Chapter 6

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Chapter 6

Postby MattT » Thu Apr 12, 2012 10:01 am

EDIT: Sorry, had a new insight since writing so my question changed slightly.

Mark,

In chapter 6, page 149 of the second edition, you suppose you have a tree with n periods and write:

At each node in the tree the stock can move up or down to the next node with some non-zero probability. Each move is independent of how the stock arrived at that node. We let our sample space be the space of paths of stock through the probability tree...The probability of a path is therefore obtained by taking the product of the probabilities of the moves along the path: this is implied by the independence of the moves.


Question 1) Are you just saying that if you imagine a representation of your stock paths as a tree, then the probability that would be attached to the line going from a node S_N to the node corresponding to an up move say (where I'm using a conventional representation of binomial trees/probabilities as in school maths) is just the conditional probability of being at S_{N_U} given that you are at node S_N (rather than the conditional probability given an entire specific path)? More precisely, if w_1 and w_2 are two (possibly different) paths which both go through node N at time t, then the the probability that i am at some node N+1 at time t+1, given i know the path w_1 up to time N is equal to the probability that i am at the node N+1 given i know the path w_2 up to time N?

Question 1b) Related to question 1, am I correct in saying that the condition you write about is NOT the same condition as requiring that up/down moves at some time, t, are independent of up/down moves at all other times (as was the case in chapter 3 where the log of a stock price went up or down at each time, independent of whether or not it had gone up or down at all other times).

Question 2) I should confess I haven't got to reading the stuff on continuous martingales yet. Nevertheless, in the discrete setting, I don't really understand the argument you give (on page 146) that if every tradable asset is a martingale we get that arbitrage is impossible. All you appear to be using is that the E[P] = E[P | F_0 ] = P_0 which isn't even close to using the full strength of the martingale property. It's not clear to me how this argument is dealing with self-financing portfolios.

I think that to take into consideration self-financing portfolios etc a more precise argument should go something like this (I assume just 3 assets S,B, and C but it's clear how this would generalise) : The idea is a kind of inductive argument on the time step k. Suppose at some time k < T, we set up a portfolio P_k at time k which is = a_k S_k + b_k B_k +c_k C_k of value 0. Here a_k, b_k and c_k are going to change based on our trading strategy which can't depend on future knowledge - in other words, I think a_k, b_k and c_k should be F_k measurable random variables. Now, at time k+1, this portfolio will have value a_k S_{k+1} + b_k B_{k+1} + c_k C_{k+1}. If this has zero conditional probability of being < 0 and positive conditional probability of being > 0 (where the probabilities have been conditioned on F_k), then we get as a consequence (I think) that E[a_k S_{k+1} + b_k B_{k+1} + c_k C_{k+1} | F_k ] > 0. On the other hand, by the martingale property, (and some basic facts on conditional expectations meaning we can pull the a_k, b_k and c_k out of the cond. exp.) we have:

E[a_k S_{k+1} + b_k B_{k+1} + c_k C_{k+1} | F_k ] = a_k S_k + b_k B_k + c_k C_k = 0.

So we have a contradiction, and therefore arbitrage is impossible at each time step, therefore arbitrage is impossible on any self-financing portfolio. Am I making this overly complicated or is this right?

Thanks in advance (and sorry for the long post).
MattT
 
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Re: Chapter 6

Postby mj » Mon Apr 16, 2012 12:04 am

Question 1) well what you say is true but it's not quite what I am saying. I am trying to shift your viewpoint from thinking about moves one by one to drawing entire paths at once.

The tree process is Markovian in the current value of spot so how you got there doesn't matter.



1b) we still have independence here

2) you need the fact that a trading strategy in a martingale is a martingale without this you don't get
E( P_t) = P_0 for dynamic portfolios.

It does hold for static portfolios, however, and you can't arbitrage using them if E(P_t) = P_0.
mj
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Re: Chapter 6

Postby MattT » Mon Apr 16, 2012 6:33 pm

Thanks for your reply. All makes more sense.

One more question about the martingale pricing method in discrete time. I don’t see where/how we use the fact that the option is European; could you explain why this method only works for European claims?
MattT
 
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Re: Chapter 6

Postby mj » Tue May 01, 2012 2:03 am

well it can be made to work for other options but there are fiddlinesses. These are discussed later in the book.
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