7.1 Black-Scholes Differential Equation

Simple generally accepted economic assumptions are insufficient to develop a rational option pricing theory. Assuming a perfect financial market in Section 2.1 lead to elementary arbitrage relations which options have to fulfill. While these relations can be used as a verification tool for sophisticated mathematical models, they do not provide an explicit option pricing function depending on parameters such as time and the stock price as well as the options underlying parameters $ K, \,
T.$ To obtain such a pricing function the value of the underlying financial instrument (stock, currency, ...) has to be modelled. In general, the underlying instrument is assumed to follow a stochastic process either in discrete or in continuous time. While the latter are analytically easier to handle, the former, which we will consider as approximations of continuous time processes for the time being, are particularly useful for numerical computations. In the second part of this text, the discrete time version will be discussed as financial time series models.

A model for stock prices which is frequently used and is also the basis of the classical Black-Scholes approach, is the so-called geometric Brownian motion. In this model the stock price $ S_t$ is a solution of the stochastic differential equation

$\displaystyle dS_t = \mu S_t dt + \sigma S_t dW_t .$ (7.1)

Equivalently, the process of stock price returns can be assumed to follow a standard Brownian motion, i.e.

$\displaystyle \frac{dS_t}{S_t} = \mu dt + \sigma dW_t .$ (7.2)

The drift $ \mu$ is the expected return on the stock in the time interval $ dt.$ The volatility $ \sigma$ is a measure of the return variability around its expectation $ \mu.$ Both parameters $ \mu$ and $ \sigma$ are dependent on each other and are important factors of the investors' risk preferences involved in the investment decision: The higher the expected return $ \mu,$ the higher, in general, the risk quantified by $ \sigma.$

Modelling the underlying as geometric Brownian motion provides a useful approximation to stock prices accepted by practitioners for short and medium maturity. In real practice, numerous model departures are known: in some situations the volatility function $ \sigma(x,t)$ of the general model (5.8) is different from the linear specification $ \sigma \cdot x$ of geometric Brownian motion. The Black-Scholes' approach is still used to approximate option prices. The basic idea to derive option prices can be applied to more general stock price models.

Black-Scholes' approach relies on the idea introduced in Chapter 2, i.e. duplicating the portfolio consisting of the option by means of a second portfolio consisting exclusively of financial instruments whose values are known. The duplicating portfolio is chosen such that both portfolios have equal values at options maturity $ T.$ Then, it follows from the assumption of a perfect financial market, and in particular of no-arbitrage opportunities, that both portfolios must have equal values at any time prior to time $ T.$ The duplicating portfolio can be created in two equivalent ways which we illustrate in an example of a call option on a stock with price $ S_t$:

1. Consider a portfolio consisting of one call of which the price is to be computed. The duplicating portfolio is composed of stocks and risk-less zero bonds of which the quantity adapts continuously to changes in the stock price. Without loss of generality, the zero bond's face value can be set equal to one since the number of zero bond's in the duplicating portfolio is free parameter. At time $ t$ the two portfolios consist of:

Portfolio $ A$:
One call option (long position) with delivery price $ K$ and maturity date $ T.$
Portfolio $ B$:
$ n_t = n(S_t, t)$ stocks and $ m_t = m(S_t,t)$ zero bonds with face value $ B_T = 1$ and maturity date $ T.$
2. Consider a perfect hedge-portfolio, which consists of stocks and written calls (by means of short selling). Due to a dynamic hedge-strategy the portfolio bears no risk at any time, i.e. profits due to the calls are neutralized by losses due to the stocks. Correspondingly, the duplicating portfolio is also risk-less and consists exclusively of zero bonds. Again, the positions are adjusted continuously to changes in the stock price. At time $ t$ the two portfolios are composed of:
Portfolio $ A$:
One stock and $ n_t = n(S_t, t)$ (by means of short selling) written call options on the stock with delivery price $ K$ and maturity date $ T.$
Portfolio $ B$:
$ m_t = m(S_t,t)$ zero bonds with face value $ B_T = 1$ and maturity dates $ T.$
Let $ T^*=T$ be the time when the call option expires worthless, and otherwise let $ T^*$ be the time at which the option is exercised. While for a European call option it holds $ T^*=T$ at any time, an American option can be exercised prior to maturity. We will see that both in 1. the call value is equal to the value of the duplicating portfolio, and in 2. the hedge-portfolio's value equals the value of the risk-less zero bond portfolio at any time $ t \leq T^*,$ and thus the same partial differential equation for the call value results, which is called Black-Scholes equation.

The Black-Scholes approach can be applied to any financial instrument $ \cal{U}$ contingent on an underlying with price $ S_t$ if the latter price follows a geometric Brownian motion, and if the derivatives price $ F_t$ is a function only of the price $ S_t$ and time: $ F_t = F(S_t,t).$ Then, according to the theorem below, a portfolio duplicating the financial instrument exists, and the approach illustrated in 1. can be applied to price the instrument. Pricing an arbitrary derivative the duplicating portfolio must have not only the same value as the derivative at exercising time $ T^*,$ but also the same cash flow pattern, i.e. the duplicating portfolio has to generate equal amounts of withdrawal profits or contributing costs as the derivative. The existence of a perfect hedge-portfolio of approach 2. can be shown analogously.

Theorem 7.1  
Let the value $ S_t$ of an object be a geometric Brownian motion (6.1). Let $ \cal{U}$ be a derivative contingent on the object and maturing in $ T.$ Let $ T^* \leq T$ be the time at which the derivative is exercised, or rather $ T^*=T$ if it is not. Let the derivative's value at any time $ t \leq T^*$ be given by a function $ F(S_t,t)$ of the object's price and time.
a)
It exists a portfolio consisting of the underlying object and risk-less bonds which duplicates the derivative in the sense that it generates up to time $ T^*$ the same cash flow pattern as $ \cal{U},$ and that it has the same time $ T^*$ value as $ \cal{U}.$
b)
The derivatives value function $ F(S,t)$ satisfies Black-Scholes partial differential equation

$\displaystyle \frac{\partial F(S,t) }{\partial t} - rF(S,t) + bS \frac{\partial...
...1}{2}\sigma^2 S^2\frac{\partial^2 F(S,t)}{\partial S^2} = 0 , \quad t \leq T^*.$ (7.3)

Proof:
To simplify we proceed from the assumption that the object is a stock paying a continuous dividend yield $ d,$ and thus involving costs of carry $ b=r-d$ with $ r$ the continuous compounded risk-free interest rate. Furthermore, we consider only the case where $ \cal{U}$ is a derivative on the stock, and that $ \cal{U}$ does not generate any payoff before time $ T^*.$

We construct a portfolio consisting of $ n_t = n(S_t, t)$ shares of the stock and $ m_t = m(S_t,t)$ zero bonds with maturity date $ T$ and a face value of $ B_T = 1.$ Let

$\displaystyle B_t = B_Te^{-r(T-t)} = e^{-r(t-T)}$

be the zero bond's value discounted to time $ t.$ We denote the time $ t$ portfolio value by

$\displaystyle V_t \stackrel{\mathrm{def}}{=}V(S_t,t) = n(S_t,t) \cdot S_t + m(S_t,t) \cdot B_t .$

It is to show that $ n_t$ and $ m_t$ can be chosen such that at exercise time respectively at maturity of $ \cal{U}$ the portfolio value is equal to the derivative's value, i.e.  $ V(S_{T^*},T^*) =
F(S_{T^*},T^*).$ Furthermore, it is shown that the portfolio does not generate any cash flow prior to $ T^*,$ i.e. it is neither allowed to withdraw nor to add any money before time $ T^*.$ All changes in the positions must be realized by buying or selling stocks or bonds, or by means of dividend yields.

First of all, we investigate how the portfolio value $ V_t$ changes in a small period of time $ dt.$ By doing this, we use the notation $ dV_t = V_{t+dt} - V_t, \; dn_t = n_{t+dt} - n_t$ etc.

$\displaystyle dV_t$ $\displaystyle =$ $\displaystyle n_{t+dt} S_{t+dt} + m_{t+dt}B_{t+dt} - n_t S_t - m_t B_t$  
  $\displaystyle =$ $\displaystyle dn_t S_{t+dt} + n_t dS_t + dm_t B_{t+dt} + m_t dB_t ,$  

and thus

$\displaystyle dV_t = dn_t (S_t+dS_t) + n_t dS_t + dm_t (B_t+dB_t) + m_t dB_t .$ (7.4)

Since the stochastic process $ S_t$ is a geometric Brownian motion and therefore an Itô-process (5.8) with $ \mu(x,t) = \mu
x$ and $ \sigma(x,t) = \sigma x,$ it follows from the generalized Itô lemma (5.10) and equation (6.1)

$\displaystyle dn_t = \frac{\partial n_t}{\partial t}dt + \frac{\partial n_t}{\partial S}dS_t + \frac{1}{2}\frac{\partial^2 n_t}{\partial S^2}\sigma ^2 S_t^2 dt ,$ (7.5)

and an analogous relation for $ m_t.$ Using

$\displaystyle (dS_t)^2 = ( \mu S_t dt + \sigma S_t dW_t )^2 = \sigma^2 S_t^2 (d...
...ptstyle \mathcal{O}}(dt) = \sigma^2 S_t^2 dt + {\scriptstyle \mathcal{O}}(dt) ,$

$\displaystyle dB_t = rB_t dt , \; dS_t\cdot dt = {\scriptstyle \mathcal{O}}(dt) \;$   and$\displaystyle \; dt^2 ={\scriptstyle \mathcal{O}}(dt)$

and neglecting terms of size smaller than $ dt$ it follows:

$\displaystyle dn_t (S_t + dS_t) = \Big(\displaystyle\frac{\partial n_t }{\parti...
...sigma ^2 S_t^2 dt \Big)S_t + \frac{\partial n_t}{\partial S}\sigma^2 S_t^2 dt ,$ (7.6)

$\displaystyle dm_t (B_t + dB_t) = \Big(\displaystyle\frac{\partial m_t }{\parti...
...}\frac{\partial^2 m_t}{\partial S^2}\sigma ^2 S_t^2 dt \Big)B_t . \hspace*{2cm}$ (7.7)

The fact that neither the derivative nor the duplicating portfolio generates any cash flow before time $ T^*$ means that the terms $ dn_t (S_t + dS_t)$ and $ dm_t (B_t + dB_t)$ of $ dV_t$ in equation (6.4) which correspond to purchases and sales of stocks and bonds have to be financed by the dividend yields. Since one share of the stock pays in a small time interval $ dt$ a dividend amount of $ d\cdot S_t \cdot dt,$ we have

$\displaystyle d \cdot n_t S_t \cdot dt = (r-b) \cdot n_t S_t \cdot dt = dn_t (S_t + dS_t) + dm_t (B_t + dB_t).$

Substituting equations (6.6) and (6.7) in the latter equation, it holds:
0 $\displaystyle =$ $\displaystyle (b-r)n_tS_tdt +\Big(\displaystyle\frac{\partial m_t
}{\partial t}...
...S_t +
\frac{1}{2}\frac{\partial^2 m_t}{\partial S^2}\sigma ^2 S_t^2 dt \Big)B_t$  
$\displaystyle *[+3mm]$   $\displaystyle + \Big(\displaystyle\frac{\partial n_t }{\partial t}dt + \frac{\p...
...igma ^2 S_t^2 dt \Big)S_t +
\frac{\partial n_t}{\partial S}\sigma ^2 S_t^2 dt .$  

Using equation (6.1) and summarizing the stochastic terms with differential $ dW_t$ as well as the deterministic terms with differential $ dt$ containing the drift parameter $ \mu,$ and all other deterministic terms gives:
0 $\displaystyle =$ $\displaystyle \quad \Big(\displaystyle\frac{\partial n_t
}{\partial S}S_t + \frac{\partial m_t }{\partial S} B_t\Big) \mu
S_t dt$  
$\displaystyle *[+3mm]$   $\displaystyle + \,
\Big\{\Big(\displaystyle\frac{\partial n_t }{\partial t} +
\...
...ma ^2 S_t^2
\Big)S_t +
\frac{\partial n_t}{\partial S}\sigma ^2 S_t^2 \nonumber$  
    $\displaystyle + \Big(\displaystyle\frac{\partial m_t }{\partial t} + \frac{1}{2...
...{\partial^2 m_t}{\partial S^2}\sigma ^2 S_t^2 \Big)B_t +
(b-r)n_tS_t \Big \} dt$  
$\displaystyle *[+3mm]$   $\displaystyle + \, \Big(\displaystyle\frac{\partial n_t }{\partial S}S_t + \frac{\partial m_t }{\partial S} B_t\Big)\sigma S_t
dW_t.$ (7.8)

This is only possible if the stochastic terms disappear, i.e.

$\displaystyle \frac{\partial n_t }{\partial S}S_t + \frac{\partial m_t }{\partial S} B_t = 0 .$ (7.9)

Thus the first term in (6.8) is neutralized as well. Hence the middle term must also be zero:
$\displaystyle \Big(\frac{\partial n_t }{\partial
t}+\frac{1}{2}\frac{\partial^2...
...l S^2}
\sigma ^2 S_t^2 \Big)S_t + \frac{\partial n_t}{\partial S}\sigma^2 S_t^2$      
$\displaystyle + \Big(\frac{\partial m_t }{\partial
t}+\frac{1}{2}\frac{\partial^2 m_t}{\partial S^2} \sigma^2 S_t^2
\Big)B_t + (b-r)n_tS_t = 0 .$     (7.10)

To further simplify we compute the partial derivative of equation (6.9) with respect to $ S:$

$\displaystyle \frac{\partial^2 n_t }{\partial S^2}S_t + \frac{\partial n_t }{\partial S} + \frac{\partial^2 m_t }{\partial S^2} B_t = 0$ (7.11)

and substitute this in equation (6.10). We then obtain

$\displaystyle \frac{\partial n_t }{\partial t}S_t + \frac{\partial m_t }{\parti...
... + \frac{1}{2}\frac{\partial n_t}{\partial S}\sigma^2 S_t^2 + (b-r)n_tS_t = 0 .$ (7.12)

Since the stock price $ S_t$ does not depend explicitly on time, i.e.  $ \partial S_t / \partial t = 0,$ the derivative of the portfolio value $ V_t = n_tS_t + m_tB_t$ with respect to time gives:

$\displaystyle \frac{\partial V_t }{\partial t} = \frac{\partial n_t }{\partial ...
...tial n_t
}{\partial t} S_t + \frac{\partial m_t }{\partial t} B_t + m_t r
B_t .$

This implies

$\displaystyle \frac{\partial n_t }{\partial t} S_t + \frac{\partial m_t }{\part...
...\partial t} - r m_t B_t
= \frac{\partial V_t }{\partial t} - r(V_t - n_t S_t) .$

Substituting this equation in equation (6.12) we eliminate $ m_t$ and obtain

$\displaystyle \frac{1}{2}\sigma^2 S_t^2\frac{\partial n}{\partial S} + \frac{\partial V_t }{\partial t} +bn_tS_t - rV_t = 0 .$ (7.13)

Since the zero bond value $ B_t$ is independent of the stock price $ S_t,$ i.e.  $ \partial B_t / \partial S = 0,$ the derivative of the portfolio value $ V_t = n_tS_t + m_tB_t$ with respect to the stock price gives (using equation (6.9))

$\displaystyle \frac{\partial V_t }{\partial S} = \frac{\partial n_t }{\partial S} S_t + n_t + \frac{\partial m_t }{\partial S} B_t = n_t ,$

and thus

$\displaystyle n_t = \frac{\partial V_t }{\partial S} .$ (7.14)

That is, $ n_t$ is equal to the so-called delta or hedge-ratio of the portfolio (see Section 6.3.1). Since

$\displaystyle m_t = \frac{V_t - n_t S_t}{B_t} $

we can construct a duplicating portfolio if we know $ V_t =
V(S_t,t).$ We can obtain this function of stock price and time as a solution of the Black-Scholes differential equation

$\displaystyle \frac{\partial V(S,t) }{\partial t} - rV(S,t) + bS \frac{\partial...
...\partial S} + \frac{1}{2}\sigma^2 S^2\frac{\partial^2 V(S,t)}{\partial S^2} = 0$ (7.15)

which results from substituting equation (6.14) in equation (6.13). To determine $ V$ we have to take into account a boundary condition which is obtained from the fact that the cash flows at exercising time respectively maturity, i.e. at time $ T^*,$ of the duplicating portfolio and the derivative are equal:

$\displaystyle V(S_{T^*},T^*) = F(S_{T^*},T^*) .$ (7.16)

Since the derivative has at any time the same cash flow as the duplicating portfolio, $ F(S,t)$ also satisfies the Black-Scholes differential equation, and at any time $ t \leq T^*$ it holds $ F_t
= F(S_t,t) = V(S_t,t) = V_t.$ $ {\Box}$

Black-Scholes' differential equation fundamentally relies on the assumption that the stock price can be modelled by a geometric Brownian motion. This assumption is justified, however, if the theory building on it reproduces the arbitrage relations derived in Chapter 2. Considering an example we verify this feature. Let $ V(S_t,t)$ be the value of a future contract with delivery price $ K$ and maturity date $ T$. The underlying object involves costs of carry at a continuous rate $ b$. Since $ V(S_t,t)$ depends only on the price of the underlying and time it satisfies the conditions of Theorem 6.1. From Theorem 2.1 and substituting $ \tau = T - t$ for the time to maturity it follows

$\displaystyle V(S,t) = S e^{(r-b)(t-T)} - Ke^{r(t-T)} .$

Substituting the above equation into equation (6.3) it can be easily seen that it is the unique solution of Black-Scholes' differential equation with boundary condition $ V(S,T) = S - K.$ Hence, Black-Scholes' approach gives the same price for the future contract as the model free no-arbitrage approach.

Finally, we point out that modelling stock prices by geometric Brownian motion gives reasonable solutions for short and medium terms. Applying the model to other underlyings such as currencies or bonds is more difficult. Bond options typically have significant longer time to maturity than stock options. Their value does not only depend on the bond price but also on interest rates which have to be considered stochastic. Modelling interest rates reasonably involves other stochastic process, which we will discuss in later chapters.

Generally exchange rates cannot be modelled by geometric Brownian motion. Empirical studies show that the performance of this model depends on the currency and on the time to maturity. Hence, applying Black-Scholes' approach to currency options has to be verified in each case. If the model is used, the foreign currency has to be understood as the option underlying with a continuous foreign interest rate $ d$ corresponding to the continuous dividend yield of a stock. Thus, continuous costs of carry with rate $ b=r-d$ equal the interest rate differential between the domestic and the foreign market. If the investor buys the foreign currency early, then he cannot invest his capital at home any more, and thus he looses the domestic interest rate $ r$. However, he can invest his capital abroad and gain the foreign interest rate $ d$. The value of the currency option results from solving Black-Scholes' differential equation (6.3) respecting the boundary condition implied by the option type.