Pricing Options with a Volatility Regime Change at a Barrier

Summary

The document describes an option-pricing problem in which volatility changes when the underlying crosses a specified barrier. It suggests this setup may represent a post-crash environment where near-term option volatility is elevated and expected to move toward a longer-run level. The model therefore depends on the current price, barrier, two volatility levels, and the time spent under the first volatility before expiry.

The author reports that an attempted formula reduces to Black-Scholes when the volatility levels match, but can produce inconsistent results in edge cases, including calls valued above the underlying price. The discussion also proposes qualitative effects: a low-volatility initial period can lower value relative to using the long-run volatility throughout, while an initially high-volatility period can add value. It gives no validated derivation or numerical evidence, so these claims should be treated as an exploratory model description rather than a pricing method ready for use.

Key ideas

• The proposed model switches volatility when the underlying crosses a barrier.
• The setup distinguishes an initial volatility period from the remaining time to expiry.
• When both volatility levels match, the author says the formula reduces to Black-Scholes.
• The attempted method has unresolved edge cases and can violate the call-price upper bound.

Tags

# How to price an option with two volatilities?


# How to price an option with two volatilities?












Imagine you have two volatilities, the second which is "activated" when the stock crosses a barrier called $p_b$. The present price is $p_1$. ($p_b>p_1$).

This can be used to price options after a crash when it's assumed that options are more expensive in the short term and are reverting over the long term.

There is also a time parameter for the ratio of time for the duration first volatility ($t_1$) with the time duration until expiration ($t_2$).

I've tried pricing such an option and it's quite messy and involves some improvisation for tricky situations such as when the barrier is the same as the present price or when the time duration $t_1$ is very small relative to $t_2$.

This is the best I have done so far. It still has inconsistencies where the call exceeds the present price for very specific instances.

When the volatilities are equal, the equation reduces to Black-Scholes.

If the first volatility is very flat and the second is greater, the stock --- as $t$ progresses --- will eventually assume the longer-run volatility, but the option is slightly cheaper than simply pricing it with the long-run because the lesser short-run volatility is like friction. Likewise, if the finite frame volatility is greater than the longer run, then we should get a small "boost" for the call price from the short-run even as the option assumes the long-run lesser volatility.

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.