Browsing by Author "Mvanda, Edwin Valentino"

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    Pricing of a perpetual American put option when stock prices are mean reverting
    (University of Dar es Salaam, 2012) Mvanda, Edwin Valentino
    This work is about pricing of a perpetual American put option when stock prices are mean reverting with the assumption that the Brownian motion ~Bt is already a Q- artingale. We started with this assumption to avoid the conversion process through the Girsanov theorem. The transformation process would have brought us back to the Geometric Brownian motion model which has already dealt with by (Shreve, 2004). The focus of this study is to _nd the optimal exercising price L_ at time t and the maximum value g(L_) of a perpetual American put option respectively. The stock price at time t was then solved directly from the model by using Ito formula. The theorem of Laplace transform for _rst passage time of drifted Ito integral have been stated which was the fundamental solution in obtaining the discounting factor which lead to getting the optimal exercising price L_ and the maximum value g(L_) of a perpetual American put option. We also assumed that there is a stopping time _L which is the _rst time stock price St reaches the barrier level of tolerance L. The _rst derivative with respect to barrier level of tolerance L of the value function was then found and set it equal to zero so as to be in a position of solving the value of L which later denoted by optimal exercising price L_. The price L_ was then substituted into the original value function to obtain the maximum value of a perpetual American put option g(L_). Lastly the analysis was done and we obtained interesting results as follows; basing on the stated objectives, results show that the best time to exercise the perpetual American put option when stock prices are mean reverting is when degree of convergence is zero or when time becomes very large, that is when time goes to in_nity. We considered when time goes to in_nity because perpetual American put option has no time limit.