We saw that the two Ito equations—for S(/) and for lnS(/)—are different, and that the difference is not exactly what would be expected from the application of ordinary calculus to the transformation of variables from S(t) to In S(t); an additional term ~o2 is required. This extra term arises because the random variables have order and hence their squares produce first-order, rather than second-order, effects. There is a systematic method for making such transformations in general, and this is encapsulated in Ito's lemma:

Ito lemma Suppose that the random process x is defined by the Ito process

where z is a standard Wiener process Suppose also that the process y(t) is defined by y(t) — F(x, t). Then v(t) satisfies the Ito equation dy(,) , ( dJLa + BF + d, + (11.22)

\ d.v at 2 9,v- / ax where z is the same Wiener process as in Eq (II 21)

Proof: Ordinary calculus would give a formula similar to (11 22), but without the term with |

We shall sketch a rough proof of the full formula We expand y with respect to a change Ay. In the expansion we keep terms up to first order in A/, but since Ax is of order \/a7, this means that we must expand to second order in Ax We find dF dF 1 d2F

3x 31 2 Ox2

= F(x,t) 4 -—(a At 4- b Az) 4 — At 4--~(c; At 4 b Az)~■

The quadratic expression in the last term must be treated in a special way When expanded, it becomes cr{At)2 4 2ab At Az 4 b2(Az)2 The first two terms of this expression are of order higher than 1 in At, so they can be dropped. The term b2(Az)2 is all that remains. However, Az has expected value zero and variance At, and hence this last term is of order At and cannot be dropped. Indeed, it can be shown that, in the limit as At goes to zero, the term (Az)2 is nonstochastic and is equal to At Substitution of this into the previous expansion leads to

(dF dF 1 d2F>1\ dF, v 4 Ay = F(x, t) 4 —a 4 — 4- --—-ir At 4- —b Az V 3..V dt 2 dx2 ) dx

Taking the limit and using y = F(x, t) yields Ito's equation, (11,22), 1

Example 11.4 (Stock dynamics) Suppose that S(t) is governed by the geometric Brownian motion dS ~ ßS dt o S Az.

Let us use Ito's lemma to find the equation governing the process F(S(t)) = In S(t) We have the identifications et ~ ¡xS and b — aS, We also have 9F/3S = 1 fS and d2F/dS2 = —l/S2 Therefore according to (11,22),

which agrees with our earlier result

Let us consider again the binomial lattice model shown in Figure 11.8 (which is identical to Figure 11.1). The model is analogous to the multiplicative model discussed earlier in this chapter, since at each step the price is multiplied by a random variable

Su3d

Sukl2

Su'' FIGURE 11 „0 Binomial lattice stock model At each step the stock price 5 either increases to u.S or decreases to cJS

Su3d

Sukl2

In this case, the random variable takes only the two possible values u and d We can find suitable values for ¿i, d, and p by matching the multiplicative model as closely as possible This is done by matching both the expected value of the logarithm of a price change and the variance of the logarithm of the price change/'

To carry out the matching, it is only necessary to ensure that the random variable 5[, which is the price after the first step, has the correct properties since the process is identical thereafter . Taking 5(0) = 1, we find by direct calculation that

E(In S\) = p In u + (1 - p)\nd var (In S{) = p(In if)2 + (1 - p)(hi d)1 - [p In u + (1 - p) In d]2 — p(\ — p)(\nu ~ 1 nd)2. Therefore the appropriate parameter matching equations are where V = In» and D = ln</

Notice that three parameters are to be chosen: U, D, and /;; but there are only two requirements. Therefore there is one degree of freedom. One way to use this freedom is to set D — —U (which is equivalent to setting d ~ \/u) In this case the

6For the lattice, the probability of attaining the various end nodes of the lattice is given by the binomial distribution Specifically, the probability of reaching the value Sukd"~k is ^^ //'(l - p)"~k, where

— -—- ¡s the binomial coefficient This distribution approaches {in a certain sense) a normai distribution for large n The logarithm oi the final prices is of the form k Inn + {/i -k) In d. which is linear in k Hence the distribution of the end point prices can be considered to be nearly lognormal pU + (1 ~ p)D = u At p( I - p)(U - D)2 = a2 At

equations (11 23) and (11 24) reduce to

If we square the first equation and add it to the second, we obtain

Substituting this in the first equation, we may solve for p directly, and then U = In u can be determined. The resulting solutions to the parameter matching equations are

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