I wonder if some one can help me with the solution to this question from Björk's "Arbitrage theory in continuous time":

At date of maturity $T_2$ the holder of a financial contract will obtain the amount: $$ \frac{1}{T_2 - T_1 } \int_{T_1}^{T_2} S(u) du $$ where $T_1$ is some time point before $T_2$. Determine the arbitrage free price of the contract at time $t$. Assume you live in a Black-Scholes world and that $t<T_1$.

Earlier in the book he states this theorem that I think one might use:

The arbitrage free price of a claim $\Phi(S(T))$ is given by: $$ \Pi(t,\Phi)=F(s,t) $$ where $F(\cdot,\cdot)$ is given by the formula $$ F(s,t)=e^{-r(T-t)}E_{s,t}^Q [\Phi(S(T))] $$ where the $Q$-dynamics of $S(t)$ are given by $$ dS(t)=rS(t)dt + S(t)\sigma(t,S(t))dW(t) $$

However I'm not really sure how to apply it in this case. Can anybody help me out here?


1 Answer 1


As stated in the theorem you mention, the price $\pi_t$ at $t$ of a financial contract which pays $\Phi(S_T)$ at maturity $T>t$ $-$ where $\Phi(\cdot)$ is the payoff function and $(S_t)_{t \geq 0}$ is the underlying asset $-$ is given by the conditional risk-neutral expectation of its discounted payoff:

$$ \pi_t = \mathbb{E}^{\mathbb{Q}}\left[e^{-\int_t^Tr_udu}\Phi(S_T)|\mathcal{F}_t\right]$$

Assuming the risk-free rate $(r_t)_{t \geq 0}$ is constant for all $t$, the price of your financial contract is given by:

$$ \pi_t = \frac{e^{-r(T_2-t)}}{T_2-T_1}\mathbb{E}^{\mathbb{Q}}\left[\int_{T_1}^{T_2}S_udu|\mathcal{F}_t\right] $$


$$ \Phi(S_{T_2}) = \frac{1}{T_2-T_1}\int_{T_1}^{T_2}S_udu $$

By linearity of the risk-neutral expectation operator $\mathbb{E}^{\mathbb{Q}}[\cdot]$ and the risk-free return of the asset $S_t$ under the measure $\mathbb{Q}$, we have:

$$ \begin{align} \pi_t & = \frac{e^{-r(T_2-t)}}{T_2-T_1}\int_{T_1}^{T_2}\mathbb{E}^{\mathbb{Q}}\left[S_u|\mathcal{F}_t\right]du \\[6pt] & = \frac{e^{-r(T_2-t)}}{T_2-T_1}\int_{T_1}^{T_2}S_te^{r(u-t)}du \\[6pt] & = \frac{e^{-r(T_2-t)}S_t}{T_2-T_1}\int_{T_1}^{T_2}e^{r(u-t)}du \\[6pt] & = e^{-r(T_2-t)}S_t\frac{e^{r(T_2-t)}-e^{r(T_1-t)}}{r(T_2-T_1)} \end{align}$$

Letting $D(t,T)$ be the discount-factor

$$ D(t,T) = e^{-r(T-t)}$$

and $\text{For}_S(t,T)$ the forward price of asset $S_t$

$$ \text{For}_S(t,T) = e^{r(T-t)}S_t $$

a convenient representation of the result is:

$$ \pi_t = D(t,T_2)\frac{\text{For}_S(t,T_2)-\text{For}_S(t,T_1)}{r(T_2-T_1)}$$


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