Option analysis

Question

Assume zero dividend and that the strike price for a European call option on a stock at a fixed maturity T and strike price K is given by C(K).Suppose that $C(K)=e^{-k}$ for all $K\geq 0$ ,then, I want to find out the following

1.What must the present value of stock be?

2.What is the risk neutral probability that the stock price will lie in the interval [5,10] at maturity

3.What is the present value of contract that pays $X^2$ at maturity if the stock price at maturity is X

Solution: I don't know answer to this question. I know Black-Scholes formula, binomial option pricing,VaR, mean-variance portfolio optimisation and black-litterman model.How should I proceed to answer these questions?

Answer 1

(1). We consider a call option with strike $K=0$. Then $S_0=C(0)=1$.

(2). We assume zero interest rate. Then, for any $K\ge 0$, \begin{align*} 1_{S_T \ge K} &=\lim_{\varepsilon \rightarrow 0}\frac{(S_T-K)^+ - (S_T-K-\varepsilon)^+}{\varepsilon}. \end{align*} That is, \begin{align*} P(S_T \ge K) &= -\frac{\partial C(K)}{\partial K}\\ &= e^{-K}. \end{align*} Therefore, \begin{align*} P(5 \le S_T < 10) &= P(S_T \ge 5) - P(S_T \ge 10)\\ &= e^{-5}-e^{-10}. \end{align*}

(3). Note that \begin{align*} S_T^2 = 2\int_0^{\infty}(S_T-K)^+ dK. \end{align*} Then \begin{align*} e^{-rT} E\big(S_T^2\big) &= 2\int_0^{\infty}e^{-rT}E\big((S_T-K)^+\big) dK\\ &=2\int_0^{\infty} C(K) dK\\ &=2\int_0^{\infty} e^{-K} dK\\ &=2. \end{align*}