91Ó°ÊÓ

Consider the standard Black-Scholes model. An innovative company, F\& H INC, has produced the derivative "the Golden Logarithm", henceforth abbreviated as the GL. The holder of a GL with maturity time T, denoted as \(G L(T)\), will, at time \(T\), obtain the sum \(\ln S(T) .\) Note that if \(S(T)<1\) this means that the holder has to pay a positive amount to \(F \& H I N C\). Determine the arbitrage free price process for the \(G L(T)\).

Consider the standard Black-Scholes model. Derive the Black-Scholes formula for the European call option.

Consider the standard Black-Scholes model. Derive the arbitrage free price process for the \(T\)-claim \(\chi\) where \(\chi\) is given by \(\chi=\\{S(T)\\}^{\beta}\). Here \(\beta\) is a known constant. Hint: For this problem you may find Exercises \(4.5\) and \(3.4\) useful.

A so called binary option is a claim which pays a certain amount if the stock price at a certain date falls within some prespecified interval. Otherwise nothing will be paid out. Consider a binary option which pays \(K\) SEK to the holder at date \(T\) if the stock price at time \(T\) is in the inerval \([\alpha, \beta]\). Determine the arbitrage free price. The pricing formula will involve the standard Gaussian cumulative distribution function \(N\).

Consider the standard Black-Scholes model. Derive the arbitrage free price process for the claim \(\chi\) where \(\chi\) is given by \(\chi=\frac{S\left(T_{1}\right)}{S\left(T_{0}\right)} .\) The times \(T_{0}\) and \(T_{1}\) are given and the claim is paid out at time \(T_{1}\).

Derive a formula for the value, at \(s\), of a forward contract on the \(T\)-claim \(X\), where the forward contract is made at \(t\), and \(t