Pricing Other Option Strikes with Constant Implied Volatility

Summary

The document explains how to estimate prices for options at strikes that are not quoted when constant volatility across strikes is assumed. Under Black–Scholes, first infer implied volatility from a known option price along with the underlying price, time to maturity, and risk-free rate. Since the volatility input cannot generally be isolated in a closed-form expression, it must be found numerically with a root-finding procedure.

Once implied volatility is estimated, use it in the pricing model for the other strikes. The answer cautions that real option markets typically show strike-dependent implied volatility, often described as a volatility smile. Therefore, carrying one volatility estimate across strikes is a simplifying assumption and may produce prices that differ from market quotes.

Key ideas

• Black–Scholes implied volatility can be inferred numerically from a known option price and required market inputs.
• The inferred volatility can be inserted into the model to price options at other strikes.
• A constant volatility assumption ignores the strike dependence commonly seen in market implied volatilities.
• Prices extrapolated across strikes are only as realistic as the constant volatility assumption.

Tags

# how to use known premium of options to determine premium of options with another strike?


# how to use known premium of options to determine premium of options with another strike?












Assuming constant volatility across all strikes, how to use known premium of options to determine premium of options with another strike? e.g. suppose we know premium of \$40 call and put, \$50 call and put, how to determine premium of \$30 and \$60 options?

## Answer by jaamor (score 2)

https://quant.stackexchange.com/a/15093

Under the Black-Scholes framework, you can calculate the implied volatility, given the option's price, underlying's price, time to maturity and the risk free rate. To calculate the implied volatility you have to use a root finding method, since there is not a closed form of the inverse of the B-S option pricing equation for volatility.

In the real world implied volatility varies across maturities (volatility smile). However, if you want to assume constant volatility as stated in the question you can use it to calculate option prices for other strikes.

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