option pricing - Calculate $E^{mathbb{Q}}left[e^{-int_{0}^{T_2}r_t,dt} frac{Sleft(T_2right)}{Sleft(T_1right)}right]$

Let $S\left(t\right)$ be a tradable financial security that doesn't generate cash flow (eg no dividend). $S\left(t\right)$ follows an unknown stochastic process.

We now have a financial derivative that pays $\frac{S\left(T_2\right)} {S\left(T_1\right)}$ at $t=T_2$, where $0

Assume interest rate $r_t$ is not constant.

What's the present value of this financial derivative at $t=0$ ?

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My attempt so far:

$V\left(0\right)=E^{\mathbb{Q}}\left[e^{-\int_{0}^{T_2}r_t\,dt} \frac{S\left(T_2\right)}{S\left(T_1\right)}\right]$

I believe my next step should be to get rid of the discount factor term. Any idea how can I do that?

Answer

We assume a Black-Scholes world except the dynamics of the stock price, namely:

• No arbitrage opportunities.


• No dividend payments from the stock.


• Existence of a riskless asset yielding the risk free rate $-$ which here we assume non-constant, $(r_t)_{t \geq 0}$.


• Possibility to borrow and lend infinitely at the risk-free rate.



• Possibility to buy and sell infinitely the stock $-$ even fractional amounts.


• No transaction costs.

We also assume that the stock is tradable and that the derivative is attainable $-$ we basically assume we are in the standard pricing setting except for the stock price dynamics.

Then the price at time $t=0$, $V(0)$, of the derivative is given by:

$$V(0) = P(0,T_1)$$

where $P(0,T_1)$ is the price of a riskless zero-coupon contracted at time $t=0$ and maturing at time $t=T_1$ $-$ which is effectively a function of the rate $r_t$ and is independent of $S(t)$.

Financial proof: the financial derivative you describe delivers a quantity $w$ of the stock at time $T_2$, where:

$$w = \frac{1}{S(T_1)}$$

Thus $w$ will only be known at time $T_1$, when you will buy $w$ shares of the stock. But at that time, the value of such a position is trivially equal to $\$1$. Thus you only need to have $\$1$ at time $T_1$ to settle the trade at maturity $T_2$; no further transactions are needed. The value today of $\$1$ at $T_1$ is simply equal to the value of a zero-coupon bond contracted at $t=0$ and maturing at $T_1$. Hence:

$$V(0) = P(0,T_1)$$

Mathematical proof: under the assumptions listed at the beginning, by the law of iterated expectations, adaptedness of the stock price with respect to a suitable filtration $(\mathcal{F})_{t \geq 0}$ and the martingality property of discounted stock prices under the risk-neutral measure $\mathbb{Q}$, we obtain:

$$\begin{align} V(0) & = E^{\mathbb{Q}}\left[e^{-\int_0^{T_2}r_t\,dt} \frac{S(T_2)}{S(T_1)}\right] \\[6pt] & = E^{\mathbb{Q}}\left[E^{\mathbb{Q}}\left[e^{-\int_0^{T_2}r_t\,dt} \frac{S(T_2)}{S(T_1)}|\mathcal{F}_{T_1}\right]\right] \\[6pt] & = E^{\mathbb{Q}}\left[e^{-\int_0^{T_1}r_t\,dt}\frac{1}{S(T_1)}E^{\mathbb{Q}}\left[e^{-\int_{T_1}^{T_2}r_t\,dt} S(T_2)|\mathcal{F}_{T_1}\right]\right] \\[6pt] & = E^{\mathbb{Q}}\left[e^{-\int_0^{T_1}r_t\,dt}\frac{1}{S(T_1)}S(T_1)\right] \\[9pt] & = P(0,T_1) \end{align}$$