Choosing Derivative Models by Market Fit and Complexity
Summary
The document considers how to model a path-dependent payoff whose number of vanilla calls increases when the underlying exceeds a threshold on periodic observation dates. It asks whether local volatility, stochastic volatility, or another model is appropriate, and proposes that model choice should reflect the calibration data and the accuracy required rather than the payoff alone.
The suggested workflow is to begin with a simple diffusion that can be implemented and, in this case, may allow a closed-form solution. Fit it to available market quotes, or to quotes for similar products when direct quotes are unavailable. If its fit is insufficient, add complexity incrementally, potentially including jumps, stochastic volatility, or stochastic interest rates when relevant. The key limitation is that added complexity is useful only if the model can be calibrated; a simple model may provide a practical baseline but fit volatility poorly. The response offers guidance, not a comparison of models or empirical results.
Key ideas
- Model choice should balance market fit against complexity rather than follow from payoff shape alone.
- Start with a simple diffusion and assess whether it meets the implementation and pricing needs.
- Use quotes for the product or comparable instruments to calibrate the model.
- Add features such as jumps, stochastic volatility, or stochastic rates only when they improve the needed fit.
- An uncalibrated complex model may not produce workable results.
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Full text
# Adequate model to payoff # Adequate model to payoff 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 ? ## Answer by rrnl (score 1, accepted) https://quant.stackexchange.com/a/71365 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.
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