Pricing an Averaged-Price Contract in Black–Scholes
Summary
The document derives a risk-neutral price for a contract that pays the average underlying price over an interval ending at maturity. It discounts the payoff at the final payment date and uses linearity to move the expectation inside the time integral. Under constant interest rates and risk-neutral Black–Scholes dynamics, the conditional expected asset price at each future time is its current price grown at the risk-free rate.
Integrating those expected prices gives a closed-form expression in terms of the current spot, interval endpoints, and rate. The result is also expressed using discount factors and forward prices. The derivation assumes the observation interval begins after the pricing time, a constant risk-free rate, and an asset with the stated risk-neutral growth. It does not address dividends, stochastic rates, or other market frictions, and the displayed formula divides by the rate, so a zero-rate case would need its limiting form.
Key ideas
- Discount the averaged payoff from its payment date under the risk-neutral measure.
- Linearity lets the expected time average be written as the integral of conditional expected prices.
- With constant rates, each conditional expected future spot equals current spot grown at the risk-free rate.
- The price can be written using discount factors and forward prices at the averaging interval endpoints.
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Full text
# Determine price of financial contract
# Determine price of financial contract
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?
## Answer by Daneel Olivaw (score 4)
https://quant.stackexchange.com/a/36002
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:
$$ \pi_t = \frac{e^{-r(T_2-t)}}{T_2-T_1}\mathbb{E}^{\mathbb{Q}}\left[\int_{T_1}^{T_2}S_udu|\mathcal{F}_t\right] $$
where:
$$ \Phi(S_{T_2}) = \frac{1}{T_2-T_1}\int_{T_1}^{T_2}S_udu $$
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:
$$ \pi_t = D(t,T_2)\frac{\text{For}_S(t,T_2)-\text{For}_S(t,T_1)}{r(T_2-T_1)}$$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.