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September 2003 Ques 10

M

MindFull

Ton up Member
Hi Everyone.

I had a ques re state prices and state price deflators. I've noticed that when calculating state prices, when we have the up and down step, we multiply by 2 since it could be up then down or down then up. With state price deflators, we only use the 2 when calculating the derivative value, ie, we do not multiply by 2 when calculating the deflators, A. In this question, in order to find the real world probabilities, we multiply the state prices by the deflators given. I'm wondering what happens to the 2 that we used to get the second state price, ie, the one with the up and down step. It wouldn't cancel with the deflator value because the deflator doesn't usually multiply by 2.

Any help please.
Thanks.
 
The deflator for the middle node on the right of a recombining 2-step tree is the discounted ratio of the probabilities, ie:

A(ud) = A(du) = exp(-2r)[2q(1-q)]/[2p(1-p)] = exp(-2r)[q(1-q)]/[p(1-p)]

So the 2's cancel here.

The state price for this node is found in the usual way using risk-neutral valuation, ie:

State price(ud) = exp(-2r)xEQ[1] = exp(-2r)x2q(1-q)

Using SPDs, we get:

State price(ud) = EP[Ax1] = 2p(1-p)xA(ud) = exp(-2r)x2q(1-q)

So the 2 still comes in the probability part of the expectation for whichever measure we're using. The SPD just swaps the p's for q's and doesn't need to adjust the 2 factor.

To get the real-world probs, we can divide the state price by the deflator. Doing this for the middle node, suing the expressions above, we see the discount factors and the q's cancel out, leaving:

state price(ud) / A(ud) = 2p(1-p)

This contains the 2-factor, as required.

Hope that helps.
 
Hi Mike,

Thanks again for the quick reply. I understand what you're saying about the 2 being left but in the Sept 2003 question, we're asked to find the real world probability. We find it to be 0.494999. I was just wondering if this value would include the 2 making it 2p(1-p). Additionally, this answer is used to find the expected value of the share price at the end of day 2. Am I correct in assuming that for the middle node, a 2 would need to be present since it could be ud or du?

Thanks again.
 
Yes, the 2 is needed for the middle node at the end of a 2-step recombining tree.

Working out the RW prob was what I was doing at the end of my previous post and you can see the 2 is there, as we want. Since this prob already includes the 2, we don't need to put in another one.
 
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