Spot and Forward Rates under Continuous Compounding
• The pricing formula:
Ce−iS(i) + F e−nS(n).
• The market discount function:
d(n) = e−nS(n).
• The spot rate is an arithmetic average of forward rates, S(n) = f (0, 1) + f (1, 2) + · · · + f (n − 1, n)
Spot and Forward Rates under Continuous Compounding (concluded)
• The formula for the forward rate:
f (i, j) = jS(j) − iS(i) j − i .
• The one-period forward rate:
f (j, j + 1) = − ln d(j + 1) d(j) .
f (T ) ≡ lim
∆T →0f (T, T + ∆T ) = S(T ) + T ∂S
• f (T ) > S(T ) if and only if ∂S/∂T > 0.
Unbiased Expectations Theory
• Forward rate equals the average future spot rate,
f (a, b) = E[ S(a, b) ]. (14)
• Does not imply that the forward rate is an accurate predictor for the future spot rate.
• Implies the maturity strategy and the rollover strategy produce the same result at the horizon on the average.
Unbiased Expectations Theory and Spot Rate Curve
• Implies that a normal spot rate curve is due to the fact that the market expects the future spot rate to rise.
– f (j, j + 1) > S(j + 1) if and only if S(j + 1) > S(j) from Eq. (12) on p. 116.
– So E[ S(j, j + 1) ] > S(j + 1) > · · · > S(1) if and only if S(j + 1) > · · · > S(1).
• Conversely, the spot rate is expected to fall if and only if the spot rate curve is inverted.
• The theory has been rejected by most empirical studies with the possible exception of the period prior to 1915.
• Since the term structure has been upward sloping about 80% of the time, the theory would imply that investors have expected interest rates to rise 80% of the time.
• Riskless bonds, regardless of their different maturities, are expected to earn the same return on the average.
• That would mean investors are indifferent to risk.
A “Bad” Expectations Theory
• The expected returns on all possible riskless bond strategies are equal for all holding periods.
(1 + S(2))2 = (1 + S(1)) E[ 1 + S(1, 2) ] (15) because of the equivalency between buying a two-period bond and rolling over one-period bonds.
• After rearrangement, 1
E[ 1 + S(1, 2) ] = 1 + S(1) (1 + S(2))2.
A “Bad” Expectations Theory (continued)
• Now consider two one-period strategies.
– Strategy one buys a two-period bond and sells it after one period.
– The expected return is
E[ (1 + S(1, 2))−1 ] (1 + S(2))2.
– Strategy two buys a one-period bond with a return of 1 + S(1).
• The theory says the returns are equal:
1 + S(1)
(1 + S(2))2 = E
1 + S(1, 2)
A “Bad” Expectations Theory (concluded)
• Combine this with Eq. (15) on p. 131 to obtain E
1 + S(1, 2)
E[ 1 + S(1, 2) ].
• But this is impossible save for a certain economy.
– Jensen’s inequality states that E[ g(X) ] > g(E[ X ]) for any nondegenerate random variable X and
strictly convex function g (i.e., g00(x) > 0).
– Use g(x) ≡ (1 + x)−1 to prove our point.
Local Expectations Theory
• The expected rate of return of any bond over a single period equals the prevailing one-period spot rate:
(1 + S(1, n))−(n−1) ¤
(1 + S(n))−n = 1 + S(1) for all n > 1.
• This theory is the basis of many interest rate models.
• To handle more general types of spot rate curve changes, define a vector [ c1, c2, . . . , cn ] that characterizes the
perceived type of change.
– Parallel shift: [ 1, 1, . . . , 1 ].
– Twist: [ 1, 1, . . . , 1, −1, . . . , −1 ].
– · · ·
• Let P (y) ≡ P
i Ci/(1 + S(i) + yci)i be the price associated with the cash flow C1, C2, . . . .
• Define duration as
−∂P (y)/P (0)
Fundamental Statistical Concepts
There are three kinds of lies:
lies, damn lies, and statistics.
— Benjamin Disraeli (1804–1881) One death is a tragedy, but a million deaths are a statistic.
— Josef Stalin (1879–1953)
• The variance of a random variable X is defined as Var[ X ] ≡ E £
(X − E[ X ])2 ¤ .
• The covariance between random variables X and Y is Cov[ X, Y ] ≡ E [ (X − µX)(Y − µY ) ] ,
where µX and µY are the means of X and Y , respectively.
• Random variables X and Y are uncorrelated if Cov[ X, Y ] = 0.
• The standard deviation of X is the square root of the variance,
σX ≡ p
Var[ X ] .
• The correlation (or correlation coefficient) between X and Y is
ρX,Y ≡ Cov[ X, Y ] σXσY ,
provided both have nonzero standard deviations.
Variance of Sum
• Variance of a weighted sum of random variables equals Var
" n X
aiaj Cov[ Xi, Xj ].
• It becomes
a2i Var[ Xi ] when Xi are uncorrelated.
• “X | I” denotes X conditional on the information set I.
• The information set can be another random variable’s value or the past values of X, say.
• The conditional expectation E[ X | I ] is the expected value of X conditional on I; it is a random variable.
• The law of iterated conditional expectations:
E[ X ] = E[ E[ X | I ] ].
• If I2 contains at least as much information as I1, then E[ X | I1 ] = E[ E[ X | I2 ] | I1 ]. (16)
The Normal Distribution
• A random variable X has the normal distribution with mean µ and variance σ2 if its probability density
• This is expressed by X ∼ N (µ, σ2).
• The standard normal distribution has zero mean, unit variance, and the distribution function
Prob[ X ≤ z ] = N (z) ≡ 1
Moment Generating Function
• The moment generating function of random variable X is
θX(t) ≡ E[ etX ].
• The moment generating function of X ∼ N (µ, σ2) is θX(t) = exp
µt + σ2t2 2
Distribution of Sum
• If Xi ∼ N (µi, σi2) are independent, then X
Xi ∼ N
• Let Xi ∼ N (µi, σi2), which may not be independent.
• Then Xn
tiXi ∼ N
ti µi, Xn
titj Cov[ Xi, Xj ]
• Xi are said to have a multivariate normal distribution.
Generation of Univariate Normal Distributions
• Let X be uniformly distributed over (0, 1 ] so that Prob[ X ≤ x ] = x for 0 < x ≤ 1.
• Repeatedly draw two samples x1 and x2 from X until ω ≡ (2x1 − 1)2 + (2x2 − 1)2 < 1.
• Then c(2x1 − 1) and c(2x2 − 1) are independent standard normal variables where
c ≡ p
−2(ln ω)/ω .
A Dirty Trick and a Right Attitude
• Let ξi are independent and uniformly distributed over (0, 1).
• A simple method to generate the standard normal variable is to calculate
ξi − 6.
• But “this is not a highly accurate approximation and should only be used to establish ballpark estimates.”a
aJ¨ackel, Monte Carlo Methods in Finance (2002).
A Dirty Trick and a Right Attitude (concluded)
• Always blame your random number generator last.a
• Instead, check your programs first.
a“The fault, dear Brutus, lies not in the stars but in ourselves that we are underlings.” William Shakespeare (1564–1616), Julius Caesar.
Generation of Bivariate Normal Distributions
• Pairs of normally distributed variables with correlation ρ can be generated.
• X1 and X2 be independent standard normal variables.
U ≡ aX1, V ≡ ρU + p
1 − ρ2 aX2.
• U and V are the desired random variables with Var[ U ] = Var[ V ] = a2 and Cov[ U, V ] = ρa2.
The Lognormal Distribution
• A random variable Y is said to have a lognormal distribution if ln Y has a normal distribution.
• Let X ∼ N (µ, σ2) and Y ≡ eX.
• The mean and variance of Y are
µY = eµ+σ2/2 and σY2 = e2µ+σ2
eσ2 − 1
– They follow from E[ Y n ] = enµ+n2σ2/2.
The shift toward options as the center of gravity of finance [ . . . ]
— Merton H. Miller (1923–2000)
Calls and Puts
• A call gives its holder the right to buy a number of the underlying asset by paying a strike price.
• A put gives its holder the right to sell a number of the underlying asset for the strike price.
• How to price options?
• When a call is exercised, the holder pays the strike price in exchange for the stock.
• When a put is exercised, the holder receives from the writer the strike price in exchange for the stock.
• An option can be exercised prior to the expiration date:
American and European
• American options can be exercised at any time up to the expiration date.
• European options can only be exercised at expiration.
• An American option is worth at least as much as an
otherwise identical European option because of the early exercise feature.
• C: call value.
• P : put value.
• X: strike price.
• S: stock price.
• D: dividend.
• A call will be exercised only if the stock price is higher than the strike price.
• A put will be exercised only if the stock price is less than the strike price.
• The payoff of a call at expiration is C = max(0, S − X).
• The payoff of a put at expiration is P = max(0, X − S).
• At any time t before the expiration date, we call max(0, St − X) the intrinsic value of a call.
• At any time t before the expiration date, we call max(0, X − St) the intrinsic value of a put.
• A call is in the money if S > X, at the money if S = X, and out of the money if S < X.
• A put is in the money if S < X, at the money if S = X, and out of the money if S > X.
• Options that are in the money at expiration should be exercised.a
• Finding an option’s value at any time before expiration is a major intellectual breakthrough.
a11% of option holders let in-the-money options expire worthless.
20 40 60 80 Price Long a put
10 20 30 40 50 Payoff
20 40 60 80 Price
Short a put
-50 -40 -30 -20 -10
20 40 60 80 Price
Long a call
10 20 30 40 Payoff
20 40 60 80 Price
Short a call
-40 -30 -20 -10
80 85 90 95 100 105 110 115 Stock price
0 5 10 15 20
80 85 90 95 100 105 110 115 Stock price
0 2 4 6 8 10 12 14
• Exchange-traded stock options are not cash dividend-protected (or simply protected).
– The option contract is not adjusted for cash dividends.
• The stock price falls by an amount roughly equal to the amount of the cash dividend as it goes ex-dividend.
• Cash dividends are detrimental for calls.
• The opposite is true for puts.
Stock Splits and Stock Dividends
• Options are adjusted for stock splits.
• After an n-for-m stock split, the strike price is only
m/n times its previous value, and the number of shares covered by one contract becomes n/m times its
• Exchange-traded stock options are adjusted for stock dividends.
• Options are assumed to be unprotected.
• Consider an option to buy 100 shares of a company for
$50 per share.
• A 2-for-1 split changes the term to a strike price of $25 per share for 200 shares.
• Short selling (or simply shorting) involves selling an asset that is not owned with the intention of buying it back later.
– If you short 1,000 XYZ shares, the broker borrows them from another client to sell them in the market.
– This action generates proceeds for the investor.
– The investor can close out the short position by buying 1,000 XYZ shares.
– Clearly, the investor profits if the stock price falls.
• Not all assets can be shorted.
Payoff of Stock
20 40 60 80 Price
Long a stock
20 40 60 80 Payoff
20 40 60 80 Price
Short a stock
-80 -60 -40 -20
Covered Position: Hedge
• A hedge combines an option with its underlying stock in such a way that one protects the other against loss.
• Protective put: A long position in stock with a long put.
• Covered call: A long position in stock with a short call.a
• Both strategies break even only if the stock price rises, so they are bullish.
aA short position has a payoff opposite in sign to that of a long position.
85 90 95 100 105 110Stock price Protective put
-2.5 2.5 5 7.5 10 12.5
85 90 95 100 105 110Stock price Covered call
-12 -10 -8 -6 -4 -2 2 Profit
Solid lines are current value of the portfolio.
Covered Position: Spread
• A spread consists of options of the same type and on the same underlying asset but with different strike prices or expiration dates.
• We use XL, XM, and XH to denote the strike prices with XL < XM < XH.
• A bull call spread consists of a long XL call and a short XH call with the same expiration date.
– The initial investment is CL − CH.
– The maximum profit is (XH − XL) − (CL − CH).
– The maximum loss is CL − CH.
85 90 95 100 105 110Stock price Bull spread (call)
-4 -2 2 4 Profit
Covered Position: Spread (continued)
• Writing an XH put and buying an XL put with identical expiration date creates the bull put spread.
• A bear spread amounts to selling a bull spread.
• It profits from declining stock prices.
• Three calls or three puts with different strike prices and the same expiration date create a butterfly spread.
– The spread is long one XL call, long one XH call, and short two XM calls.
85 90 95 100 105 110Stock price Butterfly
-1 1 2 3 Profit
Covered Position: Spread (concluded)
• A butterfly spread pays off a positive amount at
expiration only if the asset price falls between XL and XH.
• A butterfly spread with a small XH − XL approximates a state contingent claim, which pays $1 only when a
particular price results.
• The price of a state contingent claim is called a state price.
Covered Position: Combination
• A combination consists of options of different types on the same underlying asset, and they are either all
bought or all written.
• Straddle: A long call and a long put with the same strike price and expiration date.
• Since it profits from high volatility, a person who buys a straddle is said to be long volatility.
• Selling a straddle benefits from low volatility.
• Strangle: Identical to a straddle except that the call’s strike price is higher than the put’s.
85 90 95 100 105 110Stock price Straddle
-5 -2.5 2.5 5 7.5 10 Profit
85 90 95 100 105 110Stock price Strangle
-2 2 4 6 8 10 Profit
Arbitrage in Option Pricing
All general laws are attended with inconveniences, when applied to particular cases.
— David Hume (1711–1776)
• The no-arbitrage principle says there is no free lunch.
• It supplies the argument for option pricing.
• A riskless arbitrage opportunity is one that, without any initial investment, generates nonnegative returns under all circumstances and positive returns under some.
• In an efficient market, such opportunities do not exist (for long).
• The portfolio dominance principle says portfolio A
should be more valuable than B if A’s payoff is at least as good under all circumstances and better under some.
• A portfolio yielding a zero return in every possible scenario must have a zero PV.
– Short the portfolio if its PV is positive.
– Buy it if its PV is negative.
– In both cases, a free lunch is created.
The PV Formula Justified
P = Pn
i=1 Cid(i) for a certain cash flow C1, C2, . . . , Cn.
• If the price P∗ < P , short the zeros that match the security’s n cash flows and use P∗ of the proceeds P to buy the security.
• Since the cash inflows of the security will offset exactly the obligations of the zeros, a riskless profit of P − P∗ dollars has been realized now.
• If the price P∗ > P , a riskless profit can be realized by reversing the trades.
6 6 6 6 -
C1 C2 C3
· · · Cn
? ? ? ?
C1 C2 C3
· · ·
Two More Examples
• An American option cannot be worth less than the intrinsic value.
– Otherwise, one can buy the option, promptly exercise it and sell the stock with a profit.
• A put or a call must have a nonnegative value.
– Otherwise, one can buy it for a positive cash flow now and end up with a nonnegative amount at expiration.
Relative Option Prices
• These relations hold regardless of the probabilistic model for stock prices.
• Assume, among other things, that there are no
transactions costs or margin requirements, borrowing and lending are available at the riskless interest rate, interest rates are nonnegative, and there are no
• Let the current time be time zero.
• PV(x) stands for the PV of x dollars at expiration.
• Hence PV(x) = xd(τ ) where τ is the time to expiration.
Put-Call Parity (Castelli, 1877)
C = P + S − PV(X). (19)
• Consider the portfolio of one short European call, one long European put, one share of stock, and a loan of PV(X).
• All options are assumed to carry the same strike price and time to expiration, τ .
• The initial cash flow is therefore C − P − S + PV(X).
• At expiration, if the stock price Sτ ≤ X, the put will be worth X − Sτ and the call will expire worthless.
The Proof (concluded)
• After the loan, now X, is repaid, the net future cash flow is zero:
0 + (X − Sτ) + Sτ − X = 0.
• On the other hand, if Sτ > X, the call will be worth Sτ − X and the put will expire worthless.
• After the loan, now X, is repaid, the net future cash flow is again zero:
−(Sτ − X) + 0 + Sτ − X = 0.
• The net future cash flow is zero in either case.
• The no-arbitrage principle implies that the initial
Consequences of Put-Call Parity
• There is only one kind of European option because the other can be replicated from it in combination with the underlying stock and riskless lending or borrowing.
– Combinations such as this create synthetic securities.
• S = C − P + PV(X) says a stock is equivalent to a
portfolio containing a long call, a short put, and lending PV(X).
• C − P = S − PV(X) implies a long call and a short put amount to a long position in stock and borrowing the PV of the strike price (buying stock on margin).
Lemma 1 An American call or a European call on a non-dividend-paying stock is never worth less than its intrinsic value.
• The put-call parity implies
C = (S − X) + (X − PV(X)) + P ≥ S − X.
• Recall C ≥ 0.
• It follows that C ≥ max(S − X, 0), the intrinsic value.
• An American call also cannot be worth less than its intrinsic value.
Intrinsic Value (concluded)
A European put on a non-dividend-paying stock may be worth less than its intrinsic value (p. 159).
Lemma 2 For European puts, P ≥ max(PV(X) − S, 0).
• Prove it with the put-call parity.
• Can explain the right figure on p. 159 why P < X − S when S is small.
Early Exercise of American Calls
European calls and American calls are identical when the underlying stock pays no dividends.
Theorem 3 (Merton (1973)) An American call on a non-dividend-paying stock should not be exercised before expiration.
• By an exercise in text, C ≥ max(S − PV(X), 0).
• If the call is exercised, the value is the smaller S − X.
• The above theorem does not mean American calls should be kept until maturity.
• What it does imply is that when early exercise is being considered, a better alternative is to sell it.
• Early exercise may become optimal for American calls on a dividend-paying stock.
– Stock price declines as the stock goes ex-dividend.
Early Exercise of American Calls: Dividend Case
Surprisingly, an American call should be exercised only at a few dates.
Theorem 4 An American call will only be exercised at expiration or just before an ex-dividend date.
In contrast, it might be optimal to exercise an American put even if the underlying stock does not pay dividends.
Convexity of Option Prices
Lemma 5 For three otherwise identical calls or puts with strike prices X1 < X2 < X3,
CX2 ≤ ωCX1 + (1 − ω) CX3 PX2 ≤ ωPX1 + (1 − ω) PX3 Here
ω ≡ (X3 − X2)/(X3 − X1).
(Equivalently, X2 = ωX1 + (1 − ω) X3.)
Option on Portfolio vs. Portfolio of Options
An option on a portfolio of stocks is cheaper than a portfolio of options.
Theorem 6 Consider a portfolio of non-dividend-paying assets with weights ωi. Let Ci denote the price of a
European call on asset i with strike price Xi. Then the call on the portfolio with a strike price X ≡ P
i ωiXi has a value at most P
i ωiCi. All options expire on the same date.
The same result holds for European puts.