Black&Scholes Market pricing

Question

Given a Black&Scholes market and a derivative contract with payoff at the maturity T given by $(K_{2} −2*S_{T})*\mathbb{1}_{K_{1}<S_{T}<K_{2}}$,

How can I compute the price of the contract at any time $t ∈ [0, +∞)$?

Answer 1

$\newcommand{\Q}{\Bbb{Q}}\newcommand{\1}{\mathbf{1}}$Some hints: you will want to find the discounted $\Q$-expectation of the given payoff function. To do this, note that the $K_2\1_{K_1 < S_T < K_2}$ part is relatively easy to handle, assuming you know the (lognormal) distribution of $S_T$ (and remember that the $\Q$-expectation of an indicator is just the $\Q$-probability of the event). For finding the discounted $\Q$-expectation of the $S_T\1_{K_1<S_T<K_2}$ part, note that $\1_{K_1<S_T<K_2} = \1_{S_T < K_2}-\1_{S_T \le K_1}$.

So you essentially have to find discounted $\Q$-expectations of things of the form $S_T \1_{S_T < K}$. To do this, note that $$S_T \1_{S_T < K} = \color{blue}{\left(S_T - K\right)\1_{S_T < K}}+K\1_{S_T < K},$$ and use the formula for the price of a put to help find the discounted $\Q$-expectation of the first part.