A Conditional-Volatility Approximation for American Options
Summary
The note proposes approximating American option values under stochastic volatility by averaging prices from American option problems with fixed volatility paths. It assumes the underlying follows a diffusion with constant interest and dividend rates, while volatility follows its own stochastic process, and the price and volatility shocks have zero correlation. Conditioning on the volatility path turns the inner valuation into an American option with a specified time-varying volatility term structure.
The argument uses the fact that the optimal stopping value is at least the payoff from any particular stopping rule, then compares this with the expected value of the conditional optimal stopping problems. The author suggests these quantities may be approximately equal, but offers no accuracy analysis or numerical evidence. The key unresolved issue is whether optimizing after conditioning on a full volatility path overstates what can be achieved when that future path is not known at the actual exercise decision.
Key ideas
- Conditioning on a volatility path produces an American option problem with a fixed volatility term structure.
- The proposed estimate averages the conditional optimal stopping values across stochastic volatility paths.
- The setup assumes zero correlation between the underlying and volatility shocks.
- The note does not establish the approximation's accuracy or provide numerical validation.
Tags
Full text
# An approximation for American options under stochastic volatility
# An approximation for American options under stochastic volatility
I have a little approximation which I think could make sense, but I'd be very grateful if somebody here with more mathematical prowess could check:
My assumption is (initially) quite simple, namely
\begin{gather} dS_t = (r-q)S_t dt + \sigma_t S_t dW_t \\ d\sigma_t = a(\sigma_t)dt + b(\sigma_t) dZ_t \\ dW_t dZ_t = 0 \end{gather} and $r,q$ are constants.
Let $\tau$ be an optimal stopping time and define $F_\tau := E_0 [(S_\tau - K)_+]$. Then clearly, $$ \sup_{\tau} F_\tau \geq F_\tau = E_0 [(S_\tau - K)_+] $$ and similarly for puts.
In the presence of stochastic volatility, and with my assumption of zero correlation, I can also write \begin{align*} F_\tau &= E_0 [(S_\tau - K)_+] = E_0 [E_0 (S_\tau - K)_+ | \{\sigma_t\}_{t\in[0,T]}] \leq E_0 [\sup_\tau E_0 (S_\tau - K)_+ | \{\sigma_t\}_{t\in[0,T]}] \end{align*}
Note that the inner expectation is just an American option with a given volatility term structure.
On the other hand I also have $\sup_{\tau} F_\tau \geq F_\tau = E_0 [(S_\tau - K)_+]$, stochastic volatility or no stochastic volatility.
Hence, $$ \sup_{\tau} E_0 [(S_\tau - K)_+] \approx E_0 [\sup_\tau E_0 (S_\tau - K)_+ | \{\sigma_t\}_{t\in[0,T]}] $$ In other words, in the presence of stoch vol with zero correlation I can "mix" deterministic vol American option prices to approximate the price under stochastic vol.
I don't know how accurate the approx. is, that's the next step. I just want to check I have not made a grave error at this stage already.
[Oh, and since it's February 14th: Happy Volatility Day!]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.