Delta Hedging a Black–Scholes Claim from Its Pricing Function

# Replicating strategy in the Black-Scholes model


# Replicating strategy in the Black-Scholes model












I have a two-asset Black-Scholes model for a financial market:

$dB_t=B_t r dt$

$dS_t=S_t(\mu dt+\sigma dW_t)$

I introduce a European claim $\xi=max(K,S_T)$ with maturity $T$, for some fixed $K$. I have calculated what the no-arbitrage price of this claim should be at each time $t<T$ by computing expectations under the equivalent martingale measure, which is a function of $S_t$, $t$, and the fixed parameters in the model. and I am now asked to find a replicating portfolio in the original 2 asset market for this claim.

I know that if $V(t,S)$ is a solution to the Black-Scholes PDE subject to the terminal condition $V(T,S)=\max(K,S)$, then $V(t,St)$ is a no-arbitrage time-$t$ price for $\xi$, and that the trading strategy given by taking initial wealth to be $V(0,S_0)$ and the time-$t$ holding in the stock to be $\frac{\partial V}{\partial S}$ is a replicating strategy for the claim.

If I view the pricing function I originally found (by computing expectations) as a function $\xi(t,St)$, is it necessarily true taking initial wealth to be $\xi(0,S_0)$ and taking time-t holding in the stock to be $\frac{\partial \xi}{\partial S}$ will give a replicating portfolio? It should be, simply due to the fact there is a unique equivalent martingale measure in this market, so there must be a unique no-arbitrage time -$t$ cost for the claim at each time $t$, and so $\xi(t,S_t)$ must solve the Black-Scholes PDE.

My question is, is it possible to prove that this trading strategy does replicate the claim without appealing to the fact that the pricing function solves the Black-Scholes PDE?

## Answer by Vince (score 1)

https://quant.stackexchange.com/a/8035

these kinds of questions usually require careful attention to details: if it's a hw question of some kind, consult shreve's lecture notes, he has a whole section on this precise topic in all its glory. as for intuition, since holding $\frac{\partial \xi}{\partial S}$ at any point in time eliminates the dW term, in the context of a discrete time period model of the evolution of your wealth and whether it hedges the derivative in each admissible state of the world, by hedging the 'brownian' randomness it is exposed to at each time step (there are two unknowns at each time step, how much to hold of the stock and how much to invest in the money market, which depend on the realization of the brownian), by induction on $n$, where n is the number of time steps used to discretize time, the result follows.