Black–Scholes

6D.3 Stochastic Differential Equations and Feynman–Kac–6D.5 Diffusions on Manifolds and Curvature built diffusions, their densities and their geometry. Book 6D now turns to the application that made stochastic calculus an industry. An option is a contract giving the right, but not the obligation, to buy (a call) or sell (a put) an asset at a fixed strike price $K$ at a fixed expiry $T$. At expiry a call is worth $(S_T - K)^+$, where $S_T$ is the asset price then. What is it worth today? The naive answer, "the expected payoff, discounted", requires the expected return of the asset, which nobody knows, and a judgement about risk, which differs between people.

The answer found by Fischer Black, Myron Scholes and Robert Merton in 1973 needs neither. If the asset follows geometric Brownian motion, the option's payoff can be replicated exactly by continuously trading the asset and a bank account, and the cost of the replicating strategy is the only price that does not allow riskless profit (arbitrage). The cost satisfies a parabolic PDE, the Black–Scholes equation, which is the heat equation in disguise (6A.3 The Heat Equation on ℝⁿ), and the expected return of the asset drops out. Equivalently, the price is a discounted expectation, but under an artificial risk-neutral probability in which the asset earns the riskless rate, which is the Feynman–Kac formula of 6D.3 Stochastic Differential Equations and Feynman–Kac combined with Girsanov's theorem (6D.2 Martingales and the Itô Integral).

By the end of this chapter you will be able to:

• set up the Black–Scholes market, define self-financing strategies and arbitrage, and derive the Black–Scholes PDE by replication;
• prove that a classical solution of the PDE gives a replicating strategy, and hence the arbitrage-free price;
• transform the Black–Scholes equation into the heat equation, and derive the Black–Scholes formula by risk-neutral expectation;
• prove put–call parity, compute the Greeks, and check the PDE in terms of them;
• explain implied volatility, the volatility smile, and how Dupire's equation recovers a local volatility from option prices.

6.1A formula and a market

6.2The market and replication

The Black–Scholes market has two assets: a bank account $\beta_t = e^{rt}$ with constant interest rate $r$, and a stock whose price follows geometric Brownian motion (6D.2 Martingales and the Itô Integral, 6D.3 Stochastic Differential Equations and Feynman–Kac)

with drift $\mu$ and volatility $\sigma > 0$.

A trading strategy holds $\Delta_t$ shares of stock and the rest of its wealth in the bank, with $\Delta$ adapted (decided on present information). Its wealth $V_t$ is self-financing if all changes in wealth come from changes in prices, with no money added or withdrawn:

An arbitrage is a self-financing strategy with $V_0 = 0$, $V_T \geq 0$ almost surely and $\mathbb{P}(V_T > 0) > 0$: a free lottery ticket. A pricing rule is arbitrage-free if no arbitrage can be built from trading the stock, the bank and the priced contracts. Competition removes arbitrage from real markets quickly, so this is the basic economic assumption.

Suppose the price of the option at time $t$ is a smooth function $v(t, S_t)$. By Itô's formula,

Compare with a self-financing portfolio holding $\Delta_t = \partial_Sv(t, S_t)$ shares: its $dS$ terms match, and the $dt$ terms match exactly when $\partial_tv + \frac12\sigma^2S^2\partial_{SS}v = r(v - S\partial_Sv)$. This is the Black–Scholes equation

for a contract paying $h(S_T)$ at expiry ($h(S) = (S - K)^+$ for a call).

The drift $\mu$ does not appear in (BS). The option is priced relative to the stock, by replication, and two investors who disagree about the stock's expected return but agree on its volatility must agree on the option price. That is the conceptual breakthrough of 1973. The hedge ratio $\Delta = \partial_Sv$ is the number of shares that makes the portfolio locally riskless: the "delta hedge".

6.3Reduction to the heat equation

(BS) is a backward parabolic equation: it is solved from terminal data at $t = T$ backwards in time, which is the well-posed direction for an equation with $+\frac12\sigma^2S^2\partial_{SS}$ and $+\partial_t$ on the same side. Its coefficients vary with $S$, but they are those of the Euler-type operator $S\partial_S$, $S^2\partial_{SS}$, which logarithmic coordinates turn into constants.

So the solution is $w = \Phi_\tau * w(\cdot, 0)$ with the heat kernel of 6A.3 The Heat Equation on ℝⁿ, and everything known about the heat equation transfers: existence by the kernel, uniqueness in the class of functions of moderate growth (Tychonoff's example shows growth conditions are needed, 6A.3 The Heat Equation on ℝⁿ), smoothing (the option price is smooth in $S$ before expiry even though the payoff has a kink), and the maximum principle (if one payoff dominates another, so do their prices). The initial data grow like $e^{(k+1)|x|/2}$, an admissible rate for uniqueness.

6.4The Black–Scholes formula

Convolving with the heat kernel works, but the risk-neutral route is shorter and more illuminating. By the Feynman–Kac formula (6D.3 Stochastic Differential Equations and Feynman–Kac) applied to (BS), whose generator is $L = rS\partial_S + \frac12\sigma^2S^2\partial_{SS}$ and whose discount rate is $r$,

where $\tilde S$ is a geometric Brownian motion with drift $r$ in place of $\mu$. (The Feynman–Kac theorem as proved assumed bounded coefficients; for this equation it is justified by the change of variables above, or directly by the Itô argument with the explicit $\tilde S$.)

Risk-neutral pricing. The formula $v = e^{-r\tau}\mathbb{E}[h(\tilde S_T)]$ says the price is a discounted expectation in a world where the stock earns the riskless rate. Girsanov's theorem (6D.2 Martingales and the Itô Integral) makes this precise: with the market price of risk $\theta = \frac{\mu - r}{\sigma}$ and $d\mathbb{Q} = e^{-\theta B_T - \theta^2T/2}d\mathbb{P}$, the process $W_t = B_t + \theta t$ is a $\mathbb{Q}$-Brownian motion and $dS = rS\,dt + \sigma S\,dW$. Under $\mathbb{Q}$, discounted prices $e^{-rt}S_t$ and $e^{-rt}V_t$ (for every self-financing strategy) are martingales, so no strategy can turn nothing into something on average, and hence (with positivity) not at all: the first fundamental theorem of asset pricing, that the existence of such an equivalent martingale measure is equivalent to the absence of arbitrage (Harrison and Kreps 1979, Harrison and Pliska 1981; in full generality Delbaen and Schachermayer 1994). Its second fundamental theorem says the market is complete (every payoff can be replicated) exactly when this measure is unique, which in the Black–Scholes model follows from the martingale representation theorem (Øksendal, section 4.3).

Parity uses no model at all, only the absence of arbitrage, which is why it is the first thing traders check.

6.5The Greeks

The sensitivities of the price to its inputs are the Greeks. For the call, with $\varphi$ the standard normal density:

The derivations all rest on one identity, $S\varphi(d_1) = Ke^{-r\tau}\varphi(d_2)$ (Exercise 6.8), which makes the terms coming from differentiating $d_1$ and $d_2$ cancel. For the example above, $\Delta \approx 0.637$, $\Gamma \approx 0.0188$, $\mathcal V \approx 37.5$ (per unit of volatility, so $0.375$ per volatility point) and $\Theta \approx -6.41$ per year.

In terms of the Greeks, the Black–Scholes equation reads

which the numbers satisfy to rounding (Exercise 6.9). It is the trader's statement of the PDE: a delta-hedged option position loses value over time ($\Theta < 0$) at exactly the rate needed to pay for the gains from convexity ($\Gamma > 0$) when the stock moves. Hedging discretely, the hedger earns $\frac12\Gamma S^2\big((\Delta S/S)^2 - \sigma^2\Delta t\big)$ each period: the difference between realised and implied variance, weighted by gamma. This is the practical reason why the quadratic variation of 6D.1 Brownian Motion is the quantity that matters.

6.6Implied volatility and the smile

Every input to the formula is observable except $\sigma$. Since $C$ is strictly increasing in $\sigma$ ($\mathcal V > 0$), a market price determines a unique implied volatility, the $\sigma$ that reproduces it. If the model were exact, implied volatility would be the same for every strike and expiry. It is not. For equity indices, implied volatility is higher for low strikes than for high ones (a skew), a pattern that became pronounced after the stock market crash of October 1987, as option buyers started paying more for protection against large falls. For currencies it is often U-shaped (a smile). The market uses the Black–Scholes formula as a quoting convention, a nonlinear change of units from prices to implied volatilities, and the smile measures how the true distribution of returns departs from the lognormal one: heavier tails and asymmetry.

Local volatility. One response is to let the volatility depend on time and price, $dS = rS\,dt + \sigma(t, S)S\,dW$, and ask whether some function $\sigma(t, S)$ reproduces all observed call prices $C(K, T)$ at once. Bruno Dupire showed in 1994 that it does, and gave a formula.

Option prices across strikes therefore reveal the risk-neutral density of the future price (Douglas Breeden and Robert Litzenberger, 1978). In the local volatility model, $p$ satisfies the Fokker–Planck equation of 6D.3 Stochastic Differential Equations and Feynman–Kac, $\partial_Tp = -\partial_K(rKp) + \frac12\partial_{KK}(\sigma^2(T, K)K^2p)$. Integrating it twice in $K$ against the payoff and using the proposition gives Dupire's equation (derivation outline; Exercise 6.11):

a forward parabolic equation in the strike and expiry variables, so that

Local volatility is read off from the surface of call prices. In practice the surface must be interpolated and differentiated twice, an ill-posed inverse problem that needs regularisation, and the dynamics the model implies for the smile are known to be unrealistic; stochastic volatility models (Heston, 1993) and their combinations are the industry's next step.

6.7History

Louis Bachelier priced options with Brownian motion in 1900 (6D.1 Brownian Motion). Paul Samuelson proposed geometric Brownian motion for stock prices in 1965. Fischer Black and Myron Scholes, and Robert Merton, derived the replication argument and the formula in 1973. J. Michael Harrison, David Kreps and Stanley Pliska founded the martingale approach in 1979–81. Douglas Breeden and Robert Litzenberger's identity dates from 1978, Steven Heston's stochastic volatility model from 1993, and Bruno Dupire's local volatility equation from 1994 (with related work by Emanuel Derman and Iraj Kani).

6.8Exercises

Solution

(a) $\Phi(-0.15) \approx 0.4404$, $\Phi(-0.35) \approx 0.3632$: $P \approx 95.12(0.4404) - 100(0.3632) \approx 41.89 - 36.32 = 5.57$, matching parity. (b) Its price is $e^{-r\tau}\mathbb{Q}(\tilde S_T > K) = e^{-r\tau}\mathbb{P}(Z > -d_2) = e^{-r\tau}\Phi(d_2) \approx 0.9512 \times 0.5596 \approx 0.532$.

Solution

(a) $d_1 - d_2 = \sigma\sqrt\tau$ and $d_1 + d_2 = \frac{2\log(S/K) + 2r\tau}{\sigma\sqrt\tau}$, so $\frac{d_1^2 - d_2^2}{2} = \log\frac SK + r\tau$ and $\frac{\varphi(d_2)}{\varphi(d_1)} = e^{(d_1^2 - d_2^2)/2} = \frac{S}{K}e^{r\tau}$. (b) $\partial_SC = \Phi(d_1) + S\varphi(d_1)\partial_Sd_1 - Ke^{-r\tau}\varphi(d_2)\partial_Sd_2 = \Phi(d_1)$, by (a).

Solution

$rS\Delta \approx 0.05\times100\times0.63683 = 3.1842$; $\frac12\sigma^2S^2\Gamma \approx 0.5\times0.04\times10000\times0.018762 = 3.7524$; sum with $\Theta$: $-6.4140 + 3.1842 + 3.7524 = 0.5226$; and $rC \approx 0.05\times10.4506 = 0.5225$. They agree to rounding.

Solution

$C \leq S$: one share always pays at least as much as the call. $C \geq 0$: the payoff is non-negative. $C \geq S - Ke^{-r\tau}$: from parity, $C = S - Ke^{-r\tau} + P$ with $P \geq 0$. As $\sigma \to 0$, $d_1, d_2 \to \pm\infty$ according to the sign of $\log(S/K) + r\tau$, giving $(S - Ke^{-r\tau})^+$. As $\sigma \to \infty$, $d_1 \to +\infty$ and $d_2 \to -\infty$, so $C \to S$.

Solution

$\partial_TC = \int(s - K)^+\partial_Tp\,ds = \frac12\int(s - K)^+\partial_{ss}(\sigma^2s^2p)\,ds = \frac12\int\partial_{ss}(s - K)^+\,\sigma^2s^2p\,ds = \frac12\sigma^2(T, K)K^2p(T, K)$, and $p(T, K) = \partial_{KK}C$ by Breeden–Litzenberger with $r = 0$.

Solution

$w(x, \tau) = \int_0^\infty\frac{e^{-(x - y)^2/(4\tau)}}{\sqrt{4\pi\tau}}\big(e^{(k+1)y/2} - e^{(k-1)y/2}\big)dy$. Each term is the integral of a Gaussian times an exponential over a half-line; completing the square turns it into $e^{(\cdot)}$ times a Gaussian integral over a half-line, i.e. an exponential factor times $\Phi$ of an argument linear in $x$ divided by $\sqrt{2\tau}$. Undoing the change of variables, the two arguments become $d_1$ and $d_2$, and the exponential factors become $S$ and $Ke^{-r(T-t)}$.

Solution

(a) Feynman–Kac with $h = \delta_K$: $v = e^{-r(T-t)}\mathbb{E}^{\mathbb{Q}}[\delta(S_T - K)] = e^{-r(T-t)}p^{\mathbb{Q}}(T, K\mid t, S)$, and linearity gives the integral formula. (b) In $(t, S)$, $G$ is the solution of (BS) with terminal data $\delta_K$; in $(T, K)$ the density is the solution of the forward equation started at $\delta_S$ at time $t$. (c) The conjugate heat kernel is the analogue of $G$ as a function of $(t, S)$: a solution of a backward equation with a delta function as terminal data. The delta function at the final time plays the role of the payoff of an Arrow–Debreu security, and integrating the kernel against a function at the final time, like pricing a payoff, gives the solution with that terminal data.