# Valuate the European option that pays the absolute difference between the share price $S_{T}$ at maturity and spot price $S_{0}$.

In the framework of the Black-Scholes model, valuate the European option that pays the absolute difference between the share price $S_{T}$ at maturity and spot price $S_{0}$. Assume that the stock does not pay dividend. Simplify the formula.

My idea is to use the call-put parity and the Black-Scholes formula for the price of a call function.

I came in this situation: $$f(S_T)=|S_T -k|=S_0(2\Phi(d_1)-1)+ke^{-rT}({1-2\Phi(d_2)})$$

where: $d_1=\frac{log(\frac{S_o}{k})+(r+\frac{1}{2}\sigma^2)T}{\sigma \sqrt{T}}$ and $d_2=d_1-\sigma \sqrt{T}$

• Often such questions are supposed to be answered by decomposing your portfolio into calls and puts and other simpler options, and then using your knowledge of what the prices are in these cases. – Jaood Feb 27 '17 at 15:05

You can write $\lvert S_T - K\rvert = \max(S_T - K,0) + \max(K - S_T,0)$. So you have a linear portfolio consisting of a call and a put option with the same strike and maturity. The value of this portfolio must then be the sum of these two options. In your case options are at the money, i.e. $K = S_0$. So just sum the prices given by the Black-Scholes formula for these parameters.