Why Lower Rates Can Raise Implied Volatility for a Fixed Option Price

Summary

The document explains a Black–Scholes comparative effect: when the risk-free rate is lowered, the present value of the strike paid at maturity rises. Holding other inputs fixed, this reduces the model value of a call option. The question concerns implied volatility, which is calculated by finding the volatility input that makes the model price match the observed market price.

If the market option price is held fixed while the rate assumption falls, the model’s price decline must be offset by a higher volatility input. This follows from the positive sensitivity of a call price to volatility, known as vega. The explanation is about how model inputs interact in implied-volatility calculation; it does not claim that falling rates cause market volatility to increase, nor does it address changes in the observed option price or other inputs.

Key ideas

• A lower risk-free rate increases the present value of a call’s strike payment and lowers its modeled price, all else equal.
• Implied volatility is the volatility input that reconciles a model price with the observed option price.
• If the observed option price is fixed, a lower rate assumption requires higher implied volatility to offset the modeled price decrease.
• The effect is a pricing relationship and does not show that falling rates cause market volatility to rise.

Tags

# Why does implied volatility increase when we lower the risk-free interest rate?


# Why does implied volatility increase when we lower the risk-free interest rate?












I don't understand it, when I calculate it I see it, but I can't explain it. Plus, $\frac{\partial C}{\partial \sigma}$ is positive so, could you explain me please ? Is it because the market is more volatile when the price is decreasing ?

Thanks.

## Answer by siou0107 (score 2, accepted)

https://quant.stackexchange.com/a/49958

If you lower the risk-free rate, the NPV of the strike that you will pay at maturity is higher, thus reducing the PV of your payoff and so the price of your option.

Yet, your option price is fixed: it is the market price of your option, from which you want to imply the Black-Scholes volatility. Your lower risk-free rate is supposed to lower the price, so it will have to "get higher" through the only free parameter of the formula, i.e. the implied volatility.

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