Applying Dupire’s Local Volatility Formula to Put Prices

Summary

The document addresses whether Dupire’s local volatility formula can be expressed using put prices as well as call prices. Under the stated zero-rate, zero-dividend setup, or when working with undiscounted options on forwards, the formula uses the option’s time derivative and its curvature with respect to strike. The answer notes that calls and puts share the same implied risk-neutral density, so their second strike derivatives are equal.

Put-call parity then provides the corresponding expression in terms of put prices, and the answer says parity also supports extending the derivation to more general settings. This establishes that put prices can supply the same local volatility information under the relevant assumptions. The explanation is compact and does not discuss practical estimation from sparse or noisy market quotes, boundary conditions, or numerical stability, all of which can matter when differentiating option prices.

Key ideas

• Call and put prices have the same second derivative with respect to strike under the stated framework.
• The shared strike curvature corresponds to the same implied density.
• Dupire local volatility can be written using put prices as well as call prices.
• Put-call parity connects the call-based and put-based expressions.
• Using market quotes requires care because numerical differentiation can be sensitive to data quality.

Tags

# Is there a Dupire's Formula for put options?


# Is there a Dupire's Formula for put options?












Generally, Dupire's formula is taking derivatives on the call option prices. Here it only uses information of the call options.

If now we have the data including both call and put options, is there a mathematical formula using all the information? Or is there a corresponding Dupire's formula for put options?

## Answer by jherek (score 4)

https://quant.stackexchange.com/a/59495

It depends what you exactly call Dupire's formula. If you take the original formula, valid under zero interest rates and dividends (or equivalently, considering undiscounted option prices on the forwards), which reads $$\sigma_L^2 = 2 \frac{ \frac{\partial C}{\partial T} }{K^2 \frac{\partial^2 C}{\partial K^2}}\,.$$

Then the formula for a put is the same, as the implied density is the same: $\frac{\partial^2 C}{\partial K^2} = \frac{\partial^2 P}{\partial K^2}$. Using the Put-Call parity formula $C-P = F-K$, which also allows to derive the equations in the more general case, we obtain

$$\sigma_L^2 = 2 \frac{ \frac{\partial P}{\partial T} }{K^2 \frac{\partial^2 P}{\partial K^2}}\,.$$

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.