Question

Assume an asset price St follows the geometric Brownian motion, dSt = u Stdt+oS+dZt, So =s > 0 where u and o are constants, r is the risk-free rate, and Zt is the Brownian motion. 1. Using the Ito's Lemma find to the stochastic differential equation satisfied by the process X+ = St. 2. Compute E[Xt] and Var[Xt].

Answer 1

Given,

dse = p.Sedt + 0 Sędzz, So=s > 0

Where $\mu$ and $\sigma$are constant, r is the risk-free rate, and $Z_t$ is the Brownian motion.

stochastic differential equation is

dX+ = al Xt, t)dt + b(Xt, t)dW:

Here

X+ = SE

1)

We use Ito’s lemma for the process S_t .

Assume

$f(s)=s^{n}$

a, f(s) = ns-1

of(s) = n(n − 1)sn-4

Then,

dX4 = df (S)

('sp)('s)fe; + 'SP('s)*C = 'XP

$dX_{t}=nS_{t}^{n-1}dS_{t}+\frac{n(n-1)}{2}S_{t}^{n-2}(dS_{t})^{2}$

dX4 = n 5-1 (u.Sedt +oS¢dW) + 7-20²(S1)dt

_X = ns (+"?•*) de + nost aw

$dX_{t}=nX(t)\left ( \mu +\frac{n-1}{2}\sigma ^{2} \right )dt+n\sigma X(t)dW_{t}$

where

$a(X_{t},t)=nX(t)\left ( \mu +\frac{n-1}{2}\sigma ^{2} \right )$

$b(X_{t},t)=n\sigma X(t)$

2)

E(X) = n(n+"=100

$Var(X_{t})=n^{2}\sigma^{2}$

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