Abstract

In these notes we price an option contract using the no-arbitrage pricing theory.

1 Black Scholes Option Pricing

Definition 1 (Black Scholes Model).

Black Scholes model is model for the market where the risk-free interest rate r is constant and tradable asset price process follows Geometric Brownian motion i,e.

$D_t$ $\sim e^{-rt}$

${\displaystyle \frac{d{S}_t}{S_t}}$ $=\mu dt+\sigma d{W}_t$

We assume that the no-arbitrage condition holds in the market and that there is a risk neutral probability measure $\mathbb{P}$ such that discounted asset process is a local martingale under this measure.

$d{(D_t{S}_t)}_{\text{drift}}$ $=0$

${\left(-r{D}_t{S}_{t}dt+D_t{S}_t(\mu dt+\sigma d{W}_t^{\mathbb{P}})+0\right)}_{\text{drift}}$ $=0$

$(\mu -r)D_t{S}_t$ $=0$

Therefore we have $\mu =r$.

We know the solution to the Geometric Brownian motion

$$S_t=S_0{e}^{(r-\frac{1}{2}{\sigma }^2)t+\sigma W_t^{\mathbb{P}}}$$

Let’s consider a call option on the stock S with expiration at time T and strike K. The value of this option at time T is

$$V_T=\max (S_T-K,0)$$

From the no-arbitrage pricing theory we have $D_t{V}_t$ is a local martinage and under bounded regularity conditions we have

$$D_t{V}_t={𝔼}^{\mathbb{P}}[D_T{V}_T|{ℱ}_t]$$ (1)

i.e. $D_t{V}_t$ is a martingale.

Solving we have,

$V_t$ $={\displaystyle \frac{1}{D_t}}{𝔼}^{\mathbb{P}}[D_T{V}_T|{ℱ}_t]$

$=e^{rt}{𝔼}^{\mathbb{P}}[e^{-rT}\max (S_T-K,0)|{ℱ}_t]$

$=e^{-r(T-t)}{𝔼}^{\mathbb{P}}[\max (S_t{e}^{(r-\frac{1}{2}{\sigma }^2)(T-t)+\sigma W_{T-t}^{\mathbb{P}}}-K,0)|{ℱ}_t]$

$=e^{-r(T-t)}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}\max (S_t{e}^{(r-\frac{1}{2}{\sigma }^2)(T-t)+\sigma \sqrt{T-t}x}-K,0){\displaystyle \frac{1}{\sqrt{2}}}{e}^{-x^2}𝑑x$

Denoting $\tau =T-t$ and $d_1=\frac{1}{\sigma \sqrt{\tau }}\left[\log (\frac{K}{S_t})-(r-\frac{1}{2}{\sigma }^2)\tau \right]$, we have

$V_t$ $=e^{-r\tau }{\displaystyle {\int }_{d_1}^{\mathrm{\infty }}}\left(S_t{e}^{(r-\frac{1}{2}{\sigma }^2)\tau +\sigma \sqrt{\tau }x}-S_t{e}^{(r-\frac{1}{2}{\sigma }^2)\tau +\sigma \sqrt{\tau }{d}_1}\right){\displaystyle \frac{1}{\sqrt{2}}}{e}^{\frac{-x^2}{2}}𝑑x$

$=S_t{e}^{-\frac{1}{2}{\sigma }^2\tau }{\displaystyle {\int }_{d_1}^{\mathrm{\infty }}}\left(e^{\sigma \sqrt{\tau }x}-e^{\sigma \sqrt{\tau }{d}_1}\right){\displaystyle \frac{1}{\sqrt{2}}}{e}^{\frac{-x^2}{2}}𝑑x$

$=S_t{e}^{-\frac{1}{2}{\sigma }^2\tau }\left[{\displaystyle {\int }_{-\mathrm{\infty }}^{-d_1}}{e}^{-\sigma \sqrt{\tau }x}{\displaystyle \frac{1}{\sqrt{2}}}{e}^{\frac{-x^2}{2}}𝑑x-{\displaystyle {\int }_{-\mathrm{\infty }}^{-d_1}}{e}^{\sigma \sqrt{\tau }{d}_1}{\displaystyle \frac{1}{\sqrt{2}}}{e}^{\frac{-x^2}{2}}𝑑x\right]$

$=S_t{e}^{-\frac{1}{2}{\sigma }^2\tau }\left[e^{\frac{1}{2}{\sigma }^2\tau }{\displaystyle {\int }_{-\mathrm{\infty }}^{-d_1}}{\displaystyle \frac{1}{\sqrt{2}}}{e}^{\frac{-{(x+\sigma \sqrt{\tau })}^2}{2}}𝑑x-e^{\sigma \sqrt{\tau }{d}_1}{\displaystyle {\int }_{-\mathrm{\infty }}^{-d_1}}{\displaystyle \frac{1}{\sqrt{2}}}{e}^{\frac{-x^2}{2}}𝑑x\right]$

$=S_t{\displaystyle {\int }_{-\mathrm{\infty }}^{-d_1+\sigma \sqrt{\tau }}}{\displaystyle \frac{1}{\sqrt{2}}}{e}^{-y^2}𝑑y-K{e}^{(-r-\frac{1}{2}{\sigma }^2)\tau }{\displaystyle {\int }_{-\mathrm{\infty }}^{-d_1}}{\displaystyle \frac{1}{\sqrt{2}}}{e}^{\frac{-x^2}{2}}𝑑x$

$=S_{t}N(-d_1+\sigma \sqrt{\tau })-K{e}^{-r\tau }N(-d_1)$

where N is the cumulative distribution function of the standard normal distribution.