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I've been studying volatility modelling over past the few days; in particular, the connections between local vol, stochastic vol, implied vol. I've been reading Gatheral's book "The volatility surface". I'm finding it pretty tough to follow. I've also read relevant chapters in Willmott's Quant Finance book. Overall, what are some other great resources to learn about this topic? I am not looking for the most advanced papers, but rather some introductory/intermediate material. My background is in applied maths and finance. I know this paper is fairly popular: https://arxiv.org/pdf/1204.0646.pdf. Looking for any help I can get on the topic, thanks.

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Along with Gatheral's book, I'd recommend reading Lorenzo Bergomi's "Stochastic Volatility Modelling". The first 2 chapters are available for download on his website. That being said, let me try to give you the basic picture.

Below we assume that the equity forward curve $F(0,t)=\Bbb{E}_0^\Bbb{Q}[S_t]$ is given for all $t$ smaller than some relevant maturity $T$. The risk-neutral drift $\mu(t)$ is inferred from that forward curve considering a pure diffusion framework (i.e. no jumps). The same goes for the discounting curve and the risk-free rate $r(t)$. We also assume that there is no arbitrage in the market.

Implied volatility

As Rebonato puts it: implied volatility is the wrong number to put in the wrong formula to get the right price.

You should thus consider it as an alternative and equivalent way to describe vanilla option prices: instead of using the market price of an option, you find the volatility figure $\sigma$ to be plugged inside the Black-Scholes pricing formula - along with the appropriate forward price and discount factor - to recover the former price. There is only 1 such figure because the BS price is a one-to-one mapping from the volatility to price space, all other things equal.

If you do this for all listed expiries $T=\{T_1,\dots,T_N\}$ and all listed strikes $K=\{K_1,\dots,K_M\}$, you'll end up with the so-called (discrete) implied volatility surface $\Sigma(T,K)$ corresponding to the prices $V(T,K)$.

The paper you mention is one of the many which studies how you could move from this discrete $T \times K$ specification of the IV surface to a continuous one by relying on some specific parametrisation (here SVI), which aims at precluding arbitrage opportunities while interpolating/extrapolating the original data points.

Local volatility

Is the function $\sigma_{LV}(t,S)$ such that when the following 1D Markovian diffusion model for the equity spot price is used $$ \frac{dS_t}{S_t} = \mu(t) dt + \sigma(t,S_t) dW_t $$ the prices for any $(T,K)$ returned by the model perfectly coincide with the market prices $V(T,K)$, or equivalently as we've just seen, allow to fall-back on the exact same volatility surface $\Sigma(T,K)$.

Dupire's seminal work shows that from the previous definition the local volatility function should verify (see also derivation here) $$ \sigma_{LV}^2(t=T, S=K) = \frac{\Sigma^2 + 2\Sigma T \left( \frac{\partial \Sigma}{\partial T} + \mu(T)K \frac{\partial \Sigma}{\partial K} \right)} {\left( 1-\frac{Ky}{\Sigma} \frac{\partial \Sigma}{\partial K} \right)^2 + K \Sigma T \left( \frac{\partial \Sigma}{\partial K} - \frac{1}{4} K \Sigma T \left( \frac{\partial \Sigma}{\partial K} \right) ^2 + K \frac{\partial^2 \Sigma}{\partial K^2} \right)} \tag{1} $$ where $y = \text{ln}(K/F(0,T))$ and $\Sigma = \Sigma(T,K)$.

Reciprocally, in Chapter 2 of Bergomi's book but also in Gatheral's, it is shown how implied volatilities can be seen as a weighted expectation of local volatilities, see for instance equation (2.32) and (2.33) in Bergomi's book.

Stochastic volatility

Amounts to considering a diffusion model of the form \begin{align} \frac{dS_t}{S_t} = \mu(t) dt + \sigma_t dW_t^S \\ d\sigma_t = \alpha(t,\sigma_t) dt + \beta(t,\sigma_t) dW_t^\sigma \end{align} with some correlation between the 2 driving Brownian motions. The instantaneous volatility of log-returns $\sigma_t$ is this time stochastic, hence the name. There is no direct built-in connection to vanilla option prices. That being said, this modelling framework allows to generate volatility smiles and could therefore be calibrated to the market.

If you want to calibrate the model so that vanilla option prices are recovered you can appeal to the Gyöngy's theorem (also discussed in both Gatheral and Bergomi) and the dynamics of $\sigma_t$ should satisfy $$ \Bbb{E}_0^\Bbb{Q} \left[ \sigma_t^2 \vert S_t = S \right] = \sigma^2_{LV}(t,S) \tag{2}$$

Again pros and cons are examined in details in Bergomi's book. To go even further you could have a look at forward variance models and local-stochastic volatility models.

[TL;DR]

  • Implied volatility is a one-to-one mapping between a vanilla option price and a specific model quantity -- the Black-Scholes volatility.
  • Local volatility goes one step further by creating a one-to-one mapping with not one but all the vanilla option prices. This relies on a 1D Markovian representation of the spot dynamics where the instantaneous variance of the log-returns deterministically depends on the spot price level at all times (complete model).
  • Stochastic volatility amounts to considering a 2D diffusion framework where the instantaneous variance of the equity log-returns is a separate stochastic factor (incomplete model). There is no 'built-in' connection with the market although this model allows to generate IV smiles. When a stochastic volatility model is calibrated to the market, it is usually impossible to match all vanilla option prices due to the finite number of model parameters at hand.
  • Equations $(1)$ and $(2)$ establish links between the 3 relevant quantities.
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  • $\begingroup$ How is stochastic volatility an incomplete model? I really dont understand mathematically what you mean by that (and why local vol is complete) $\endgroup$ – Permian Mar 25 '18 at 15:37
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    $\begingroup$ In stochastic volatility models the additional state variable (which corresponds to the instanteneous volatility or variance) is not a marketed security. Hence the model is incomplete. In local volatility models there is no such additional state variable, the spot price remains the only stochastic factor. Hence the model remains complete. $\endgroup$ – Quantuple Mar 25 '18 at 15:44
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Gatheral's book is one of the best reference around so it's worth bearing with it, especially as he covers the relationship between implied, local, and stochastic volatility:

  • local volatility computed from implied volatility using Dupire's formula;

  • square local volatility as conditional expectation of square stochastic volatility

Once you have understood these concepts, most of the relevant literature deals with specific modeling/parameterization/interpolation of the implied volatility surface, popular models being the SABR developed by Hagan and al., https://en.wikipedia.org/wiki/SABR_volatility_model used a lot for interest rate derivatives, or the Vanna-Volga model for FX options https://www.researchgate.net/profile/Fabio_Mercurio/publication/285662078_The_Vanna-Volga_method_for_implied_volatilities/links/56e6cd9308aedb4cc8af896a/The-Vanna-Volga-method-for-implied-volatilities.pdf.

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