bad. Options solve this problem. No wonder that active options trading started only a year or

two after the ¬rst stocks started trading.

The huge beta also means that options are very useful for hedging. If you have a large

illiquid portfolio, you can offset the risks very cheaply with options.

Finally, options allow you to shape the distribution of returns in interesting and sometimes

dangerous ways. For example, if you buy a 20% out of the money put option as well as a

stock, you have bought “catastrophe insurance” for your stock portfolio, at what turns out to

be a remarkably small price. You cut off the left tail of the return distribution, at a small cost

to the mean of the overall distribution.

On the other side, by writing out of the money put options, you can earn a small fee year

in and year out, only once in a while experiencing a huge loss. You have a large probability of

a small gain and a small probability of a large loss. You are providing catastrophe insurance

to the market, and it works much like, say, writing earthquake insurance. The distribution

of returns from this strategy is extremely non-normal, and thus statistical evaluation of its

properties will be dif¬cult. This strategy is tempting to a portfolio manager who is being

evaluated only by the statistics of his achieved return. If he writes far out of the money

options in addition to investing in an index, the chance of beating the index for one or even

¬ve years is extremely high. If the catastrophe does happen and he loses a billion dollars or

so, the worst you can do is ¬re him. (Your contract with the manager is always a call option.)

This is why portfolio management contracts are not purely statistical, but also write down

what kind of investments can and cannot be made.

Straddle

Call payoff

Put payoff

P+C

Straddle profit

Stock Price

Make money if stock ends up here

Figure 32. Payoff diagram for a straddle.

288

SECTION 17.1 BACKGROUND

Portfolios of put and call options are called strategies, and have additional interesting

properties. Figure 32 graphs the payoff of a straddle, which combines a put and call at the

same strike price. This strategy pays off if the stock goes up or goes down. It loses money

if the stock does not move. Thus the straddle is a bet on volatility. Of course, everyone

else understands this, and will bid the put and call prices up until the straddle earns only an

equilibrium rate of return. Thus, you invest in a straddle if you think that stock volatility is

higher than everyone else thinks it will be. Options allow ef¬cient markets and random walks

to operate on the second and higher moments of stocks as well as their overall direction! You

can also see quickly that volatility will be a central parameter in option prices. The higher

the volatility, the higher both put and call prices.

More generally, by combining options of various strikes, you can buy and sell any piece

of the return distribution. A complete set of options “ call options on every strike price “ is

equivalent to complete markets, i.e., it allows you to form payoffs that depend on the terminal

stock price in any way; you can form any payoff of the form f (ST ).

289

CHAPTER 17 OPTION PRICING

17.1.2 Prices: one period analysis

I use the law of one price “ existence of a discount factor “ and no-arbitrage “ existence

of a positive discount factor “ to characterize option prices. The results are: 1) Put-call

parity: P = C ’ S + X/Rf . 2) Arbitrage bounds, best summarized by Figure 34 3) The

proposition that you should never exercise an American call option early on a stock that pays

no dividends. The arbitrage bounds are a linear program, and this procedure can be used to

¬nd them in more complex situations where clever identi¬cation of arbitrage portfolios may

fail.

We have a set of interesting payoffs. Now what can we say about their prices “ their

values at dates before expiration? Obviously, p = E(mx) as always. We have learned about

x, now we have to think about m.

We can start by imposing little structure “ the law of one price and the absence of arbi-

trage, or, equivalently, the existence of some discount factor or a positive discount factor. .

In the case of options, these two principles actually do tell you a good deal about the option

price.

Put-Call parity

The law of one price, or the existence of some discount factor that prices stock, bond, and

a call option, allows us to deduce the value of a put in terms of the price of the stock, bond,

and call. Consider the following two strategies: 1) Hold a call, write a put, same strike price.

2) Hold stock, promise to pay X. The payoffs of these two strategies are the same, as shown

in Figure 33.

Equivalently, the payoffs are related by

PT = CT ’ ST + X.

Thus, so long as the law of one price holds, the prices of left and right hand sides must be

equal. Applying E(m·) to both sides for any m,

P = C ’ S + X/Rf .

(The price of ST is S. The price of the payoff X is X/Rf .)

Arbitrage bounds

If we add the absence of arbitrage, or equivalently the restriction that the discount factor

must be positive, we can deduce bounds on the call option price without needing to know the

put price. In this case, it is easiest to cleverly notice arbitrage portfolios “ situations in which

290

SECTION 17.1 BACKGROUND

Buy call Buy stock

+ = +

=

X ST Write put Borrow strike

-X

Figure 33. Put call parity.

portfolio A dominates portfolio B. Then, either directly from the de¬nition of no-arbitrage or

from A > B, m > 0 ’ E(mA) > E(mB), you can deduce that the price of A must be

greater than the price of B. The arbitrage portfolios are

1. CT > 0 ’ C > 0. The call payoff is positive so the call price must be positive.

2. CT ≥ ST ’ X ’ C ≥ S ’ X/Rf . The call payoff is better than hold stock - pay strike

for sure, so the call price is greater than holding the stock and borrowing the strike, i.e.

promising to pay it for sure.

3. CT ¤ ST ’ C ¤ S. The call payoff is worse than stock payoff (because you have to

pay the strike price. Thus, the call price is less than stock price.

Figure 34 summarizes these arbitrage bounds on the call option value. We have gotten

somewhere “ we have restricted the range of the option prices. However, the arbitrage bounds

are too large to be of much use. Obviously, we need to learn more about the discount factor

than pure arbitrage or m > 0 will allow. We could retreat to economic models, e.g. use the

CAPM or other explicit discount factor model. Option pricing is famous because we don™t

have to do that. Instead, if we open up dynamic trading“the requirement that the discount

factor price the stock and bond at every date to expiration“it turns out that we can sometimes

determine the discount factor and hence the option value precisely.

This presentation is unsettling for two reasons. First, you may worry that you will not be

clever enough to dream up dominating portfolios in more complex circumstances. Second,

you may worry that we have not dreamed up all of the arbitrage portfolios in this circum-

stance. Perhaps there is another one lurking out there, which would reduce the unsettling

large size of the bounds. It leaves us hungry for a constructive technique for ¬nding arbi-

291

CHAPTER 17 OPTION PRICING

C

Call value

Today

Call value C

In here

S

X/Rf

Stock value today

Figure 34. Arbitrage bounds for a call option

trage bounds that would be guaranteed to work in general situations, and to ¬nd the tightest

arbitrage bound.

We want to know Ct = E(mt,T xc ) wherexc = max(ST ’X, 0) denotes the call payoff,

T T

we want to use information in the observed stock and bond prices to learn about the option

price, and we want to impose the absence of arbitrage. We can capture this search with the

following problem:

(214)

max Ct = Et (mxC ) s.t. m > 0, St = Et (mST ), 1 = Et (mRf )

T

m

and the corresponding minimization. The ¬rst constraint implements absence of arbitrage.

The second and third use the information in the stock and bond price to learn what we can

about the option price.

Write 17.214 out in state notation,

X X X

π(s)m(s)xC (s) s.t. m(s) > 0, St = π(s)m(s)Rf

max Ct = π(s)m(s)ST (s), 1 =

T

{m(s)}

s s s

This is a linear program “ a linear objective and linear constraints. In situations where you

do not know the answer, you can calculate arbitrage bounds “ and know you have them all

“ by solving this linear program. I don™t know how you would begin to check that for every

portfolio A whose payoff dominates B, the price of A is greater than the price of B. The

discount factor method lets you construct the arbitrage bounds.

292

SECTION 17.2 BLACK-SCHOLES FORMULA

Early exercise

By applying the absence of arbitrage, we can show quickly that you should never exercise

an American call option on a stock that pays no dividends before the expiration date. This is a

lovely illustration because such a simple principle leads to a result that isn™t initially obvious.

Follow the table:

Payoffs CT = max(ST ’ X, 0) ≥ ST ’ X

Price S ’ X/Rf

C ≥

Rf > 1 C ≥ S’X

S ’ X is what you get if you exercise now. The value of the call is greater than this value,

because you can delay paying the strike, and because exercising early loses the option value.

Put-call parity let us concentrate on call options; this fact lets us concentrate on European

options.

17.2 Black-Scholes formula

Write a process for stock and bond, then use Λ— to price the option. The Black-Scholes

formula 17.220 results. You can either solve for the ¬nite-horizon discount factor ΛT /Λ0

and ¬nd the call option price by taking the expectation C0 = E0 (ΛT /Λ0 xC ), or you can ¬nd

T

a differential equation for the call option price and solve it backward.

Our objective, again, is to learn as much as we can about the value of an option, given

the value of the underlying stock and bond. The one-period analysis led only to arbitrage

bounds, at which point we had to start thinking about discount factor models. Now, we allow

intermediate trading, which means we really are thinking about dynamic multiperiod asset

pricing.

The standard approach to the Black-Scholes formula rests on explicitly constructing port-

folios: at each date we cleverly construct a portfolio of stock and bond that replicates the

instantaneous payoff of the option; we reason that the price of the option must equal the

price of the replicating portfolio. Instead, I follow the discount factor approach. The law of

one price is the same thing as the existence of a discount factor. Thus, rather than construct

law-of-one-price replicating portfolios, construct at each date a discount factor that prices the

stock and bond, and use that discount factor to price the option. the discount factor approach

shows how thinking of the world in terms of a discount factor is equivalent in the result and

as easy in the calculation as other approaches.

This case shows some of the interest and engineering complexity of continuous time

models. Though at each instant the analysis is trivial law of one price, chaining it together

293

CHAPTER 17 OPTION PRICING

over time is not trivial.

The call option payoff is

CT = max(ST ’ X, 0)

where X denotes the strike price X and ST denotes the stock price on the expiration date T .

The underlying stock follows

dS

= µdt + σdz.

S

There is also a money market security that pays the real interest rate rdt.

We want a discount factor that prices the stock and bond. All such discount factors are of

the form m = x— + w, E(xw) = 0. In continuous time, all such discount factors are of the

form

dΛ (µ ’ r)

= ’rdt ’ dz ’ σ w dw; E(dwdz) = 0.

Λ σ

(You can check that this set of discount factors does in fact price the stock and interest rate,

or take a quick look back at section 4.3.)

Now we price the call option with this discount factor, and show that the Black-Scholes

equation results. Importantly, the choice of discount factor via choice of σ w dw turns out to

have no effect on the resulting option price. Every discount factor that prices the stock and

interest rate gives the same value for the option price. The option is therefore priced using

the law of one price alone.

There are two paths to follow. Either we solve the discount factor forward, and then ¬nd

the call value by C = E(mxC ), or we characterize the price path and solve it backwards

from expiration.

17.2.1 Method 1: Price using discount factor

Let us use the discount factor to price the option directly:

½ ¾Z

ΛT ΛT

C0 = Et max (ST ’ X, 0) = max (ST ’ X, 0) df (ΛT , ST )

Λt Λt

where ΛT and ST are solutions to

dS

(17.215)

= µdt + σdz

S

dΛ µ’r

= ’rdt ’ dz ’ σ w dw.

Λ σ

I simplify the algebra by setting σ w dw to zero, anticipating that it does not matter. You can

294

SECTION 17.2 BLACK-SCHOLES FORMULA

reason that since S does not depend on dw, CT depends only on ST , so C will only depend

on S, and dw will have no effect on the answer. If this isn™t good enough, a problem asks

you to include the dw, trace through the remaining steps and verify that the answer does not

in fact depend on dw.

“Solving” a stochastic differential equation such as (17.215) means ¬nding the distribu-

tion of the random variables ST and ΛT , using information as of date 0. This is just what we

do with difference equations. For example, if we solve xt+1 = ρxt + µt+1 with µ normal

P

forward to xT = ρT x0 + T ρT ’j µj , we have expressed xT as a normally distributed ran-

j=1

P

dom variable with mean ρ x0 and standard deviation T ρ2(T ’j) . In the continuous time

T

j=1

case, it turns out that we can solve some nonlinear speci¬cations as well. Integrals of dz give

us shocks, as integrals of dt give us deterministic functions of time.

We can ¬nd analytical expressions for the solutions equations of the form (17.215). Start

with the stochastic differential equation

dY

(216)

= µY dt + σ Y dz.

Y

Write

µ ¶

dY 11 12

2

d ln Y = ’ dY = µY ’ σY dt + σ Y dZ

2Y2

Y 2

Integrating from 0 to T , (17.216) has solution

µ ¶

σ2

(217)

Y

ln YT = ln Y0 + µY ’ T + σY (zT ’ z0 )

2

zT ’ z0 is a normally distributed random variable with´ mean zero and variance T . Thus, ln Y

³

σ2

is conditionally normal with mean ln Y0 + µY ’ 2 T and variance σ2 T. You can check

Y

Y

this solution by differentiating it “ don™t forget the second derivative terms.

Applying the solution (17.217) to (17.215), we have

µ ¶

√

σ2

(17.218)

ln ST = ln S0 + µ ’ T + σ Tµ

2

Ã ¶2 !

µ

µ ’ r√

1 µ’r

ln ΛT = ln Λ0 ’ r + T’ Tµ

2 σ σ

where the random variable µ is

zT ’ z0

√

µ= ∼ N (0, 1) .

T

Having found the joint distribution of stock and discount factor, we evaluate the call

295

CHAPTER 17 OPTION PRICING

option value by doing the integral corresponding to the expectation,

Z ∞

ΛT

(17.219)

C0 = (ST ’ X) df (ΛT , ST )

Λt

ST =X

Z∞

ΛT (µ)

= (ST (µ) ’ X) df (µ)

Λt

ST =X

We know the joint distribution of the terminal stock price ST and discount factor ΛT

on the right hand side, so we have all the information we need to calculate this integral.

This example has enough structure that we can ¬nd an analytical formula. In more general

circumstances, you may have to resort to numerical methods. At the most basic level, you

can simulate the Λ, S process forward and then take the integral by summing over many such

simulations.

Doing the integral

Start by breaking up the integral (17.219) into two terms,

Z Z

∞ ∞

ΛT (µ) ΛT (µ)

C0 = ST (µ) df (µ) ’ X df (µ) .

Λt Λt

ST =X ST =X

ST and ΛT are both exponential functions of µ. The normal distribution is also an exponential

function of µ. Thus, we can approach this integral exactly as we approach the expectation of

a lognormal; we can merge the two exponentials in µ into one term, and express the result

as integrals against a normal distribution. Here we go. Plug in (17.218) for ST , ΛT , and

simplify the exponentials in terms of µ,

Z ³ ´ √

∞ √

2

’ r+ 1 ( µ’r ) T ’ µ’r T µ 12

S0 e(µ’ 2 σ )T +σ Tµ

C0 = e f (µ)dµ

2 σ σ

ST =X

Z∞ ³ ´ √

2

’ r+ 1 ( µ’r ) T ’ µ’r T µ

’X e f (µ) dµ

2 σ σ

ST =X

Z h ³ ´i √

∞ 2

µ’r’ 1 σ2 +( µ’r ) T +(σ’ µ’r ) T µ

= S0 e f(µ)dµ

2 σ σ

ST =X

Z∞ ³ ´ √

2

’ r+ 1 ( µ’r ) T ’ µ’r T µ

’X e f (µ)

2 σ σ

ST =X

Now add the normal distribution formula for f(µ),

1 ’ 1 µ2

f (µ) = e2.

2π

296

SECTION 17.2 BLACK-SCHOLES FORMULA

The result is

Z∞ h ³ ´i √

1 2

µ’r’ 1 σ 2 +( µ’r ) T +(σ’ µ’r ) T µ’ 1 µ2

= √ S0

C0 e dµ

2 σ σ 2

2π ST =X

Z∞ h i √

1 2

’ r+ 1 ( µ’r ) T ’ µ’r T µ’ 1 µ2

’√ X e dµ

2 σ σ 2

2π S =X

Z ∞T √2

1 µ’r

e’ 2 [µ’(σ’ σ ) T ] dµ

1

= √ S0

2π ST =X

Z∞ √2

1 ’ 1 (µ+ µ’r T )

’rT

’ √ Xe e dµ.

2 σ

2π ST =X

Notice that the integrals have the form of a normal distribution with nonzero mean. The lower

bound ST = X is, in terms of µ,

µ ¶ √

σ2

ln X = ln ST = ln S0 + µ ’ T + σ Tµ

2

³ ´

σ2

ln X ’ ln S0 ’ µ ’ T

2

√

µ= .

σT

Finally, we can express de¬nite integrals against a normal distribution by the cumulative

normal,

Z∞

1 2

1

e’ 2 (µ’µ) dµ = ¦ (µ ’ a)

√

2π a

i.e., ¦() is the area under the left tail of the normal distribution.

³ ´

«

ln X ’ ln S0 ’ µ ’ 2 T µ ¶

σ2

µ’r √

= S0 ¦ ’ √

C0 + σ’ T

σ

σ T ’t

³ ´

«

σ2

ln X ’ ln S0 ’ µ ’ 2 T µ ’ r√

’Xe’r(T ’t) ¦ ’ √ ’ T

σ

σT

Simplifying, we get the Black-Scholes formula

Ã ¤! Ã ¤!

£ £

12 12

ln S0 /X + r + 2 σ T ln S0 /X + r ’ 2 σ T

. (220)

’ Xe’rT ¦

√ √

C0 = S0 ¦

σT σT

297

CHAPTER 17 OPTION PRICING

17.2.2 Method 2: Derive Black-Scholes differential equation

Rather than solve the discount factor forward and then integrate, we can solve the price

backwards from expiration. The instantaneous or expected return formulation of a pricing

model amounts to a differential equation for prices.

Guess that the solution for the call price is a function of stock price and time to expiration,

Ct = C(S, t). Use Ito™s lemma to ¬nd derivatives of C(S, t),

1

dC = Ct dt + CS dS + CSS dS 2

2

· ¸

1 22

dC = Ct + CS Sµ + CSS S σ dt + CS Sσdz

2

Plugging into the basic asset pricing equation

0 = Et (dΛC) = CEt dΛ + ΛEt dC + Et dΛdC,

using Et (dΛ/Λ) = ’rdt and canceling Λdt, we get

1

0 = ’rC + Ct + CS Sµ + CSS S 2 σ2 ’ S (µ ’ r) CS

2

or,

1

(221)

0 = ’rC + Ct + SrCS + CSS S 2 σ 2 .

2

This is the Black-Scholes differential equation for the option price.

We now know a differential equation for the price function C(S, t). We know the value of

this function at expiration, C(ST , T ) = max(ST ’ X, 0). The remaining task is to solve this

differential equation backwards through time. Conceptually, and numerically, this is easy.

Express the differential equation as

‚C(S, t) 1 ‚ 2 C(S, t) 2 2

‚C(S, t)

’ = ’rC(S, t) + Sr + Sσ.

2 ‚S 2

‚t ‚S

At any point in time, you know the values of C(S, t) for all S “ for example, you can store

them on a grid for S. Then, you can take the ¬rst and second derivatives with respect to S

and form the quantity on the right hand side at each value of S. Now, you can ¬nd the option

price at any value of S, one instant earlier in time.

This differential equation, solved with boundary condition

C = max {ST ’ X, 0}

has an analytic solution “ the familiar formula (17.220). One standard way to solve differ-

ential equations is to guess and check; and by taking derivatives you can check that (17.220)

298

SECTION 17.3 PROBLEMS

does satisfy (17.221). Black and Scholes solved the differential equation with a fairly com-

plicated Fourier transform method. The more elegant Feynman-Kac solution amounts to

showing that solutions of the partial differential equation (17.221) can be represented as in-

tegrals of the form that we already derived independently as in (17.219). (See Duf¬e 1992

p.87)

17.3 Problems

1. We showed that you should never exercise an American call early if there are no

dividends. Is the same true for American puts, or are there circumstances in which it is

optimal to exercise American puts early?

2. Retrace the steps in the integral derivation of the Black-Scholes formula and show that

the dw does not affect the ¬nal result.

299

Chapter 18. Option pricing without

perfect replication

18.1 On the edges of arbitrage

The Black-Scholes formula is justly famous and launched a thousand techniques for option

pricing. The principle of no-arbitrage pricing is obvious, but its application leads to many

subtle and unanticipated pricing relationships.

However, in many practical situations, the law of one price arguments that we used in

the Black-Scholes formula break down. If options really were redundant, it is unlikely that

they would be traded as separate assets. It really is easy to synthesize forward rates from

zero-coupon bonds, and forward rates are not separately traded or quoted.

We really cannot trade continuously, and trying to do so would drown a strategy in trans-

actions costs. As a practical example, at the time of the 1987 stock market crash, several

prominent funds were trying to follow “portfolio insurance” strategies, essentially synthe-

sizing put options by systematically selling stocks as prices declined. During the time of

the crash, however, they found that the markets just dried up “ they were unable to sell as

prices plummeted. We model this situation mathematically as a Poisson jump, a discontin-

uous movement in prices. In the face of such jumps the call option payoff is not perfectly

hedged by a portfolio of stock and bond, and cannot be priced as such.

Generalizations of the stochastic setup lead to the same result. If the interest rate or

stock volatility are stochastic, we do not have securities that allow us to perfectly hedge the

corresponding shocks, so the law of one price again breaks down.

In addition, many options are written on underlying securities that are not traded, or not

traded continually and with suf¬cient liquidity. Real options in particular “ the option to

build a factory in a particular location “ are not based on a tradeable underlying security, so

the logic behind Black-Scholes pricing does not apply. Executives are speci¬cally forbidden

to short stock in order to hedge executive options.

Furthermore, applications of option pricing formulas to trading activities seem to suffer a

strange inconsistency. We imagine that the stock and bond are perfectly priced and perfectly

liquid “ available for perfect hedging. Then, we search for options that are priced incorrectly

as trading opportunities. If the options can be priced wrong, why can™t the stock and bond

be priced wrong? We should treat all assets symmetrically in evaluating trading opportuni-

ties. Trading opportunities also involve risk, and a theory that pretends they are arbitrage

opportunities does not help much to quantify that risk.

In all of these situations, an unavoidable “basis risk” creeps in between the option payoff

and the best possible hedge portfolio. Holding the option entails some risk, and the value

of the option depends on the “market price” of that risk “ the covariance of the risk with an

300

SECTION 18.2 ONE-PERIOD GOOD DEAL BOUNDS

appropriate discount factor.

Nonetheless, we would like not to give up and go back to the consumption-based model,

factor models, or other “absolute” methods that try to price all assets. We are still willing to

take as given the prices of lots of assets in determining the price of an option, and in particular

assets that will be used to hedge the option. We can form an “approximate hedge” or portfolio

of basis assets “closest to” the focus payoff, and we can hedge most of the option™s risk with

that approximate hedge. Then, the uncertainty about the option payoff is reduced only to

¬guring out the price of the residual.

In addition, since the residuals are small, we might be able to say a lot about option

prices with much weaker restrictions on the discount factor than those suggested by absolute

models.

In this chapter, I survey “good deal” option price bounds, a technique that Jesus Saá-

Requejo and I (1999) advocated for this situation. The good deal bounds amount to system-

atically searching over all possible assignments of the “market price of risk” of the residual,

constraining the total market price of risk to a reasonable value, and imposing no arbitrage

opportunities, to ¬nd upper and lower bounds on the option price. It is not equivalent to pric-

ing options with pure Sharpe ratio arguments. The concluding section of this chapter surveys

some alternative and additional techniques.

18.2 One-period good deal bounds

We want to price the payoff xC , for example, xC = max(ST ’ K, 0) for a call option. We

have in hand a N ’dimensional vector of basis payoffs x, whose prices p we can observe, for

example the stock and bond. The good deal bound ¬nds the minimum and maximum value of

xC by searching over all positive discount factors that price the basis assets and have limited

volatility:

C = max E(mxC ) s.t. p = E(mx), m ≥ 0, σ2 (m) ¤ h/Rf (222)

{m}

The corresponding minimization yields the lower bound C. This is a one-period discrete-

time problem. The Black-Scholes formula does not apply because you can™t trade between

the price and payoff periods.

The ¬rst constraint on the discount factor imposes the price of the basis assets. We want

to do as much relative pricing as possible; we want to extend what we know about the prices

of x to price xC , without worrying about where the prices of x come from. The second

constraint imposes the absence of arbitrage. This problem without the last constraint yields

the arbitrage bounds that we studied in section 2. In most situations, the arbitrage bounds are

too wide to be of much use.

The last is an additional constraint on discount factors, and the extra content of good-

deal vs. arbitrage bounds. It is a relatively weak restriction. We could obtain closer bounds

301

CHAPTER 18 OPTION PRICING WITHOUT PERFECT REPLICATION

on prices with more information about the discount factor. In particular, if we know the

correlation of the discount factor with the payoff xC we could price the option a lot better!

As m > 0 means that no portfolios priced by m may display an arbitrage opportunity,

σ (m) ¤ h/Rf means that no portfolio priced by m may have a Sharpe ratio greater than h.

2

Recall E(mRe ) = 0 implies E(m)E(Re ) = ’ρσ(m)σ(Re ) and |ρ| ¤ 1.

It is a central advantage of a discount factor approach that we can easily impose both the

discount factor volatility constraint and positivity, merging the lessons of factor models and

option pricing models. The prices and payoffs generated by discount factors that satisfy both

m ≥ 0 and σ(m) ¤ h/Rf do more than rule out arbitrage opportunities and high Sharpe

ratios.

I™ll treat the case that there is a riskfree rate, so we can write E(m) = 1/Rf . In this case,

it is more convenient to express the volatility constraint as a second moment, so the bound

(18.222) becomes

¡¢

(223)

C = min E (m xc ) s.t. p = E (mx) , E m2 ¤ A2 , m ≥ 0,

{m}

where A2 ≡ (1 +h2 )/Rf 2 . The problem is a standard minimization with two inequality con-

straints. Hence we ¬nd a solution by trying all the combinations of binding and nonbinding

constraints, in order of their ease of calculation. 1) Assume the volatility constraint binds and

the positivity constraint is slack. This one is very easy to calculate, since we will ¬nd ana-

lytic formulas for the solution. If the resulting discount factor m is nonnegative, this is the

solution. If not, 2) assume that the volatility constraint is slack and the positivity constraint

binds. This is the classic arbitrage bound. Find the minimum variance discount factor that

generates the arbitrage bound. If this discount factor satis¬es the volatility constraint, this is

the solution. If not, 3) solve the problem with both constraints binding.

18.2.1 Volatility constraint binds, positivity constraint is slack

If the positivity constraint is slack, the problem reduces to

¡¢

C = min E(m xc ) s.t. p = E (mx) , E m2 ¤ A2 . (224)

{m}

We could solve this problem directly, choosing m in each state with Lagrange multipliers

on the constraints. But as with the mean-variance frontier it is much more elegant to set up

orthogonal decompositions and then let the solution pop out.

Figure 35 describes the idea. X denotes the space of payoffs of portfolios of the basis

assets x, a stock and a bond in the classic Black-Scholes setup. Though graphed as a line, X

is typically a larger space. We know all prices in X, but the payoff xc that we wish to value

does not lie in X.

302

SECTION 18.2 ONE-PERIOD GOOD DEAL BOUNDS

Start by decomposing the focus payoff xc into an approximate hedge xc and a residual w,

ˆ

xc = xc + w,

ˆ

xc ≡ proj(xc |X) = E(xc x0 )E(xx0 )’1 x,

ˆ

≡ xc ’ xc . (18.225)

w ˆ

We know the price of xc . We want to bound the price of the residual w to learn as much as

ˆ

we can about the price of xc .

All discount factors that price x “ that satisfy p = E(mx)“ lie in the plane through x— .

As we sweep through these discount factors, we generate any price from ’∞ to ∞ for the

residual w and hence payoff xc . All positive discount factors m > 0 lie in the intersection

of the m plane and the positive orthant “ the triangular region. Discount factors m in this

range generate a limited range of prices for the focus payoff “ the arbitrage bounds. Since

second moment de¬nes distance in Figure 35, the set of discount factors that satis¬es the

volatility constraint E(m2 ) ¤ A2 lies inside a sphere around the origin. The circle in Figure

35 shows the intersection of this sphere with the set of discount factors. This restricted range

of discount factors will produce a restricted range of values for the residual w and hence a

restricted range of values for the focus payoff xc . In the situation I have drawn, the positivity

constraint is slack, since the E(m2 ) ¤ A2 circle lies entirely in the positive orthant.

We want to ¬nd the discount factors in the circle that minimize or maximize the price

of the residual w. The more a discount factor points in the w direction, the larger a price

E(mw) it assigns to the residual. Obviously, the discount factors that maximize or minimize

the price of w point as much as possible towards and away from w. If you add any movement

µ orthogonal to w, this increases discount factor volatility without changing the price of w.

Hence, the discount factor that generates the lower bound is

(226)

m = x— ’ vw

where

s

A2 ’ E(x—2 )

(227)

v=

E(w2 )

is picked to just satisfy the volatility constraint. The bound is

(228)

C = E(mxc ) = E(x— xc ) ’ vE(w2 )

The upper bound is given by v = ’v

The ¬rst term in equation (18.228) is the value of the approximate hedge portfolio, and

can be written several ways, including

(229)

E(x— xc ) = E(x— xc ) = E(mˆc )

ˆ x

303

CHAPTER 18 OPTION PRICING WITHOUT PERFECT REPLICATION

xc X

w x

xc

ˆ

x * + vw

E (m 2 ) < A2

x*

m>0

x * ’vw

Figure 35. Construction of a discount factor to solve the one-period good deal bound when

the positivity constraint is slack.

for any discount factor m that prices basis assets. (Don™t forget, E(xy) = E[x proj(y|X)].)

The second term in equation (18.228) is the lowest possible price of the residual w consistent

with the discount factor volatility bound:

vE(w2 ) = E(vw w) = E[(x— + vw)w] = E(mw).

For calculations you can substitute the de¬nitions of x— and w in equation (18.228) to

obtain an explicit, if not very pretty, formula:

p p

C = p0 E(xx0 )’1 E(xxc ) ’ A2 ’ p0 E(xx0 )’1 p E(xc2 ) ’ E(xc x0 )E(xx0 )’1 E(xxc ).

(230)

304

SECTION 18.2 ONE-PERIOD GOOD DEAL BOUNDS

The upper bound C is the same formula with a + sign in front of the square root.

Using (18.226), check whether the discount factor is positive in every state of nature. If

so, this is the good-deal bound, and the positivity constraint is slack. If not, proceed to the

next step.

If you prefer an algebraic and slightly more formal argument, start by noticing that any

discount factor that satis¬es p = E(mx) can be decomposed as

m = x— + vw + µ

where E(x— w) = E(x— µ) = E(wµ). Check these properties from the de¬nition of w and µ;

this is just like R = R— + wRe— + n. Our minimization problem is then

min E(mxc ) s.t. E(m2 ) ¤ A2

{v,µ}

¡ —2 ¢

min E [((x— + vw + µ) (ˆc + w)] s.t. E x + v2 E(w2 ) + E(µ2 ) ¤ A2

x

{v,µ}

¡¢

min E(x— xc ) + vE(w2 ) s.t. E x—2 + v2 E(w2 ) + E(µ2 ) ¤ A2

ˆ

{v,µ}

q2

’E(x—2

The solution is µ = 0 and v = ± A E(w2 ) ) .

18.2.2 Both constraints bind

Next, I ¬nd the bounds when both constraints bind. Though this is the third step in the pro-

cedure, it is easiest to describe this case ¬rst. Introducing Lagrange multipliers, the problem

is

δ £ ¡ 2¢ ¤

max E (m xc ) + »0 [E (mx) ’ p] + E m ’ A2

C = min

2

{m>0} {»,δ>0}

The ¬rst order conditions yield a discount factor that is a truncated linear combination of the

payoffs,

µc ¶·c ¸+

x + »0 x x + »0 x

(231)

m = max ’ ,0 = ’ .

δ δ

The last equality de¬nes the []+ notation for truncation. In ¬nance terms, this is a call option

with zero strike price.

You can derive (18.231) by introducing a Kuhn-Tucker multiplier π(s)ν(s) on m > 0

305

CHAPTER 18 OPTION PRICING WITHOUT PERFECT REPLICATION

and taking partial derivatives with respect to m in each state,

" #

X X

π(s)m(s)xc (s) + »0

C = min π(s)m(s)x(s) ’ p

{m}

s s

" #

δX X

2 2

+ π(s)m(s) ’ A + π(s)ν(s)m(s)

2s s

1‚

: xc (s) + »0 x(s) + δm(s) + ν(s) = 0 (232)

π(s) ‚s

If the positivity constraint is slack, the Kuhn-Tucker multiplier ν(s) is zero,

xc (s) + »0 x(s)

m(s) = ’ .

δ

If the positivity constraint binds, then m(s) = 0, and ν(s) is just enough to make (18.232)

hold. In sum, we have (18.231).

We could plug expression (18.231) into the constraints, and solve numerically for La-

grange multipliers » and δ that enforce the constraints. Alas, this procedure requires the

solution of a system of nonlinear equations in (», δ), which is often a numerically dif¬cult or

unstable problem.

Hansen, Heaton and Luttmer (1995) show how to recast the problem as a maximization,

which is numerically much easier. Interchanging min and max,

δ £ ¡ 2¢ ¤

min E(m xc ) + »0 [E (mx) ’ p] + E m ’ A2 . (233)

C = max

2

{»,δ>0} {m>0}

The inner minimization yields the same ¬rst order conditions (18.231). Plugging those ¬rst-

order conditions into the outer maximization of (18.233) and simplifying, we obtain

( )

·c 0 ¸+2

δ x +»x δ

’ »0 p’ A2 . (234)

C = max E ’ ’

2 δ 2

{»,δ>0}

You can search numerically over (», δ) to ¬nd the solution to this problem. The upper bound

is found by replacing max with min and replacing δ > 0 with δ < 0.

18.2.3 Positivity binds, volatility is slack

If the volatility constraint is slack and the positivity constraint binds, the problem reduces to

(235)

C = min E (m xc ) s.t. p = E (mx) , m > 0.

{m}

306

SECTION 18.2 ONE-PERIOD GOOD DEAL BOUNDS

These are the arbitrage bounds. We found these bounds in section 2 for a call option by just

being clever. If you can™t be clever, (18.235) is a linear program.

We still have to check that the discount factor volatility constraint can be satis¬ed at the

arbitrage bound. Denote the lower arbitrage bound by Cl . The minimum variance (second

moment) discount factor that generates the arbitrage bound Cl solves

· ¸ µ· ¸¶

p x

2 2

E(m )min = min E(m ) s.t =E m , m > 0.

xc

Cl

{m}

Using the same conjugate method, this problem is equivalent to

n o

+2

2 c 0

’ 2v0 p’2µCl .

E(m )min = max ’E [’ (µx + v x)]

{v,µ}

Again, search numerically for (v,µ) to solve this problem. If E(m2 )min ¤ A, Cl is the

solution to the good-deal bound; if not we proceed with the case that both constraints are

binding described above.

18.2.4 Application to Black-Scholes

The natural ¬rst exercise with this technique is to see how it applies in the Black-Scholes

world. Keep in mind, this is the Black-Scholes world with no intermediate trading; compare

the results to the arbitrage bounds, not to the Black-Scholes formula. Figure 36, taken from

Cochrane and Saá-Requejo (1999) presents the upper and lower good-deal bounds for a call

option on the S&P500 index with strike price K = $100, and three months to expiration. We

used parameter values E(R) = 13%, σ(R) = 16% for the stock index return and an riskfree

rate Rf = 5%. The discount factor volatility constraint is twice the historical market Sharpe

ratio, h = 2 — E(R ’ Rf )/σ(R) = 1.0. To take the expectations required in the formula,

we evaluated integrals against the lognormal stock distribution.

The ¬gure includes the lower arbitrage bounds C ≥ 0, C ≥ K/Rf . The upper arbitrage

bound states that C ¤ S, but this 45—¦ line is too far up to ¬t on the vertical scale and still see

anything else. As in many practical situations, the arbitrage bounds are so wide that they are

of little use. The upper good-deal bound is much tighter than the upper arbitrage bound. For

example, if the stock price is $95, the entire range of option prices between the upper bound

of $2 and the upper arbitrage bound of $95 is ruled out.

The lower good-deal bound is the same as the lower arbitrage bound for stock prices less

than about $90 and greater than about $110. In this range, the positivity constraint binds and

the volatility constraint is slack. This range shows that it is important to impose both volatility

and positivity constraints. Good deal bounds are not just the imposition of low Sharpe ratios

on options. (I emphasize it because this point causes a lot of confusion.) The volatility bound

alone admits negative prices. A free out of the money call option is like a lottery ticket: it is

an arbitrage opportunity, but its expected return/standard deviation ratio is terrible, because

the standard deviation is so high. A Sharpe ratio criterion alone will not rule it out.

307

CHAPTER 18 OPTION PRICING WITHOUT PERFECT REPLICATION

Figure 36. Good deal option price bounds as a function of stock price. Options have

three months to expiration and strike price K = $100. The bounds assume no trading until

expiration, and a discount factor volatility bound h = 1.0 corresponding to twice the market

Sharpe ratio. The stock is lognormally distributed with parameters calibrated to an index

option.

In between $90 and $110, the good-deal bound improves on the lower arbitrage bound.

It also improves on a bound that only imposes only the volatility constraint. In this region,

both positivity and volatility constraints bind. This fact has an interesting implication: Not all

values outside the good-deal bounds imply high Sharpe ratios or arbitrage opportunities. Such

values might be generated by a positive but highly volatile discount factor, and generated by

another less volatile but sometimes negative discount factor, but no discount factor generates

these values that is simultaneously nonnegative and respects the volatility constraint.

It makes sense rule out these values. If we know that an investor will invest in any ar-

bitrage opportunity or take any Sharpe ratio greater than h, then we know that his unique

marginal utility satis¬es both restrictions. He would ¬nd a utility-improving trade for values

outside the good-deal bounds, even though those values may not imply a high Sharpe ratio,

an arbitrage opportunity, or any other simple portfolio interpretation.

The right thing to do is to intersect restrictions on the discount factor. Simple portfolio

308

SECTION 18.3 MULTIPLE PERIODS AND CONTINUOUS TIME

interpretations, while historically important, are likely to fall by the wayside as we add more

discount factor restrictions or intersect simple ones.

18.3 Multiple periods and continuous time

Now, on to the interesting case. Option pricing is all about dynamic hedging, even if imper-

fect dynamic hedging. Good deal bounds would be of little use if we could only apply them

to one-period environments.

18.3.1 The bounds are recursive

The central fact that makes good deal bounds tractable in dynamic environments is that the

bounds are recursive. Today™s bound can be calculated as the minimum price of tomorrow™s

bound, just as today™s option price can be calculated as the value of tomorrow™s option price.

To see that the bounds are recursive, consider a two-period version of the problem,

E0 (m1 m2 xc ) s.t.

C0 = min 2

{m1 , m2 }

(236)

pt = Et (mt+1 pt+1 ); Et (m2 ) ¤ A2 , mt+1 > 0, t = 0, 1.

t+1 t

This two period problem is equivalent to a series of one period problems, in which the C0

problem ¬nds the lowest price of the C1 lower bound,

C 1 = min E1 (m2 xc ) ; C 0 = min E0 (m1 C 1 )

2

{m2 } {m1 }

subject to (18.236). Why? The solution to the two-period problem min E0 (m1 E1 (m2 xc ))

must minimize E1 (m2 xc ) in each state of nature at time 1. If not, you could lower E1 (m2 xc )

without affecting the constraints, and lower the objective. Note that this recursive property

only holds if we impose m > 0. If m1 < 0 were possible we would want to maximize

E1 (m2 xc ) in some states of nature.

18.3.2 Basis risk and real options

The general case leads to some dense formulas, so a simple example will let us understand

the idea most simply. Let™s value a European call option on an event V that is not a traded

asset, but is correlated with a traded asset that can be used as an approximate hedge. This

situation is common with real options and non¬nancial options and describes some ¬nancial

options on illiquid assets.

309

CHAPTER 18 OPTION PRICING WITHOUT PERFECT REPLICATION

The terminal payoff is

xc = max(VT ’ K, 0).

T

Model the joint evolution of the traded asset S and the event V on which the option is written

as

dS

= µS dt + σS dz,

S

dV

= µV dt + σ V z dz + σ V w dw.

V

The dw risk cannot be hedged by the S asset, so the market price of dw risk “ its correlation

with the discount factor “ will matter to the option price.

We are looking for a discount factor that prices S and rf , has instantaneous volatility

A, and generates the largest or smallest price for the option. Hence, it will have the largest

loading on dw possible. By analogy with the one period case (18.226), you can quickly see

that the discount factor will have the form

q

dΛ—

dΛ

± A2 ’ h2 dw

= S

—

Λ Λ

dΛ—

= ’rdt ’ hS dz

Λ—

µS ’ r

hS = .

σS

dΛ— /Λ— is the familiar analogue to x— that prices stock and bond. We add a loading on the

orthogonal shock dw just suf¬cient to satisfy the constraint Et (dΛ2 /Λ2 ) = A2 . One of ±

will generate the upper bound, and one will generate the lower bound.

Now that we have the discount factor, the good deal bound is given by

· ¸

ΛT

C t = Et max(VT ’ K) .

Λt

St , Vt , and Λt are all diffusions with constant coef¬cients. Therefore, ST , VT and ΛT are

jointly lognormally distributed, so the double integral de¬ning the expectation is straightfor-

ward to perform, and works very similarly to the integral we evaluated to solve the Black-

Scholes formula in section 2.1. (If you get stuck, see Cochrane and Saá-Requejo 1999 for the

algebra.)

The result is

µ ¶ µ ¶

1√ 1√

C or C = V0 e·T φ d + σ V T ’ Ke’rT φ d ’ σV T (237)

2 2

310

SECTION 18.3 MULTIPLE PERIODS AND CONTINUOUS TIME

where φ(·) denotes the left tail of the normal distribution and

dV 2

σ2≡ Et 2 = σ 2 z + σ2 w

V V V

V

ln(V0 /K) + (· + r) T

√

d≡

σV T

s

" Ã !#

p

2

A 2

·≡ hV ’ hS ρ ’ a 2 ’1 1’ρ σV

hS

µS ’ r µ ’r

; hV ≡ V

hS ≡

σS σ

¶V

µ

dV dS σV z

ρ ≡ corr , =

VS σV

½

+1 upper bound

.

a=

’1 lower bound

This expression is exactly the Black-Scholes formula with the addition of the · term.

µV enters the formula because the event V may not grow at the same rate as the asset

S. Obviously, the correlation ρ between V shocks and asset shocks enters the formula, and as

this correlation declines, the bounds widen. The bounds also widen as the volatility constraint

A becomes larger relative to the asset Sharpe ratios hS .

Market prices of risk

Continuous-time pricing problems are often speci¬ed in terms of “market prices of risk”

rather than discount factors. This is the instantaneous Sharpe ratio that an asset must earn if

it loads on a speci¬c shock. If an asset has a price process P that loads on a shock σdw, then

its expected return must be

µ ¶

dP dΛ

’ rf dt = ’σEt

Et dw

P Λ

with Sharpe ratio

µ ¶

Et dP ’ rf dt dΛ

P

»= = ’Et dw .

σ Λ

I have introduced the common notation » for the market price of risk. Thus, problems are

often attacked by making assumptions about » directly and then proceeding from

dP

’ rf dt = »σ.

Et

P

In this language, the market price of stock risk is hS and can be measured by observing

the stock, and does not matter when you can price by arbitrage (notice it is missing from

the Black-Scholes formula). Our problem comes down to choosing the market price of dw

311

CHAPTER 18 OPTION PRICING WITHOUT PERFECT REPLICATION

risk, which cannot be measured by observing a traded asset, in such a way as to minimize or

q

h2 + »2 ¤ A.

maximize the option price, subject to a constraint that the total price of risk S

18.3.3 Continuous time

Now, a more systematic expression of the same ideas in continuous time. As in the option

pricing case in the last chapter and the term structure case in the next chapter, we will obtain

a differential characterization. To actually get prices, we have either to solve the discount

factor forward, or to ¬nd a differential equation for prices which we solve backward.

Basis assets

In place of E(x), E(xx0) etc., model the price processes of an nS -dimensional vector of

basis assets by a diffusion,

dS

(238)

= µS (S, V, t)dt + σS (S, V, t)dz; E(dz dz 0 ) = I

S

Rather than complicate£the notation, understand division to operate element by element on

¤

vectors, e.g., dS/S = dS1 /S1 dS2 /S2 · · · . The basis assets may pay dividends at

rate D(S, V, t)dt.

V represents an nV -dimensional vector of additional state variables that follow

dV = µV (S, V, t)dt + σ V z (S, V, t)dz + σV w (S, V, t)dw; E(dw dw0 ) = I; E(dw dz 0 ) = 0.

(239)

This could include a stochastic stock volatility or stochastic interest rate “ classic cases in

which the Black-Scholes replication breaks down. Again, I keep it simple by assuming there

is a risk free rate r(S, V, t)dt.

The problem

We want to value an asset that pays continuous dividends at rate xc (S, V, t)dt and with a

terminal payment xc (S, V, T ). Now we must choose a discount factor process to minimize

T

the asset™s value

µ ¶

Z T

Λs c ΛT c

(240)

Ct = min Et x ds + Et x

Λt s Λt T

{Λs , t<s¤T } s=t

subject to the constraints that 1) the discount factor prices the basis assets S, r at each

moment in time, 2) the instantaneous volatility of the discount factor process is less than a

prespeci¬ed value A2 and 3) the discount factor is positive Λs > 0, t ¤ s ¤ T .

One period at a time; differential statement

312

SECTION 18.3 MULTIPLE PERIODS AND CONTINUOUS TIME

Since the problem is recursive, we can study how to move one step back in time,

Z t+∆t ¡ ¢

Λs xc ds + Et Λt+∆t C t+∆t

C t Λt = min Et s

{Λs } s=t

or, for small time intervals,

C t Λt = min Et {xc ∆t + (C t + ∆C) (Λt + ∆Λ)} .

{∆Λ}

Letting ∆t ’ 0, we can write the objective in differential form,

xc Et [d (ΛC)]

0 = t dt + min (241)

,

C ΛC

{dΛ}

subject to the constraints. We can also write (18.241) as

µ ¶

dC xc dΛ dC

+ t dt ’ rf dt = ’ min Et (242)

Et .

C C ΛC

{dΛ}

Since the second and third terms on the left hand side are ¬xed, the condition sensibly tells

us to ¬nd the lowest value C by maximizing the drift of the bound at each date. You should

recognize the form of (18.241) and (18.242) as the basic pricing equations in continuous time,

relating expected returns to covariance with discount factors.

Constraints

Now we express the constraints. As in the discrete time case, we orthogonalize the dis-

count factor in m = x— + µ form, and then the solution pops out. Any discount factor that

prices the basis assets is of the form

dΛ—

dΛ

(243)

= — ’ vdw

Λ Λ

where

dΛ—

≡ ’rdt ’ µ0 Σ’1 σ S dz

˜S S

Λ—

D

’ r; ΣS = σS σ0 .

µS

˜ ≡ µS + S

S

and v is a 1 — nV matrix. We can add shocks orthogonal to dw if we like, but they will have

no effect on the answer; the minimization will say to set such loadings to zero.

The volatility constraint is

dΛ2

1

Et 2 ¤ A2

dt Λ

313

CHAPTER 18 OPTION PRICING WITHOUT PERFECT REPLICATION

and hence, using (18.243),

1 dΛ—2

Et —2 = A2 ’ µ0 Σ’1 µS . (244)

vv0 ¤ A2 ’ ˜S S ˜

dt Λ

By expressing the constraints via (18.243) and (18.244), we have again reduced the prob-

lem of choosing the stochastic process for Λ to the choice of loadings v on the noises dw with

unknown values, subject to a quadratic constraint on vv0 . Since we are picking differentials

and have ruled out jumps, the positivity constraint is slack so long as Λ > 0.

Market prices of risk

Using equation (18.243), v is the vector of market prices of risks of the dw shocks “ the

expected return that any asset must offer if its shocks are dw:

µ ¶

1 dΛ

’E dw = v.

dt Λ

Thus, the problem is equivalent to: ¬nd at each date the assignment of market prices of risk

to the dw shocks that minimizes (maximizes) the focus payoff value, subject to the constraint

that the total (sum of squared) market price of risk is bounded by A2 .

Now, we™re ready to follow the usual steps. We can characterize a differential equation for

the option price that must be solved back from expiration, or we can try to solve the discount

factor forward and take an expectation.

Solutions: the discount factor and bound drift at each instant

We can start by characterizing the bound™s process, just as the basis assets follow (18.238).

This step is exactly the instantaneous analogue of the one-period bound without a positivity

constraint, so remember that logic if the equations start to get a bit forbidding.

Guess that lower bound C follows a diffusion process, and ¬gure out what the coef¬cients

must look like. Write

dC

(245)

= µC (S, V, t)dt + σCz (S, V, t)dz + σCw (S, V, t)dw.

C