Finding Zero Crossings in a Diffusion-Based Derivative Payoff
Summary
The document asks how to find the time horizon at which the expected change in a derivative payoff reaches zero. It models two underlying quantities as drifting diffusion processes, with one carrying Brownian variance, and expresses the expected payoff change using normal cumulative distribution terms and a fixed fee. The resulting equation is a nonlinear root-finding problem in time.
The author notes that graphical or recursive methods can locate roots but asks for an analytical or closed-form solution. The document provides no solution, numerical example, or evidence that a closed form exists. Its stated setup also leaves assumptions and payoff notation unclear, so the displayed expectation and diffusion formula would need validation before using them in pricing. It is best read as a problem formulation rather than a complete valuation method.
Key ideas
- The target is the time horizon at which the expected payoff change equals zero.
- The proposed model represents the underlying quantities as drifting processes with diffusion uncertainty.
- The expected payoff equation uses normal distribution functions and includes a fixed fee.
- The document asks for a closed-form root but does not provide one or demonstrate that one exists.
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Full text
# Solving for roots of a stochastic pay-off function
# Solving for roots of a stochastic pay-off function
I have a pay-off function for a derivative which is defined by the Heaviside difference between $G$ and $B$ shifted by $-F$. To find the value of $V_{t=0}$, I need to find $\tau$ when $\frac{dV}{dt} = 0$.
I start by setting up as a classic diffusion argument for the expected value of $\frac{dV}{dt}$.
$\mathbb{E}\left[\frac{dV}{dt}\right] = \max{(-F,\, G_t -B_t-F)} \approx \mathbb{H}\left(G_t - B_t\right) - F = \mathbb{E}\left[G_t\right]^+ - \mathbb{E}\left[B_t\right]^- -F$
where $\mathbb{H}$ is the Heaviside step function: $f(\frac{dV}{dt}, \,\tau) =0 \quad \forall \; G_t < B_t$
Assume $G_t$ and $B_t$ are both drifting proccess with drift $= -d$; $G_t$ is a $\mathbb{P}$ Brownian Motion (i.e, Wiener process) with variance $= \sigma^2$. Therefore, we can set up set up the diffusion process as follows:
$\mathbb{E}\left[\frac{dV}{dt}\right] =\varPhi\left(\frac{\log{\left(\frac{G}{B}\right)}+\frac{\sigma ^2 \tau }{2}}{\sigma \sqrt{\tau}}\right)G e^{-d \tau } - \varPhi\left(\frac{\log{\left(\frac{G}{B}\right)}-\frac{\sigma ^2 \tau }{2}}{\sigma \sqrt{\tau}}\right)B e^{-d \tau } - F $
$\mathbb{E}\left[\frac{dV}{dt}\right] = 0 \to \frac{e^{-d \tau}}{\sqrt{2 \pi}} \left[G\left(\int_{-\infty }^{\frac{\log \left(\frac{G}{B}\right)+\frac{\sigma ^2 \tau }{2}}{\sigma \sqrt{\tau }}} e^{-\frac{Z^2}{2}} \, dZ\right)-{B \left(\int_{-\infty }^{\frac{\log \left(\frac{G}{B}\right)-\frac{\sigma ^2 \tau }{2}}{\sigma \sqrt{\tau }}} e^{-\frac{Z^2}{2}} \, dZ\right)}\right] = F$
where $\varPhi(X)$ is the CDF of the standard normal (Gaussian) distribution for i.i.d. $X_i$.
I can solve for the roots of $\frac{dV}{dt}$ graphically and/or recursively, but I require an analytical and/or closed-form approach.
How can I find $\tau$ when $\frac{dV}{dt} = 0$?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.