Pricing Discontinuous Option Payoffs with PDEs

# Pricing of Binary or Digital Options or more generally options with discontinuous payoffs using PDEs


# Pricing of Binary or Digital Options or more generally options with discontinuous payoffs using PDEs












I am trying to find references (books, papers, etc.) for calculating $\mathbb E f(X_T)$, where $X_T$ is a diffusion and $f$ is a real function that is not continuous, by means of solving a PDE or Feynman-Kac equation.

Edit adressing the comments: Even if the PDE has a solution it can only be shown to equal the expectation under certain conditions. That is why I am asking for a reference for the caclulation of the expectation as solution to a PDE and not about the PDE and its solutions in itself.

Any such "verification theorem" basically uses the Ito formula for the value function and thus requires twice differentiability. This can only be ensured for the PDE solotion if the end data is continuous. So I am not interested in "it should just work" arguments but rather in answers or references to the "when" and "why.

Thank you

## Answer by Mark Joshi (score 1)

https://quant.stackexchange.com/a/16952

if we take a digital option and price under BS then you can do the whole thing by direct verification.

i.e. $N(d_2)$ solves the PDE and converges to the final pay-off pointwise.

So if the final pay-off has a finite number of jump discontinuities then subtract a linear combination of digitals to reduce to the continuous case.

## Answer by jensa (score 0)

https://quant.stackexchange.com/a/16943

$E\{f(X_T)\}$ can still exist even if $f$ is not continuous.

For example, $$ \begin{equation} f: x \mapsto \begin{cases} 1, \, x >= 3 \\ 0, \text{ otherwise} \end{cases} \end{equation} $$

Then $$ \begin{equation} E\{f(X_T)\} = P(\{X_T >= 3\}) \end{equation} $$

So, if you're example is a binary option which pays 1 if $X_T >= B$ and zero otherwise, and $\{X_t\}$ follows a standard diffusion, $$ \begin{equation} dX_t = X_t(r dt + \sigma dW_t) \end{equation} $$ you'll get $$ \begin{eqnarray} V_t &=& e^{-r\tau}\mathbb{E}\{1_{\{X_T >= B\}}|\mathcal{F_t}\} \\ &=& e^{-r\tau}P(\{X_T >= B\}) \\ &=& e^{-r\tau}\Phi\left(\frac{\log\left(\frac{X_t}{B}\right) + (r - \frac{\sigma^{2}}{2})\tau}{\sqrt{\tau}\sigma}\right) \end{eqnarray} $$, where $\tau := T - t$.

To calculate $\mathbb{E}\{f(X_T) | \mathcal{F_t}\}$ you would find the stochastic differential of $$\begin{equation} e^{-rT}\mathbb{E}\{f(X_T) | \mathcal{F_t}\} = e^{-rt}V_t. \end{equation} $$ Set the "dt-term" to zero (which gives the Black-Scholes PDE) and set appropriate boundary conditions. The temporal end boundary condition is your payoff function. For upper and lower bounds in the spatial dimension you would need to set "suitably large" values. Actual numerical schemes for solving PDEs can be found in standard textbooks on PDEs.