Black and Scholes option pricing

Question

I have to solve the following problem in the Black and scholes model: find the price at anty $t\in[0,T)$ for an option whose payoff at the maturity is: \begin{equation} 0 \ \ \ \text{if} \ S_T<K_1\\ K_2-S_T \ \ \ \text{if} \ K_1<S_T<K_2\\ K_2-K_1 \ \ \ \text{if} \ K_2< S_T \end{equation}

SOLUTION I have rewritten the payoff as: \begin{equation} Payoff_T=(K_2-S_T)\textbf{1}_{\{K_1<S_T<K_2\}}+(K_2-K_1)\textbf{1}_{\{S_T>K_2\}} \end{equation} Since $S$ evolves under martingale measure $\mathbb{Q}$ as a geometric Brownian Motion whit dynamic: \begin{equation} S_t=S_se^{(R-\frac{\sigma^2}{2})(t-s)+\sigma Y\sqrt{t-s}} \end{equation} where $Y\sim N(0,1)$ then I whant to compute for which values of $Y$: \begin{equation} K_1<S_T\Rightarrow K_1< S_te^{(R-\frac{\sigma^2}{2})(T-t)+\sigma Y\sqrt{T-t}}\Rightarrow\dfrac{K_1}{S_t}e^{-(R-\frac{\sigma^2}{2})(T-t)}<e^{\sigma Y\sqrt{T-t}}\\ \Rightarrow y_1=\dfrac{\ln(\frac{K_1}{S_t})-(R-\frac{\sigma^2}{2})(T-t)}{\sigma\sqrt{T-t}}<Y \end{equation} similarly I get: \begin{equation} S_T<K_2\Rightarrow Y<\dfrac{\ln(\frac{K_2}{S_t})-(R-\frac{\sigma^2}{2})(T-t)}{\sigma\sqrt{T-t}}=y_2 \end{equation} Now applying the formula for the price in B$\&$S market: \begin{equation} price_t=e^{-R(T-t)}E^{\mathbb{Q}}(Payoff_T|\mathcal{F}_t)\\ =e^{-R(T-t)}\bigg(\int_{y_1}^{y_2}(K_2-S_te^{(R-\frac{\sigma^2}{2})(T-t)+\sigma y\sqrt{T-t}})\frac{1}{\sqrt{2\pi}}e^{-y^2/2}dy+\int_{y_2}^{\infty}(K_2-K_1)\frac{1}{\sqrt{2\pi}}e^{-y^2/2}dy\bigg) \end{equation} Now i omit the computation of this integrals (not difficult) and I have the final formula where I denote with $\Phi(x)=P(X\leq x)$ with $X\sim N(0,1)$: \begin{equation} price_t=e^{-R(T-t)}(K_2-K_1)(1-\Phi(y_2))+K_2e^{-R(T-t)}(\Phi(y_2)-\Phi(y_1))-S_t(\Phi(y_2-\sigma\sqrt{T-t})-\Phi(y_1-\sigma\sqrt{T-t})) \end{equation} At this point my questions are:

1. is this computation fine?
2. Since the second question of the exercise is to compute the delta of the contract (Derivative w.r.t the underlying S) is it possible to express the payoff in terms of Call/put options for which I know an explicit expression of the delta?

Answer 1

Your computations seem correct to me.

As for your second question, I think the only viable solution is to compute the derivative. If I'm not misteken, we can add and subtract some terms to write at the end the price of a put, but we add additional terms which we have to derive anyway.

So I believe the fastest solution is to just use the definition of Delta and compute the derivative wrt the price.