# Boundary Conditions for Call Option in Black Scholes Model

Let $$C(t,S)$$ be the value function of a call option. I want to price that option using (explicit) finite differences and the Black Scholes PDE. I consider the grid $$0=t_0 and $$S_0.

I impose the boundary conditions

• Payoff: $$C(t_N,S_j)=(S_j-K)^+$$ for all $$j=0,...,M$$,
• Low stock price: $$C(t_i,S_0)=0$$ for all $$i=0,...,N-1$$,
• High stock price: $$C(t_i,S_M)=S_M$$ for all $$i=0,...,N-1$$.

But isn't there a jump in the option value in the top right corner? At expiry, we use the payoff $$C(t_N,S_M)=S_M-K$$ but then we use $$C(t_{N-1},S_M)=S_M$$ as upper stock price boundary condition for all other time points? But that means that over $$\Delta t$$, the option price jumps by \$$$K$$. The conditions for $$S=0$$ and $$t=T$$ match in the $$(t_N,S_0)$$ point but there seems to be a mismatch for $$(t_N,S_M)$$? ## 2 Answers Note that: $$C(t,S) =S-K{\rm e}^{-r(T-t)}$$ as $$S\rightarrow \infty$$, for all $$t$$. Basically because one can easily accept $$P(t,S) =0$$ as $$S\rightarrow \infty$$, for all $$t$$, and one still expects the put-call parity to hold: $$C(t,S) - P(t,S) = S-K{\rm e}^{-r(T-t)}$$ for all $$S$$. • I always associated$S-Ke^{-r(T-t)}$(i.e. the futures value) with a lower bound for$C(t,S)$. But I see the relationship with the put call parity. So, you'd suggest to use$S_j-Ke^{-r(T-t_i)}\$ as upper stock price condition for finite differences?
– Alex
Mar 10 at 21:16
• Yes. No other way. :) You'll need to insert the dividends too.
– ir7
Mar 10 at 21:20
• Thanks very much, the link to the put-call parity was really nice :)
– Alex
Mar 10 at 21:29

I would argue your assumption is a bit confusing. C(T, Smax) is the payoff Smax-K. But in this case, whatever your Maturity time is, Smax can be really large. However when you say C(t,Smax) this is bad because at the starting point you dont know that it is Smax, you only know the current/underlying value of the asset. And in the case of a payoff you are pricing for the end of the time period, T. So C(t,Smax) would not be Smax. At that point the call option does not have value since with European options the best payoff would be at time of execution. But to know that American payoff you would have to know the discounted payoff. I think your boundaries are close but I'm not sure why you high stock price is written as C(t,Smax). That little "t" is the beginning time when you start, where neither the call payoff nor the underlying should be maxed out.

• Thanks for your answer. I clarified my question and stated the grid I'm using. Perhaps that makes my question clearer?
– Alex
Mar 10 at 20:48
• @Alex the comment above me regarding Put-Call Parity is spot on. Thats the way to think about it Mar 11 at 21:34