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 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 \(N\).