Lecture 24 Math 50051, Topics in Probability Theory and Stochastic Processes

Ito formula

As discussed earlier, in stochastic environments a formal notion of derivative does not exist. Shocks to asset prices are assumed to be unpredictable, and in continuous time they become “too erratic”. The resulting asset prices may be continuous, but they are not smooth. Stochastic differentials need to be used in place of derivatives.

Ito’s rule provides an analytical formula that simplifies handling stochastic differentials and leads to explicit computations.

We begin by discussing various types of derivatives.

Suppose we have a function F (St, t) depending on two variables St and t, where St itself varies with time t. Further, assume that St is a random process.

In standard calculus, where all variables are deterministic, there are three sorts of derivatives that one can talk about.

The first are the partial derivatives of F (St, t), denoted by

∂F (St, t) ∂F (St, t) Fs = ,Ft = . (1) ∂St ∂t The second is the total derivative dealing with differentials:

dF = FsdSt + Ftdt. (2)

The third is the chainrule: dF (S , t) dS t = F t + F . (3) dt s dt t A financial market participant may be interested in these derivatives for various reasons.

The partial derivation has no direct real-life counterpart, but gives “multipliers” that can be used in evaluating responses of asset prices to observed changes in risk factors. For example, Fs measures the response of F (St, t) to a small change in St only. As such, Fs is a hypothetical concept, since the only way a continuous random variable St can change is if some time passes. Hence, in reality, t has to change as well. Partial derivatives abstract from such questions. Because they are simple multipliers, there is no difference between the way stochastic and deterministic environments define partial derivatives.

A classical example of the use of partial derivation occurs in delta hedging. Suppose a market participant knows the functional form of F (St, t). Then, this mathematical formula can be differ- entiated only with respect to St, in order to find the partial derivative Fs. This Fs is a measure of how much the derivative asset price will change per unit change in St. In this sense, one does not have any of the difficulties encountered in defining a time derivative for Wiener processes. What

1 is under investigation is not how F (St, t) moves over time, but how F (·) responds to a “small” hypothetical change in St, with time fixed.

The total derivative is a more “realistic” notion. It is assumed that both time t and the underlying security price St change, and then the total response of F (St, t) is calculated. The result is the (stochastic) differential dFt. This is clearly a very useful quantity to the market participant. It represents the observed change in the price of the derivative asset during an interval dt.

The chain rule is quite similar to the total derivative. In classical calculus, the chain rule expresses the rate of change of a variable as a chain effect of some initial variation. In stochastic calculus, we know that operations such as dFt/dt, dSt/dt cannot be defined for continuous-time square integrable martingales, or Brownian motion. But a stochastic equivalent of the chain rule can be formulated in terms of absolute changes such as dFt,dSt, dt, and the Ito integral can be used to justify these terms. Thus, in stochastic calculus, the term “chain rule” will refer to the way stochastic differentials relate to one another. In other words, a stochastic version of total differentiation is developed.

Example

We discuss a simple example before going into Ito’s formula. The example will help clarify the mechanics of taking various derivatives. Let F (rt, t) be the price of a T-bill that matures at time T , and let rt be a fixed, continuously compounding risk-free rate. Then

−rt(T −t) F (rt, t) = e 100. (4)

Let us calculate the partial derivatives Fr,Ft:

∂F −rt(T −t) Fr = = −(T − t)[e 100] (5) ∂rt and ∂F F = = r [e−rt(T −t)100]. (6) t ∂t t

Note that these partials will be the same regardless of whether rt is deterministic or random. By taking these partial derivatives, we are simply calculating the rate of change of F (·) with respect to small hypothetical changes in rt or in t.

On the other hand, the total derivative relates to the actual occurrence of random events. In standard calculus, with nonrandom rt, the total derivative of this particular F (·) will be given by

−rt(T −t) −rt(T −t) dF (rt, t) = −(T − t)[e 100]drt + rt[e 100]dt. (7)

This example suggests that when rt is random, we may be able to define the counterpart of total derivative, using the Ito integral, which gives a meaning to stochastic differentials such as drt. This intuition is correct, and the result is Ito’s formula. However, with stochastic rt, not only does the interpretation of drt change, but the formula will also be different.

2 Ito’s Lemma; Let X be a semimartingale defined by

dX(t) = µ(t)dt + σ(t)dB(t) and let f ∈ C2. Then

Z t 1 Z t f(X(t)) = f(X(0)) + f 0(X(s))dX(s) + f 00(X(s))d < X > (s) 0 2 0 Z t 1 Z t = f(X(0)) + [f 0(X(s))µ(s) + f 00(X(s))σ2(s)]ds + f 0(X(s))σ(s)dB(s) 0 2 0

In particular, if X(t) = B(t) (B.m.) then

Z t 1 Z t f(B(t)) = f(B(0)) + f 0(B(s))dB(s) + f 00(B(s))d < B > (s) 0 2 0

In conclusion, remember (dB(s))2 = ds, d < X > (s) := (dX(s))2 = σ2(s)ds,(dt)2 = 0

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