The Black-Scholes Equation

The pioneering work of Black and Scholes[7] started the serious study of the theory of option pricing. All further advances in this field have been extensions and refinements of the original idea expressed in that paper. Considering its immense importance in the field of option pricing, we will study the theory behind the Black-Scholes equation in some detail. We will also introduce the quantum mechanical formulation of the Black-Scholes theory which will be of great use in the following chapters as we consider stochastic volatility.

The Assumptions Behind the Black-Scholes Equation

There are several assumptions involved in the derivation of the Black-Scholes equation. It is important to understand these properly so as to see the limitations of the theory. These assumptions are summarised below :

1. The efficient market hypothesis is assumed to be satisfied. In other words, the markets are assumed to be liquid, have price-continuity, be fair and provide all players with equal access to available information. This implies that zero transaction costs are assumed in the Black-Scholes analysis.
2. It is assumed that the underlying security is perfectly divisible and that short selling with full use of proceeds is possible.
3. Constant risk-free interest rates are assumed. In other words, we assume that there exists a risk-free security which returns $1 at time $T$ when $$e^{-r(T-t)}$ is invested at time $t$.
4. As we will see later, the Black-Scholes analysis requires continuous trading. This is, of course, not possible in practice as the more frequently one trades, the larger the transaction costs.
5. The prinicple of no arbitrage is assumed to be satisfied.
6. The price of the underlying security is assumed to follow a geometric Brownian process of the form $dS = (\phi S + \sigma SW)dt$ where $W$ is white noise.

It should be obvious that none of these principles can be perfectly satisfied. Transaction costs exist in all markets, all securities come in discrete units, short selling with full use of proceeds is very rare, interest rates vary with time and we will later see that there is evidence that the price of most stocks do not precisely follow a geometric Brownian process.

Considering the above, it may seem strange that we pay so much importance to the Black-Scholes theory. There are several reasons for this. The most important one is that the concepts behind the Black-Scholes analysis provide the framework for thinking about option pricing. All the research in option pricing since the Black-Scholes analysis has been done either to extend it or to generalize it. Another important reason for studying the Black-Scholes theory is that the financial world uses it as a standard. In fact, traders quote Black-Scholes volatility to each other, not the actual price of the options! Further, Black-Scholes prices still give very good approximations to the prices of options.

Self-financing, Replicating Hedging Strategies

One common way of pricing a derivative is to form a self-financing, replicating hedging strategy for it. (A strategy is basically an algorithm which allows us to form a portfolio with certain properties.) To understand what this means, we will have to look at each term in turn. Self-financing means that the portfolio produced by the strategy must not itself take up any money apart from a possible initial investment. As we will soon see, this initial investment will be the price of the security that the strategy is replicating. The term replicating means that the strategy must replicate the payoff of the security we are trying to price. Further, the portfolio produced by the strategy should always produce the same result regardless of price changes in the underlying security. In other words, the value of the portfolio generated by the strategy should be deterministic and cannot have a stochastic component (except for the stochastic components of the underlying securities of the derivative). This explains the term ``hedging'' used for the strategy.

For example, suppose we have a 2-year coupon bond with payoffs of $1 every half a year and $11 at the maturity. We can replicate this bond using the following portfolio : one zero-coupon bond (we assume that all zero-coupon bonds have a payoff of $1 at maturity) each of maturity 0.5 year, 1 year, 1.5 years and 11 zero-coupon bonds of maturity 2 years. As each of the zero-coupon bonds mature, they replicate the payoff of the coupon-bearing bond. The portfolio in this example is perfectly hedged since the underlying price of the bonds is known at maturity. The present value of the bonds is not important for our purposes since our aim is only to replicate the payoff of the coupon-bearing bond.

By the principle of no arbitrage, any portfolio that can replicate the payoff of an option (or, for that matter, any derivative) must have the same value as that of the option. In the above example, the price of the coupon bond must be the same as the combination of zero-coupon bonds. The Black-Scholes analysis makes a portfolio which replicates the payoff of an option, hence solving the valuation problem.

Risk-Neutral Valuation

This is the most important principle in derivative valuation. It states that the value of a derivative is its expected future value discounted at the risk-free interest rate. This is exactly the same result that we would obtain if we assumed that the world was risk-neutral. In such a world, investors would require no compensation for risk. This means that the expected return on all securities would be the risk-free interest rate. This is a very useful principle as it states that we can assume that the world is risk-neutral when calculating option prices. The result would still be correct in the real world even if (as is most probably the case) it is not risk-neutral.

We derive the put-call parity relation (1.4) using this principle. To do so, use the fact that the sum of the payoffs of a long call and a short put option with the same strike price and maturity (and, of course, on the same underlying security)is given by $S-K$. Hence, the value of the call and put option is given by $e^{-r(T-t)} E[S-K] = e^{-r(T-t)}(E[S] - K) = S-Ke^{-r(T-t)}$, the last equality coming from risk-neutral valuation. This gives us $C-P = S - Ke^{-r(T-t)}$ or $C+Ke^{-r(T-t)} = P+S$ which is the same as (1.4).

Ito's Lemma

We need one more piece of information before we can derive the Black-Scholes equation. This is Ito's lemma which is an important result in the theory of stochastic processes. We will also provide a simple though non-rigorous derivation of the lemma.

Ito's lemma states that if a variable $x$ follows a stochastic process of the form

$$dx = a(x, t)\,dt + b(x, t)\,Wdt$$ (43)

where $W$ is white noise, then any smooth function $G(x,t)$ follows the process

$$dG = \left(\frac{\partial G}{\partial x}a + \frac{\partial G... ...rtial x^2}b^2 \right) dt + \frac{\partial G}{\partial x} bWdt.$$ (44)

We now present a non-rigorous definition of Ito's lemma using the Taylor series formula. For a smooth function $G(x, t)$, the normal Taylor series expansion goes as

$$\Delta G = \pdif{G}{x} \Delta x + \pdif{G}{t} \Delta t + \fr... ...elta x \Delta t + \frac{1}{2}\pdiftwo{G}{t} \Delta t^2 + \dots$$ (45)

For a non-stochastic process, when $\Delta x \tendsto 0, \Delta t \tendsto 0$, the above equation becomes

$$dG = \pdif{G}{x}dx + \pdif{G}{t}dt.$$ (46)

However, as $x$ follows the process $dx = a(x, t)dt + b(x, t)Wdt$, the discretised form of which is $\Delta x = a(x, t)\Delta t + b(x, t)\epsilon \sqrt{\Delta t}$ where $\epsilon$ is a standard normal variable, we see that

$$\Delta x^2 = a^2\Delta t^2 + 2ab\epsilon\Delta t^{\frac{3}{2... ...^2\Delta t = b^2\epsilon^2\Delta t + O(\Delta t^{\frac{3}{2}})$$ (47)

Now, since $\epsilon$ is a standard normal variable, we know that $E(\epsilon^2) = 1$. Thus, $E(\Delta x^2) = b^2\Delta t$ which is a first order term! Further, the variance of $\epsilon^2\Delta t$ is $E(\epsilon^4\Delta t^2) - E^2(\epsilon^2\Delta t) = O(\Delta t^2)$. Hence, as $\Delta t \tendsto 0$, $\Delta x^2$ becomes deterministic and equal to its expected value which is $b^2\Delta t$. Hence, we get (3.2) which is Ito's lemma. While this is far from a rigorous proof which can be found in any advanced book on stochastic processes such as Ross[4], it gives us a good idea of why (3.2) is correct.

The Black-Scholes Equation

We are now in a position to present a derivation of the Black-Scholes equation. We consider a general derivative $f$ whose value is a function of the value of the underlying security $S$. $S$ is assumed to follow the stochastic process

$$dS = \phi Sdt + \sigma SWdt$$ (48)

where $\phi$ (the average growth rate of the underlying security) and $\sigma$ (the volatility) are constants. Using Ito's lemma, we see that

$$df = \pdif{f}{S}dS + \left(\pdif{f}{t} + \frac{\sigma^2S^2}{... ...a^2S^2}{2} \pdiftwo{f}{S} \right) dt + \sigma SW\pdif{f}{S} dt$$ (49)

We cannot value this directly as there is a stochastic term. To eliminate the stochastic term, we consider the portfolio $\Pi = f - \pdif{f}{S}S$. We see that

$$d\Pi = df - \pdif{f}{S}dS = \left(\pdif{f}{t} + \frac{\sigma... ...f}{S} \right) dt = r\Pi dt = r\left(f - \pdif{f}{S}S\right) dt$$ (50)

with the last equality following from the no-arbitrage condition (since there is no stochastic term, $\Pi$ is a risk-free investment and hence must offer the same return as any other risk-free investment). Simplifying the above equation, we obtain the Black-Scholes equation

$$\pdif{f}{t} + rS\pdif{f}{S} + \frac{1}{2}\sigma^2S^2 \pdiftwo{f}{S} =rf.$$ (51)

The initial (or, in finance, usually final) conditions determine the kind of derivative that we are pricing. For a call option, the final condition we have to use is $f = \max(S-K, 0)$. We see that the principle of risk-neutral valuation is clearly satisfied in this case since the Black-Scholes equation is independent of $\phi$, the expected rate of growth of the underlying security price.

It is important to note that the portfolio $\Pi$ represents a self-financing, replicating, hedging strategy. It replicates a risk-free investment and it is hedged since it has no stochastic component.

Solution of the Black-Scholes Equation

There are several ways of solving the Black-Scholes equation. I will present two solutions, one using the principle of risk-neutral valuation and the other using the quantum mechanical formulation. These are two of the most elegant solutions using minimal mathematics.

Solution Using the Principle of Risk-Neutral Valuation

We can try to solve equation (3.9) directly but a simpler way exists to get the solution. This involves analysing the assumed process for the stock prices (3.6) using Ito's lemma and applying the principle of risk-neutral valuation to the result. Applying Ito's lemma to (3.6) gives us

$$d(\ln S) = \left(\phi - \frac{\sigma^2}{2}\right) dt + \sigma W$$ (52)

Now, the time integral of the white noise $W$ will give us a random walk whose distribution we know to be normal. In fact, it can be easily seen from the above equation that

$$\ln S - \ln S_0 \sim N\left[\left(\phi - \frac{\sigma^2}{2}\right)(T-t), \sigma\sqrt{T-t}\right]$$ (53)

(where $S$ and $S_0$ are the prices of the underlying security at time $T$ and $t$ respectively) or

$$\ln S \sim N\left[\ln S_0 + \left(\phi - \frac{\sigma^2}{2}\right)(T-t), \sigma\sqrt{T-t}\right]$$ (54)

which shows that $S$ follows a lognormal distribution.

We are now almost done. The principle of risk-neutral valuation implies that the present value of the option is the expected final value $E[\max(S-K, 0)]$ of the option discounted at the risk-free interest rate. So, we have

$$c = e^{-r(T-t)}E[\max(S-K, 0)] = e^{-r(T-t)}\int_K^\infty (S-K)g(S)dS$$ (55)

where $g(S)$, the probability density function of $S$ is given by (3.12) which can be explicitly written as

$$g(S) = \frac{1}{\sigma S\sqrt{2\pi(T-t)}} \exp\left(- \frac... ...ac{\sigma^2}{2}\right) (T-t)\right)^2}{2\sigma^2 (T-t)}\right)$$ (56)

where $\phi$ has been replaced by $r$ in accordance with the principle of risk-neutral valuation. We can easily verify that this solution satisfies the principle of risk-neutral valuation by evaluating $E[S] = \int_0^\infty Sg(S) dS = S_0e^{r(T-t)}$.

The value of the integral (3.13) can be found with a bit of algebraic manipulation and is

$$c = SN(d_1) - Ke^{-r(T-t)}N(d_2)$$ (57)

where

$$d_1 = \frac{\ln\left(\frac{S}{K}\right) + \left(r + \frac{\s... ...}{2}\right)(T-t)}{\sigma \sqrt{T-t}} = d_1 - \sigma \sqrt{T-t}$$ (58)

and $N(x)$ is the cumulative standard normal distribution.

While the result looks very complicated, it has an intuitive interpretation. Equation (3.15) can be written as

$$c = e^{-r(T-t)}[e^{r(T-t)}SN(d_1) - KN(d_2)].$$ (59)

$N(d_2)$ is the probability that the final stock price will be above $K$ (in other words, that the option will be exercised) in a risk-neutral world so that $KN(d_2)$ is the strike price times the probability that the strike price will be paid. The expression $SN(d_1) e^{r(T-t)}$ is the expected value of a variable that equals $S$ if $S>K$ and 0 otherwise in a risk-neutral world. In other words, $e^{r(T-t)}SN(d_1) - KN(d_2)$ is the expected value of the option at maturity. The above result is therefore just an expression of the principle of risk-neutral valuation.

We can also form a self-financing, replicating hedging strategy for the call option. The portfolio in this case is made up of $N(d_1)$ units of the underlying security and $KN(d_2)$ zero-coupon bonds with the same maturity as the option. This approach is discussed in great detail in Chriss[1]. We do not consider this approach here as it is unintuitive since the result must be known in advance.

Solution Using the Quantum Mechanical Formalism

The following solution is adapted from Baaquie [8].

We can recast the Black-Scholes equation (3.9) into a Schrödinger-like equation. We identify $f$, the value of the option as a wave function dependent on time and the price of the underlying security. Then, the Schrödinger equation becomes

$$\pdif{f}{t} = (r + \hat{H}_{BS})f$$ (60)

where $\hat{H}_{BS}$ is the Hamiltonian. From (3.9), we see that the Hamiltonian is given by

$$\hat{H}_{BS} = -\frac{1}{2}\sigma^2S^2 \pdiftwo{f}{S} - rS\pdif{f}{S}.$$ (61)

For subsequent simplification, we introduce the variable $x = \ln S$. The Hamiltonian is then given by

$$\hat{H}_{BS} = -\frac{\sigma^2}{2}\pdiftwo{}{x} + (\frac{\sigma^2}{2} - r)\pdif{}{x}$$ (62)

The price of the European call option is then given by

$$f(t, x) = e^{-r(T-t)}\int_{-\infty}^\infty dx' \matel{x} {e^{-\hat{H}_{BS}(T-t)}}{x'} h(x')$$ (63)

where $h(x') = \max(e^{x'}-K, 0)$ is the final wave function (note that time runs backward in this formulation).

We now change the basis to the momentum basis where $\hat{H}_{BS}$ is diagonal. The transformation from the ``position'' basis to the ``momentum'' basis is defined by

$$\langle x \mid x' \rangle = \delta(x-x') = \int_{-\infty}^\i... ...{-\infty}^\infty \frac{dp}{2\pi}\innprod{x} {p}\innprod{p}{x'}$$ (64)

The Hamiltonian $\hat{H}_{BS}$ in the momentum basis is given by

$$\matel{p}{\hat{H}_{BS}}{p'} = \left[\frac{\sigma^2p^2}{2} + ip\left(\frac{\sigma^2}{2} - r\right)\right]\delta(p-p')$$ (65)

Hence,
$$\begin{align} \matel{x}{e^{-\tau \hat{H}_{BS}}}{x'} &= \int_{-\infty}^\infty \f... ... \left[x-x' + \tau\left(r-\frac{\sigma^2}{2}\right)\right]^2\right) \end{align}$$
where $\tau = T-t$. After changing variables from $x$ to $S$ and noting that time runs backward here, we see that this distribution is the same as (3.14) as it should be. Hence, the solution is the same as the one using the risk-neutral valuation method as it should be.

While the advantages of the quantum mechanical method may not be very obvious here, they will become obvious when we consider stochastic volatility (i.e., when we allow $\sigma$ to follow a stochastic process).

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