Valuing a Payoff That Depends on Two Dates in Vasicek GBM
Summary
This discussion addresses a payoff paid at a later date that uses the underlying price at both an earlier and a later date. The standard risk-neutral valuation formula still applies when the payoff is measurable at the payment date: by then, both observed prices are known, so their ratio is a valid terminal payoff. The value is the discounted conditional expectation of that payoff.
For a constant-coefficient geometric Brownian motion, the ratio of prices across the interval is independent of information at the earlier date, and its distribution depends on the interval’s length. This can simplify the calculation. That simplification does not generally extend to arbitrary functions of prices at both dates; the general valuation formula remains applicable, but the special distributional argument may fail.
Key ideas
- A payoff may depend on multiple observation dates and still be valued at its payment date if it is measurable then.
- Risk-neutral valuation discounts the conditional expectation of the terminal payoff.
- Under constant-coefficient geometric Brownian motion, the price ratio over an interval is independent of information at its start.
- The ratio’s distribution depends on the interval length in this model.
- The simplification need not hold for a general payoff involving both prices.
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# Bjork exercise 7.6: Claim that depends on $T_1$ and $T_0$
# Bjork exercise 7.6: Claim that depends on $T_1$ and $T_0$
See the solution to Exercise 7.6 here.
The solution calculates $E^Q (S(T_1)/S(T_0))$ and then just plugs that into the risk neutral valuation formula. But why? The risk neutral valuation formula holds for claims that depend on $T$ alone. Why can it be used also for this case where the claim depends on one more date?
My solution was similar, but I still don't trust it. How can I use a theorem which solves a different problem than this one, where the claim depends on multiple dates?
## Answer by Quantuple (score 2)
https://quant.stackexchange.com/a/31539
The valuation formula for a contingent claim delivering a payoff at $T$, as seen of today $t$ knowing that the underlying is currently worth $s$ reads $$ \Pi(t,s) = e^{-r(T-t)} \Bbb{E}^\Bbb{Q} \left[ f(S_T) \mid \mathcal{F}_t \right] $$ where $f(S_T)$ is the $\mathcal{F}_T$-measureable payoff of your contingent claim.
In the exercise you mention, we have $t < T_0 < T_1=T$ along with $f(S_{T_1}) = \frac{S_{T_1}}{S_{T_0}}$.
Because $f(S_{T_1})$ is $\mathcal{F}_{T_1}$ measurable (at $T_1$ you "known" both $S_{T_0}$ and $S_{T_1}$) then you can seamlessly use the standard valuation formula to write:
$$ \Pi(t,s) = e^{-r(T_1-t)} \Bbb{E}^\Bbb{Q} \left[ \frac{S_{T_1}}{S_{T_0}}\mid \mathcal{F}_t \right] $$
## Answer by LocalVolatility (score 1)
https://quant.stackexchange.com/a/31538
The spot price process is driven by a constant coefficient geometric Brownian motion. Thus, the ratio $S \left( T_1 \right) / S \left( T_0 \right)$ is
- independent of $\mathcal{F} \left( T_0 \right)$ and
- its distribution only depends on the length of the time interval $T_1 - T_0$.
It follows that
\begin{equation} S \left( T_1 \right) / S \left( T_0 \right) \sim S \left( T_1 - T_0 \right) / S(0), \end{equation}
where $\sim$ denotes equality in distribution. I.e. the solution is the same as the one that only involves the single date $T_1 - T_0$. Note that in the general case where the payoff function is given by $f \left( S \left( T_0 \right), S \left( T_1 \right) \right)$, this might no longer hold.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.