DISCRETE BLACK-SCHOLES THEORY

TERENCE TAO

Abstract. Some notes on the discrete version of Black-Scholes theory.

1. Introduction

Suppose we have a stock S whose dollar value S(t) at time t varies stochastically with time. There are many models of how S(t) varies, but we shall choose a very simple one, namely a discrete random walk. More precisely, we suppose that t takes values in the integer multiples of dt for some small interval dt, and we have

S(t + dt) = S(t) ± σdt^{1/2}  (1)

where σ > 0 is a known constant called the volatility. One can also put in a drift term μdt to represent the long-term rate of return of the stock, but it turns out that such a term would have only a negligible effect since dt ≪ dt^{1/2}.

Of course, this type of random walk permits S to take upon negative values; in practice one has to modify the model somewhat to avoid this (e.g. use a log-random walk instead of a random walk). I’ll ignore this technicality.

I’ll the interest rate for borrowing or depositing money is 0%. This implies that the value of a single monetary unit at time t_{1} is equal to the value of the same unit at time t_{0} for any t_{0}, t_{1}. In practice, of course, this is not true, but this can be easily fixed by valuing everything in real dollars rather than nominal dollars (assuming that interest rates are constant). I’ll also assume that the time value for borrowing stock is also 0%.

I’ll also assume zero transaction costs in buying or borrowing the stock S or units of money. Related to this, I’ll assume that there are no time delays in trading stock; if one decides at time t to buy or sell some stock, this transaction can be completed before time t + dt.

One consequence of these two assumptions is that one is allowed to have negative amounts of money or negative amounts of stock, by borrowing money or stock from other parties.

A basic investment strategy is to buy α units of stock at time t and sell it at time t + dt, for a total profit of α(S(t + dt) − S(t)). Note that α can be negative, since we are allowed to have negative amounts of stock; this technique is known as short selling.

Definition 1.1. A derivative D with expiry date t_{0} is defined to be a function D(S) with one real variable S. At the expiration date t_{0}, the monetary value of this derivative is D(S(t_{0})).

For instance, a call option at strike price S_{0} (the right, but not the obligation, to buy the stock S at price S_{0} at time t_{0}), has a function

D(S) = max(0, S − S_{0}).

Similarly, a put option at strike price S_{0} (the right, but not the obligation, to sell the stock S at price S_{0} at time t_{0}) has the function

D(S) = max(0, S_{0} − S).

Fancier derivatives are possible, but they can all be fabricated by linear combinations (with positive or negative co-efficients) of the above two basic types.

Suppose D is a derivative with expiry date t_{0}. Black-Scholes theory addresses the problem of determining the value of this derivative at times t < t_{0}. Surprisingly, the answer to this problem is deterministic. More precisely, the value of D at time t is given by

D(S(t), t)  (2)

where the function D(S, t) is given by solving the backwards recurrence relation

D(S, t) = \frac{D(S + σdt^{1/2}, t + dt) + D(S − σdt^{1/2}, t + dt)}{2}  (3)

with initial data

D(S, t_{0}) = D(S).  (4)

A Taylor expansion of (3) gives (assuming D is smooth, which is of course a bit illegal for put and call options, but lets brush this technicality aside)

D(S, t) = D(S, t) + D_{t}(S, t)dt + \frac{σ^{2}}{2} D_{SS}(S, t)dt + O(dt^{2}),

which (if dt → 0) converges to the backwards heat equation

D_{t} = −\frac{σ^{2}}{2} D_{S}S.

(The fact that the heat equation is smoothing allows one to go back and fix up the problem of performing the Taylor expansion of a non-smooth derivative). Thus D(S, t) is given by convolving the original function D(S) with a suitable Gaussian. If one uses a log-random walk instead of a random walk, then one ends up with the logarithmic heat equation instead of the ordinary heat equation (i.e. the heat equation for log S rather than S).

It remains to explain why the value of the derivative at time t is given by (2), (3), (4). This is done by backwards induction. By (4), (2) is the correct value of D at time t_{0}. Now suppose inductively that (2) is the correct value of D at time t + dt.

We want to show why (2) is also correct at time t. From (3) and (1) we observe the key identity

D(S(t + dt), t + dt) = D(S(t), t) + α(S(t), t)(S(t + 1) − S(t))  (5)

regardless of the value of the sign ±, where

α(S, t) = \frac{D(S − σdt^{1/2}, t + dt) − D(S + σdt^{1/2}, t + dt}{2σdt^{1/2}}.

Now we show that D(S(t), t) is the value of the derivative at time t. To do this we demonstrate that we can both buy and sell this derivative at time t for a risk-free price of D(S(t), t).

To buy this derivative at time t, we first buy α(S(t), t) units of stock. At time t+dt we sell this stock and then buy the derivative D, which has value D(S(t+dt), t+dt) by induction. By (5), our total expenditure is D(S(t), t) as desired. (Recall our assumption of zero interest rates).

To sell this derivative at time t, swap the words “buy” and “sell” in the above sentence.

The fundamental reason why Black-Scholes theory gives a deterministic price is that (1) has only two possible values for the stock price S(t + dt) at any time t. If there were more than two possible outcomes then it would usually be impossible to make the equation (5) hold for all possible values of S(t + dt) since we have only one parameter, α(S(t), t), to play with.

Department of Mathematics, UCLA, Los Angeles CA 90095-1555

E-mail address: tao@math.ucla.edu