Stochastic Processes and Financial Mathematics
(part two)

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Chapter 15 The Black-Scholes model

Our discussion of finance, in continuous time, will be centred around the Black-Scholes model. The Black-Scholes model is, in some sense, the continuous time version of the binomial model from Section 5.4. Moving into continuous time has one big advantage: we can make our stock price process more realistic.

15.1 The Black-Scholes market

The Black-Scholes market contains two assets, cash and stock. In analogy to our discrete time model, cash earns interest at a deterministic rate, whereas the value of stock fluctuates randomly. As in discrete time, the model has some parameters: $r,\mu $ and $\sigma $, all real valued deterministic constants.

Here is the model:

• The value of a unit of stock at time $t$ is $S_t$, where $S_t$ is a geometric Brownian motion (from Section 13.2) given by the SDE
$\seteqnumber{0}{15.}{0}$
\begin{equation} \label {eq:stock_price} dS_t=\mu S_t\,dt+\sigma S_t\,dB_t, \end{equation}

with initial value $S_0$. Here, of course, $B_t$ is a Brownian motion. From (13.9) we know that the (unique) solution of this SDE is $S_t=S_0e^{(\mu -\frac 12\sigma ^2)t+\sigma B_t}$.
• If we hold $x$ units of cash at the start of a time interval of length $t$, its final value will be $xe^{rt}$. This is the definition of ‘cash earning interest at continuous rate $r>0$ for time $t$’.

A neater way of representing it is that we think of cash as an asset whose value changes over time. That is, the value of a ‘unit of cash’ at time $t$ is given by
$\seteqnumber{0}{15.}{1}$
\begin{equation} \label {eq:cash_price} dC_t=r C_t\,dt. \end{equation}

with initial condition $C_0=1$, and (unique) solution $C_t=e^{rt}$.

It might be appealing to ‘divide by $dt$’ and write (15.2) as an ODE, in the form $\frac {dC_t}{dt}=rC_t$ (with solution $C_t=C_0e^{rt}$). Justifying this step rigorously requires an application of the fundamental theorem of calculus. Whilst it would be mathematically correct, it is not what we want; (15.2) is better because its form is more compatible with the SDE (15.1).

Remark 15.1.1 We write the initial value of the stock as $S_0$. We will use $s$ as a variable. Note that this is different to our use of $s$ in discrete time, in which we set $s=S_0$.

As usual, we work over a filtered space $(\Omega ,\mc {F},(\mc {F}_t),\P )$ where the filtration $\mc {F}_t$ is generated by the Brownian motion $B_t$, that is

\[\mc {F}_t=\sigma (B_u\-u\leq t).\]

Here, $B_t$ is the same Brownian motion that drives the random stock price in (15.1). Within the Black-Scholes model, $B_t$ is the only source of randomness that we’ll need.

As in the binomial model, we assume that we can borrow both cash and stock, and hold real valued amounts in each case. Thus, our definition of a portfolio remains the same as before:

Definition 15.1.2 A portfolio is a pair $h=(x,y)$ where $x\in \R $ denotes an amount of cash and $y\in \R $ denotes a number of units of stock. The value of this portfolio at time $t$ is

\[V^h_t=xC_t+yS_t.\]

However, our definition of a portfolio strategy needs to be upgraded. Previously, at each time $t\in \N $ we had a round of buying/selling, and then in between times $t\mapsto t+1$ we had stock/cash changing in value. Now, both these process must occur together, continuously. Happily, we have already developed the theoretical framework to do this:

Definition 15.1.3 A portfolio strategy is a pair of continuous stochastic processes $(h_t)=(x_t,y_t)$ where both $x_t$ and $y_t$ are adapted to the filtration $\mc {F}_t$. The value of $(h_t)$ at time $t$ is

\[V^h_t=x_tC_t+y_tS_t.\]

Note that here we break our usual convention of writing random quantities in capitals and deterministic quantities in lower case. The amounts of cash $x_t$ and stock $y_t$ that we hold at a given time are stochastic processes. (Just like in discrete time.)

We have required that our portfolio strategies be continuous. This assumption is helpful from a mathematical point of view, because we have only developed Ito integration for continuous processes, but it is not entirely realistic. In reality, it is possible to buy/sell large amounts of stock in a single transaction. We’ll put this issue to one side for now, but we will return to discuss discontinuities and related matters (such as transaction costs) in Sections 16.1 and 17.3.

Definition 15.1.4 A portfolio strategy $(h_t)$ is said to be self-financing if
$\seteqnumber{0}{15.}{2}$
\begin{equation} \label {eq:sf} dV^h_t=x_t\,dC_t+y_t\,dS_t. \end{equation}

We will need a little thought to understand why this definition captures the concept of a portfolio being self-financing. To do so, we need to think of a stochastic differential $dX_t$ as ‘the change in $X$ over a short time interval. That is, if we choose time limits $[t,t+\delta ]$ where $\delta $ is small then $Y_t dX_t$ represents $\int _{t}^{t+\delta }Y_u\,dX_u\approx Y_t(X_{t+\delta }-X_t)$. Note that we use $Y_t$ and not $Y_{t+\delta }$ here to match the definition of the Ito integral in (12.12). So, approximately, (15.3) means that

\[V^h_{t+\delta }-V^h_t=x_t(C_{t+\delta }-C_t)+y_t(S_{t+\delta }-S_t)\]

This means that $x_t$ and $y_t$ are chosen in such a way as the changes in $C$ and $S$ entirely explain the variation in the value $V^h$ during the time interval $[t,t+\delta ]$. In other words, we haven’t injected any value in, nor taken any value out. Formally, imposing this condition for all $t$ in a limit as $\delta \downarrow 0$ results in (15.3).

Finally, and exactly as before:

Definition 15.1.5 We say that a self-financing portfolio strategy $(h_t)$ is an arbitrage possibility if
$\seteqnumber{0}{15.}{3}$
\begin{align*} V^h_0&=0\\ \P [V^h_t\geq 0]&=1\\ \P [V^h_t> 0]&>0\\ \end{align*}

Definition 15.1.6 A contingent claim with date of exercise $T$ is any random variable of the form $X=\Phi (S_T)$, where $\Phi :[0,\infty )\to \R $ is a deterministic function.

We say that a portfolio strategy $h_t=(x_t,y_t)$ hedges (or replicates) $X$ if $(h_t)$ is self-financing and $V^h_T=X$.

For now, we restrict ourselves to portfolios $h_t=(x_t,y_t)$ containing only cash and stock. Later on, in Chapter 16, we will also consider portfolios that include financial derivatives (such as call/put options).

Stochastic Processes and Financial Mathematics (part two)

Chapter 15 The Black-Scholes model

15.1 The Black-Scholes market

Stochastic Processes and Financial Mathematics
(part two)