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March 15th, 2012, 11:51 PM   #1
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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.
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March 22nd, 2012, 05:30 PM   #2
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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 arbitrage-free, 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.
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March 22nd, 2012, 05:33 PM   #3
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Re: Binomial Option pricing

formula typo correction:
?max:=inf(t>=0 : Ep*(Ut+1| Ft) ? Ut under p*)?T
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March 22nd, 2012, 05:39 PM   #4
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Re: Binomial Option pricing

formula typo correction:
?min:=min( t>=0 : Ut under p* =Ht)
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