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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?

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    $\begingroup$ In this special case, can't you just use the fact that $\xi = \eta +K$ where $\eta = \max(0,S_T - K)$ is a claim for European call? Also, computing the Ito differential of the discounted portfolio can lead you to the strategy (just take a look on the martingale term). $\endgroup$
    – Ilya
    May 21, 2013 at 11:20
  • $\begingroup$ Thanks for your response - I'm not quite sure what you mean by calculating the Ito differential of the discounted portfolio - do you mean the Ito differential of the discounted value of the portfolio? $\endgroup$
    – Mark
    May 21, 2013 at 11:42
  • $\begingroup$ yes, I mean that if you replication portfolio is of the form $$ \mathrm dX_t = \beta_t\mathrm dB_t + \gamma_t\mathrm dS_t $$ and it is self-financing, then $\frac{X_t}{B_t}$ has to be a martingale. $\endgroup$
    – Ilya
    May 21, 2013 at 11:51
  • $\begingroup$ I understand that the discounted wealth process of a general portfolio is a local martingale under the equivalent local martingale measure (the measure under which S/B is a local martingale) - unless I'm mistaken, I don't see why it must be a true martingale for a replicating strategy, since if $\pi$ is a replicating strategy, and $\phi$ is an admissible strategy with initial wealth 1 and terminal wealth 0, then whilst both $\pi$ and $\pi+\phi$ are replicating strategies, at most one can be a martingale. $\endgroup$
    – Mark
    May 21, 2013 at 12:02
  • $\begingroup$ EDIT: I see that if we want a strategy $\pi$ such that the associated wealth process is equal to $\xi_t$ for all $0 \leq t \leq T$, then the discounted wealth process must be a martingale under the equivalent local martingale measure, but I'm not sure how to bring $\frac{\partial \xi}{\partial S}$ into this. $\endgroup$
    – Mark
    May 21, 2013 at 12:10

1 Answer 1

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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.

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  • $\begingroup$ Thanks for your response - do you know where I can find Shreve's notes? $\endgroup$
    – Mark
    May 21, 2013 at 11:51
  • $\begingroup$ amazon.com/Stochastic-Calculus-Finance-Binomial-Textbooks/dp/… $\endgroup$
    – Veeken
    May 21, 2013 at 16:25
  • $\begingroup$ @Mark, I think his pre-publishing notes are still available for free as pdf download somewhere. Google and you shall find. Btw, unless I am mistaken those are his lecture notes which he compiled to his two books, one on Discrete the other on Continuous time series. And the notes are not copyright protected and can thus be legally obtained for free. But if you intend to make a career in this field I highly recommend to get his books. In my opinion the best treatise of introductory stochastic calculus $\endgroup$
    – Matt Wolf
    May 23, 2013 at 3:18

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