MTH 4500: Introductory Financial Mathematics

Black-Scholes Model

Introduction

Recall that in the discrete time setting we made an assumption that in each period the stock price goes up by factor \(u\) or down by factor \(d\). This way we defined a very precise mathematical model in which we were able to calculate the prices of options. The questions that one naturally asks is whether this model is a good approximation to reality. And we won‘t answer it since this is not a mathematical question, because the word ``reality‘‘ makes no sense in mathematics.

We will now develop a continuous model for movements of the stock prices. In the same way as in the case of the binomial model, it is not of a concern to mathematicians whether that continuous model corresponds to the real world, and to what extent it does, if it does at all. As with the case of the discrete model, we are again going to make some assumptions. Different people will have different opinions on how realistic the model is once we introduce those assumptions. However, the following model is a building block of the stochastic calculus and the theories behind the pricing of securities.

The stock price at time \(t\) will be denoted by \(S_t\). We will assume that the risk-free rate is \(r\) and that the interest is compounded continuously. We are looking for a formula for pricing a derived security with payoff function \(\varphi\). For example, the European call option with strike \(K\) has the payoff \(\varphi(S_t)=\left(S_t-K\right)^+\).

Motivation behind the definition of the Black-Scholes Model

The model that we will discuss is called the Black-Scholes Model and it is constructed from the \(n\) period binomial model by taking the limit as \(n\) goes to infinity. Before giving the definition of the model, and before listing all the assumptions that we will be making, let us first analyze what kind of restrictions we need to make if we are about to recycle the smart ideas we came by when studying the \(n\)-period binomial model.

Let us divide the continuous time interval \([0,t]\) into \(n\) sub-intervals of length \(t/n\). We will assume that in each of the sub-intervals the stock can either go up by factor \(u\) or down by factor \(d\) - so we did make our first crazy assumption. In each of the time intervals the stock behaves in the same way. Notice that the interest accumulated over one of these intervals is \(e^{\frac{rt}n}\). Recall that in the discrete case, the interest accumulated over one period was \(1+r\).

If we denote by \(k\) the total number of up movements, then the total number of down movements is \(n-k\) and the stock price can be represented as \[S_t= S_0\cdot u^k\cdot d^{n-k}=S_0 e^{k\ln u+(n-k)\ln d}.\] We want to express the above formula using the notation from the theory of random walks. For each \(i\in\{1,2,\dots, n\}\), let \(\xi_i\) be either \(1\) or \(-1\) depending on whether the stock went up or down in the \(i\)-th interval. Let us denote \(W_n=\xi_1+\cdots+\xi_n\). We can express \(W_n\) in terms of the number \(k\) of the up-movements of the stock as \(W_n=k-(n-k)=2k-n\). This is equivalent to \(k=\frac{W_n+n}2\). We now have \begin{eqnarray*}S_t&=&S_0e^{k\left(\ln u-\ln d\right)+n\ln d} = S_0e^{\frac{W_n+n}2\left(\ln u-\ln d\right)+n\ln d}= S_0e^{\frac{\ln u+\ln d}2\cdot n+\frac{\ln u-\ln d}2\cdot W_n} \newline &=& S_0e^{\frac{\left(\ln u+\ln d\right)n}{2t}\cdot t+ \frac{\left(\ln u-\ln d\right)\sqrt n}{2\sqrt{t}}\cdot \sqrt t \frac{W_n}{\sqrt n}}. \end{eqnarray*} We will introduce the following notation \begin{eqnarray*} \mu=\frac{\left(\ln u+\ln d\right)n}{2t}\quad\quad\mbox{ and }\quad\quad\sigma= \frac{\left(\ln u-\ln d\right)\sqrt n}{2\sqrt{t}}. \quad\quad\quad\quad\quad\quad (1)\end{eqnarray*} The last formula for \(S_t\) can be written as \begin{eqnarray*} S_t=S_0e^{\mu t+\sigma \sqrt t \frac{W_n}{\sqrt n}}. \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad (2)\end{eqnarray*}

Notice that if we assume that the probability of each up-movement is the same as the probability of each down movement, and that the movements are independent, then according to the central limit theorem we know that \(\frac{W_n}{\sqrt n}\) converges in distribution to \(N(0,1)\). Subsequently we have that \(\sqrt t \cdot \frac{W_n}{\sqrt n}\) converges to \(N(0,t)\). This is why the above equation is often written as \begin{eqnarray*} S_t=S_0\cdot e^{\mu t+\sigma B_t}, \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad (3)\end{eqnarray*} where \(B_t\sim N(0,t)\). Such \(B_t\) is called Brownian motion.

Definition of the Black-Scholes model

We can approximate the model with \(n\)-period binomial model where in each period the stock moves up by factor \(u\) or down by factor \(d\). The values \(u\) and \(d\) are expressed in terms of \(\mu\) and \(\sigma\) by solving the system (1). The solutions to the system are \[u=e^{\mu\frac tn+\sigma \sqrt{\frac tn}},\quad\quad d=e^{\mu\frac tn-\sigma \sqrt{\frac tn}}.\]

Approximation by the binomial model

The model is defined in the following way: For given \(\mu\in\mathbb R\) and \(\sigma > 0\), assume that the stock price \(S_t\) follows the formula (3), where \(B_t\) is the standard Brownian motion.

If we want to price the option with pay-off function \(\varphi(S_t)\) in our \(n\)-step binomial model we can use the formula \begin{eqnarray*} C_0= e^{-rt}\cdot\mathbb E_*\left[\varphi(S_t)\right]=e^{-rt}\mathbb E_*\left[\varphi\left(S_0e^{\mu t+\sigma B_t}\right)\right], \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad (4) \end{eqnarray*} where \(\mathbb E_*\) denotes the expected value with respect to the probability space in which \(\xi_1\), \(\dots\), \(\xi_n\) are independent, identically distributed and satisfy \[\mathbb P_*\left(\xi_i=1\right)=p^*=\frac{e^{\frac{rt}n}-d}{u-d}\quad\quad \mbox{and}\quad\quad \mathbb P_*\left(\xi_i=-1\right)=1-p^*=\frac{u-e^{\frac{rt}n}}{u-d}.\] The random variable \(B_t\) has normal distribution with parameters \(0\) and \(t\). However, this random variable is not normally distributed under the probability \(\mathbb P_*\) and its expectation can not be calculated in a straight-forward way.

We will express \(B_t\) in terms of another random variable that is normally distributed. Recall that we defined \(B_t\) in the following way: \[B_t=\sqrt t\cdot \frac{W_n}{\sqrt n}= \sqrt t\cdot \frac{\xi_1+\xi_2+\cdots+\xi_n}{\sqrt n}.\] We can use the Central Limit Theorem to conclude that \begin{eqnarray*} \frac{\xi_1+\cdots+\xi_n-n\left(p^*-q^*\right)}{\sqrt n}\Rightarrow N\left(0,1-\left(p^*-q^*\right)^2\right), \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad (5) \end{eqnarray*} because \(\mathbb E_*\left[\xi_1\right]=p^*-q^*\) and \(\mbox{var}_*\left(\xi_1\right)=1-\left(p^*-q^*\right)^2\).

We will now express \(p^*\) and \(q^*\) in terms of \(\mu\), \(\sigma\), and \(r\). The denominator can be simplified using the Taylor expansion \(e^x=1+x+\frac{x^2}2+O\left(x^3\right)\). \begin{eqnarray*} u-d&=&e^{\mu \frac tn+\sigma \sqrt{\frac tn}}-e^{\mu \frac tn-\sigma \sqrt{\frac tn}}=e^{\mu\frac tn}\left(e^{\sigma\sqrt{\frac tn}}-e^{-\sigma\sqrt{\frac tn}}\right) \newline&=&e^{\mu\frac tn}\left(1+\sigma\sqrt{\frac tn}+\frac12\sigma^2\cdot \frac tn+O\left(\sqrt{\frac tn}^3\right)-\left( 1-\sigma\sqrt{\frac tn}+\frac12\sigma^2\cdot \frac tn+O\left(\sqrt{\frac tn}^3\right)\right)\right)\newline &=&2\sigma\sqrt{\frac tn}e^{\mu\frac tn} \left(1+O\left(\frac tn\right)\right). \end{eqnarray*}

Using a similar expansion for the numerator we derive: \begin{eqnarray*} e^{r\cdot\frac tn}-e^{\mu\cdot\frac tn-\sigma\sqrt\frac tn}&=&e^{\mu\cdot\frac tn}\cdot \left(e^{(r-\mu)\cdot\frac tn}-e^{-\sigma\sqrt\frac tn}\right)\newline&=&e^{\mu\cdot\frac tn}\left(1+(r-\mu)\frac tn+O\left(\sqrt{\frac tn}^3\right)- 1+\sigma\sqrt{\frac tn}-\frac12\sigma^2\cdot\frac tn+O\left(\sqrt{\frac tn}^3\right)\right)\newline &=&e^{\mu\cdot\frac tn}\left(\sigma\sqrt{\frac tn}+\left(r-\mu-\frac{\sigma^2}2\right)\frac tn+O\left(\sqrt{\frac tn}^3\right)\right)\newline &=& \sigma\sqrt{\frac tn}e^{\mu\frac tn}\left(1+\frac{r-\mu-\frac{\sigma^2}2}{\sigma}\sqrt{\frac tn}+O\left(\frac tn\right)\right). \end{eqnarray*}

Therefore we get \begin{eqnarray*}p^*&=&\frac12\left(1+\frac{r-\mu-\frac{\sigma^2}2}{\sigma}\sqrt{\frac tn}+O\left(\frac tn\right)\right)\newline q^*&=&\frac12\left(1-\frac{r-\mu-\frac{\sigma^2}2}{\sigma}\sqrt{\frac tn}+O\left(\frac tn\right)\right). \end{eqnarray*} The formula (5) can be now simplified by calculating \(p^*-q^*=\frac{r-\mu-\frac{\sigma^2}2}{\sigma}\sqrt\frac tn\) and \(1-\left(p^*-q^*\right)^2=1-\left(\frac{r-\mu-\frac{\sigma^2}2}{\sigma}\right)^2\cdot \frac tn \approx 1\). The random variable \(B_t\) can be represented in terms of a standard normal random variable \(Z\sim N(0,1)\) in the following way \[B_t=\sqrt t\cdot \frac{\xi_1+\xi_2+\cdots+\xi_n}{\sqrt n}=\frac{\sqrt t}{\sqrt n}\cdot \left(n\cdot \frac{r-\mu-\frac{\sigma^2}2}{\sigma}\sqrt\frac tn+\sqrt n\cdot Z\right) =t\cdot \frac{r-\mu-\frac{\sigma^2}2}{\sigma}+\sqrt t\cdot Z .\] Therefore the formula (4) becomes \begin{eqnarray*}C_0&=& e^{-rt}\mathbb E_*\left[\varphi\left(S_0e^{\mu t+\sigma B_t}\right)\right] =e^{-rt}\mathbb E_*\left[\varphi\left(S_0e^{\left(r-\frac{\sigma^2}2\right)t+\sigma \sqrt t Z}\right)\right]\newline &=&e^{-rt}\mathbb E_*\left[\varphi\left(S_0e^{\left(r-\frac{\sigma^2}2\right)t+\sigma B^*_t}\right)\right], \end{eqnarray*} where \(B^*_t\) is a standard Brownian motion with respect to the risk-neutral probability \(\mathbb P_*\).

Black-Scholes formula for the price of European call option

In the case of European call option we assume that \(\varphi(x)=\left(x-K\right)^+\). The above formula becomes \begin{eqnarray*}C_0&=& e^{-rt}\mathbb E_*\left[\left(S_0e^{\left(r-\frac{\sigma^2}2\right)t+\sigma B_t^*}-K\right)^+\right]=e^{-rt}\mathbb E_*\left[\left(S_0e^{\left(r-\frac{\sigma^2}2\right)t+\sigma \sqrt t Z}-K\right)^+\right], \end{eqnarray*} where \(Z\sim N(0,1)\).

By further simplifying the result one derives \begin{eqnarray*}C_0&=&\frac{e^{-rt}}{\sqrt{2\pi}}\int_{-\infty}^{+\infty} \left(S_0e^{\left(r-\frac{\sigma^2}2\right)t+\sigma \sqrt t z}-K\right)^+ \cdot e^{-\frac{z^2}2}\,dz =\frac{e^{-rt}}{\sqrt{2\pi}}\int_{S_0e^{\left(r-\frac{\sigma^2}2\right)t+\sigma \sqrt t z}>K}\left(S_0e^{\left(r-\frac{\sigma^2}2\right)t+\sigma \sqrt t z}-K\right) \cdot e^{-\frac{z^2}2}\,dz\newline &=&\frac{e^{-rt}}{\sqrt{2\pi}}\int_{\frac{\ln\frac{K}{S_0}-\left(r-\frac{\sigma^2}2\right)t}{\sigma \sqrt t}}^{+\infty} S_0e^{\left(r-\frac{\sigma^2}2\right)t+\sigma \sqrt t z} \cdot e^{-\frac{z^2}2}\,dz - \frac{Ke^{-rt}}{\sqrt{2\pi}}\int_{\frac{\ln\frac{K}{S_0}-\left(r-\frac{\sigma^2}2\right)t}{\sigma \sqrt t}}^{+\infty} e^{-\frac{z^2}2}\,dz. \end{eqnarray*} Let us denote by \(\Phi\) the cumulative distribution function of the standard normal random variable, i.e. \(\Phi(x)=\frac1{\sqrt{2\pi}} \int_{-\infty}^xe^{-\frac{z^2}2}\,dz\). The cost of the European call option can be expressed in terms of \(\Phi\) as \begin{eqnarray*}C_0&=& \frac{S_0e^{- \frac{\sigma^2}2t}}{\sqrt{2\pi}}\int_{\frac{\ln\frac{K}{S_0}-\left(r-\frac{\sigma^2}2\right)t}{\sigma \sqrt t}}^{+\infty} e^{\sigma \sqrt t z} \cdot e^{-\frac{z^2}2}\,dz - Ke^{-rt}\Phi\left(\frac{\ln\frac{S_0}{K}+\left(r-\frac{\sigma^2}2\right)t}{\sigma\sqrt t}\right). \end{eqnarray*} Let us introduce the following notation \[d_{\pm}=\frac{\ln\frac{S_0}K+\left(r\pm\frac{\sigma^2}2\right)t}{\sigma \sqrt t}.\] The price of the call option can be now further simplified to \begin{eqnarray*}C_0&=& \frac{S_0}{\sqrt{2\pi}}\int_{\frac{\ln\frac{K}{S_0}-\left(r-\frac{\sigma^2}2\right)t}{\sigma \sqrt t}}^{+\infty} e^{-\frac{\sigma^2 t}2+\sigma \sqrt t z-\frac{z^2}2}\,dz - Ke^{-rt}\Phi\left(d_-\right)\newline &=& \frac{S_0}{\sqrt{2\pi}}\int_{\frac{\ln\frac{K}{S_0}-\left(r-\frac{\sigma^2}2\right)t}{\sigma \sqrt t}}^{+\infty} e^{-\frac{\left(z-\sigma\sqrt t\right)^2}2}\,dz - Ke^{-rt}\Phi\left(d_-\right). \end{eqnarray*} The integral from the right-hand side of the last equation can be simplified by using the substitution \(w=z-\sigma\sqrt t\). The integral becomes \begin{eqnarray*}C_0&=& \frac{S_0}{\sqrt{2\pi}}\int_{\frac{\ln\frac{K}{S_0}-\left(r+\frac{\sigma^2}2\right)t}{\sigma \sqrt t}}^{+\infty} e^{-\frac{w^2}2}\,dw - Ke^{-rt}\Phi\left(d_-\right)\newline &=&S_0\Phi\left(d_+\right)-Ke^{-rt}\Phi\left(d_-\right). \end{eqnarray*}

Black-Scholes formula for the price of European put option

In an analogous way we can derive the formula for the price of European put with strike \(K\). Or, alternatively, we may use the put-call parity \(C-P=S-Ke^{-rt}\) to conclude that \[P=Ke^{-rt}\Phi\left(-d_-\right)-S_0\Phi\left(-d_+\right).\]