MMLU-Pro / 212 / Assume the Black-Scholes framework. For t≥0, let S(t) be the time-t…

Assume the Black-Scholes framework. For t≥0, let S(t) be the time-t price of a nondividend-paying stock. You are given:
(i) S(0)=0.5(ii)Thestockpriceprocessis\frac{dS(t)}{S(t)} = 0.05dt+0.2dZ(t)whereZ(t)isastandartBrownianmotion.(iii)E[S(1)^\alpha]=1.4,where\alphaisanegativeconstant.(iv)Thecontinuouslycompoundedrisk−freeinterestrateis3%.ConsideracontingentclaimthatpaysS(1)^\alpha$ at time 1. What is the time-0 price of the contigent claim?

Assume the Black-Scholes framework. For $t \ge 0$, let $S(t)$ be the time-$t$ price of a nondividend-paying stock. You are given:
(i) $S(0)=0.5
(ii) The stock price process is $\frac{dS(t)}{S(t)} = 0.05dt+0.2dZ(t)$ where $Z(t)$ is a standart Brownian motion.
(iii) $E[S(1)^\alpha]=1.4$, where $\alpha$ is a negative constant.
(iv) The continuously compounded risk-free interest rate is $3%$.
Consider a contingent claim that pays $S(1)^\alpha$ at time 1. What is the time-0 price of the contigent claim?