Need help with Shreve Vol 1 Ch 4

[retro active]

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?

Thanks a ton!

 

[Onegin]

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?

 

[Michsund]

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.

 

[IntoDarkness]

plz google asian option

 

[Onegin]

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.