I am trying to find a closed form solution to a stochastic integral -- which is really just a generalized expression for the expected net present value, $E^*[V_t]$, of an annuity (or perpetuity if $T \to \infty$), when cash flows are the result of some stochastic process. $V_t$ is defined by:

(1) $$V_t =Max(L_t, \int_t^{T} (G_0e^{(m-d+W_t\sigma)t}-B_0e^{-d*t}-F)\frac{1}{e^{r*t}}dt)$$


$T$ is arbitrarily set to the value which maximizes expected value, $E^*[V_t]$.

$0 \le t \le \infty$

$G_0$, $B_0$, and $F$ are constants $\gt 0$

$L_t$ is some arbitrary value representing the liquidation value of $V_t$; assume it's $0$.

$m$ and $d$ are constant rates $\gt 0$

$\sigma^2$ is variance

$W_t$ is Brownian Motion such that $\frac{dG}{G} = (\mu - d)dt + \sigma dW_t$


$G_t = G_0*e^{((\mu-\frac{\sigma^2}{2})-d)t+\sigma W_t}$

subject to the boundary conditions:

(condition i) $-\infty \ge T \ge 0$

(condition ii) $\frac{dV}{dt} \to G_t$ as $G_t \to \infty$.

(condition iii) $\frac{dV}{dt} \to 0$ as $G \to 0$

(condition iv) $\frac{dV}{dt} \to 0$ as $t \to \infty$

(condition v) $V_t \ge 0$ for all $t$

As $t \to \infty$, $E^*[\frac{dV}{dt}] \to -F*e^{-r*t} \to 0$. So I am fairly confident that the integral will converge and that a finite $T$ will satisfy the constraint.

I am also assuming that $G_t$ can be delta-hedged -- I am treating $\frac{dG_t}{dt}$ as a martingale with $\mu = 0$.

I tried to evaluate this problem with standard option pricing models, but they all assume that pay-offs are a one-time deal -- even perpetual American options are built on the assumption of a single period pay-off. The assumption that pay-offs accrue continuously results in risk-neutral probabilities (vis-a-vis $N(d_1)$ and $N(d_2)$ in common derivations of Black-Scholes) which instantaneously change as a function of $dt$.

It seems fairly straightforward to solve this problem using discrete, iterative methods (i.e., binomial trees, Monte Carlo simulations, etc) which discretize risk-neutral probabilities. However, I am not able to use any of those methods in the current setting.

What am I missing? Your help is much appreciated. Let me know if you'd like to get more background on what equation (1) is attempting to model.

  • $\begingroup$ Note: I would also accept the following assumptions as convergence to the answer.......... (1') $V_t =\int_t^T \ (G_0e^{(m-d+W_t\sigma)t}-B_0e^{-d*t}-F)\frac{1}{e^{r*t}}dt$ ........... subject to the following additional boundary condition: (condition v) $V_t \ge 0$ for any $t$. $\endgroup$ Mar 3, 2017 at 19:07
  • $\begingroup$ Last thing: under equation (1) or (1'), in the preceding comment, $T$ may be arbitrarily set to the value which maximizes the expected present value, $E^*[V_t]$. $\endgroup$ Mar 3, 2017 at 20:18
  • $\begingroup$ Keep in mind that Black and Scholes are able to find the value of a call option because the delta-hedged option is riskless. The general case, where the cash flows are risky I believe has not been solved yet, nor has it been proven that the solution is unique. (Maybe I would pay a different price for this annuity that you would). $\endgroup$
    – nbbo2
    Mar 3, 2017 at 21:44
  • $\begingroup$ Thanks for your response, @noob2. You make a good point about delta-hedging... and you may have helped me significantly. Suppose that $G_t$ is a function of an underlying stochastic process, $P_t$ (i.e., commodity prices???). Why can't we, for any $t$, take a short position in $q_t^{\prime}$ shares of $P_t$, where $q_t^{\prime} = \frac{G_t - B_t - F}{P_t}$ .......... such that the $\frac{dV}{dt} = 0$ for all $t$? $\endgroup$ Mar 3, 2017 at 23:30
  • $\begingroup$ $V_{t}$ as defined by the integral over t is dependent on the path of $W_{t}$, which means it's random. Is it possible, that you mean the expected value of the integral? Also you seem give a solution for $V_{t}$ in your other post quant.stackexchange.com/questions/32747/… $\endgroup$
    – Ami44
    Mar 4, 2017 at 0:19

1 Answer 1


I believe I can answer my own question... mostly. Thank you to all who gave valuable input!

I was able to demonstrate sufficiently to myself that convolution of an expected range of outcomes around a known probability distribution of outcomes converges with conventional asset and options pricing theory. I am pleased to learn that the methodology applies generally to all process in which a stochastic variable is present.

Similar approaches are enumerated in several papers:

  1. T. Artikis, A. Voudouri. A stochastic integral arising in discounting continuous cash flows and certain transformed characteristic functions

  2. M. Schmelzle. Option Pricing Formulae using Fourier Transform: Theory and Application

and more.

Basically, if you have a known probability density function, such as a normal distribution:

$\phi(Z) = \frac{1}{\sqrt{2\pi}}e^{-\frac{Z^2}{2}} $


$\int_{-\infty}^{\infty}\frac{1}{\sqrt{2\pi}}e^{-\frac{Z^2}{2}} = 1$


$ Z = \frac{1}{\sigma\sqrt{\Delta T}}(-r+\mu+\frac{(\sigma^2)}{2})(\Delta T)$


$\sigma^2$ is the variance

$\Delta T$ is the change in time, $t$.

$\mu$ is instanteous drift, and;

$r$ is the force of interest

It can be shown that for some stochastic price process, $P_t$, with a known a distribution, that convolution around a transformed risk-neutral probability density function, $\theta$, equals the expected future value at $P_T$. Let the expected $P$ at time, $T$, equal:

(2)$$E^*[P_T] =f(P * \theta)(T) = \int_{-\infty}^{\infty}\theta(Z(T-\tau)) (P_0e^{\sigma*Z(T-\tau)}) {d \tau} \to P_0e^{r*T}$$


$\theta$ is the transformation of the risk-neutral probability density function of the normal distribution such that:

$\theta_t = \phi(Z) (\int_{-x}^{x}{\phi(Z)dt})^{-1}$

Convolution under the risk-free distribution converges therefore with the no-arbitrage theory of asset pricing under the correct assumptions regarding the future disribution. It's a chicken and egg thing, I know, but it's still neat!

I believe that this approach can be adopted for pricing general quantities under uncertainty. Applying this concept to equation (1) in the original question would then yield the following relationship:

$$E^*[V_t] = \int_{t=0}^{T}\theta(Z(t-\tau)) \frac{dV_t}{dt}e^{r*t} d\tau $$

In practice, this approach has provided me with satisfactory results that converge with a modified Black-Scholes and discrete binomial-probability models. The convolution approach can, in fact, be used to derive risk-neutral probabilities of the cumulative density function, $\Phi$, which satisfy parameters used in closed-form solutions resembling the Black-Scholes.

It should be noted that although the distribution of $\phi$ is normal, the resultant convolution around an exponential process is log-normal. Due to the resultant log-normal distribution of outcomes, this approach will result in higher present valuations when cost functions are fixed than straight DCFs. It seems that the convolution approach could be adapted for mean reverting Ornstein–Uhlenbeck processes. This would temper the upward bias somewhat, but would still assign significantly higher values to marginal projects and/or positive values to unprofitable projects in which a straight DCF would assign $0$ or negative value.

Now, my only problem is choosing an optimal $T$. If I choose $T$ such that:

$G_0e^{-d*t} - B_0^{-d*t} - F = 0 \to T = \frac{Ln(\frac{G_0-B_0}{F})}{d}$

I feel like I am leaving something out because there is often some residual value to gained by increasing the $T$, yet setting $T \to \infty$ results in value destruction. Another way to see this problem is at the terminal time-node, there is still a expected positive value for running time forward. I could discretely iterate through $t$s... I could also estimate $T$ by optimizing $V_t$,.. but all that is yucky. Any ideas for a closed-form approach?


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