# Pricing Exotic options

I am stuck at a assignment problem where I have to compute the price of an exotic option.

I am given the values the prices of option $C(X;k) = E[max(0,X_T - k)]$ for different strike prices $k$ and I have to compute the exotic price

$D(X;k) = E[I(X_T > k)]$ for the same set of $k$s.

I did this using difference of sum and got

$D(X;k) = C(X;k) - C(X;k+1)$

Now using these I have to compute another exotic price:

$P(X;k) = E[max(X_T - k,0)^2]$

I tried using the same method to get :

$P(X;k) - P(X;k-1) = 2C(X;k) - D(X;k+1)$

but to numerically calculate $P(X;k)$ at some k, I need the value of $P(X;k+1)$ and I have no initial value for the pricing $P$.

Any help would be great

• What means the definition E[ ]? expected value? – Nick Oct 2 '16 at 11:29
• The price of this option can be written in closed form. See the accepted answer by Gordon or the other one which discusses the use of power numéraire with a link to a nice paper by Mark Joshi: quant.stackexchange.com/questions/26240/…. – Quantuple Oct 2 '16 at 14:32
• To use the link I posted observe that $\max(X_T-K,0)^2 = (X_T-K)^2 1(X_T > K) = X_T^2 1(X_T > K) - 2 K X_T 1(X_T > K) - K^2 1(X_T > K)$. The evluation of the first term is discusses in the linked, the second is similar to the $N(d_1)$ term in standard BS formula, while the last is similar to $N(d_2)$. – Quantuple Oct 2 '16 at 14:40
• @Quantuple thanks a lot. I also got a interesting way to actually compute the distribution of the underliers from $C(X;k)$ here – stochastic_zeitgeist Oct 2 '16 at 19:00
• @rekcaH-Xunil, unfortunately, i could not find your chat room profile, i'd like to invite you in the chat room, chat.stackexchange.com/rooms/46241/requared-payoff-function – Nick Oct 3 '16 at 9:01