Black–Scholes Call Option Boundary Conditions and Replication

Summary

The document asks how to justify the boundary conditions for the Black–Scholes equation for a European call. It states the pricing PDE and lists a zero call value at zero underlying price, an asymptotic call value matching the underlying price as that price becomes very large, and the terminal payoff equal to the positive part of price minus strike.

The question specifically asks whether these conditions can be proved from the value of a replicating portfolio, or by induction. No derivation or answer is included. The material identifies the conditions to justify but does not explain their mathematical basis or distinguish terminal payoff from spatial boundary behavior, so it is a prompt for a proof rather than a complete lesson.

Key ideas

• The document states the Black–Scholes PDE for a call option.
• It lists zero value at zero underlying price and an asymptotic condition at very high prices.
• The terminal condition is the call’s payoff at maturity.
• It asks whether replication or induction can establish the conditions but supplies no proof.

Tags

# Can we proof the boundary condition for the Black Scholes derived from a replicating Portfolio?


# Can we proof the boundary condition for the Black Scholes derived from a replicating Portfolio?












So for Black Scholes we know that the PDE is the follwing: ${\frac {\partial V}{\partial t}}+{\frac {1}{2}}\sigma ^{2}S^{2}{\frac {\partial ^{2}V}{\partial S^{2}}}=rV-rS{\frac {\partial V}{\partial S}}$. The boundary conditions for that equations then are the following:

\begin{aligned}C(0,t)&=0{\text{ for all }}t\\C(S,t)&\rightarrow S{\text{ as }}S\rightarrow \infty \\C(S,T)&=\max\{S-K,0\}\end{aligned}

How do we proof the boundary condition exactly? Can we do it by induction (e.g. that the value of the replicating portfolio should be equal to the call option?) or by some thing else ? Thank you in advance for helping out!

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.