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Pricing Non-Option Derivatives with a Local Volatility Model

Article Quant Q&A · Author: econmajorr
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Summary

The document explains how to use a calibrated local volatility model to price a derivative whose payoff is not a standard call or put. In Monte Carlo pricing, the model supplies volatility as a function of the underlying price and time. A simulation evolves the underlying with an appropriate discretization, looking up the local volatility at each step using the current state, then applies the target derivative’s payoff to the simulated paths.

The key point is that the model describes the underlying dynamics, while the payoff determines the instrument being priced; the same simulated paths can therefore support different payoff types. Local volatility is calibrated to reproduce market option prices, but the document does not specify calibration details, discretization choices, or numerical safeguards. It briefly notes stochastic local volatility as an extension that adds volatility randomness, so the basic method’s assumptions and limitations are not examined in depth.

Key ideas

  • A local volatility model provides volatility as a function of underlying price and time.
  • Monte Carlo simulation can use that state-dependent volatility at each step to generate underlying paths.
  • A non-option derivative can be priced by evaluating its payoff on paths generated under the model.
  • Local volatility aims to calibrate simulated dynamics to observed market option prices.
  • Stochastic local volatility adds randomness to volatility beyond the local volatility function.

Tags

Derivatives pricingOptionsVolatilityStatistics
Full text
# Pricing with local volatility for derivatives beside options


# Pricing with local volatility for derivatives beside options












Say I have calibrated an local volatility mode to market data on a forward on stock X. Say I want to price a derivative Y that is NOT a call/put option. What is the (or one of many) general strategy to compute the price of the new derivative?

I am used to experiment with stochastic volatility and simulatons based derivative pricing.

My knowledge of Local Vol is this:

http://sp-finance.e-monsite.com/pages/volatility/volatility-models/local-volatility-models/dupire-equation-uses.html

notes (as the link above) and slides are welcome, scientific paper are not as they more often look at specifics rather the overall picture.

## Answer by Slade (score 1, accepted)

https://quant.stackexchange.com/a/46596

Reposting comments as an answer:

If you are doing Monte Carlo you'd have a new volatility function to use (rather than a constant vol like in black scholes) for each time and stock price from the local volatility model (It's usually written $\sigma (S, t)$). So you could use this local volatility function during a simulation regardless of the type of payoff.

So using the local volatility function, you know what the volatility 'should be' as a function of stock price and time. So when you do a simulation of an SDE, (however you decide to discretize), you can use the volatility for the next time step as the local volatility function evaluated at the current time and stock price. And repeat this as the simulation proceeds.

The point of the local volatility function is just to ensure that the volatility calibrates to market prices. There's also stochastic local volatility models that both calibrate to market prices but have volatility of the volatility involved as well.

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.