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Need help with Shreve Vol 1 Ch 4

Hi,

I am reading Shreve's Stochastic Calculus and finance Volume 1 and I am finding Chapter 4 to be hard. Can someone link me to some sources that touch the same topics but are easier to grasp?

Specifically, I'm finding it difficult to grasp concepts where Shreve tries to create a pricing algorithm for General American Derivatives i.e. including derivatives for which the intrinsic value is permitted to be path dependent. Specifically, sections 4.3, 4.4 and 4.5 are the ones I need help with. Any kind of material or source that can explain the concepts in simpler fashion would be helpful.

Thanks a ton!
 
I think the trick for me was realizing there were two functions- first, the discounted value of the call starting from the terminal end point ( t=3). This gives you the expected value of a call from the up case and the down case. Same as for European call.

Except you can also exercise the call at time 2. So look at your current price at time 2. Subtract the strike price, and there’s your payoff.

Now you have two numbers, the discounted expected value of the call exercised at t3 under the risk neutral measure and the payoff if you exercise right now.

We are all greedy bastards in finance, so as greedy bastards, a natural question is - which is bigger? First value, discounted expected payoffs under RN, or second value, pulling the trigger right now?

So t2 value is max of either pull trigger now, or wait. Fix that guy as your T2 value for the “up” case. Now go and do the same for the T2 “down” case. Keep going back until you get to time zero.

Does that help?
 
I think the trick for me was realizing there were two functions- first, the discounted value of the call starting from the terminal end point ( t=3). This gives you the expected value of a call from the up case and the down case. Same as for European call.

Except you can also exercise the call at time 2. So look at your current price at time 2. Subtract the strike price, and there’s your payoff.

Now you have two numbers, the discounted expected value of the call exercised at t3 under the risk neutral measure and the payoff if you exercise right now.

We are all greedy bastards in finance, so as greedy bastards, a natural question is - which is bigger? First value, discounted expected payoffs under RN, or second value, pulling the trigger right now?

So t2 value is max of either pull trigger now, or wait. Fix that guy as your T2 value for the “up” case. Now go and do the same for the T2 “down” case. Keep going back until you get to time zero.

Does that help?
I also believe if you look over the options book by Hull there’s some good examples in there which might be helpful.
 
I think the trick for me was realizing there were two functions- first, the discounted value of the call starting from the terminal end point ( t=3). This gives you the expected value of a call from the up case and the down case. Same as for European call.

Except you can also exercise the call at time 2. So look at your current price at time 2. Subtract the strike price, and there’s your payoff.

Now you have two numbers, the discounted expected value of the call exercised at t3 under the risk neutral measure and the payoff if you exercise right now.

We are all greedy bastards in finance, so as greedy bastards, a natural question is - which is bigger? First value, discounted expected payoffs under RN, or second value, pulling the trigger right now?

So t2 value is max of either pull trigger now, or wait. Fix that guy as your T2 value for the “up” case. Now go and do the same for the T2 “down” case. Keep going back until you get to time zero.

Does that help?


Sorry if I wasn't clear in my post. Although I did understand your explanation and it was helpful, but what you're talking about is for the non path dependent American Derivatives.

I'm more specifically looking for help with the path dependent American Derivatives. So what you were basically saying is that
Value of Derivative at time n = max { g(s) , discounted expected value of derivative at (n+1) under RN measure}
where g(s) is the payoff if we exercise the derivative at time n itself.

So I understand this case pretty well. I'm rather not clear about the path dependent case. I'm adding the page from Shreve below:

1586144744232.png



So I don't follow how is 4.4.1 derived, as in what's the intuition behind it. Specifically, the path dependent case should be a more general case, such that even the non path dependent cases get accounted for in that general formulation.

Another thing that I dont get is, why wouldnt the formulation V[n]= max{ g(s), RN expected value of derivative at n+1} not be valid for all path dependent case too. Like I want to see an example of a derivative (path dependent) for which this formulation, which we use for non path dependent case, doesnt stand valid.

I hope I'm clear in my question. Thanks!
 
This is a problem in optimal stopping time; the tau, the optimal time to stop, is path dependent. If you have an american call, and it's out of the money, then tau goes to infinity because you never exercise it.

The example I described still holds; for the max (exercise now, value of option at n+1), the value of the option at n+1 takes the same max function. That is, n+1 valuation is the greater of either the current exercise or the discount of an exercise at a future date.

For 4.4.1, the indicator function refers to the optimal stopping time. If tau, when you can optimally exercise, is < N (maturity), then discount the expected value of that option under the risk neutral measure back from the optimal exercise step.

I'll double check the section you mention a bit later.
 
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