MODERN QUANTITATIVE FINANCE
Introduction to quantitative finance Part 5: How would you teach Gen Alphas about different Option Valuation models?
This article is a precursor on one of my other articles titled, “On path-dependent Financial Options and path-dependent Volatility”
Timeline of the classical CEV model, the classical Heston model, and the classical SABR model
In the classical Black-Scholes (B-S) model, the (underlying) asset price follows a geometric Brownian motion. The log-return of the asset follows a Gaussian distribution. This is very demure, very modest; but not true in practice!
The CEV (Constant Elasticity of Variance) model was first introduced by economist John Cox in 1975 in his seminal paper Notes on Option Pricing I: Constant Elasticity of Variance Diffusions. Cox proposed the model as a more realistic alternative to the B-S framework for options valuation. While the B-S assumes a constant volatility, the CEV model allows volatility to be dependent on the asset price.
Then the 1993 classical Heston Stochastic Volatility model of Steve Heston was the first stochastic volatility model compatible with efficient valuation of options via characteristic functions: This Heston model differs from the classical B-S model in its diffusion coefficient for the stock price process SDE. In the Heston model, this diffusion coefficient (and thus the volatility) is stochastic.
The 2002 SABR (Stochastic Alpha, Beta, Rho model of Patrick Shawn Hagan et al.) can value vanilla equity options in special cases — although it was meant for FX options and IR derivatives: Now, note that the CEV model does not assume that volatility is stochastic in the usual ‘separate stochastic process’ sense (like in Heston or SABR). Instead, the CEV model makes volatility a deterministic function of the asset price — close to how a local volatility model treats volatility as a deterministic function of both time and price.
The provision of alternative financial tools was the foundation for what Steve Heston came up with (If you only have a hammer, everything looks like a nail)
The Heston model models volatility stochastically in an attempt to better capture the volatility dynamics exhibited by the financial market. Modelling volatility stochastically offers an improvement over the B-S by capturing important observed phenomena like non-lognormal asset returns (i.e. fat tails in returns), mean-reverting volatility, vol clustering, and a somewhat of a vol smile/skew.
- While the Heston model wasn’t the first stochastic volatility model to be proposed, it was the first to result in a closed-form characteristic function of the log-asset price under stochastic volatility. This means it can be used alongside the characteristic function valuation methods, along with some approximations, to estimate values for vanilla European options. Thus, the Heston model is the first practical model that allowed for stochastic volatility, while still capable of being used for valuation via numerical integration.
- But, unlike the B-S model, you don’t get the elementary formula for an option’s value. So, no closed-form valuation solution exists for even vanilla calls and puts under the Heston model. But, as previously seen, the Heston model is better in capturing what happens in the market than the B-S model since it results in somewhat of a volatility smile (The B-S model is incapable of capturing this). So, since the volatility surfaces generated by Heston’s model somewhat look like empirical implied volatility surfaces, the model offers a better calibration choice than the B-S model.
Then there’s the 1997 general model framework of Bakshi, Cao and Chen which incorporates stochastic volatility, stochastic interest rate, and jumps. Their option valuation model is abbreviated as SVSI-J model. The setup of the model contains the B-S model, the stochastic interest rate (SI) model, the stochastic volatility (SV) model, the stochastic volatility & stochastic interest rate (SVSI) model, and the stochastic volatility random jump (SVJ) model.
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But conventional stochastic models generate vol surfaces that are still a bit inconsistent with the observed vol surfaces. The conventional solutions are to introduce more volatility factors (e.g. two-factor Heston) or jump processes or local-stochastic vol hybrids like the SABR model or the local-Heston model (For local-stochastic vol hybrids, the local vol is good at calibration, while stochastic vol is somewhat good at capturing dynamics). But even these don’t fully match empirical volatility behaviour, especially the extreme short-term roughness. Empirical volatility exhibits high-frequency rough behavior!
Now, the classical Brownian motion has independent increments i.e. the change in volatility over one time period tells you nothing about the next. But rough volatility models relax the assumption of volatility increments being independent — by assuming volatility evolves like a fractional Brownian motion (fBm) with a Hurst exponent value H of less than 0.5. Keeping in mind that empirical volatility paths have H of around 0.1, the fBm is also cool because it has dependent increments; and specifically, when fBM’s H < 0.5, the increments are negatively correlated. Now, roughness often implies this anti-persistence property of volatility.
Microstructural foundation for rough vol — Microstructure meets Macro modelling
Order flow is highly autocorrelated (i.e. it has long memory property — as buy order stimulate more buy orders, and sell orders stimulate more sell orders). As such, price movements induced by these trades accumulate slowly and have memory. This naturally leads to long-memory effects in volatility. Now, if you aggregate many microstructural signals with long memory, you end up with fBm–like effects in volatility.
Then, since the rough volatility model is a consistent model for both the historical and implied volatility, Jim Gatheral nicknamed it as the God’s model. This is the opposite of usable market models — those which approximate the reality of the world regardless of what goes on underneath or what’s truly underlying from God’s point of view! Thus came the rough Heston model — the rough version of the classical Heston model (a not-so-good model of the market) which has been extended to the Volterra-Heston model. Another example of a rough volatility model is the rough Bergomi (rBergomi) model of Bayer, Friz and Gatheral — which a simple case of the rough fractional stochastic volatility (RFSV) model. The core rBergomi doesn’t include jumps; instead, it relies on the ‘roughness’ of the volatility process to capture observed market behaviours, such as the term structure of at-the-money volatility skew.
Conclusion
The majority of market movements is sourced from the interactions between traders! Now, anyone who starts trading is likely going to lose money for some years. This is because you’re competing against extremely smart people who are trying to take money from you. So, make use of objective tools, like the classical B-S model — but this is demure. The conventional stochastic volatility models are mid, but the rough volatility God models ate (in their comparison competition)!
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