Stochastic Deflator Results

[Delvesy888]

Hi all,

I am having trouble deriving two of the results of Chapter 13 on page 41 (which are apparently straightforward.

There are 3 identities on this page:

1. Ep[An] = exp(-rn).

I can arrive at this solution quite comfortably.

2. Ep[AnSn] = S_0

Having trouble here, going through the algebra, I end up with Ep[AnSn] = S_0 * exp(-rn) * Summation

Where I am pretty sure the Summation does not equal exp(rn).

Any help? Perhaps i'm going a long-winded way.

3. Vt = (Ep[A_T V_T])/At

I really can't see how we can arrive at this identity.
The numerator is V_0, no? Or at least, is deterministic.
Vt depends on the function f.
At does not depend on the function f.
So i'm not sure how this equality holds? (although it does make sense given the interpretation of the stochastic deflator).

Any help would be greatly appreciated.

Thanks

 

[Graham Aylott]

Hi,

I've attached a derivation of the first result you mention.

The second result re. Vt is is just the general formula, which allows for the fact that VT is the state price deflator (SPD) to deflate or discount a cashflow from time T all the way back to time 0. So, if instead we only wish to discount the cashflow back to time t, because that's the time at which we're valuing the asset in question, then each cashflow needs to be discounted using the ratio of VT (at the relevant node in the tree) to Vt (again at the relevant node in the tree).

So, valuing a two-period derivative at one of the nodes at time 1, we need to use the ratio of the SPD at the relevant time 2 node to the SPD at the relevant time 1 node.

We do this implicitly with risk-neutral valuation too. When valuing a two-period derivative at time 1, the risk-neutral discount factor we use is exp(-2r)/exp(-r). However, we normally just write it as exp(-r), as the discount factors are the same at all of the nodes at any date.

Graham