When call warrants are issued by a corporation on its own stock, exercise will lead to new stock being issued
Executive Stock Options
Option issued by a company to executives
When the option is exercised the company issues more stock
Usually at-the-money when issued
become vested after a period of time
Can’t be sold
Last as long as 10 or 15 years
Convertible Bonds
Convertible bonds are regular bonds that can be exchanged for equity at certain times in the future according to a predetermined exchange ratio
Convertible Bonds
(continued)
Very often a convertible is callable
The call provision is a way in which the issuer can force conversion at a time earlier than the holder might otherwise choose
Chapter 7
Properties of Stock Option Prices
Notation
c : European call option price
p : European put option price
C : American Call option price
P : American Put option price
S0 :Stock price today
ST :Stock price at time T
X : Strike price
T : Life of option
s: Volatility of stock price
D : Present value of dividends during option’s life
r : Risk-free rate for maturity T with cont comp
期权价格的特性
期权价格(或者说价值)等于期权的内在价值加上时间价值。
期权的内在价值
期权的内在价值（Intrinsic Value）是指多方行使期权时可以获得的收益的现值。
无收益资产欧式看涨期权的内在价值等于S0-Xe-rT.
有收益资产欧式看涨期权的内在价值等于S0-D- Xe-rT。
当标的资产市价低于协议价格时，期权多方是不会行使期权的，因此期权的内在价值应大于等于0。
期权的时间价值
期权的时间价值（Time Value）是指在期权有效期内标的资产价格波动为期权持有者带来收益的可能性所隐含的价值。显然，标的资产价格的波动率越高，期权的时间价值就越大。
此外，期权的时间价值还受期权内在价值的影响。以无收益资产看涨期权为例，当S=X e-rT时，期权的时间价值最大。当S-X e-rT的绝对值增大时，期权的时间价值是递减的，如下图所示。
Xe-rT
S0
时间价值
Question
假设A股票（无红利）的市价为元，A股票有两种看涨期权，其协议价格分别为X1=10元，X2=8元，它们的有效期都是1年，1年期无风险利率为10%（连续复利）。这两种期权的内在价值分别为0和元。那么这两种期权的时间价值谁高呢？
假设这两种期权的时间价值相等，都等于2元，则第一种期权的价格为2元，第二种期权的价格为元。那么让读者从中挑一种期权，你们愿意挑哪一种呢？
分析
为了比较这两种期权，我们假定1年后出现如下三种情况：
情况一：ST=14元。则期权持有者可从期权1中获利（）=元，可从期权2中获利（）=元。期权1获利金额等于期权2。
情况二：ST=10元。则期权1亏=元，期权2也亏-2=元。期权1亏损等于期权2。
情况三：ST=8元。则期权1亏=元，而期权2亏 =元。期权1亏损少于期权2。
由此可见，无论未来A股票价格是涨是跌还是平，期权1均优于或等于期权2。显然，期权1的时间价值应高于期权2。
另一种情形
我们再来比较如下两种期权。X1=10元，X3=12元。其它条件与上例相同。显然，期权1的内在价值为0，期权3的内在价值虽然也等于0，但S-X e-rT却等于元。通过同样的分析，我们也可以得出期权1 的时间价值应高于期权3的结论。综合这三种期权，我们就可以得出无收益资产看涨期权的时间价值在S=X e-rT点最大的结论。
Effect of Variables on Option Pricing (Table , page 141)
c
p
C
P
+
+
–
+
+
+
+
+
+
+
+
+
+
–
+
–
–
–
–
+
–
–
Variable
S0
X
T(无红利)
r(静态)
D
+
+
边际时间价值
在一般情况下（即剔除标的资产支付大量收益这一特殊情况），由于有效期越长，标的资产的风险就越大，空头亏损的风险也越大，因此即使是欧式期权，有效期越长，其期权价格也越高，即期权的边际时间价值（Marginal Time Value）为正值。
但随着时间的延长，期权时间价值的增幅是递减的,这边际时间价值递减规律。换个角度说，对于到期日确定的期权来说，在其它条件不变时，随着时间的流逝，其时间价值的减小是递增的。这意味着，当时间流逝同样长度，期限长的期权的时间价值减小幅度将小于期限短的期权时间价值的减小幅度。
无风险利率与期权价格(一)
首先我们可以从比较静态的角度考察，即比较不同利率水平下的两种均衡状态。如果状态1的无风险利率较高，则标的资产的预期收益率也应较高，这意味着对应于标的资产现在特定的市价（So），未来预期价格[E(ST)]较高。同时由于贴现率较高，未来同样预期盈利的现值就较低。这两种效应都将减少看跌期权的价值。但对于看涨期权来说，前者将使期权价格上升，而后者将使 期权价格下降。由于前者的效应大于后者，因此对应于较高的无风险利率，看涨期权的价格也较高。
无风险利率与期权价格(二)
其次我们可从动态的角度考察，即考察一个均衡被打破到另一个均衡的过程。在标的资产价格与利率呈负相关时（如股票、债券等），当无风险利率提高时，原有均衡被打破，为了使标的资产预期收益率提高，均衡过程通常是通过同时降低标的资产的期初价格和预期未来价格，只是前者的降幅更大来实现的。同时贴现率也随之上升。对于看涨期权来说，两种效应都将使期权价格下降，而对于看跌期权来说，前者效应为正，后者为负，由于前者效应通常大于后者，因此其净效应是看跌期权价格上升。
大家应注意到，从两个角度得到的结论刚好相反。因此我们在具体运用时要注意区别分析的角度。
American vs European Options
An American option is worth at least as much as the corresponding European option
C ³ c
P ³ p
看涨期权价格的上限
看跌期权价格的上限
pXe-rT
无收益资产欧式看涨期权价格的下限
为了推导出期权价格下限，我们考虑如下两个组合：
组合A：一份欧式看涨期权加上金额为Xe-rT的现金
组合B：一单位标的资产
在组合A中，如果现金按无风险利率投资则在T时刻将变为X，即等于协议价格。此时多头要不要执行看涨期权，取决于T时刻标的资产价格（ST）是否大于X。若ST>X，则执行看涨期权，组合A的价值为ST；若STX，则不执行看涨期权，组合A 的价值为X。因此，在T时刻，组合A 的价值为：max(ST, X)
而在T时刻，组合B的价值为ST。由于max(ST, X) ST ，因此，在t时刻组合A的价值也应大于等于组合B，即:
c+Xe-rT≥S
c≥S-Xe-rT 内在价值
由于期权的价值一定为正，因此无收益资产欧式看涨期权价格下限为：
c≥max[S-Xe-rT,0]
有收益资产欧式看涨期权价格的下限
c≥max[S-D-Xe-rT,0]
无收益资产欧式看跌期权价格的下限
组合C：一份欧式看跌期权加上一单位标的资产
组合D：金额为的现金
在T时刻，如果ST<X，期权将被执行，组合C价值为X；如果ST>X，期权将不被执行，组合C价值为ST，即在组合C的价值为：max（ST，X）
假定组合D的现金以无风险利率投资，则在T时刻组合D的价值为X。由于组合C的价值在T时刻大于等于组合D，因此组合C的价值在t时刻也应大于等于组合D，即：
p+S ≥Xe-rT
p≥max[Xe-rT –S,0]
有收益资产欧式看跌期权价格的下限
p≥max[D+Xe-rT –S,0]
Early Exercise
Usually there is some chance that an American option will be exercised early
An exception is an American call on a non-dividend paying stock
This should never be exercised early
An Extreme Situation
For an American call option:
S0 = 100; T = ; X = 60; D = 0
Should you exercise immediately?
What should you do if
You want to hold the stock for the next 3 months?
You do not feel that the stock is worth holding for the next 3 months?
Reasons For Not Exercising a Call Early
(No Dividends )
No income is sacrificed
We delay paying the strike price
Holding the call provides insurance against stock price falling below strike price
Should Puts Be Exercised
Early ?
Are there any advantages to exercising an American put when
S0 = 60; T = ; r=10%
X = 100; D = 0
提前执行无收益资产美式看跌期权的合理性
组合A：一份美式看跌期权加上一单位标的资产
组合B：金额为Xe-rT的现金
若不提前执行，则到T时刻，组合A的价值为max（X，ST），组合B的价值为X，因此组合A的价值大于等于组合B。
若在时刻提前执行，则组合A的价值为X，组合B的价值为Xe- (T-τ)，因此组合A的价值也高于组合B。
比较这两种结果我们可以得出结论：是否提前执行无收益资产的美式看跌期权，主要取决于期权的实值额（X-S）、无风险利率水平等因素。一般来说，只有当S相对于X来说较低，或者r较高时，提前执行无收益资产美式看跌期权才可能是有利的。
提前执行有收益资产美式看涨期权的合理性 (1)
由于提前执行有收益资产的美式期权可较早获得标的资产，从而获得现金收益，而现金收益可以派生利息，因此在一定条件下，提前执行有收益资产的美式看涨期权有可能是合理的。
我们假设在期权到期前，标的资产有n个除权日，t1，t2……，tn为除权前的瞬时时刻，在这些时刻之后的收益分别为D1，D2，……，Dn，在这些时刻的标的资产价格分别为S1，S2，……Sn。
由于在无收益的情况下，不应提前执行美式看涨期权，我们可以据此得到一个推论：在有收益情况下，只有在除权前的瞬时时刻提前执行美式看涨期权方有可能是最优的。因此我们只需推导在每个除权日前提前执行的可能性。
提前执行有收益资产美式看涨期权的合理性 (2)
我们先来考察在最后一个除权日（tn）提前执行的条件。如果在tn时刻提前执行期权，则期权多方获得Sn-X的收益。若不提前执行，则标的资产价格将由于除权降到Sn-Dn。
在tn时刻期权的价值（Cn）
因此，如果：
即:
则在tn提前执行是不明智的。
相反，如果
则在tn提前执行有可能是合理的。实际上，只有当tn时刻标的资产价格足够大时，提前执行美式看涨期权才是合理的。
提前执行有收益资产美式看涨期权的合理性 (3)
同样，对于任意i<n,在ti时刻不能提前执行有收益资产的美式看涨期权条件是：
由于存在提前执行更有利的可能性，有收益资产的美式看涨期权价值大于等于欧式看涨期权，其下限为：
Arbitrage Opportunities
Suppose that
c = 3 S0 = 31
T = r = 10%
X =30 D = 0
What are the arbitrage possibilities when
p = ?
p = 1 ?
Put-Call Parity; No Dividends
Consider the following 2 portfolios:
Portfolio A: European call on a stock + PV of the strike price in cash
Portfolio B: European put on the stock + the stock
Both are worth MAX(ST , X ) at the maturity of the options
They must therefore be worth the same today
This means that c + Xe -rT = p + S0
American Put-Call Parity
1. P>p=c+Xe-rT-S0=C +Xe-rT-S0
C-P< S0- Xe-rT (1)
2.组合A：一份欧式看涨期权加上金额为X的现金
组合B：一份美式看跌期权加上一单位标的资产
如果美式期权没有提前执行，则在T时刻组合B的价值为max(ST,X)，而此时组合A的价值为max(ST,X) +Xer(T-t)-X。因此组合A的价值大于组合B。
如果美式期权在 时刻提前执行，则在 时刻，组合B的价值为X，而此时组合A的价值大于等于Xer（τ-t）。因此组合A的价值也大于组合B。
无论美式期权是否提前执行，组合A的价值都高于组合B，即：c+X>P+S0 由于c=C, 所以
C-P>S0-X (2)
S0-X<C-P< S0- Xe-rT
看涨期权价格曲线
欧式看跌期权价格曲线
美式看跌期权价格曲线
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Introduction to Binomial Trees
Chapter 9
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A Simple Binomial Model
A stock price is currently $20
In three months it will be either $22 or $18
Stock Price = $22
Stock Price = $18
Stock price = $20
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*
Stock Price = $22
Option Price = $1
Stock Price = $18
Option Price = $0
Stock price = $20
Option Price=?
A Call Option
A 3-month call option on the stock has a strike price of 21.
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Consider the Portfolio: long D shares short 1 call option
Portfolio is riskless when 22D – 1 = 18D or D =
22D – 1
18D
Setting Up a Riskless Portfolio
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Valuing the Portfolio
( Risk-Free Rate is 12% )
The riskless portfolio is:
long shares short 1 call option
The value of the portfolio in 3 months is 22 × – 1 = 18 × =
The value of the portfolio today is – × =
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Valuing the Option
The portfolio that is
long shares short 1 option
is worth
The value of the shares is (= × 20 )
The value of the option is therefore (= – )
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Generalization
A derivative lasts for time T and is dependent on a stock
Su
ƒu
Sd
ƒd
S
ƒ
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Generalization
(continued)
Consider the portfolio that is long D shares and short 1 derivative
The portfolio is riskless when SuD – ƒu = Sd D – ƒd or
SuD – ƒu
SdD – ƒd
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Generalization
(continued)
Value of the portfolio at time T is Su D – ƒu
Value of the portfolio today is (Su D – ƒu )e–rT
Another expression for the portfolio value today is S D – f
Hence ƒ = S D – (Su D – ƒu )e–rT
*
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Generalization
(continued)
Substituting for D we obtain
ƒ = [ p ƒu + (1 – p )ƒd ]e–rT
where
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Risk-Neutral Valuation
ƒ = [ p ƒu + (1 – p )ƒd ]e-rT
The variables p and (1 – p ) can be interpreted as the risk-neutral probabilities of up and down movements
The value of a derivative is its expected payoff in a risk-neutral world discounted at the risk-free rate
Su
ƒu
Sd
ƒd
S
ƒ
p
(1 – p )
*
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Irrelevance of Stock’s Expected Return
When we are valuing an option in terms of the underlying stock the expected return on the stock is irrelevant
*
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Original Example Revisited
Since p is a risk-neutral probability
= 22p + 18(1 – p ); p =
Alternatively, we can use the formula
Su = 22
ƒu = 1
Sd = 18
ƒd = 0
S
ƒ
p
(1 – p )
*
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Valuing the Option
The value of the option is
e–× [ × 1 + × 0]
=
Su = 22
ƒu = 1
Sd = 18
ƒd = 0
S
ƒ
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A Two-Step Example
Each time step is 3 months
20
22
18
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Valuing a Call Option
Value at node B = e– × ( × + × 0) =
Value at node A = e– × ( × + × 0)
=
20
22
18
A
B
C
D
E
F
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A Put Option Example; X=52
50
60
40
72
0
48
4
32
20
A
B
C
D
E
F
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What Happens When an Option is American
50
60
40
72
0
48
4
32
20
A
B
C
D
E
F
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Delta
Delta (D) is the ratio of the change in the price of a stock option to the change in the price of the underlying stock
The value of D varies from node to node
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Choosing u and d
One way of matching the volatility is to set
where s is the volatility and Dt is the length of the time step. This is the approach used by Cox, Ross, and Rubinstein
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Financial Engineering
Chapter 10
Model of the Behavior
of Stock Prices
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Stochastic Processes
A stochastic process is a variable that evolves over time in a way that is at least in part random. . temperature and IBM stock price
A stochastic process is defined by a probability law for the evolution xt of a variable over time t. For given times, we can calculate the probability that the corresponding values x1,x2, x3,etc. lie in some specified range.
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Categorization of Stochastic Processes
Discrete time; discrete variable
Random walk:
if can only take on discrete values
Discrete time; continuous variable
is a normally distributed random variable with zero mean.
Continuous time; discrete variable
Continuous time; continuous variable
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Modeling Stock Prices
We can use any of the four types of stochastic processes to model stock prices
The continuous time, continuous variable process proves to be the most useful for the purposes of valuing derivative securities
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Markov Processes
In a Markov process future movements in a variable depend only on where we are, not the history of how we got where we are
We will assume that stock prices follow Markov processes
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Weak-Form Market Efficiency
The assertion is that it is impossible to produce consistently superior returns with a trading rule based on the past history of stock prices. In other words technical analysis does not work.
A Markov process for stock prices is clearly consistent with weak-form market efficiency
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Example of a Discrete Time Continuous Variable Model
A stock price is currently at $40
At the end of 1 year it is considered that it will have a probability distribution of f(40,10) where f(m,s) is a normal distribution with mean m and standard deviation s.
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Questions
What is the probability distribution of the stock price at the end of 2 years?
½ years?
¼ years?
Dt years?
Taking limits we have defined a continuous variable, continuous time process
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Variances & Standard Deviations
In Markov processes changes in successive periods of time are independent
This means that variances are additive
Standard deviations are not additive
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Variances & Standard Deviations (continued)
In our example it is correct to say that the variance is 100 per year.
It is strictly not correct to say that the standard deviation is 10 per year.
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A Wiener Process (Brownian Motion)
We consider a variable z whose value changes continuously
The change in a small interval of time Dt is Dz
The variable follows a Wiener process if
1.
2. The values of Dz for any 2 different (non-overlapping) periods of time are independent
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Financial Engineering
Properties of a Wiener Process
Mean of [z (T ) – z (0)] is 0
Variance of [z (T ) – z (0)] is T
Standard deviation of [z (T ) – z (0)] is
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Taking Limits . . .
What does an expression involving dz and dt mean?
It should be interpreted as meaning that the corresponding expression involving Dz and Dt is true in the limit as Dt tends to zero
In this respect, stochastic calculus is analogous to ordinary calculus
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Financial Engineering
Generalized Wiener Processes
A Wiener process has a drift rate (ie average change per unit time) of 0 and a variance rate of 1
In a generalized Wiener process the drift rate & the variance rate can be set equal to any chosen constants
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Generalized Wiener Processes
(continued)
The variable x follows a generalized Wiener process with a drift rate of a & a variance rate of b2 if
dx=adt+bdz
or: x(t)=x0+at+bz(t)
Financial Engineering
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Generalized Wiener Processes
(continued)
Mean change in x in time T is aT
Variance of change in x in time T is b2T
Standard deviation of change in x in time T is
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The Example Revisited
A stock price starts at 40 & has a probability distribution of f(40,10) at the end of the year
If we assume the stochastic process is Markov with no drift then the process is
dS = 10dz
If the stock price were expected to grow by $8 on average during the year, so that the year-end distribution is f(48,10), the process is
dS = 8dt + 10dz
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Financial Engineering
Why ?(1)
It’s the only way to make the variance of (xT-x0)depend on T and not on the number of steps.
time up into n discrete periods of length Δt, n=T/Δt. In each period the variable x either moves up or down by an amount Δh with the probabilities of p and q respectively.
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Financial Engineering
Why ?(2)
distribution for the future values of x:
E(Δx)=(p-q) Δh
E[(Δx)2]= p(Δh)2+q(-Δh)2
So, the variance of Δx is:
E[(Δx)2]-[E(Δx)]2=[1-(p-q)2](Δh)2=4pq(Δh)2
3. Since the successive steps of the random walk are independent, the cumulated change(xT-x0)is a binomial random walk with mean:
n(p-q) Δh=t (p-q) Δh/Δt
and variance: n [1-(p-q)2](Δh)2= 4pqt(Δh)2 /Δt
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Why ?(3)
When let Δt go to zero, we would like the mean and variance of (xT-x0) to remain unchanged, and to be independent of the particular choice of p,q, Δh and Δt.
The only way to get it is to set:
and
then
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Financial Engineering
Why ?(4)
When Δt goes to zero,the binomial distribution converges to a normal distribution, with mean
and variance
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Sample path(a= per year,b= per year)
Taking a time interval of one month, then calculating a trajectory for xt using the equation:
A trend of per year implies a trend of per month. A standard deviation of per year implies a variance of per year, and hence a variance of per month, so that the standard deviation in monthly terms is .
See Investment under uncertainty, p66
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Forecast using generalized Brownian Motion
Given the value of x(t)for Dec. 1974,X1974 , the forecasted value of x for a time T months beyond Dec. 1974 is given by:
See Investment under uncertainty, p67
In the long run, the trend is the dominant determinant of Brownian Motion, wheras in the short run, the volatility of the process dominates.
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Financial Engineering
Why a Generalized Wiener Processis not Appropriate for Stocks
For a stock price we can conjecture that its expected proportional change in a short period of time remains constant not its expected absolute change in a short period of time
We can also conjecture that our uncertainty as to the size of future stock price movements is proportional to the level of the stock price
The price of a stock never fall below zero.
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Ito Process
In an Ito process the drift rate and the variance rate are functions of time
dx=a(x,t)dt+b(x,t)dz
or:
The discrete time equivalent
is only true in the limit as Dt tends to
zero
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An Ito Process for Stock Prices
where m is the expected return s is the volatility.
The discrete time equivalent is
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Monte Carlo Simulation
We can sample random paths for the stock price by sampling values for e
Suppose m= , s= , and Dt = , then
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Monte Carlo Simulation – One Path
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Ito’s Lemma
If we know the stochastic process followed by x, Ito’s lemma tells us the stochastic process followed by some function G (x, t )
Since a derivative security is a function of the price of the underlying & time, Ito’s lemma plays an important part in the analysis of derivative securities
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Taylor Series Expansion
A Taylor’s series expansion of G(x , t ) gives
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Ignoring Terms of Higher Order Than Dt
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Substituting for Dx
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The e2Dt Term
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Taking Limits
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Application of Ito’s Lemma
to a Stock Price Process
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Examples
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Ito’s Lemma for several Ito processes
Suppose F=F(x1,x2,…,xm,xt) is a function of time and of the m Ito process x1,x2,…,xm,
where dxi=ai(x1,x2,…,xm,t)dt+bi(x1,x2,…,xm,t)dzi,i=1,…,m,with E(dzidzj)= ρ Ito’s Lemma gives the defferential dF as
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Examples
Suppose F(x,y)=xy, where x and y each follow geometric Brownian motions:
dx=axxdt+bxxdzx
dy=ayydt+byydzy
with E(dzxdzy)=ρdt. What’s the process followed by F(x,y) and by G=logF?
dF=xdy+ydx+dxdy
=(ax+ay+ ρbxby)Fdt+(bxdzx+bydzy)F
dG= (ax+ay-1/2bx2-1/2by2)dt+bxdzx+bydzy
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复习
几何布朗运动与效率市场假说
马尔科夫过程与鞅过程
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The Black-Scholes
Model
Chapter 11
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The Stock Price Assumption
Consider a stock whose price is S
In a short period of time of length Dt the change in the stock price is assumed to be normal with mean mSdt and standard deviation
m is expected return and s is volatility
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The Lognormal Property
It follows from this assumption that
Since the logarithm of ST is normal, ST is lognormally distributed
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The Lognormal Distribution
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Continuously Compounded Return, h
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Estimating Volatility from Historical Data
1. Take observations S0, S1, . . . , Sn at intervals of t years
2. Define:
3. Calculate the standard deviation, s , of the ui ´s
4. The historical volatility estimate is:
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The Concepts Underlying Black-Scholes
The option price & the stock price depend on the same underlying source of uncertainty
We can form a portfolio consisting of the stock and the option which eliminates this source of uncertainty
The portfolio is instantaneously riskless and must instantaneously earn the risk-free rate
This leads to the Black-Scholes differential equation
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Assumptions of BS Formula
The short-term interest rate is known and is constant through time.
The stock price follows a random walk in continuous time with a variance rate proportional to the square of the stock the distribution of stock prices is lognormal. The variance rate of the return on the stock is constant.
The sock pays no dividends.
The option is “European”.
There are no transaction costs.
It’s possible to borrow money to buy stocks.
There are no penalties to short selling.
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1 of 3: The Derivation of the
Black-Scholes Differential Equation
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2 of 3: The Derivation of the
Black-Scholes Differential Equation
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3 of 3: The Derivation of the
Black-Scholes Differential Equation
The return on the portfolio must be the risk-free rate. Hence
We substitute for and in these equations to get the Black-Scholes differential equation:
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The Differential Equation
Any security whose price is dependent on the stock price satisfies the differential equation
The particular security being valued is determined by the boundary conditions of the differential equation
In a forward contract the boundary condition is ƒ = S – K when t =T
The solution to the equation is
ƒ = S – K e–r (T – t )
Financial Engineering
*
Financial Engineering
*
Risk-Neutral Valuation
The variable m does not appear in the Black-Scholes equation
The equation is independent of all variables affected by risk preference
The solution to the differential equation is therefore the same in a risk-free world as it is in the real world
This leads to the principle of risk-neutral valuation
Financial Engineering
*
Financial Engineering
*
Applying Risk-Neutral Valuation
1. Assume that the expected return from the stock price is the risk-free rate
2. Calculate the expected payoff from the option
3. Discount at the risk-free rate
Financial Engineering
*
Financial Engineering
*
The Black-Scholes Formulas
Financial Engineering
*
Financial Engineering
*
BS公式的推导（1）
Financial Engineering
*
Financial Engineering
*
BS公式的推导（2）
将上述对ST的积分转换成对Q的积分，有：
Financial Engineering
*
Financial Engineering
*
BS公式的推导（3）
Financial Engineering
*
Financial Engineering
*
BS公式的解释
S0N(d1)是Asset-or-noting call option的价值，-e-rTXN(d2)是X份cash-or-nothing看涨期权空头的价值。
N(d2)是在风险中性世界中期权被执行的概率，或者说ST大于X的概率， e-rTXN(d2)是X的风险中性期望值的现值。 S0N(d1)是得到ST的风险中性期望值的现值。
是复制交易策略中股票的数量，S0N（d1)就是股票的市值， -e-rTXN(d2)则是复制交易策略中负债的价值。
Financial Engineering
*
Financial Engineering
*
Implied Volatility
The implied volatility of an option is the volatility for which the Black-Scholes price equals the market price
The is a one-to-one correspondence between prices and implied volatilities
Traders and brokers often quote implied volatilities rather than dollar prices
Financial Engineering
*
Financial Engineering
*
Causes of Volatility
Volatility is usually much greater when the market is open (. the asset is trading) than when it is closed
For this reason time is usually measured in “trading days” not calendar days when options are valued
Financial Engineering
*
Financial Engineering
*
Warrant Valuation
The analysis of warrants is much more complicated than that of options, because:
The life of a warrant is typically measured in years, rather than in months, so the variance rate may change substantially.
The Exercise price of the warrant is usually not adjusted at all for dividends.
The exercise price of a warrant sometimes changes on specified dates.
If the company is involved in a merger, the adjustment that is made in the terms of the warrant may change its value.
The exercise of a large number of warrants may sometimes result in a significant increase in the number of common shares outstanding.
Financial Engineering
*
Financial Engineering
*
Warrants & Dilution
When a regular call option is exercised the stock that is delivered must be purchased in the open market
When a warrant is exercised new stock is issued by the company
This will dilute the value of the existing stock
One valuation approach is to assume that all equity (warrants + stock) follows geometric Brownian motion
Financial Engineering
*
Financial Engineering
*
Adjustment for the increase in the number of common shares
某公司有N股普通股和M份欧式认股权证,每份权证可以在T时刻按每股X价格购买b股股票.令V表示公司股票和认股权证的总价值,则认股权证被行使后股票的除权价格为:
认股权证持有者的赢利为:
Financial Engineering
*
Financial Engineering
*
Warrant Valuation(2)
So,the value of the warrant is the value of Nb/(N+Mb)regular call option on V/N.
Where W is the price of the warrant,so:
所以如果我们用上式右边代替BS公式的S0,用公司股票加认股权证的波动率代替股票的波动率,并乘以Nb/(N+Mb),BS公式就变成了欧式认股权证的定价公式.
Financial Engineering
*
Financial Engineering
*
Dividends
European options on dividend-paying stocks are valued by substituting the stock price less the present value of dividends into Black-Scholes
Only dividends with ex-dividend dates during life of option should be included
The “dividend” should be the expected reduction in the stock price expected
Financial Engineering
*
Financial Engineering
*
American Calls
An American call on a non-dividend-paying stock should never be exercised early
An American call on a dividend-paying stock should only ever be exercised immediately
prior to an ex-dividend date
Financial Engineering
*
Financial Engineering
*
Black’s Approach to Dealing with
Dividends in American Call Options
Set the American price equal to the maximum of two European prices:
1. The 1st European price is for an option maturing at the same time as the American option
2. The 2nd European price is for an option maturing just before the final ex-dividend date
Financial Engineering
*
*
Options on
Stock Indices, Currencies, and Futures
Chapter 12
*
*
European Options on Stocks
Paying Continuous Dividends
We get the same probability distribution for the stock price at time T in each of the following cases:
1. The stock starts at price S0 and provides a continuous dividend yield = q
2. The stock starts at price S0e–q T and provides no income
*
*
European Options on Stocks
Paying Continuous Dividends
continued
We can value European options by reducing the stock price to S0e–q T and then behaving as though there is no dividend
*
*
Extension of Chapter 7 Results
(Equations to )
Lower Bound for calls:
Lower Bound for puts
Put Call Parity
*
*
Extension of Chapter 11 Results (Equations and )
*
*
The Binomial Model(Risk-neutral world)
S0u
ƒu
S0d
ƒd
S0
ƒ
p
(1 – p )
f=e-rT[pfu+(1-p)fd ]
*
*
The Binomial Model
continued
In a risk-neutral world the stock price grows at r-q rather than at r when there is a dividend yield at rate q
The probability, p, of an up movement must therefore satisfy
pS0u+(1-p)S0d=S0e (r-q)T
so that
*
*
Index Options
Option contracts are on 100× the index
The most popular underlying indices are
the S&P 100 (American) OEX
the S&P 500 (European) SPX
Contracts are settled in cash
*
*
Index Option Example
Consider a call option on an index with a strike price of 560
Suppose 1 contract is exercised when the index level is 580
What is the payoff?
*
*
Using Index Options for Portfolio Insurance
Suppose the value of the index is S0 and the strike price is X
If a portfolio has a b of , the portfolio insurance is obtained by buying 1 put option contract on the index for each 100S0 dollars held
If the b is not , the portfolio manager buys b put options for each 100S0 dollars held
In both cases, X is chosen to give the appropriate insurance level
*
*
Example 1
Portfolio has a beta of
It is currently worth $5 million
The index currently stands at 1000
What trade is necessary to provide insurance against the portfolio value falling below $ million?
*
*
Example 2
Portfolio has a beta of
It is currently worth $1 million and index stands at 1000
The risk-free rate is 12% per annum(每3个月计一次复利）
The dividend yield on both the portfolio and the index is 4%(每3个月计一次复利）
How many put option contracts should be purchased for portfolio insurance?
*
*
If index rises to 1040, it provides a 40/1000 or 4% return in 3 months
Total return (incl. dividends)=5%
Excess return over risk-free rate=2%
Excess return for portfolio=4%
Increase in Portfolio Value=4+3-1=6%
Portfolio value=$ million
Calculating Relation Between Index Level and Portfolio Value in 3 months
*
*
Determining the Strike Price
An option with a strike price of 960 will provide protection against a 10% decline in the portfolio value
*
*
Valuing European Index Options
We can use the formula for an option on a stock paying a continuous dividend yield
Set S0 = current index level
Set q = average dividend yield expected during the life of the option
*
*
Currency Options
Currency options trade on the Philadelphia Exchange (PHLX)
There also exists an active over-the-counter (OTC) market
Currency options are used by corporations to buy insurance when they have an FX exposure
*
*
The Foreign Interest Rate
We denote the foreign interest rate by rf
When a . company buys one unit of the foreign currency it has an investment of S0 dollars
The return from investing at the foreign rate is dollars
This shows that the foreign currency provides a “dividend yield” at rate rf
*
*
Valuing European Currency Options
A foreign currency is an asset that provides a continuous “dividend yield” equal to rf
We can use the formula for an option on a stock paying a continuous dividend yield :
Set S0 = current exchange rate
Set q = rƒ
*
*
Formulas for European Currency Options
*
*
Alternative Formulas
Using
*
*
Mechanics of Call Futures Options
Most of Futures options are American.
The maturity date is usually on, or a few days before, the earliest delivery date of the underlying futures contract.
When a call futures option is exercised the holder acquires
1. A long position in the futures
2. A cash amount equal to the excess of
the futures price over the strike price
*
*
Mechanics of Put Futures Option
When a put futures option is exercised the holder acquires
1. A short position in the futures
2. A cash amount equal to the excess of
the strike price over the futures price
*
*
The Payoffs
If the futures position is closed out immediately:
Payoff from call = F0-X
Payoff from put = X-F0
where F0 is futures price at time of exercise
*
*
Why Futures Option instead of Spot Option?
Futures is more liquid and easier to get the price information
Can be settled in cash
Lower transaction cost
*
*
Put-Call Parity for Futures Option
Consider the following two portfolios:
1. European call plus Xe-rT of cash
2. European put plus long futures plus cash equal to F0e-rT
They must be worth the same at time T so that
c+Xe-rT=p+F0 e-rT
*
*
Futures Price = $33
Option Price = $4
Futures Price = $28
Option Price = $0
Futures price = $30
Option Price=?
Binomial Tree Example
A 1-month call option on futures has a strike price of 29.
*
*
Consider the Portfolio: long D futures short 1 call option
Portfolio is riskless when 3D – 4 = -2D or D =
3D – 4
-2D
Setting Up a Riskless Portfolio
*
*
Valuing the Portfolio
( Risk-Free Rate is 6% )
The riskless portfolio is:
long futures short 1 call option
The value of the portfolio in 1 month is
The value of the portfolio today is – =
*
*
Valuing the Option
The portfolio that is
long futures short 1 option
is worth
The value of the futures is zero
The value of the option must therefore be
*
*
Generalization of Binomial Tree Example
A derivative lasts for time T & is dependent on a futures
F0u
ƒu
F0d
ƒd
F0
ƒ
*
*
Generalization
(continued)
Consider the portfolio that is long D futures and short 1 derivative
The portfolio is riskless when
F0u D - F0 D – ƒu
F0d D- F0D – ƒd
*
*
Generalization
(continued)
Value of the portfolio at time T is F0u D –F0D – ƒu
Value of portfolio today is – ƒ
Hence ƒ = – [F0u D –F0D – ƒu]e-rT
*
*
Generalization
(continued)
Substituting for D we obtain
ƒ = [ p ƒu + (1 – p )ƒd ]e–rT
where
*
*
Growth Rates For Futures Prices
A futures contract requires no initial investment
In a risk-neutral world the expected return should be zero
The expected growth rate of the futures price is therefore zero
The futures price can therefore be treated like a stock paying a dividend yield of r
*
*
Futures price vs. expected future spot ptice
In a risk-neutral world, the expected growth rate of the futures price is zero,so
Because FT=ST, so
*
*
Valuing European Futures Options
We can use the formula for an option on a stock paying a continuous dividend yield
Set S0 = current futures price (F0)
Set q = domestic risk-free rate (r )
Setting q = r ensures that the expected growth of F in a risk-neutral world is zero
*
*
Black’s Formula
The formulas for European options on futures are known as Black’s formulas
*
*
European Futures Option Prices vs Spot Option Prices
If the European futures option matures at the same time as the futures contract, then the two options are in theory equivalent.
If the European call future option matures before the futures contract, it is worth more than the corresponding spot option in a normal market, and less in an inverted market.
*
*
American Futures Option Prices vs Spot Option Prices
There is always some chance that it will be optimal to exercise an American futures option early. So, if futures prices are higher than spot prices (normal market), an American call on futures is worth more than a similar American call on spot. An American put on futures is worth less than a similar American put on spot
When futures prices are lower than spot prices (inverted market) the reverse is true
*
*
Summary of Key Results
We can treat stock indices, currencies, & futures like a stock paying a continuous dividend yield of q
For stock indices, q = average dividend yield on the index over the option life
For currencies, q = rƒ
For futures, q = r
*
*
The Greek Letters
Chapter 13
*
*
Example
A bank has sold for $300,000 a European call option on 100,000 shares of a nondividend paying stock
S0 = 49, X = 50, r = 5%, s = 20%,
T = 20 weeks, m = 13%
The Black-Scholes value of the option is $240,000
How does the bank hedge its risk?
*
*
Naked & Covered Positions
Naked position
Take no action
Covered position
Buy 100,000 shares today
Both strategies leave the bank exposed to significant risk
*
*
Stop-Loss Strategy
This involves:
Buying 100,000 shares as soon as price reaches $50
Selling 100,000 shares as soon as price falls below $50
This deceptively simple hedging strategy does not work well
*
*
Delta
Delta (D) is the rate of change of the option price with respect to the underlying
Option
price
A
B
Slope = D
Stock price
*
*
Delta Hedging
This involves maintaining a delta neutral portfolio
The delta of a European call on a stock paying dividends at rate q is N (d 1)e– qT
The delta of a European put is
e– qT [N (d 1) – 1]
*
*
Delta Hedging
continued
The hedge position must be frequently rebalanced
Delta hedging a written option involves a “buy high, sell low” trading rule
See Tables and for examples of delta hedging
*
*
Using Futures for Delta Hedging
The delta of a futures contract is e(r-q)T times the delta of a spot
The position required in futures for delta hedging is therefore e-(r-q)T times the position required in the corresponding spot
*
*
Theta
Theta (Q) of a derivative (or portfolio of derivatives) is the rate of change of the value with respect to the passage of time
See Figure for the variation of Q with respect to the stock price for a European call
*
*
Gamma
Gamma (G) is the rate of change of delta (D) with respect to the price of the underlying asset
See Figure for the variation of G with respect to the stock price for a call or put option
*
*
Gamma Addresses Delta Hedging Errors Caused By Curvature
S
C
Stock price
S’
Call
price
C’
C’’
*
*
Interpretation of Gamma
For a delta neutral portfolio,
DP » Q Dt + ½GDS 2
DP
DS
Positive Gamma
*
*
Relationship Among Delta, Gamma, and Theta
For a portfolio of derivatives on a stock paying a continuous dividend yield at rate q
*
*
Vega
Vega (n) is the rate of change of the value of a derivatives portfolio with respect to volatility
See Figure for the variation of n with respect to the stock price for a call or put option
*
*
Managing Delta, Gamma, & Vega
D can be changed by taking a position in the underlying
To adjust G & n it is necessary to take a position in an option or other derivative
See example (p327).
*
*
Rho
Rho is the rate of change of the value of a derivative with respect to the interest rate
For currency options there are 2 rhos
*
*
Hedging in Practice
Traders usually ensure that their portfolios are delta-neutral at least once a day
Whenever the opportunity arises, they improve gamma and vega
As portfolio becomes larger hedging becomes less expensive
*
*
Scenario Analysis
A scenario analysis involves testing the effect on the value of a portfolio of different assumptions concerning asset prices and their volatilities
*
*
Hedging vs Creation of an Option Synthetically
When we are hedging we take positions that offset D, G, n, etc.
When we create an option synthetically we take positions that match D, G, & n
*
*
Portfolio Insurance
In October of 1987 many portfolio managers attempted to create a put option on a portfolio synthetically
This involves initially selling enough of the portfolio (or of index futures) to match the D of the put option
*
*
Portfolio Insurance
continued
As the value of the portfolio increases, the D of the put becomes less negative & some of the original portfolio is repurchased
As the value of the portfolio decreases, the D of the put becomes more negative & more of the portfolio must be sold
*
*
Portfolio Insurance
continued
The strategy did not work well on October 19, 1987...
14.*
Value at Risk
Chapter 14
14.*
The Question Being Asked in VaR
“What loss level is such that we are X% confident it will not be exceeded in N business days?”
14.*
VaR and Regulatory Capital
Regulators require banks to keep capital for market risk equal to the average of VaR estimates for past 60 trading days using X=99 and N=10, times a multiplication factor.
(Usually the multiplication factor equals 3)
14.*
Advantages of VaR
It captures an important aspect of risk
in a single number
It is easy to understand
It asks the simple question: “How bad can things get?”
14.*
Daily Volatilities
In option pricing we express volatility as volatility per year
In VaR calculations we express volatility as volatility per day
14.*
Daily Volatility continued
Strictly speaking we should define sday as the standard deviation of the continuously compounded return in one day
In practice we assume that it is the standard deviation of the proportional change in one day
14.*
IBM Example (See page 343)
We have a position worth $10 million in IBM shares
The volatility of IBM is 2% per day (about 32% per year)
We use N=10 and X=99
14.*
IBM Example continued
The standard deviation of the change in the portfolio in 1 day is $200,000
The standard deviation of the change in 10 days is
14.*
IBM Example continued
We assume that the expected change in the value of the portfolio is zero (This is OK for short time periods)
We assume that the change in the value of the portfolio is normally distributed
Since N()=, the VaR is
14.*
AT&T Example
Consider a position of $5 million in AT&T
The daily volatility of AT&T is 1% (approx 16% per year)
The per 10 days is
The VaR is
14.*
Portfolio (See page 344)
Now consider a portfolio consisting of both IBM and AT&T
Suppose that the correlation between the returns is
14.*
. of Portfolio
A standard result in statistics states that
In this case sx = 632,456 and sY=158,114 and r = . The standard deviation of the change in the portfolio value over a 10-day period is therefore 751,665
14.*
VaR for Portfolio
The VaR for the portfolio is
The benefits of diversification are
(1,473,621+368,405)-1,751,379=$90,647
What is the incremental effect of the AT&T holding on VaR?
14.*
The Linear Model
We assume
The change in the value of a portfolio is linearly related to the change in the value of market variables
The changes in the values of the market variables are normally distributed
14.*
The General Linear Model continued (Equation )
14.*
Handling Interest Rates
We do not want to define every interest rate as a different market variable
An approach is to use the duration relationship DP=-DPDy so that
sP=DPysy where sy is the volatility of yield changes and sP is as before the standard deviation of the change in the portfolio value
14.*
Alternative: Cash Flow Mapping (See page 347)
We choose as market variables bond prices with standard maturities (1mm, 3mm, 6mm, 1yr, 2yr, 5yr, 7yr, 10yr, 30yr)
Suppose that the 5yr rate is 6% and the 7yr rate is 7% and we will receive a cash flow of $10,000 in years.
The volatilities per day of the price of 5yr and 7yr bonds are % and % respectively
14.*
Example continued
We interpolate between the 5yr rate of 6% and the 7yr rate of 7% to get a rate of %
The PV of the $10,000 cash flow is
14.*
Example continued
We interpolate between the % volatility for the 5yr bond price and the % volatility for the 7yr bond price to get % as the volatility for the bond price
We allocate a of the PV to the 5yr bond and (1- a) of the PV to the 7yr bond
14.*
Example continued
Suppose that the correlation between movement in the 5yr and 7yr bond prices is
To match variances
This gives a=
14.*
Example continued
The cash flow of 10,000 in years is replaced by
6,540*=
in 5 years and by
6,540*=
in 7 years.
This cash flow mapping preserves value and variance
14.*
When Linear Model Can be Used
Portfolio of stocks
Portfolio of bonds
Forward contract on foreign currency
Interest-rate swap
14.*
The Linear Model and Options
(See page 350)
Consider a portfolio of options dependent on a single stock price, S. Define
and
14.*
Linear Model and Options continued
As an approximation
Similar when there are many underlying market variables
where di is the delta of the portfolio with respect to the ith asset
14.*
Example
Consider an investment in options on IBM and AT&T. Suppose the stock prices are 120 and 30 respectively and the deltas of the portfolio with respect to the two stock prices are 1,000 and 20,000 respectively
As an approximation
where Dx1 and Dx2 are the proportional changes in the two stock prices
14.*
Skewness
The linear model fails to capture skewness in the probability distribution of the portfolio value.
14.*
Quadratic Model
(See page 352)
For a portfolio dependent on a single stock price
this becomes
14.*
Moments of DP
(See page 354)
14.*
Quadratic Model continued
With many market variables and each instrument dependent on only one
where di and gi are the delta and gamma of the portfolio with respect to the ith variable
14.*
Quadratic Model continued
When the change in the portfolio value has the form
we can calculate the moments of DP analytically if the Dxi are assumed to be normal
14.*
Quadratic Model continued
Once we have done this we can use the Cornish Fisher expansion to calculate percentiles of the distribution of DP
14.*
Monte Carlo Simulation
(See page 355)
The stages are as follows
Value portfolio today
Sample once from the multivariate distributions of the Dxi
Use the Dxi to determine market variables at end of one day
Revalue the portfolio at the end of day
14.*
Monte Carlo Simulation
Sample once from the multivariate normal distribution of the Dxi’s
Use the value of the Dxi’s to determine the value of each market variable at the end of one day
Calculate DP in one day
Repeat many times to build up a probability distribution for DP in one day
N-day VaR is the appropriate percentile of the distribution times square root of N days
For example, with 1,000 trial the 1 percentile is the 10th worst case.
14.*
Speeding Up Monte Carlo
Use the quadratic approximation() to calculate DP
14.*
Historical Simulation
(See page 356)
Create a database of the daily movements in all market variables.
The first simulation trial assumes that the percentage changes in all market variables are as on the first day
The second simulation trial assumes that the percentage changes in all market variables are as on the second day
and so on
14.*
Stress Testing
(See page 357)
This involves testing how well a portfolio performs under some of the most extreme market moves seen in the last 10 to 20 years
14.*
Back-Testing
(See page 357)
Tests how well VaR estimates would have performed in the past
We could ask the question: How often was the loss greater than the 99%/10 day VaR?
14.*
Principal Components Analysis
(See page 357)
Suppose that a portfolio depends on a number of related variables (eg interest rates) which are affected by some factors.
Use historical data to find factor loadings
The factor loading s have the property that the sum of their squares for each factor is .
The amount of the factors moves on a particular day are known as the factor scores.
The important of a factor is measured by the standard deviation of its factor score.
The variances of the factor scores is equal to the total variance of the data.
14.*
Results for Interest Rates
(Table )
The first factor is a roughly parallel shift (% of variation explained)
The second factor is a twist (10% of variation explained)
The third factor is a bowing (% of variation explained)
14.*
Using Principle Components analysis to calculate VaR(p360)
Find the exposure of the portfolio to interest rates.
Use factor loading to calculate the exposure to some important factors.
Use the standard deviations of factors to calculate the standard deviations of the portfolio
Then to calculate VaR.
*
*
Estimating Volatilities and Correlations
Chapter 15
*
*
Standard Approach to Estimating Volatility (Equation )
Define sn as the volatility per day on day n, as estimated at end of day n-1
Define Si as the value of market variable at end of day i
Define ui= ln(Si/Si-1), an unbiased estimate is
*
*
Simplifications Usually Made
(Equation , page 369)
Define ui as (Si-Si-1)/Si-1
Assume that the mean value of ui is zero
Replace m-1 by m
This gives
*
*
Weighting Scheme
Instead of assigning equal weights to the observations we can set
*
*
ARCH(m) Model
In an ARCH(m) model we also assign some weight to the long-run variance rate, V:
*
*
EWMA Model
(Equation , page 370)
In an exponentially weighted moving average model, the weights assigned to the u2 decline exponentially as we move back through time
This leads to
*
*
Attractions of EWMA
Relatively little data needs to be stored
We need only remember the current estimate of the variance rate and the most recent observation on the market variable
Tracks volatility changes
JP Morgan use l = for daily volatility forecasting
*
*
GARCH (1,1)
(Equation , page 372)
In GARCH (1,1) we assign some weight to the long-run average variance rate
Since weights must sum to 1
g + a + b =1
*
*
GARCH (1,1) continued
Setting w = gV the GARCH (1,1) model is
and
*
*
Example
Suppose
the long-run variance rate is so that the long-run volatility per day is %
*
*
Example continued
Suppose that the current estimate of the volatility is % per day and the most recent proportional change in the market variable is 1%.
The new variance rate is
The new volatility is % per day
*
*
GARCH (p,q)
*
*
Other Models
We can design GARCH models so that the weight given to ui2 depends on whether ui is positive or negative
We do not have to assume that the conditional distribution is normal
*
*
Variance Targeting
One way of implementing GARCH(1,1) that increases stability is by using variance targeting
We set the long-run average volatility equal to the sample variance
Only two other parameters then have to be estimated.
*
*
Maximum Likelihood Methods
(Page 374)
In maximum likelihood methods we choose parameters that maximize the likelihood of the observations occurring
*
*
Example 1
We observe that a certain event happens one time in ten trials. What is our estimate of the proportion of the time, p, that it happens
The probability of the outcome is
We maximize this to obtain a maximum likelihood estimate: p=
*
*
Example 2（Suppose the variance is constant)
Estimate the variance of observations from a normal distribution with mean zero
*
*
Application to GARCH
We choose parameters that maximize
*
*
How Good is the Model?
(Table , page 378)
We compare the autocorrelation of the ui’s with the autocorrelation of the ui/si
The Ljung-Box statistic tests for autocorrelation (page 379)
*
*
Forecasting Future Volatility
(Equation , page 379)
A few lines of algebra shows that
The variance rate for an option expiring on day m is
*
*
Volatility Term Structures
The GARCH (1,1) model allows us to predict volatility term structures changes
It suggests that, when calculating vega, we should shift the long maturity volatilities less than the short maturity volatilities
*
*
Correlations
Define ui=(Ui-Ui-1)/Ui-1 and vi=(Vi-Vi-1)/Vi-1
Also
su,n: daily vol of U calculated on day n-1
sv,n: daily vol of V calculated on day n-1
covn: covariance calculated on day n-1
*
*
Correlations continued
Under GARCH (1,1)
covn = w + a un-1vn-1+b covn-1
*
*
Positive semi-definite Condition (Equation , page 384)
A variance-covariance matrix, W, is internally consistent if the positive semi-definite condition
for all vectors w is satisfied.
*
*
Example
The variance covariance matrix
is not internally consistent
*
*
Chapter 16 Numerical Procedures
*
*
Binomial Trees
Binomial trees are frequently used to approximate the movements in the price of a stock or other asset
In each small interval of time the stock price is assumed to move up by a proportional amount u or to move down by a proportional amount d
*
*
Movements in Time Dt
Su
Sd
S
p
1 – p
*
*
1. Tree Parameters for a
Nondividend Paying Stock
We choose the tree parameters p, u, and d so that the tree gives correct values for the mean & standard deviation of the stock price changes in a risk-neutral world
er Dt = pu + (1– p )d
s2Dt = pu 2 + (1– p )d 2 – [pu + (1– p )d ]2
A further condition often imposed is u = 1/ d
*
*
2. Tree Parameters for a
Nondividend Paying Stock
When Dt is small a solution to the equations is
*
*
The Complete Tree
S0
S0u
S0d
S0
S0
S0u 2
S0d 2
S0u 2
S0u 3
S0u 4
S0d 2
S0u
S0d
S0d 4
S0d 3
*
*
Backwards Induction
We know the value of the option at the final nodes
We work back through the tree using risk-neutral valuation to calculate the value of the option at each node, testing for early exercise when appropriate
*
*
Example: Put Option
S0 = 50; X = 50; r =10%; s = 40%;
T = 5 months = ;
Dt = 1 month =
The parameters imply
u = ; d = ;
a = ; p =
*
*
Example (continued)
*
*
Calculation of Delta
Delta is calculated from the nodes at time Dt
*
*
Calculation of Gamma
Gamma is calculated from the nodes at time 2Dt
*
*
Calculation of Theta
Theta is calculated from the central nodes at times 0 and 2Dt
*
*
Calculation of Vega
We can proceed as follows
Construct a new tree with a volatility of 41% instead of 40%.
Value of option is
Vega is
*
*
Trees & Continuous Dividends
When a stock price pays continuous dividends at rate q we construct the tree in the same way but set a = e(r – q )Dt
As with Black-Scholes:
For options on stock indices, q equals the dividend yield on the index
For options on a foreign currency, q equals the foreign risk-free rate
For options on futures contracts q = r
*
*
Binomial Tree for Dividend Paying Stock
Procedure:
Draw the tree for the stock price less the present value of the dividends
Create a new tree by adding the present value of the dividends at each node
This ensures that the tree recombines and makes assumptions similar to those when the Black-Scholes model is used
*
*
Extensions of Tree Approach
Time dependent interest rates
The control variate technique
fA*=fA+ fBS -fE
*
*
Alternative Binomial Tree
Instead of setting u = 1/d we can set each of the 2 probabilities to and
*
*
Trinomial Tree
S
S
Sd
Su
pu
pm
pd
*
*
Adaptive Mesh Model
This is a way of grafting a high resolution tree on to a low resolution tree
We need high resolution in the region of the tree close to the strike price and option maturity
*
*
Monte Carlo Simulation
When used to value European stock options, this involves the following steps:
1. Simulate 1 path for the stock price in a risk neutral world
2. Calculate the payoff from the stock option
3. Repeat steps 1 and 2 many times to get many sample payoff
4. Calculate mean payoff
5. Discount mean payoff at risk free rate to get an estimate of the value of the option
*
*
Sampling Stock Price Movements
In a risk neutral world the process for a stock price is
We can simulate a path by choosing time steps of length Dt and using the discrete version of this
where e is a random sample from f(0,1)
*
*
A More Accurate Approach
*
*
Extensions
When a derivative depends on several underlying variables we can simulate paths for each of them in a risk-neutral world to calculate the values for the derivative
*
*
Sampling from Normal Distribution
One simple way to obtain a sample
from f(0,1) is to generate 12 random numbers between & , take the sum, & subtract
*
*
To Obtain 2 Correlated Normal Samples
*
*
Standard Errors
The standard error of the estimate of the option price is the standard deviation of the discounted payoffs given by the simulation trials divided by the square root of the number of observations.
*
*
Application of Monte Carlo Simulation
Monte Carlo simulation can deal with path dependent options, options dependent on several underlying state variables, & options with complex payoffs
It cannot easily deal with American-style options
*
*
Determining Greek Letters
For D:
1. Make a small change to asset price
2. Carry out the simulation again using the same random number streams
3. Estimate D as the change in the option price divided by the change in the asset price
Proceed in a similar manner for other Greek letters
*
*
Variance Reduction Techniques
Antithetic variable technique
Control variate technique
Importance sampling
Stratified sampling
Moment matching
Using quasi-random sequences
*
*
Representative Sampling Through the Tree
We can sample paths randomly through a binomial or trinomial tree to value an option
An alternative is to choose representative paths
Paths are representative if the proportion of paths through each node is approximately equal to the probability of the node being reached
*
*
Finite Difference Methods
Finite difference methods aim to represent the differential equation in the form of a difference equation
Define ƒi,j as the value of ƒ at time iDt when the stock price is jDS
*
*
Finite Difference Methods
(continued)
*
*
Implicit Finite Difference Method
*
*
Explicit Finite Difference Method
*
*
Implicit vs Explicit Finite Difference Method
The explicit finite difference method is equivalent to the trinomial tree approach
The implicit finite difference method is equivalent to a multinomial tree approach
*
*
Implicit vs Explicit
Finite Difference Methods
ƒi , j
ƒi +1, j
ƒi +1, j –1
ƒi +1, j +1
ƒi +1, j
ƒi , j
ƒi , j –1
ƒi , j +1
Implicit Method
Explicit Method
*
*
Other Points on Finite Difference Methods
It is better to have ln S rather than S as the underlying variable
Improvements over the basic implicit and explicit methods:
Hopscotch method
Crank-Nicholson method
*
*
The Barone Adesi & Whaley Analytic Approximation for American Call Options
*
*
The Barone Adesi & Whaley Analytic
Approximation for American Put Options
*
*
Volatility Smiles and
Alternatives to
Black-Scholes
Chapter 17
*
*
Put-Call Parity Arguments
Put-call parity p +S0e-qT = c +X e–r T holds regardless of the assumptions made about the stock price distribution
It follows that the call option pricing error caused by using the wrong distribution is the same as the put option pricing error when both have the same strike price and maturity
*
*
Implied Volatilities
The implied volatility calculated from a European call option should be the same as that calculated from a European put option when both have the same strike price and maturity
The same is approximately true of American options
*
*
Volatility Smile
A volatility smile shows the variation of the implied volatility with the strike price
The volatility smile should be the same whether calculated from call options or put options
*
*
The Volatility Smile for Foreign Currency Options
(Figure , page 437)
Implied
Volatility
Strike
Price
*
*
Implied Distribution for Foreign Currency Options
The implied distribution is as shown in Figure , page 437
Both tails are fatter than the lognormal distribution
It is also “more peaked than the normal distribution
*
*
The Volatility Smile for Equity Options (Figure , page 439)
Implied
Volatility
Strike
Price
*
*
Implied Distribution for Equity Options
The implied distribution is as shown in Figure , page 439
The right tail is fatter and the left tail is thinner than the lognormal distribution
*
*
Other Volatility Smiles?
What is the volatility smile if
True distribution has a thin left tail and fat right tail
True distribution has both a thin left tail and a thin right tail
*
*
Possible Causes of Volatility Smile
Asset price exhibiting jumps rather than continuous change
Volatility for asset price being stochastic
(One reason for a stochastic volatility in the case of equities is the relationship between volatility and leverage)
*
*
Volatility Term Structure
In addition to calculating a volatility smile, traders also calculate a volatility term structure
This shows the variation of implied volatility with the time to maturity of the option
*
*
Volatility Term Structure
The volatility term structure tend to be downward sloping when volatility is high and upward sloping when it is low
*
*
Example of a Volatility Matrix
(Table , page 441)
*
*
Alternative Models
Time-dependent interest rates/volatility
Compound option
Displaced diffusion
Constant elasticity of variance
Pure Jump
Jump diffusion
Stochastic volatility
*
*
Time-dependent interest rates/volatility
Set the interest rate, r, in Black- Scholes equal to the yield on a zero-coupon bond maturing at the same time as the option
Set volatility equal to the average volatility during the remaining life of the option.
*
*
Compound option
The equity in a leverage firm can be viewed as a call option on the value of the firm.
An option on stock of the firm can be regarded as an option on an option on the value of the firm. This is known as a compound option and has been analyzed by Geske.
*
*
Displaced diffusion-Rubinstein
The firm is assumed to hold two categories of assets: risky and riskless asset, and it has a certain fixed amount default-free debt. Let is the initial proportion of risky asset and is the initial debt-to-equity ratio. Let
*
*
Displaced diffusion-Rubinstein(Cont.)
a=risky asset/equity
Because
risky asset+riskless asset=equity + debt
so: if a>1, then risky asset >equity, and debt > riskless asset. Netting the riskless assets off against debt, the model becomes very similar to the compound option model.
If a<1, the model is markedly different from the properties of the compound option model. Let
S=SA+SB
It follows and the volatility and stock price are positively related (the reverse of that observed in practice).
*
*
Constant elasticity of variance-Cox and Ross
The stock price has a volatility of
where
*
*
Pure Jump-CRR
In each small interval of time,Δt, the aset price has a probability λ Δt of moving from S to Su and a probability of 1- λ Δt of moving from S to Se-w Δt.
In the limit as Δt→0,jump occur according to a Poisson process at rate of λ.
*
*
Jump diffusion-Merton
Where dz is a Wiener process, dq is the Poisson process generating the jumps. Dq is equal to 0 with probability 1- λ Δt , and equal to 1 with probability λ Δt .k is the average jump size measured as a proportional increase in the asset price.
*
*
Stochastic volatility-Hull and White
*
*
Stochastic volatility-Hull and White(Cont.)
Hull and White show that when volatility is stochastic but uncorrelated with the asset price, the price of a European option is the Black-Scholes price integrated over the probability distribution of the average variance rate during the life of the option.
*
*
*
*
Exotic Options
Chapter 18
*
*
Types of Exotics
Package
Nonstandard American options
Forward start options
Compound options
Chooser options
Barrier options
Binary options
Lookback options
Shout options
Asian options
Options to exchange one asset for another
Options involving several assets
*
*
Analytic Results Available For:
Forward start options
Compound options
As you like it options
Barrier options
Binary options
Lookback options
Asian options (approximate distribution of the average stock price by a lognormal distribution)
Options to exchange one asset for another
*
*
Basic Valuation Procedures
Trees / finite difference methods
Monte Carlo simulation
*
*
Path Dependence:
The Traditional View
Backwards induction works well for American options. It cannot be used for path-dependent options
Monte Carlo simulation works well for path-dependent options; it cannot be used for American options
*
*
Extension of Backwards Induction
Backwards induction can be used for some path-dependent options
We will first illustrate the methodology using lookback options and then show how it can be used for Asian options
*
*
Lookback Example (Page 472)
Consider an American lookback put on a stock where
S = 50, s = 40%, r = 10%, Dt = 1 month & the life of the option is 3 months
Payoff is Smax-ST
We can value the deal by considering all possible values of the maximum stock price at each node
(This example is presented to illustrate the methodology. There are more efficient ways of handling lookbacks.)
*
*
Example: An American Lookback Put Option (Figure , page 472)
S = 50, s = 40%, r = 10%, Dt = 1 month,
A
*
*
Why the Approach Works
This approach works for lookback options because
The payoff depends on just 1 function of the path followed by the stock price. (We will refer to this as a “path function”)
The value of the path function at a node can be calculated from the stock price at the node & from the value of the function at the immediately preceding node
The number of different values of the path function at a node does not grow too fast as we increase the number of time steps on the tree
*
*
Extensions of the Approach
The approach can be extended so that there are no limits on the number of alternative values of the path function at a node
The basic idea is that it is not necessary to consider every possible value of the path function
It is sufficient to consider a relatively small number of representative values of the function at each node
*
*
Working Forward
First work forwards through the tree calculating the max and min values of the “path function” at each node
Next choose representative values of the path function that span the range between the min and the max
Simplest approach: choose the min, the max, and N equally spaced values between the min and max
*
*
Backwards Induction
We work backwards through the tree in the usual way carrying out calculations for each of the alternative values of the path function that are considered at a node
When we require the value of the derivative at a node for a value of the path function that is not explicitly considered at that node, we use linear or quadratic interpolation
*
*
Part of Tree to Calculate Value of an Option on the Arithmetic Average
(Figure , page 474)
S =
Average S
Option Price
S =
Average S
Option Price
S =
Average S
Option Price
X
Y
Z
S=50, X=50, s=40%, r=10%, T=1yr, Dt=. We are at time 4Dt
*
*
Part of Tree to Calculate Value of an Option on the Arithmetic Average (continued)
Consider Node X when the average of 5 observations is
Node Y: If this is reached, the average becomes . The option price is interpolated as
Node Z: If this is reached, the average becomes . The option price is interpolated as
Node X: value is
(× + ×)e–× =
*
*
A More Efficient Approach for Lookbacks (Section , page 475)
*
*
Using Trees with Barriers
Section , page 477)
When trees are used to value options with barriers, convergence tends to be slow
The slow convergence arises from the fact that the barrier is inaccurately specified by the tree
*
*
True Barrier vs Tree Barrier for a Knockout Option: The Binomial Tree Case
Barrier assumed by tree
True barrier
*
*
True Barrier vs Tree Barrier for a Knockout Option: The Trinomial Tree Case
Barrier assumed by tree
True barrier
*
*
Alternative Solutions to the Problem
Ensure that nodes always lie on the barriers
Adjust for the fact that nodes do not lie on the barriers
Use adaptive mesh
In all cases a trinomial tree is preferable to a binomial tree
*
*
Modeling Two Variables
Section , page 482)
APPROACHES:
1. Transform variables so that they are not correlated & build the tree in the transformed variables
2. Take the correlation into account by adjusting the position of the nodes
3. Take the correlation into account by adjusting the probabilities
*
*
Implied Trees
(Section , page 485)
*
*
Implied Trees: Analytic Approach
(Dupire; Andersen and Brotherton-Ratcliffe)
*
*
Implied Trees: Numerical Approach
Derman and Kani and Rubinstein provide methods for constructing a tree iteratively so that the market prices of options are matched
*
*
How Difficult is it to Hedge Exotic Options?
In some cases exotic options are easier to hedge than the corresponding vanilla options. (. Asian options)
In other cases they are more difficult to hedge (. barrier options)
*
*
Static Options Replication
(Section , page 487)
This involves approximately replicating an exotic option with a portfolio of vanilla options
Underlying principle: if we match the value of an exotic option on some boundary , we have matched it at all interior points of the boundary
Static options replication can be contrasted with dynamic options replication where we have to trade continuously to match the option
*
*
Example
A 9-month up-and-out call option on a non-dividend paying stock where S = 50, X = 50, the barrier is 60, r = 10%, and s = 30%
Any boundary can be chosen but the natural one is
c (S , ) = MAX(S – 50, 0) when S < 60
c (60, t ) = 0 when 0 £ t £
*
*
Example
(continued)
We might try to match the following points on the boundary
c (S , ) = MAX(S – 50, 0) for S < 60
c (60, ) = 0
c (60, ) = 0
c (60, ) = 0
*
*
Example continued
(See Table , page 489)
We can do this as follows:
+ call with maturity & strike 50
– call with maturity & strike 60
+ call with maturity & strike 60
+ call with maturity & strike 60
*
*
Example (continued)
This portfolio is worth at time zero compared with for the up-and out option
As we use more options the value of the replicating portfolio converges to the value of the exotic option
For example, with 18 points matched on the horizontal boundary the value of the replicating portfolio reduces to ; with 100 points being matched it reduces to
*
*
Using Static Options Replication
To hedge an exotic option we short the portfolio that replicates the boundary conditions
The portfolio must be unwound when any part of the boundary is reached
19.*
Extension of the
Theoretical Framework
for Pricing Derivatives:
Martingales and Measures
Chapter 19
19.*
Derivatives Dependent on a Single Underlying Variable
19.*
Forming a Riskless Portfolio
19.*
Market Price of Risk (Page 500)
This shows that (m – r )/s is the same for all derivatives dependent only on the same underlying variable, q, and t.
We refer to (m – r )/s as the market price of risk for q and denote it by l
19.*
Differential Equation for ƒ
(Equation , page 501)
Using Ito’s lemma() to obtain expressions for m and s in terms of m and s. The equation
m-ls=r
becomes
19.*
Risk-Neutral Valuation
Compare the equation with (), it shows that q is like a stock price paying a dividend yield of r – m + ls
This analogy shows that we can value ƒ in a risk-neutral world providing the drift rate of q is reduced from m to m – ls
Note: When q is not the price of an investment asset, the risk-neutral valuation argument does not necessarily tell us anything about what would happen with q in a risk-neutral world .
19.*
Extension of the Analysis
to Several Underlying Variables
(Equations and , page 503)
19.*
Traditional Risk-Neutral Valuation with Several Underlying Variables
A derivative can always be valued as if the would is risk neutral, provided that the expected growth rate of each underlying variable is assumed to be mi-λisi rather than mi.
The volatility of the variables and the coefficient of the correlation between variables are not changed. (CIR,1985).
19.*
Derivatives Dependent on Commodity Prices (Page 506)
For a commodity the futures price gives the expected value in the traditional risk-neutral world.
19.*
Martingales (Page 507)
A martingale is a stochastic process with zero drfit
A martingale has the property that its expected future value equals its value today
19.*
Alternative Worlds
19.*
A Key Result (Page 509)
19.*
Forward Risk Neutrality
We refer to a world where the market price of risk is the volatility of g as a world that is forward risk neutral with respect to g.
If Eg denotes a world that is FRN wrt g
19.*
Aleternative Choices for the Numeraire Security g
Money Market Account
Zero-coupon bond price
Annuity factor
19.*
Money Market Account
as the Numeraire
The money market account is an account that starts at $1 and is always invested at the short-term risk-free interest rate
The process for the value of the account is
dg=rgdt
This has zero volatility. Using the money market account as the numeraire leads to the traditional risk-neutral world
19.*
Money Market Account
continued
19.*
Zero-Coupon Bond Maturing at time T as Numeraire
19.*
Forward Prices
In a world that is FRN wrt P(0,T), the expected value of a security at time T is its forward price
F=f0/P(0,T)
So, F=ET(fT)
f0=P(0,T)F
19.*
Interest Rates(p512-513)
In a world that is FRN wrt P(0,T2) the expected value of an interest rate lasting between times T1 and T2 is the forward interest rate
19.*
Annuity Factor as the Numeraire-1
Let Sn,N(t) is the forward swap rate of a swap starting at the time Tn with payment dates at times Tn+1, Tn+2,…,TN+1. Then the value of the fixed side of the swap is
19.*
Annuity Factor as the Numeraire-2
If we add $1 at time TN+1, the floating side of the swap is worth $1 at time Tn. So, the value of the floating side is: P(t,Tn)-P(t, TN+1)
Equating the values of the fixed and floating side we obstain:
19.*
Annuity Factor as the Numeraire-3
19.*
Extension to Several Independent Factors (Page 513)
19.*
Extension to Several Independent Factors
continued
19.*
Applications
(Section , page 514)
Valuation of a European call option when interest rates are stochastic
Valuation of an option to exchange one asset for another
19.*
Valuation of a European call option when interest rates are stochastic
Assume ST is lognormal with the standard deviation of ln(ST) equal to s then:
The result is the same as BS except r replaced by R.
19.*
Valuation of an option to exchange one asset(U) for another(V)
Choose U as the numeraire, and set f as the value of the option so that fT=max(VT-UT,0), so,
19.*
Change of Numeraire
(Section , page 517)
19.*
Quantos
(Section , page 518)
Quantos are derivatives where the payoff is defined using variables measured in one currency and paid in another currency
Example: contract providing a payoff of ST – K dollars ($) where S is the Nikkei stock index (a yen number)
19.*
Diff Swap
Diff swaps are a type of quanto
A floating rate is observed in one currency and applied to a principal in another currency
19.*
Quantos continued
19.*
Quantos continued
19.*
Siegel’s Paradox
19.*
Siegel’s Paradox(2)
In the process of dS, the numeraire is the money market account in currency Y. In the second equation, the numeraire is also the money market account in currency Y.
To change the numeraire from Y to X,the growth rate of 1/S increase by ρσVσS where V=1/S and ρ is the correlation between S and 1/ this case, ρ=-1, and σv =σ follows that the change of numeraire causes the growth rate of 1/S to increase – σS2.
2002-5-30
*
Interest Rate Derivatives:
The Standard Market Models
Chapter 20
2002-5-30
*
Why Interest Rate Derivatives are Much More Difficult to Value Than Stock Options
We are dealing with the whole term structure of interest rates; not a single variable
The probabilistic behavior of an individual interest rate is more complicated than that of a stock price
2002-5-30
*
Why Interest Rate Derivatives are Much More Difficult to Value Than Stock Options
Volatilities of different points on the term structure are different
Interest rates are used for discounting as well as for defining the payoff
2002-5-30
*
Main Approaches to Pricing
Interest Rate Options
Use a variant of Black’s model
Use a no-arbitrage (yield curve based) model
2002-5-30
*
Black’s Model & Its Extensions
Black’s model is similar to the Black-Scholes model used for valuing stock options
It assumes that the value of an interest rate, a bond price, or some other variable at a particular time T in the future has a lognormal distribution
2002-5-30
*
Black’s Model & Its Extensions
(continued)
The mean of the probability distribution is the forward value of the variable
The standard deviation of the probability distribution of the log of the variable is
where s is the volatility
The expected payoff is discounted at the T-maturity rate observed today
2002-5-30
*
Black’s Model (Eqn and , p 532)
X : strike price
r : zero coupon yield for maturity T
F0 : forward value of variable
T : option maturity
s : volatility
2002-5-30
*
The Black’s Model: Payoff Later Than Variable Being Observed
X : strike price
r * : zero coupon yield for maturity T *
F0 : forward value of variable
T : time when variable is observed
T * : time of payoff
s : volatility
2002-5-30
*
Validity of Black’s Model
Black’s model appears to make two approximations:
1. The expected value of the underlying variable is assumed to be its forward price. In traditional risk neutral world, E(VT)=Futures price of VT. But forward price and the futures price are not the same when the interest rates are stochastic.
2. The stochastic behavior of interest rate is not taken into account in the way the discounting is done.
We will see that these assumptions offset each other
2002-5-30
*
European Bond Options
When valuing European bond options it is usual to assume that the future bond price is lognormal
2002-5-30
*
European Bond Options
continued
F0 : forward bond price
X : strike price
P(0,T) : price of zero coupon
bond paying $1 at time T
T : life of the option
sB : volatility of price of
underlying bond
2002-5-30
*
Yield Vols vs Price Vols
The change in forward bond price is related to the change in forward bond yield by
where D is the (modified) duration of the forward bond at option maturity
2002-5-30
*
Yield Vols vs Price Vols
continued
This relationship implies the following approximation
where sy is the yield volatility and sB is the price volatility
Often sy is quoted with the understanding that this relationship will be used to calculate sB
2002-5-30
*
Theoretical Justification for Bond Option Model
2002-5-30
*
Caps
A cap is a portfolio of caplets
Each caplet can be regarded as a call option on a future interest rate with the payoff occurring in arrears
When using Black’s model we assume that the interest rate underlying each caplet is lognormal
2002-5-30
*
Black’s Model for Caps
(Equation , p. 540)
The value of a caplet, for period [tk, tk+1] is
Fk : forward interest rate
for (tk, tk+1)
sk : interest rate volatility
L: principal
RX : cap rate
dk=tk+1-tk
2002-5-30
*
When Applying Black’s Model
To Caps We Must ...
EITHER
Use forward volatilities
Volatility different for each caplet
OR
Use flat volatilities
Volatility same for each caplet within a particular cap but varies according to life of cap
2002-5-30
*
Theoretical Justification for Cap Model
2002-5-30
*
European Swaptions
When valuing European swap options it is usual to assume that the swap rate is lognormal
Consider a swaption which gives the right to pay RX on an n -year swap starting at time T . The payoff on each swap payment date is
where L is principal, m is payment frequency and R is market swap rate at time T
2002-5-30
*
European Swaptions continued
(Equation , page 545)
The value of the swaption is
F0 is the forward swap rate; s is the swap rate volatility; ti is the time from today until the i th swap payment; and
2002-5-30
*
Theoretical Justification for Swap Option Model
2002-5-30
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Relationship Between Swaptions and Bond Options
An interest rate swap can be regarded as the exchange of a fixed-rate bond for a floating-rate bond
A swaption or swap option is therefore an option to exchange a fixed-rate bond for a floating-rate bond
2002-5-30
*
Relationship Between Swaptions and Bond Options (continued)
At the start of the swap the floating-rate bond is worth par so that the swaption can be viewed as an option to exchange a fixed-rate bond for par
An option on a swap where fixed is paid & floating is received is a put option on the bond with a strike price of par
When floating is paid & fixed is received, it is a call option on the bond with a strike price of par
2002-5-30
*
Convexity Adjustments
We define the forward yield on a bond as the yield calculated from the forward bond price
There is a non-linear relation between bond yields & bond prices
It follows that when the forward bond price equals the expected future bond price, the forward yield does not necessarily equal the expected future yield
What is known as a convexity adjustment may be necessary to convert a forward yield to the appropriate expected future yield
2002-5-30
*
Relationship Between Bond Yields and Prices (Figure , page 548)
Bond
Price
Yield
Y3
B 1
Y1
Y2
B 3
B 2
2002-5-30
*
Analytic Approximation for
Convexity Adjustment (Eqn , p. 549)
Suppose a derivative depends on a bond yield, yT observed at time T . Define:
G(yT) : price of the bond as a function of its yield
y0 : forward bond yield at time zero
sy : forward yield volatility
The convexity adjustment that should be made to the forward bond yield is
2002-5-30
*
Example (page 549)
An instrument provides a payoff in 3 years = the 1-year zero-coupon rate multiplied by $100
Volatility is 20%
Yield curve is flat at 10% (with annual compounding)
The convexity adjustment is bps so that the value of the instrument is =
2002-5-30
*
LIBOR In Arrears Swap
In a LIBOR-in-arrears swap the floating rate paid on a payment date equals the rate observed on that date
We can value a LIBOR-in arrears swap by assuming that the floating rate between T1 and T2 is
where R0 is the forward rate, sR is the forward rate volatility, and t=T2-T1
2002-5-30
*
Timing Adjustments
When a variable is observed at time T1 and the resultant payoff occurs at time T2 rather than T1 , the growth rate of the variable should be increased by
where R is the forward interest rate between T1 and T2, sR is the volatility of R, F is the forward value of the variable, and sF is the volatility of F
2002-5-30
*
CMS Swap
A CMS swap is an interest rate swap where the floating rate equals the swap rate for a swap with a certain life.
To value a CMS swap we must make both a convexity adjustment and a timing adjustment to the forward swap rates
2002-5-30
*
When is a Convexity or Timing Adjustment Necessary
A convexity or timing adjustment is necessary when the payoff from a derivative does not incorporate the natural time lags between an interest rate being set and the interest payments being made
They are not necessary for a vanilla swap, a cap or a swap option
2002-5-30
*
Accrual Swaps (Section , page 556)
Accrual swaps are swaps where the interest on one side accrues only when the floating reference rate is within a certain range.
An accrual swap can be decomposed in a regular swap and a series of binary options
2002-5-30
*
Deltas of Interest Rate Derivatives
Alternatives:
Calculate a DV01 (the impact of a 1bps parallel shift in the zero curve)
Calculate impact of small change in the quote for each instrument used to calculate the zero curve
Divide zero curve (or forward curve) into buckets and calculate the impact of a shift in each bucket
Carry out a principal components analysis. Calculate delta with respect to the first two or three factors
*
Interest Rate Derivatives:
Models of the Short Rate
Chapter 21
*
Term Structure Models
Black’s model is concerned with describing the probability distribution of a single variable at a single point in time
A term structure model describes the evolution of the whole yield curve
*
Use of Risk-Neutral Arguments
The process for the instantaneous short rate, r, in the traditional risk-neutral world defines the process for the whole zero curve in this world
*
The value of an interest-rate derivative
The value at time t of an interest-rate derivative that provides a payoff of fT at time T is:
If P(t, T ) is the price at time t of a zero-coupon bond maturing at time T
*
Relationship between R(t,T) and r
If R(t,T) is the continuously compounded interest rate at time t for a term of T-t,
So:
*
Equilibrium Models
*
Mean Reversion
(Figure , page 566)
Interest
rate
HIGH interest rate has negative trend
LOW interest rate has positive trend
Reversion
Level
*
The Vasicek Model
The risk-neutral process for r is
So,
*
Alternative Term Structures
in Vasicek & CIR
(Figure , page 568)
Zero Rate
Maturity
Zero Rate
Maturity
Zero Rate
Maturity
*
Equilibrium vs No-Arbitrage Models
In an equilibrium model today’s term structure is an output
In a no-arbitrage model today’s term structure is an input
*
Developing No-Arbitrage
Model for r
A model for r can be made to fit the initial term structure by including a function of time in the drift
*
Ho & Lee(1986)
dr = q(t )dt + sdz
q(t )=Ft(0,t)+ s2t
Advantages: analytically tractable; It’s easy to apply and provides an exact fit to the current term structure of interest rate.
Disadvantages: It gives the user very little flexibility in choosing the volatility structure; It has no mean reversion.
*
Initial Forward
Curve
Short Rate
r
r
r
r
Time
Diagrammatic Representation of Ho and Lee
*
Hull and White Model(1990)
dr = [q(t ) – ar ]dt + sdz
Many analytic results for bond prices and option prices
Two volatility parameters, a and s
Interest rates normally distributed
*
Diagrammatic Representation of Hull and White
Short Rate
r
r
r
r
Time
Forward Rate
Curve
*
Options on Coupon Bearing Bonds
A European option on a coupon-bearing bond can be expressed as a portfolio of options on zero-coupon bonds.
We first calculate the critical interest rate (Rx) at the option maturity for which the coupon-bearing bond price equals the strike price at maturity
The strike price for each zero-coupon bond is set equal to its value when the interest rate (R) equals this critical value (Rx)
*
Interest Rate Trees vs Stock Price Trees
The variable at each node in an interest rate tree is the Dt period rate
Interest rate trees work similarly to stock price trees except that the discount rate used varies from node to node
*
Two-Step Tree Example
(Figure , page 579))
Payoff after 2 years is MAX[100(r – ), 0]
pu=; pm=; pd=; Time step=1yr
**
*
3
1
0
0
0
r
P
*: (×3 + ×1 + ×0)e–×1
**: (× + × +×0)e–×1
*
Alternative Branching Processes in a Trinomial Tree
(Figure , page 580)
(a)
(b)
(c)
*
An Overview of the Tree Building Procedure
dR = [q(t ) – aR ]dt + sdz
1. To construct a tree for R* that is initially zero and follow the process:
dR*=-aR*dt+ sdz
2. Draw a trinomial tree for R* to match the mean and standard deviation of the process for R*. The branching methods used at a node must lead to all three probabilities being positive. (p582)
3. Convert the tree for R* into a tree for R.
*
Example
s =
a =
Dt = 1 year
The zero curve is as shown in Table on page 584
*
The Initial Tree
(Figure , page 582)
A
B
C
D
E
F
G
H
I
Node
A
B
C
D
E
F
G
H
I
r
%
%
%
%
%
%
%
%
%
p
u
p
m
p
d
*
The Final Tree
(Figure , Page 585)
A
B
C
D
E
F
G
H
I
Node
A
B
C
D
E
F
G
H
I
r
% %
%
%
%
%
%
%
%
p
u
p
m
p
d
*
Procedure for r tree
ri,j=R*I,j+αi,i=Δt,2Δt,3Δt,…
α0=R0(Δt), QA=1
The value of a zero-coupon bond maturing at time 2Δt calculated by Using Ri,j must equal to that calculated by using R0(2Δt). The former is calculated by the equation (). α1is so decided. So on.
*
Extensions
The tree building procedure can be extended to cover more general models of the form:
d ƒ(r ) = [q(t ) – a ƒ(r )]dt + sdz
*
Other Models
These models allow the initial volatility environment to be matched exactly
But the future volatility structure may be quite different from the current volatility structure
*
Calibration
The volatility parameters a and s are chosen so that the model fits the prices of actively traded instruments such as caps and European swap options as closely as possible
*
Forward Rates vs Futures Rates
Convexity adjustment should be made when a forward interest rate is calculated from a Eurodollar futures rate quote.
For example, in Ho-Lee the futures rate should be reduced by s2t1t 2/2 where t1 is the maturity of the futures contract & t2 is the maturity of the rate underlying the futures
Suppose s = & the 10-year Eurodollar futures rate is 5%. What is the corresponding forward rate? %
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