## Exercise 8.11

This forum is to discuss the book "the concepts and practice of mathematical finance" by Mark Joshi.

### Exercise 8.11

Hi Mark,

Perhaps I have misunderstood question 8.11 (comparing prices of out-of-the-money American and European digital calls when r =0, asset price follows BM). Please tell me where I am going wrong:

I assume you mean W_t is standard BM, so by risk-neutral pricing, at time t, the European digital has price E(1_{W_T >= K } | F_t ), which by some simple computation and facts about such conditional expectations is just E(1_{W_T >= K} | W_t = x) = P( N(x, sqrt(T-t) ) >= K ) where N(x, sigma) is used to denote a normal random variable with mean x, variance sigma^2. We're assuming that spot is out-of the money, so x < K, in which case this probability is < 0.5.

On the other hand, the price of the digital is E(1_{M_T} >= K | F_t ) and if the spot had previously gone above K, then this value is obviously 1. So I don't make the price of the European being half the price of the American. I get this only when we assume spot is *at-the-money*.

Thanks.
MattT

Posts: 22
Joined: Fri Jan 20, 2012 4:26 pm

### Re: Exercise 8.11

if the American pays off then at some point t in (0,T) S_t =K. There is then a precisely 50% chance that S_T > S_t = K so the probability
that European pays off given the American does is 0.5.

So its price is half that of the American one.
mj

Posts: 1380
Joined: Fri Jul 27, 2007 7:21 am

### Re: Exercise 8.11

Ok...I think we are saying exactly the same thing. At time t, when S_t = K I had the same conclusion that the price of the European is half the American.

What I was saying is, if then at some later time s, T > s > t, spot moves out-of-the-money for the European, i.e. S_s < K, the European price will not be half the American price, as the probability at time s that the European pays off is certainly less than 0.5, whilst the American still has value 1.

(I'm assuming this is a digital American as you define at the end of the chapter where the payoff is received at the expiry time. I suppose if you assume that the American is settled at time t, then this latter consideration is irrelevant.)
MattT

Posts: 22
Joined: Fri Jan 20, 2012 4:26 pm

### Re: Exercise 8.11

whilst that is true, I don't see what it has to do with the problem! we are pricing before the
American is triggered.
mj

Posts: 1380
Joined: Fri Jul 27, 2007 7:21 am

### Re: Exercise 8.11

Ahh...I see. I had interpreted "out-of-the-money" to mean spot price is < K (but that the American may or may not have already been triggered) so was considering the two cases. All is clear now! Thanks.
MattT

Posts: 22
Joined: Fri Jan 20, 2012 4:26 pm