Consider a payoff that pays a certain amount N of a vanilla Call (underlying: S, Maturity= T, strike:K). Every semester date Ts before T, if S>K(Ts), then N is increased by 1.

This product seems time-dependant (the amount of C(T,K) paid dépends on paths that S will take, and doesn’t look to be depending on forward volatility.

Thus, would you consider a Dupire (local volatility model) model, or a model with stochastic volatility, or some other models ?

What is your decision criteria that make you decide between local vol and stochastic vol models ?

I would not let the chosen model depend on the payoff function. For instance, consider a financial derivative where the underlying asset is a perfectly deterministic function of time. Then, your payoff function is also deterministic. So why model this underlying asset with a stochastic process?

The preferred model is often a trade-off between accuracy/market fit and complexity. Therefore, I would start with a basic model and try to fit it to market quotes. If no quotes are available, try to use similar products for which the quotes are available and fit those. When the fit does not suffice your needs, try adding little bits of complexity to increase the fit of your model to the market data.

I think in your case, start with a plain diffusion process to check if you can implement a working model. In this case, you can even derive a closed-form solution, I think. This will probably result in a poor volatility fit but gives you a good starting point to improve your model and find out which approach will work. Then maybe increase to a jump-diffusion model, and afterward include stochastic volatility. Especially, if your underlying is possibly dependent on the market interest rates, it also makes sense to think about incorporating stochastic interest rates.
Note that your model needs to be fitted to obtain workable results. Therefore, it makes no sense to include much complexity if the model cannot be fitted.

I hope you find it useful.

• In case of deterministic payoff, I would not have a stochastic model for the underlying but I would ask myself if I need to model my discounting function and eventually some credit events. On the methodology to chose the model, If I understand well, I should 1) find a representative portfolio of similar products, 2) list compatible models 3) do my best so that the models fits the calibration portfolio 4) compare the calibration quality (market consistency/repricing quality) 5) the chosen model is the one with the highest repricing quality. Do you validate this? Jun 28, 2022 at 10:01
• Yes, of course, I totally agree one would not consider a deterministic model in case of a deterministic underlying. It was just an example to illustrate the relation between the payoff and the preferred model.
– rrnl
Jun 28, 2022 at 11:23
• If you indeed follow the full approach, that is indeed the case. Form a portfolio containing the products resembling the risk factors, then use for example Feynman-Kac in combination with your chosen model for the risk factors to obtain a pricing formula and use the pricing formula to obtain implied parameters. Then test the goodness of fit on new market quotes to obtain an understanding of the fit.
– rrnl
Jun 28, 2022 at 11:27
• I often start, for the pricing part, with the fundamental theorem of asset pricing and start with a simple model that is verified in the literature. Then based on a working model, you can add complexities to obtain a possibly better model that fits the data. Sometimes, a more simple model is sufficient for the work, so then there is no need to increase the complexity :).
– rrnl
Jun 28, 2022 at 11:30