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.

• If you are doing Monte Carlo you'd have a new volatility function to use for each time and stock price from the local volatility model. So you could use the volatility function regardless of the type of payoff – Slade Jul 13 '19 at 3:32
• Can you add a little bit more to it!? So you are saying: When I do monte simulation of my SDE, then at each time step I simply update the volatility term according to a function of spot. Right? – econmajorr Jul 13 '19 at 11:12
• Yeah. 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. 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 – Slade Jul 13 '19 at 11:17
• ok thanks! I certainly think this is an answer, so I think you should post this as an answer. Can you suggest reading material on this topic? (beside scientific papers) – econmajorr Jul 13 '19 at 12:31
• This book might be difficult (it was/is for me) but a lot of how to perform Monte Carlo well with different vol models is in the book 'nonlinear option pricing' (Guyon). It also has citations with papers that may be helpful. I'd also just look up stochastic volatility modeling in general, and there'll be books on that too. One from CRC press (Bergomi) is well known but there's a few others as well. – Slade Jul 13 '19 at 16:27

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.