Let $S_t$ be a process which follows a Geometric Brownian Motion:
$\frac{dS_\tau}{S_\tau} = \mu \,d\tau + \sigma \,dW_\tau$
By Ito's lemma, we have:
$S_T = S_t e^{(\mu-{\sigma^2 \over 2})(T-t) + \sigma W_{T-t} }$
Define:
$Y_{T-t} \equiv \int_t^T e^{(r-\sigma^2/2)\tau + \sigma \, W_\tau}d\tau$
Intuitively, the problem arising from $Y_\tau$ is inherent to pricing path dependent contingent claims, such as the arithmetic Asian options. In this case, however, we are concerned with the accumulated value of $S$. Such a problem has many applications (such as discounted cash flow and annuity models where we are concerned with the expected area under the curve at $S_\tau$ goes from $S_t \to S_T$).
Question A: How is $Y_{T-t}$ distributed?
I.e., $\int_{t}^T S_{\tau} d\tau \sim \,\text{???} $
Question B: Or, if this is not analytical tractable, what is the a fast and accurate method for taking a conditional expectation of $Y_\tau$
Although the unconditional expected value is easy to obtain, how can way obtain the conditional value for some arbitrary constraints? E.g.:
- The conditional value for a standard (Dirichlet) boundary condition (i.e., $S_t\,Y_{T-t}>K \, \forall \,\tau \in(t,T]$)
- The expected value for a first hitting model (i.e., "knock out" or "stop out" barrier options).
Moreover, do we absolutely need to know the distribution of $Y_\tau$ to take a conditional expectation?
A quick derivation of the expected value of $Y_{\tau=T}$.
We can rewrite it as:
$S_t\,Y_{T-t} =\int_{t}^T S_{u} du =\int_t^T \exp \left[ \left(\ln(S_t)+\int_t^T \mu \,du+\int_t^T \sigma \,dW_s\right) \right] d u$
$ =\int_t^T S_te^{{ \mu \,u }} \, e^{\sigma W_{u}-\frac{\sigma^2}{2}u} d u$
It stands to reason that if the second exponential term above has the unconditional expectation of 1 then the randomness disappears (over the probability density interval $(-\infty,\,\infty)$) resulting in:
$\mathbb{E} \left[S_t\,Y_{T-t}\right]= \int_{t}^T S_{t}e^{\mu\,\tau} d\tau = S_t \frac{(e^{\mu \,T}-e^{\mu \,t})}{\mu}$
...which is the correct answer. We can also derive this more quickly with Fubini's theorem. But this only deals with the unconditional expectations. In order to do anything else here (i.e., price contingent claims), we need to know more about the terminal distribution of $S_t\,Y{T-t}$. Indeed, imposing the condition that reintroduces randomness into the integral.
Some Interesting Findings
There are similar questions given throughout the SE Network (e.g., "Expected value of time integral of geometric brownian motion"; "How to compute the conditional expected value of a geometric brownian motion?"; "Integral of Brownian Motion w.r.t Time"). While Asian options are well studied, to my knowledge there exists no closed form solution to the time integral of GBM.
For example, one of the more common findings is that distribution of the sum of lognormals appears to behave like (but never converges to) a lognormal as the number of time steps becomes large. As a result, Fenton-Wilkinson calibrations of the first two moments of a lognormal distribution are commonly used approximations.
Furthermore, one the more surprising findings is that "the stationary density for the arithmetic average of a geometric Brownian motion is given by a reciprocal gamma density, i.e. the reciprocal of the average has a gamma density." (Milevsky and Posner 1998) (Dufresne 1989) (De Schepper, Teunen, and Goovaerts 1994) (Yor 1992). Moreover, Dufresne (2000) has proven that a) the distribution of $\frac{1}{\left( Y_\tau \right)}$ is determined by its moments and; b) the distribution of the $\ln[\bar Y_\tau]$ is determined by its moments. However, calibration remains very clunky IMO.
Until quants figure out another kind of stochastic chain rule for these types of exponential integrals, I will probably have to settle for a fast and accurate approximation.
Bonus questions:
Additionally, I appreciate any references on the following:
- Can we formulate the solution using a pde approach?
- Are there any alternative stochastic processes (suitable for modeling securities prices) for which the time integrals of arithmetic expected values (and their terminal densities) are more tractable?
Your thoughts and references are much appreciated.
