Black Scholes replication and the pricing PDE

Definition The Black-Scholes market on a horizon $[0, T]$ has two traded assets. The money-market account is $B_t = e^{rt}$ with a constant continuously compounded rate $r$. The stock has an ex-dividend price with

under the physical measure $\mathbb{P}$, with a constant drift $\mu$, a constant volatility $\sigma > 0$, a starting price $S_0 > 0$ and a constant continuous dividend yield $q$, so that one share pays cash at the rate $q S_t$. The filtration is the one generated by the single Brownian motion $W$, and stock paths are continuous. The frictionless assumptions are that trading is continuous in time, any real number of shares may be held, short sales are allowed with full use of the proceeds, cash is lent and borrowed at the same rate $r$, and there are no transaction costs, bid-ask spreads, taxes, margin limits or price impact. Trading strategies are predictable, satisfy the integrability that their stochastic integrals need, and are admissible. Admissibility is a technical restriction that rules out doubling strategies, for example a lower bound on the value or a square-integrability condition on the discounted gains under a pricing measure. The claim to be priced is European and pays $g(S_T)$ at $T$.

Definition A trading strategy holds $\Delta_t$ shares and $\beta_t$ units of the money-market account, so its value is $\Pi_t = \Delta_t S_t + \beta_t B_t$ and its cash holding is $\beta_t B_t = \Pi_t - \Delta_t S_t$. It is self-financing when

understood in integral form, with $\Delta$ predictable and $\int_0^T \Delta_t^2 S_t^2\,dt < \infty$ almost surely. The three terms are the gain on the shares, the dividends they receive and the interest on the cash holding, which is negative when cash is borrowed. This extends the discrete accounting of the lesson on self-financing portfolios. For strategies that are themselves Itô processes, the product rule gives

and the self-financing condition says that the last three terms add up to the dividends $q \Delta_t S_t\,dt$. Shares bought at the market price are paid for out of the cash account, so rebalancing by itself creates no gain. Given a pricing function $V(t, S)$ with a continuous derivative $\partial_S V$, the delta hedge is the self-financing strategy that holds

shares and keeps the rest of its value in cash. Its Brownian part is $\sigma S_t \Delta_t\,dW_t$, the same as the Brownian part $\sigma S_t\,\partial_S V(t, S_t)\,dW_t$ of $V(t, S_t)$ in Itô’s formula. The holding is chosen from the current time and price before the next Brownian increment, so it is predictable.

Definition Work under the frictionless assumptions with $\sigma > 0$, and let $V$ have continuous derivatives $\partial_t V$, $\partial_S V$ and $\partial_{SS} V$ on $[0, T) \times (0, \infty)$. Suppose a self-financing strategy has value $\Pi_t = V(t, S_t)$ for $t < T$. Itô’s formula gives

with the derivatives at $(t, S_t)$. The decomposition of a continuous Itô process into a $dt$ part and a $dW_t$ part is unique, so the coefficients can be compared with those of the self-financing equation. The $dW_t$ coefficients give $\sigma S_t \Delta_t = \sigma S_t\,\partial_S V$, and since $\sigma S_t > 0$ the strategy must be the delta hedge. With that holding the $dt$ coefficients give

The term $\mu S\,\partial_S V$ appears on both sides and cancels, which leaves the Black-Scholes PDE along the path. For every $t > 0$ the law of $S_t$ has a positive density on all of $(0, \infty)$, so by continuity the PDE holds at every point of $[0, T) \times (0, \infty)$. Conversely, suppose $V$ is also continuous on $[0, T] \times (0, \infty)$, solves the PDE and satisfies $V(T, S) = g(S)$. Then Itô’s formula and the PDE show that the delta hedge started with $V(0, S_0)$ is self-financing with value $V(t, S_t)$ at every $t < T$, and this value tends to $g(S_T)$ as $t \to T$. If the hedge and the opposite position are both admissible, no arbitrage forces the claim’s price at time $t$ to be $V(t, S_t)$. At any other price, buying the cheaper of the claim and the hedge and selling the dearer locks in the difference with no risk, because the two pay the same at $T$. This lesson derives the PDE in this way and verifies a smooth example. A general existence theory for nonsmooth payoffs, and a general representation of every claim as a hedge, have further hypotheses and are stated as boundaries rather than proved.

Definition Under the frictionless assumptions with constants $r$, $q$ and $\sigma > 0$, the Black-Scholes PDE for a pricing function $V(t, S)$ is

for $0 \le t < T$ and $S > 0$, with $q = 0$ for a dividend-free stock. It is linear, second order and parabolic, it is solved backward from maturity, and its coefficients do not contain $\mu$. The four terms are the time derivative in calendar time, the gamma term, the carry of the stock position at the rate $r - q$ and the financing of the claim’s value at the rate $r$. In the log price $x = \ln S$, where $S\,\partial_S = \partial_x$ and $S^2\,\partial_{SS} = \partial_{xx} - \partial_x$, the equation has constant coefficients,

the form from which numerical schemes and the reduction to the heat equation start.

Definition The terminal condition of a European claim with payoff $g$ is $V(T, S) = g(S)$ for every $S > 0$, and the pricing function must approach it, $V(t, S) \to g(S)$ as $t \uparrow T$. For a call with strike $K$ the payoff is $g(S) = (S - K)^{+}$, and for a put it is $g(S) = (K - S)^{+}$. The PDE alone has many solutions, and the terminal condition selects the one that belongs to the contract. The data are given at the final time and the equation is solved toward earlier times. In the remaining time $\tau = T - t$ it becomes a forward parabolic equation with initial data $g$, while prescribing the values at $t = 0$ and solving forward in calendar time would be an ill-posed backward heat problem. A payoff need not be smooth. The call payoff has a kink at $K$, so the smooth derivation is applied for $t < T$, where the model’s call price is smooth when $\sigma > 0$, and the payoff is reached as the limit at maturity. Weak and viscosity formulations handle nonsmooth problems without claiming classical derivatives everywhere, and they are not developed here.

Definition The Black-Scholes PDE is posed on the unbounded domain $S > 0$, and a GBM price started above zero never reaches zero or infinity in finite time. No boundary value is imposed in the classical sense. Instead, the solution is required to lie in a growth class, for example $\lvert V(t, S) \rvert \le A(1 + S^m + S^{-m})$ for constants $A$ and $m$, which excludes solutions that grow too fast toward the edges of the domain. How such a growth class gives uniqueness is shown in the lesson on Feynman-Kac. The behavior near the edges then follows from the payoff. Setting $S = 0$ in the equation removes the terms with derivatives in $S$ and leaves $\partial_t V - rV = 0$, which is consistent with the limits $V(t, 0+) = e^{-r\tau} g(0)$ of a call and a put, where $\tau = T - t$. For a call, $V(t, S) \to 0$ as $S \to 0$, and as $S \to \infty$

For a put, $V(t, S) \to K e^{-r\tau}$ as $S \to 0$ and $V(t, S) \to 0$ as $S \to \infty$. A numerical method on a truncated interval $[S_{\min}, S_{\max}]$ needs conditions at the artificial edges that agree with these limits, such as $V(t, S_{\max}) = S_{\max} e^{-q\tau} - K e^{-r\tau}$ for a call, and the error from the truncation shrinks as the interval widens.