Pricing a Stock-Ratio Payoff with the Forward Measure

Summary

The document studies the time-t price of a claim paying the ratio of a stock price at a later date to its price at an earlier date, with payment at the later date. The question applies the forward measure associated with the payment date and initially obtains a surprising constant price by treating a conditional expectation involving a bond as though it were simply the reciprocal of today’s bond price.

The accepted answer corrects that step using the forward-measure martingale property. It rewrites the reciprocal bond value at the earlier date as a ratio of bond prices, then evaluates its conditional expectation as the ratio of the corresponding bond prices at time t. Multiplying by the payment-date bond price leaves the price of a bond maturing at the earlier date. The correction relies on the stated pricing-measure setup and bond martingale relation; the excerpt does not develop the result under alternative market assumptions.

Key ideas

• The claim pays the ratio of stock prices observed at two future dates.
• Pricing under a forward measure uses the discount bond associated with the payment date.
• The conditional expectation of the reciprocal bond value is not simply the reciprocal of its current price.
• Applying the bond martingale relation reduces the claim value to the price of a bond maturing at the earlier observation date.

Tags

# How to Calculate Return Option with Forward Measure


# How to Calculate Return Option with Forward Measure












I am trying to computing the price of an option at time $t$, with payoff $X = \frac{S_{T_2}}{S_{T_1}}$, at time $T_2$, where $t < T_1 < T_2$.

Here how I compute it:

Using the forward measure $Q_{T_2}$ with settlement date $T_2$, price at $t$ is $$P_{t,T_2}E_{Q_{T_2}}\left(\frac{S_{T_2}}{S_{T_1}}| \mathcal{F}_t\right),$$ where $P_{t,T_2}$ is the price of a discount bond at $t$ maturing at $T_2$.

By the law of iterated expectation, and using the fact that price process $h_t/P_{t,T2}$ is a martingale for any contingent claim with payoff $H$ at $T_2$, I get, $$P_t = P_{t,T_2}E_{Q_{T_2}}\left(\frac{1}{S_{T_1}} \frac{S_{T_1}}{P_{T_1,T2}}| \mathcal{F}_t\right) = P_{t,T_2} E_{Q_{T_2}}\left(\frac{1}{P_{T_1,T2}}| \mathcal{F}_t\right) =P_{t,T_2} \frac{1}{P_{t,T2}} =1 $$

I find this result rather surprising (maybe because it is wrong?). If anybody has another answer or can come up with an intuitive justification of such result. I would be glad to get some insight.

## Answer by Gordon (score 1, accepted)

https://quant.stackexchange.com/a/21861

There is a problem in your last step. Note that \begin{align*} P_{t, T_2}E_{Q_{T_2}}\left(\frac{1}{P_{T_1, T_2}} \mid \mathcal{F}_t \right) &= P_{t, T_2}E_{Q_{T_2}}\left(\frac{P_{T_1, T_1}}{P_{T_1, T_2}} \mid \mathcal{F}_t \right)\\ &=P_{t, T_2} \times \frac{P_{t, T_1}}{P_{t, T_2}}\\ &=P_{t, T_1}. \end{align*}

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