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Pricing Forward-Start Calls with Heston PDEs and Conditional Valuation

Article Quant Q&A · Author: mth_mad
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Summary

The document presents two ways to price a forward-start call whose payoff depends on the asset price at the start date as well as at expiry. One approach conditions on the information at the forward start date and rewrites the value using the price of an ordinary Heston European call, with strike proportional to the asset price at that date. Monte Carlo simulation then averages this conditional value, allowing existing call-pricing procedures to be reused.

A second approach discretizes an auxiliary variable for the asset level at the start date. It solves a collection of finite-difference PDEs over the option's life after the start date, followed by a PDE over the earlier interval. This higher-dimensional state representation turns the dependence on the earlier asset level into a Markovian problem. The Monte Carlo approach is described as easy to implement but slow; the PDE approach adds computational and state-space complexity. No numerical comparison or calibration evidence is provided.

Key ideas

  • Condition on the forward start date to express the payoff through an ordinary Heston call value.
  • Monte Carlo can average the conditional call values using existing pricing routines.
  • A finite-difference method can track the start-date asset level as an auxiliary state variable.
  • The auxiliary-state method converts path dependence into a higher-dimensional Markovian problem.

Tags

OptionsDerivatives pricingVolatilityBacktesting
Full text
# Pricing Forward Start Option with PDE


# Pricing Forward Start Option with PDE












I am looking for references (books and papers) or suggestions on how to price forward starting calls using a PDE approach typically in the Heston model (In the BS world, the computation is trivial), with forward payoff $$\left(\frac{S_{t+\tau}}{S_t}-K\right)^{+},$$ where $t$ and $\tau$ are positive numbers.

I feel like the only way to use a PDE approach would be to identify the fundamental solution of the PDE in order to be able to apply the tower property on the expectation of the payoff.

All I have read up to know focus computing the characteristic function, and the martingale approach.

## Answer by parsiad (score 0, accepted)

https://quant.stackexchange.com/a/21919

Here's an approach that's easy to code (but FAR from the fastest). Let $ f(T,S,v,K) $ denote the price of a European call in the Heston model with time-to-expire $T$, initial price $S$, initial volatility $v$, strike $K$. First, use the tower property to transform the pricing problem: \begin{align*} V_{0} & =\mathbb{E}\left[e^{-r\left(t+\tau\right)}\left(S_{t+\tau}/S_{t}-K\right)^{+}\right]\\ & =\mathbb{E}\left[\frac{e^{-rt}}{S_{t}}\mathbb{E}\left[e^{-r\tau}\left(S_{t+\tau}-KS_{t}\right)^{+}\mid\mathcal{F}_{t}\right]\right]\\ & =\mathbb{E}\left[\frac{e^{-rt}}{S_{t}}f(\tau,S_{t},v_{t},KS_{t})\right]. \end{align*} Now perform a Monte-Carlo simulation to approximate the above (the advantage here is that you can use existing procedures to compute $f$).

## Answer by Antoine Conze (score 1)

https://quant.stackexchange.com/a/21926

You introduce a discretized auxiliary variable which represents $S_t$ to solve $N$ PDEs on $[t, t+\tau]$ using finite differences which will give you the present value of the option at time $t$ conditional on $S_t$. Then you solve one PDE using finite differences on $[0, t]$ to obtain the the present value at time $0$.

This is the same methodology than that used for pricing path dependent options using finite differences. The general idea is to transform a non markovian problem into a markovian problem of higher dimension by adding auxiliary variables that capture the past.

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.