# Black and Scholes option pricing

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: $$$$0 \ \ \ \text{if} \ S_T

SOLUTION I have rewritten the payoff as: $$$$Payoff_T=(K_2-S_T)\textbf{1}_{\{K_1K_2\}}$$$$ Since $$S$$ evolves under martingale measure $$\mathbb{Q}$$ as a geometric Brownian Motion whit dynamic: $$$$S_t=S_se^{(R-\frac{\sigma^2}{2})(t-s)+\sigma Y\sqrt{t-s}}$$$$ where $$Y\sim N(0,1)$$ then I whant to compute for which values of $$Y$$: $$$$K_1 similarly I get: $$$$S_T Now applying the formula for the price in B$$\&$$S market: $$$$price_t=e^{-R(T-t)}E^{\mathbb{Q}}(Payoff_T|\mathcal{F}_t)\\ =e^{-R(T-t)}\bigg(\int_{y_1}^{y_2}(K_2-S_te^{(R-\frac{\sigma^2}{2})(T-t)+\sigma y\sqrt{T-t}})\frac{1}{\sqrt{2\pi}}e^{-y^2/2}dy+\int_{y_2}^{\infty}(K_2-K_1)\frac{1}{\sqrt{2\pi}}e^{-y^2/2}dy\bigg)$$$$ Now i omit the computation of this integrals (not difficult) and I have the final formula where I denote with $$\Phi(x)=P(X\leq x)$$ with $$X\sim N(0,1)$$: $$$$price_t=e^{-R(T-t)}(K_2-K_1)(1-\Phi(y_2))+K_2e^{-R(T-t)}(\Phi(y_2)-\Phi(y_1))-S_t(\Phi(y_2-\sigma\sqrt{T-t})-\Phi(y_1-\sigma\sqrt{T-t}))$$$$ 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?