Payoff and Boundary Conditions for Black-Scholes Claims

From a Contractual Payoff to PDE Data

A European claim pays a specified amount at its maturity \(T\). For a call with strike \(K>0\), the holder receives the excess of the stock price over \(K\), if that excess is positive. A European put with the same strike pays the excess of \(K\) over the stock price, if positive. These maturity payments become the terminal values of the corresponding price functions in the Black–Scholes partial differential equation (PDE).

Let \(S_t\) be the stock price, and write \(C(t,S)\) and \(P(t,S)\) for the prices at time \(t\) when the stock price is \(S>0\). Let \(r\) be a constant interest rate, \(\sigma>0\) a constant volatility, and assume there are no dividends. Under the physical probability measure \(\mathbb{P}\), the stock may have drift \(\mu\); under the risk-neutral measure \(\mathbb{Q}\), its drift is \(r\). This distinction matters: the Black–Scholes PDE uses the risk-free rate, not the physical drift. The earlier replication argument gives, for either price \(V=C\) or \(V=P\),

The derivatives are taken with respect to time \(t\) and stock price \(S\). The terminal condition specifies the contract’s actual payment at \(t=T\); it is not a boundary condition at \(S=0\) or \(S=\infty\). To avoid confusing these roles, set \(\tau=T-t\) for time remaining. When a boundary condition depends on time, its discounting is over \(\tau\), not over the elapsed time \(t\).

The Terminal Conditions, Including the Kink

The positive-part operation makes both payoffs continuous but not differentiable at \(S=K\). That kink is part of the terminal data: one must not replace it by a smooth approximation or assign a single classical derivative there. At expiry, the call has zero value for \(S\le K\) and value \(S-K\) for \(S>K\). The put has value \(K-S\) for \(S<K\) and zero for \(S\ge K\).

Proof. By definition, \(x^+=\max(x,0)\). For the call, if \(S-K\le0\), then \((S-K)^+=0\); if \(S-K>0\), then \((S-K)^+=S-K\). For the put, if \(K-S>0\), then \((K-S)^+=K-S\); if \(K-S\le0\), then \((K-S)^+=0\). At \(S=K\), both arguments of the positive-part function equal zero. These are exactly the stated terminal values. \(\square\)

The Zero-Stock Boundary

In the no-dividend Black–Scholes model, a stock starting at zero remains at zero. The PDE gives the corresponding price boundary by setting \(S=0\): the terms containing \(S\) and \(S^2\) vanish, leaving the time equation \(V_t-rV=0\). The values at \(S=0\) must also match the contracts’ zero-stock terminal payoffs.

Proof. Write \(v(t)=V(t,0)\). At \(S=0\), the PDE reduces to \(v'(t)-rv(t)=0\). Multiplying by \(e^{-rt}\) gives \(\frac{d}{dt}(e^{-rt}v(t))=0\), so \(e^{-rt}v(t)\) is constant. Therefore \(v(t)=v(T)e^{r(t-T)}=v(T)e^{-r(T-t)}\). The call’s terminal value at zero is \(C(T,0)=0\), which gives \(C(t,0)=0\). The put’s terminal value is \(P(T,0)=K\), which gives \(P(t,0)=Ke^{-r(T-t)}\). \(\square\)

Large-Stock Boundary Behavior

At very large stock prices, the call is in the money and has approximately one unit of stock exposure: its leading value grows like \(S\). The strike is a fixed cash amount payable through the payoff, so its contribution at time \(t\) is discounted over the remaining time. The put is far out of the money and its price tends to zero. These give the standard large-stock conditions for selecting the call and put solutions:

These are asymptotic conditions, not exact formulas at finite \(S\). In particular, the call condition says the difference from \(S-Ke^{-r(T-t)}\) tends to zero; it does not say that the call equals that expression for every stock price. At \(t=T\), the conditions match the terminal payoffs at large \(S\): the call is exactly \(S-K\) for \(S>K\), and the put is zero for \(S\ge K\). At earlier times the discounted strike is \(Ke^{-r(T-t)}\), which becomes \(K\) as \(t\) approaches \(T\).

Why These Conditions Select the Intended Claim

The terminal payoff alone identifies what the claim pays at expiry. The boundary conditions add information about behavior at the edges of the spatial domain \(S>0\). The zero-stock condition captures the model’s behavior when the underlying has no value. The large-stock condition controls growth and distinguishes the one-stock-unit exposure of the call from the vanishing put value at very high stock prices.

For a well-posed pricing problem, these conditions are imposed together with appropriate regularity and growth restrictions on the PDE solution. They are not optional refinements of the payoff: on an unbounded spatial domain, specifying terminal data without controlling the boundary behavior need not by itself select the intended solution. A proof of uniqueness for a chosen class of solutions requires a parabolic comparison or uniqueness argument; the boundary identities above do not replace that argument.

The risk-neutral measure \(\mathbb{Q}\) is the measure under which the no-dividend stock drift is \(r\); it is distinct from the physical measure \(\mathbb{P}\), under which the stock drift need not equal \(r\). The boundary conditions are PDE data and do not assert a pricing-expectation formula. No stochastic integral is needed to state or derive these conditions. In the underlying model, stochastic integrals are Itô integrals on the filtered probability space, with progressively measurable integrands in \(\mathcal{L}^2\) on finite horizons when the square-integrability condition \(\mathbb{E}\int_0^T H_s^2\,ds<\infty\) applies.

Common Mistakes

Using the strike without discounting at the large-stock boundary. A common incorrect condition is \(C(t,S)-(S-K)\to0\) for every \(t<T\). It uses the terminal cash amount \(K\) instead of its time-\(t\) discounted contribution. The appropriate condition is \(C(t,S)-(S-Ke^{-r(T-t)})\to0\); the two agree at \(t=T\), but generally differ before maturity.

Giving the put its undiscounted zero-stock payoff at all times. The incorrect calculation \(P(t,0)=K\) ignores the PDE boundary equation. Solving \(v'(t)-rv(t)=0\) with \(v(T)=K\) gives \(P(t,0)=Ke^{-r(T-t)}\), not \(K\) when \(r(T-t)\ne0\).

Confusing the terminal condition with a spatial boundary condition. The equality \(C(T,S)=(S-K)^+\) applies at maturity across stock prices. The equality \(C(t,0)=0\) applies at zero stock across times. They constrain different edges of the time-stock domain and should be stated separately.