would go to him and ignore you because they would be getting a better deal. But then a

third person would decide to sell the option for less than your competition but more than

W0 , say at $3.50. This would continue as long as any one would try to sell an option above

price W0 .

We will examine this problem of pricing options in more complicated contexts, and

while doing so, it will become apparent where the formulas for ∆0 and W0 came from. At

this point, we want to make a few observations.

Remark 6.1. First of all, if 1 + r > u, one would never buy stock, since one can always

do better by putting money in the bank. So we may suppose 1 + r < u. We always have

1 + r ≥ 1 > d. If we set

1+r’d u ’ (1 + r)

p= , q= ,

u’d u’d

then p, q ≥ 0 and p + q = 1. Thus p and q act like probabilities, but they have nothing to

do with P and Q. Note also that the price V0 = W0 does not depend on P or Q. It does

depend on p and q, which seems to suggest that there is an underlying probability which

controls the option price and is not the one that governs the stock price.

Remark 6.2. There is nothing special about European call options in our argument

above. One could let V1u and Vd1 be any two values of any option, which are paid out if the

26

stock goes up or down, respectively. The above analysis shows we can exactly duplicate

the result of buying any option V by instead buying some shares of stock. If in some model

one can do this for any option, the market is called complete in this model.

Remark 6.3. If we let P be the probability so that S1 = uS0 with probability p and

S1 = dS0 with probability q and we let E be the corresponding expectation, then some

algebra shows that

1

V0 = E V1 .

1+r

This will be generalized later.

Remark 6.4. If one buys one share of stock at time 0, then one expects at time 1 to

have (P u + Qd)S0 . One then divides by 1 + r to get the value of the stock in today™s

dollars. (r, the risk-free interest rate, can also be considered the rate of in¬‚ation. A dollar

tomorrow is equivalent to 1/(1 + r) dollars today.) Suppose instead of P and Q being the

probabilities of going up and down, they were in fact p and q. One would then expect to

have (pu + qd)S0 and then divide by 1 + r. Substituting the values for p and q, this reduces

to S0 . In other words, if p and q were the correct probabilities, one would expect to have

the same amount of money one started with. When we get to the binomial asset pricing

model with more than one step, we will see that the generalization of this fact is that the

stock price at time n is a martingale, still with the assumption that p and q are the correct

probabilities. This is a special case of the fundamental theorem of ¬nance: there always

exists some probability, not necessarily the one you observe, under which the stock price

is a martingale.

Remark 6.5. Our model allows after one time step the possibility of the stock going up or

going down, but only these two options. What if instead there are 3 (or more) possibilities.

Suppose for example, that the stock goes up a factor u with probability P , down a factor

d with probability Q, and remains constant with probability R, where P + Q + R = 1.

The corresponding price of a European call option would be (uS0 ’ K)+ , (dS0 ’ K)+ , or

(S0 ’ K)+ . If one could replicate this outcome by buying and selling shares of the stock,

then the “no arbitrage” rule would give the exact value of the call option in this model.

But, except in very special circumstances, one cannot do this, and the theory falls apart.

One has three equations one wants to satisfy, in terms of V1u , V1d , and V1c . (The “c” is

a mnemonic for “constant.”) There are however only two variables, ∆0 and V0 at your

disposal, and most of the time three equations in two unknowns cannot be solved.

Remark 6.6. In our model we ruled out the cases that P or Q were zero. If Q = 0,

that is, we are certain that the stock will go up, then we would always invest in the stock

if u > 1 + r, as we would always do better, and we would always put the money in the

bank if u ¤ 1 + r. Similar considerations apply when P = 0. It is interesting to note that

27

the cases where P = 0 or Q = 0 are the only ones in which our derivation is not valid.

It turns out that in more general models the true probabilities enter only in determining

which events have probability 0 or 1 and in no other way.

28

7. The multi-step binomial asset pricing model.

In this section we will obtain a formula for the pricing of options when there are n

time steps, but each time the stock can only go up by a factor u or down by a factor d.

The “Black-Scholes” formula we will obtain is already a nontrivial result that is useful.

We assume the following.

(1) Unlimited short selling of stock

(2) Unlimited borrowing

(3) No transaction costs

(4) Our buying and selling is on a small enough scale that it does not a¬ect the market.

We need to set up the probability model. „¦ will be all sequences of length n of H™s

and T ™s. S0 will be a ¬xed number and we de¬ne Sk (ω) = uj dk’j S0 if the ¬rst k elements

of a given ω ∈ „¦ has j occurrences of H and k ’ j occurrences of T . (What we are doing is

saying that if the j-th element of the sequence making up ω is an H, then the stock price

goes up by a factor u; if T , then down by a factor d.) Fk will be the σ-¬eld generated by

S0 , . . . , S k .

Let

(1 + r) ’ d u ’ (1 + r)

p= , q=

u’d u’d

and de¬ne P(ω) = pj q n’j if ω has j appearances of H and n ’ j appearances of T . We

observe that under P the random variables Sk+1 /Sk are independent and equal to u with

probability p and d with probability q. To see this, let Yk = Sk /Sk’1 . Thus Yk is the

factor the stock price goes up or down at time k. Then P(Y1 = y1 , . . . , Yn = yn ) = pj q n’j ,

where j is the number of the yk that are equal to u. On the other hand, this is equal to

P(Y1 = y1 ) · · · P(Yn = yn ). Let E denote the expectation corresponding to P.

The P we construct may not be the true probabilities of going up or down. That

doesn™t matter - it will turn out that using the principle of “no arbitrage,” it is P that

governs the price.

Our ¬rst result is the fundamental theorem of ¬nance in the current context.

Proposition 7.1. Under P the discounted stock price (1 + r)’k Sk is a martingale.

Proof. Since the random variable Sk+1 /Sk is independent of Fk , we have

E [(1 + r)’(k+1) Sk+1 | Fk ] = (1 + r)’k Sk (1 + r)’1 E [Sk+1 /Sk | Fk ].

Using the independence the conditional expectation on the right is equal to

E [Sk+1 /Sk ] = pu + qd = 1 + r.

29

Substituting yields the proposition.

Let ∆k be the number of shares held between times k and k + 1. We require ∆k

to be Fk measurable. ∆0 , ∆1 , . . . is called the portfolio process. Let W0 be the amount

of money you start with and let Wk be the amount of money you have at time k. Wk is

the wealth process. If we have ∆k shares between times k and k + 1, then at time k + 1

those shares will be worth ∆k Sk+1 . The amount of cash we hold between time k and k + 1

is Wk minus the amount held in stock, that is, Wk ’ ∆k Sk . At time k + 1 this is worth

(1 + r)[Wk ’ ∆k Sk ]. Therefore

Wk+1 = ∆k Sk+1 + (1 + r)[Wk ’ ∆k Sk ].

Note that in the case where r = 0 we have

Wk+1 ’ Wk = ∆k (Sk+1 ’ Sk ),

or

k

∆i (Si+1 ’ Si ).

Wk+1 = W0 +

i=0

This is a discrete version of a stochastic integral. Since

E [Wk+1 ’ Wk | Fk ] = ∆k E [Sk+1 ’ Sk | Fk ] = 0,

it follows that in the case r = 0 that Wk is a martingale. More generally

Proposition 7.2. Under P the discounted wealth process (1 + r)’k Wk is a martingale.

Proof. We have

(1 + r)’(k+1) Wk+1 = (1 + r)’k Wk + ∆k [(1 + r)’(k+1) Sk+1 ’ (1 + r)’k Sk ].

Observe that

E [∆k [(1 + r)’(k+1) Sk+1 ’ (1 + r)’k Sk | Fk ]

= ∆k E [(1 + r)’(k+1) Sk+1 ’ (1 + r)’k Sk | Fk ] = 0.

The result follows.

Our next result is that the binomial model is complete. It is easy to lose the idea

in the algebra, so ¬rst let us try to see why the theorem is true.

For simplicity let us ¬rst consider the case r = 0. Let Vk = E [V | Fk ]; by Propo-

sition 4.3 we see that Vk is a martingale. We want to construct a portfolio process, i.e.,

30

choose ∆k ™s, so that Wn = V . We will do it inductively by arranging matters so that

Wk = Vk for all k. Recall that Wk is also a martingale.

Suppose we have Wk = Vk at time k and we want to ¬nd ∆k so that Wk+1 = Vk+1 .

At the (k + 1)-st step there are only two possible changes for the price of the stock and so

since Vk+1 is Fk+1 measurable, only two possible values for Vk+1 . We need to choose ∆k

so that Wk+1 = Vk+1 for each of these two possibilities. We only have one parameter, ∆k ,

to play with to match up two numbers, which may seem like an overconstrained system of

equations. But both V and W are martingales, which is why the system can be solved.

Now let us turn to the details. In the following proof we allow r ≥ 0.

Theorem 7.3. The binomial asset pricing model is complete.

The precise meaning of this is the following. If V is any random variable that is Fn

measurable, there exists a constant W0 and a portfolio process ∆k so that the wealth

process Wk satis¬es Wn = V . In other words, starting with W0 dollars, we can trade

shares of stock to exactly duplicate the outcome of any option V .

Proof. Let

Vk = (1 + r)k E [(1 + r)’n V | Fk ].

By Proposition 4.3 (1 + r)’k Vk is a martingale. If ω = (t1 , . . . , tn ), where each ti is an H

or T , let

Vk+1 (t1 , . . . , tk , H, tk+2 , . . . , tn ) ’ Vk+1 (t1 , . . . , tk , T, tk+2 , . . . , tn )

∆k (ω) = .

Sk+1 (t1 , . . . , tk , H, tk+2 , . . . , tn ) ’ Sk+1 (t1 , . . . , tk , T, tk+2 , . . . , tn )

Set W0 = V0 , and we will show by induction that the wealth process at time k equals Vk .

The ¬rst thing to show is that ∆k is Fk measurable. Neither Sk+1 nor Vk+1 depends

on tk+2 , . . . , tn . So ∆k depends only on the variables t1 , . . . , tk , hence is Fk measurable.

Now tk+2 , . . . , tn play no role in the rest of the proof, and t1 , . . . , tk will be ¬xed,

so we drop the t™s from the notation. If we write Vk+1 (H), this is an abbreviation for

Vk+1 (t1 , . . . , tk , H, tk+2 , . . . , tn ).

We know (1 + r)’k Vk is a martingale under P so that

Vk = E [(1 + r)’1 Vk+1 | Fk ] (7.1)

1

= [pVk+1 (H) + qVk+1 (T )].

1+r

(See Note 1.) We now suppose Wk = Vk and want to show Wk+1 (H) = Vk+1 (H) and

Wk+1 (T ) = Vk+1 (T ). Then using induction we have Wn = Vn = V as required. We show

the ¬rst equality, the second being similar.

31

Wk+1 (H) = ∆k Sk+1 (H) + (1 + r)[Wk ’ ∆k Sk ]

= ∆k [uSk ’ (1 + r)Sk ] + (1 + r)Vk

Vk+1 (H) ’ Vk+1 (T )

Sk [u ’ (1 + r)] + pVk+1 (H) + qVk+1 (T )

=

(u ’ d)Sk

= Vk+1 (H).

We are done.

Finally, we obtain the Black-Scholes formula in this context. Let V be any option

that is Fn -measurable. The one we have in mind is the European call, for which V =

(Sn ’ K)+ , but the argument is the same for any option whatsoever.

Theorem 7.4. The value of the option V at time 0 is V0 = (1 + r)’n E V .

Proof. We can construct a portfolio process ∆k so that if we start with W0 = (1+r)’n E V ,

then the wealth at time n will equal V , no matter what the market does in between. If

we could buy or sell the option V at a price other than W0 , we could obtain a riskless

pro¬t. That is, if the option V could be sold at a price c0 larger than W0 , we would sell

the option for c0 dollars, use W0 to buy and sell stock according to the portfolio process

∆k , have a net worth of V + (1 + r)n (c0 ’ W0 ) at time n, meet our obligation to the buyer

of the option by using V dollars, and have a net pro¬t, at no risk, of (1 + r)n (c0 ’ W0 ).

If c0 were less than W0 , we would do the same except buy an option, hold ’∆k shares at

time k, and again make a riskless pro¬t. By the “no arbitrage” rule, that can™t happen,

so the price of the option V must be W0 .

Remark 7.5. Note that the proof of Theorem 7.4 tells you precisely what hedging

strategy (i.e., what portfolio process) to use.

In the binomial asset pricing model, there is no di¬culty computing the price of a

European call. We have

E (Sn ’ K)+ = (x ’ K)+ P(Sn = x)

x

and

n

pk q n’k

P(Sn = x) =

k

if x = uk dn’k S0 . Therefore the price of the European call is

n

n

(1 + r)’n (uk dn’k S0 ’ K)+ pk q n’k .

k

k=0

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The formula in Theorem 7.4 holds for exotic options as well. Suppose

V = max Si ’ min Sj .

i=1,...,n j=1,...,n

In other words, you sell the stock for the maximum value it takes during the ¬rst n time

steps and you buy at the minimum value the stock takes; you are allowed to wait until

time n and look back to see what the maximum and minimum were. You can even do this

if the maximum comes before the minimum. This V is still Fn measurable, so the theory

applies. Naturally, such a “buy low, sell high” option is very desirable, and the price of

such a V will be quite high. It is interesting that even without using options, you can

duplicate the operation of buying low and selling high by holding an appropriate number