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March 16th, 2012, 12:51 AM  #1 
Newbie Joined: Mar 2012 Posts: 1 Thanks: 0  Binomial Option pricing
Someone help please. New to option pricing, and stuck with the following question  Consider a stock price that follows a binomial process. The current stock price is 100. It goes up by 10% or down by 30% each month. The monthly interest rate is 5% for both months. An American put option has strike price K and expires in two months. Find the range of strike prices K such that the option is not exercised immediately, but is exercised in one month if the stock price goes down during that month. 
March 22nd, 2012, 06:30 PM  #2 
Newbie Joined: Mar 2012 From: a Greek in Germany Posts: 18 Thanks: 0  Re: Binomial Option pricing
I didn't try to solve the problem on paper, but I believe this is an optimal stopping problem. Draw the tree. Calculate the value of your put option at t=0,1,2 in relation to K. Let's say your equivalent martingale measure is p* (given you have proven the model is arbitragefree, meaning such a p* exists). Then you must use the Snell Envelope of your Claim. Right down the equations, again in relation to K. Now, the optimal stopping times occur when: ?min:=min( t>=0 : Ut under p* P=Ht) or ?max:=inf(t>=0 : Ep*(Ut+1 Ft ? Ut under p*)?T. where Ut: Snell envelope, Ht : discounted Payoffs of Claim, Ft: ?algebra, all at time t=0,1,2 and T ist the last period, (in this case T=2). (Practically, ?max holds when the Snell envelope loses its margingale attribute.) Now, you want these equalities to hold for t=1, and not hold for t=0. That means you need to find those Ks for which ?min or ?max hold at t=1, and exclude Ks for which ?min or ?max hold at t=0. Hope to have been helpful, although I know it's not detailed enough. Give it a try, if it doesn't work, i will try to find the time solving it myself, cause it's a good problem I have to say. 
March 22nd, 2012, 06:33 PM  #3 
Newbie Joined: Mar 2012 From: a Greek in Germany Posts: 18 Thanks: 0  Re: Binomial Option pricing
formula typo correction: ?max:=inf(t>=0 : Ep*(Ut+1 Ft) ? Ut under p*)?T 
March 22nd, 2012, 06:39 PM  #4 
Newbie Joined: Mar 2012 From: a Greek in Germany Posts: 18 Thanks: 0  Re: Binomial Option pricing
formula typo correction: ?min:=min( t>=0 : Ut under p* =Ht) 

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binomial, option, pricing 
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