Pricing a Forward-Started Asset Payoff with a Change of Measure
Summary
The note studies the value at an earlier time of a payoff equal to the difference between an asset’s value at a later date and its value at an intermediate fixing date. A total return swap is offered as an example. Under the later-date forward measure, the terminal asset value can be related to its forward price and any income paid before maturity. The question is how to handle the earlier asset value when changing to a measure associated with the intermediate date.
The answer identifies a key error in the proposed derivation: the expectation contains a product of two random quantities, the asset value and a discount factor. The expectation of that product cannot be simplified by treating both as deterministic or pulling them outside the expectation. Further progress requires a model for their joint evolution, including the relationship between the asset and discount factor. The note flags this dependency but does not provide a complete pricing formula or specify such a model.
Key ideas
- The payoff depends on asset values at two different future dates.
- A forward-measure change introduces a random discount factor into the expectation.
- The expectation of an asset value multiplied by a random discount factor cannot be split by treating either as fixed.
- Further valuation requires a model for the joint evolution of the asset and discount factor.
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Full text
# Answer by dm63 (score 1)
# How to price a forward struck contract today by changing from a $T>T'$ forward measure to $T'$ forward measure at time $t<T'<T$?
Suppose that the payoff of some contract is $V_{T}=S_{T}-S_{T'}$ where $T'<T$ and we want to value the contract at time $t<T'$ (the situation where this arises could be a total return swap, where a fixing is anticipated at time $T'$ for the performance period $[T',T]$). Then presumably from risk-neutral pricing $$M_{t}V_{t}=E_{t}[M_{T}(S_{T}-S_{T'})]$$ for a choice of nuumeraire $M$. If we use the $T$-forward measure, then $$V_{t}=P_{t}^{T}E^{T}_{t}[S_{T}]-P_{t}^{T}E_{t}^{T}[S_{T'}]$$ where $P_{t}^{T}$ is the discount factor observed at time $t$ for the period $[t,T]$.
The first expectation is just $S_{t}-I_{t}^{T}$ where $I_{t}^{T}$ is the present value of any income $S$ pays in $[t,T]$ (we are using the fact that in the $T$-forward measure $E_{t}^{T}[S_{T}]=Fwd_{t}^{T}[S]=\frac{S_{t}-I_{t}^{T}}{P_{t}^{T}}$.
My question is how to properly calculate the second expectation by (presumably) switching to the $T'$-forward measure. Basically, the conceptual difficulty that I am having is how to make sense of the quantity $P_{T}^{T'}$ that shows up when you change numeraires: $$P_{t}^{T}E_{t}^{T}[S_{T'}]=\frac{P_{t}^{T}E_{t}^{T'}[\frac{S_{T'}}{P_{T}^{T'}}]P_{t}^{T'}}{P_{t}^{T}}=P_{t}^{T'}E_{t}^{T'}[\frac{S_{T'}}{P_{T}^{T'}}].$$
To me it seems like a misapplication of the change of numeraire formula to substitute $T'$ instead of $T$ into the new numeraire process under the expectation, even though $S$ is being evaluated at $T'$ (the overall payoff is still at time $T$). Barring that possibility, I proceed formally like $$P_{t}^{T'}E_{t}^{T'}[\frac{S_{T'}}{P_{T}^{T'}}]=P_{t}^{T'}E_{t}^{T'}[S_{T'}P_{T'}^{T}]=P_{t}^{T'}E_{t}^{T'}[\frac{S_{T'}P_{t}^{T}}{P_{t}^{T'}}],$$ but it's not clear to me at the moment how to treat this. If we just take everything out of the expectation, we get $$P_{t}^{T}E_{t}^{T'}[S_{T'}]=P_{t}^{T}\frac{S_{t}-I_{t}^{T'}}{P_{t}^{T'}}$$
## Answer by dm63 (score 1)
https://quant.stackexchange.com/a/30661
The last step (taking everything outside the expectation ) is invalid. The expression $E[\frac{S_{T'}}{P^{T'}_T}]$ is the expectation of the product of 2 random variables. As such, you need a model to describe the joint evolution of the stock and the discount factor before you can go any further.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.