Pricing a European Call with Two Volatility Regimes

Summary

The document asks how to value a one-year at-the-money European call when volatility is specified as 20% for the first half-year and 90% for the second half-year. The proposed answer combines the two periods’ variances, weighted by their durations, and takes the square root to obtain an equivalent volatility over the full year. It explains this through the claim that a European option depends on the terminal distribution, so the variance accumulated over the life of the option determines its value.

The answer says no spot-price path assumption is needed when the stated volatility information is sufficient. This is a model-based explanation, not a numerical option valuation: it gives no interest rate, dividend yield, or spot level beyond the at-the-money condition. Its terminal-distribution argument presumes volatility describes the variance process and does not address path-dependent claims or uncertainty about future volatility.

Key ideas

• Combine volatility across time by adding each period’s variance contribution.
• For a European option, the answer treats terminal return distribution as the pricing input.
• A single equivalent volatility can summarize the two specified regimes under the stated assumptions.
• The document does not supply all market inputs needed for a numerical option price.

Tags

# pricing option with two volatility regimes


# pricing option with two volatility regimes












How would I price a 1-year ATM European call option, given I know that for the first 6 months, the realized volatility will be 20% and the latter six months, the realized volatilty will be 90%?

One estimation is computing the vega pnl at the six month mark when we remark the option from 20% to 90%. But this relies on an assumption about the spot price. Is there an actual solution to this problem without making too many assumptions?

## Answer by Arshdeep (score 1, accepted)

https://quant.stackexchange.com/a/79025

The implied vol should be the sqrt of realised variance over the option.

This is $sqrt(0.5*(20)^2+0.5*(90)^2)$.

This is just a result of the fact that for a European option, only the terminal distribution matters, so the vol of the terminal distribution determines everything.

Edit: You don't need to make any assumption, option is priced by the information you have provided.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.