Black Scholes Formula

Posted by shede on February 21, 2010

We give a simple derivation of the Black-Scholes formula for option pricing. Let ${r}$ be the risk-free interest rate, and let ${s_t}$ be the stock price at time ${t}$ , which follows a geometric Brownian motion with drift ${\mu}$ and volatility ${\sigma}$ . Let ${C(s, t, K, \sigma, r)}$ be the price of a call option with strike price ${K}$ and expiration time ${t}$ , and ${s=s_0}$ is the current stock price.

Since ${s_t/s}$ is a lognormal random variable with mean ${\mu t}$ and variance ${\sigma^2 t}$ , we have

$\displaystyle E \left[s_t | s\right] = s E\left[\exp(\log(s_t/s)) \right] = s \exp(\mu t + \sigma^2 t /2 ).$

By the arbitrage theorem, this must equal to ${s \exp(rt)}$ and thus we have ${\mu=r-\sigma^2/2}$ . So the geometric Brownian motion has drift parameter ${r-\sigma^2/2}$ and volatility parameter ${\sigma^2}$ , and we have

$\displaystyle s_t = s \exp\left( (r-\sigma^2/)t + \sigma \sqrt{t} Z \right)$

where ${Z}$ is a ${N(0,1)}$ random variable.

Let ${(x)_+ := \max(0, x)}$ , then by definition the call price at the expiration will be ${(s_t - K)_+}$ . Again by no arbitrage we should have

$\displaystyle C(s, t, K, \sigma, r)= E\left[ \exp(-rt) (s_t - K)_+ \right].$

Define an indicator random variable ${I}$ such that ${I=1}$ if ${s_t > K}$ and 0 otherwise, and let ${\Phi(.)}$ be the cdf of ${N(0,1)}$ . Then by definition we have ${I=1}$ iff ${s_t > K}$ iff ${(r-\sigma^2/)t +\sigma\sqrt{t}Z > \log(K/s)}$ iff ${Z > \frac{1}{\sigma\sqrt{t}} [\log(K/s) - (r-\sigma^2/2)t]}$ iff ${Z > \sigma\sqrt{t} - \omega}$ , where

$\displaystyle \omega := \frac{rt + \sigma^2 t/2 - \log(K/s)}{\sigma\sqrt{t}}.$

First, notice that ${E[I] = P(s_t > K) = P(Z > \sigma\sqrt{t} -\omega) = \Phi(\omega - \sigma\sqrt{t})}$ . Second, we have

$\displaystyle \begin{array}{rcl} E\left[ I s_t \right] & = & \int_{\sigma\sqrt{t} -\omega}^\infty s \exp\left((r-\sigma^2/2)t +\sigma\sqrt{t}x\right) \frac{1}{\sqrt{2\pi}} \exp(-x^2/2)dx \\ & = & \frac{1}{\sqrt{2\pi}} s \exp(rt) \int_{\sigma\sqrt{t}-\omega}^\infty \exp\left(-(x-\sigma\sqrt{t})^2/2 \right) dx\\ & = & s\exp(rt) \Phi(\omega). \end{array}$

Finally we have

$\displaystyle \begin{array}{rcl} C(s, t, K, \sigma, r) & = & \exp(-rt) E\left[(s_t -K)_+ \right]\\ & = & \exp(-rt) E\left[I (s_t - K) \right]\\ & = & \exp(-rt) E[I S_t] - K \exp(-rt) E[I]\\ & = & s \Phi(\omega) - K\exp(-rt) \Phi(\omega - \sigma\sqrt{t}). \end{array}$

This entry was posted on February 21, 2010 at 8:20 pm and is filed under Finance. Tagged: quant. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

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