by Chris » Sat Jan 02, 2010 11:26 am
I think I spotted a couple of minor typos in section 18.3 The Pathwise Method, on page 493:
In [18.14] the second line is \hat{\Delta}_{S_0}=0 and I think it should be \hat{\Delta}_{S_0}=1.
Same for [18.15], but also the "hat" is missing on \Delta, i.e. it reads \Delta_{S_0}=0 and should read \hat{\Delta}_{S_0}=1.
Going back a bit, though, in [18.11], (the derivative of [18.9] w.r.t. \phi), the last two terms have \hat{S}^\prime_n in them. With the primes on \mu and \sigma, this does not make sense, i.e. either drop the \hat{S}^\prime_n or write out the chain rule with \frac{\partial \mu}{\partial \hat{S}_n} \hat{S}^\prime_n. This what is done in the sentence just before [18.14] and in [18.14]:
"For example, if we denote the partial derivative with respect to the spot price and abbreviate the partial derivative..."
may I suggest that the sentence would be better as
"For example, if we denote the partial derivative with respect to the spot price with $\prime$ and abbreviate the partial derivative..."
Great book -- I am not this far along (Chap. 18) but I had a school project with the pathwise method so I was keen to read this chapter. Everyone seems to use the exact answer for the European call option when illustrating the pathwise method but to me the coolest thing is that you don't need to solve the SDE or (in this case know anything about lognormal distributions) in order to use the pathwise method... you can just differentiate the Euler scheme to get what you need.