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A general model (with continuous paths) can be written $$\frac{dS_t}{S_t} = r_t dt + \sigma_t dW_t^S$$ where the short rate $r_t$ and spot volatility $\sigma_t$ are stochastic processes. In the Black-Scholes model both $r$ and $\sigma$ are deterministic functions of time (even constant in the original model). This produces a flat smile for any expiry $... 13 To simplify the problem, let us consider normal local volatilities$ \sigma \left ( S_t, t \right) $and implied volatilities$ \sigma_i \left ( K, T \right) $such that the model is: $$dS_t = \sigma \left ( S_t, t \right) dW$$ (no rate, repo, dividends, etc.) and$ \sigma_i \left ( K, T \right) $is the normal volatility input into Bachelier's formula ... 13 Along with Gatheral's book, I'd recommend reading Lorenzo Bergomi's "Stochastic Volatility Modelling". The first 2 chapters are available for download on his website. That being said, let me try to give you the basic picture. Below we assume that the equity forward curve$F(0,t)=\Bbb{E}_0^\Bbb{Q}[S_t]$is given for all$t$smaller than some relevant ... 13 1. What does it mean by the vol surface is the current view of vol? The local volatility model is calibrated to vanillas prices (and equivalently their implied volatilities), which reflect the market's view of the volatility, in order to use it to use it to price other options that one will hedge with the vanillas. Where a Black-Scholes model (no smile) ... 9 Some Notations It's easy to get lost so let's introduce some notations and let $$\sigma : (t, S, K, \tau) \to \sigma(K,\tau; S, t)$$ denote the implied volatility smile prevailing at time$t$when the spot price is$S_t=S$for an option with strike level$K$and time to expiry$\tau=T-t$. From here onward, we drop the$t$argument to keep notations ... 9 Some points below as food for thought: Suppose you possess an implied volatility surface over a continuous strike cross time to expiry domain (how to get there from the discrete market specification is another question). Further assume that you have to price a path-dependent option, e.g. a Barrier or an Asian. If you are using Black-Scholes, what implied ... 7 I have also currently started to learn about the subject. This is some of the material I have encountered: Many people recommend the book "The Volatility Surface: A Practitioner's Guide" by Jim Gatheral. It is a standard reference in the area (even though I personally found it a bit confusing and a bit unclear at some parts). The author also have ... 6 The local vol model has exactly enough freedom to match the individual densities$X_t.$There is no additional freedom in the local vol model to match even a joint density for a pair of times$(X_t,X_s).$When you ask about the joint density across the continuum of times$t \in [0,T]$it is pretty easy to show that any local vol model differs from any ... 6 Whenever you use any model to price anything, all you need to do is make sure you model the underlying dynamics that the product you're pricing actually depends on. Any product will be dependent on numerous facets, to varying degrees - this is the same with modelling anything. The modelling that happens in pricing financial derivatives is an integration ... 6 You should not expect the local vol to be equal to the implied vol except in the trivial case where both are constant (Black-Scholes model). I haven't read the Derman articles but it is quite clear using Dupire's formula (see Gatheral's book for example). Local volatility can be computed in terms of call prices using Dupire's formula $$\sigma^2(T,K) = \... 6 The following paper is helpful for understanding the point you raise: Hagan et al.: Managing Smile Risk, January 2002, Wilmott 1:84-108 The main point is given in the paper: [...] the dynamics of the market smile predicted by local vol models is opposite of observed market behavior: when the price of the underlying decreases, local vol models ... 6 Stochastic-Local Vol (SLV) is an attempt to mix the strengths and weaknesses of both Stochastic Vol and Local Vol models. Below, I'll quickly summarise each model and their strengths and weaknesses, and then discuss how SLV tries to improve things. Although there are many stochastic vol models, I limit the discussion here to the Heston model to keep things ... 6 We can demonstrate this via a pricing experiment using QuantLib-Python. I've defined several utility functions in the code block at the bottom of the answer that you will need to replicate the work. First, let's create a Heston process, and calibrate a local vol model to match it. Up to numerical issues, these should both price vanillas the same. v0, kappa, ... 5 This is merely a question of notation, you should simply read$$ \sigma(K,T) = \sigma(S_t=K, t=T) $$For an easy to follow derivation see this excellent note from Fabrice Rouah Some intuition behind the developments: The price of a European option, for instance a call, can be written in integral form:$$ C(t, S_t, K, T) = e^{-r(T-t)} \int_0^\infty (S_T-K)^... 5 Gatheral and Jacquier discuss this issue in section 4 of the paper. Instead of using the raw parameterization of the SVI, they use the natural parameterization of the total implied variance: $$w(k) = \Delta + \frac{\omega}{2} \left\{ 1 + \zeta \rho (k - \mu) + \sqrt{(\zeta (k-\mu) + \rho)^2 + (1-\rho^2)} \right\} (\text{p. 61 of the published paper})$$ In ... 5 Here "dynamics" means the assumed future behaviour of the spot process, namely that it follows the SDE $$dS/S = r dt + \sigma_{loc}(S,t) dW_t .$$ There are various ways to see that these dynamics are unrealistic. One is to look for time homogeneity. In normal cases, you expect the market to follow the same rules in one week and in one year from today. ... 5 You can view the price of an option as the cost to dynamically replicate it. The more volatility, the more costs you will have trading the underlying to keep your delta equal to 0 (I'm assuming you sold the option, hence a negative gamma position). So, if at any spot, any date your local vol is above 0.194, rebalancing the portfolio will be constantly more ... 5 This is not quite true, in either direction. If you have an arbitrage free implied vol surface, you might not have a well-defined local vol surface. An example comes from a discrete model. Consider a spot dynamics where the spot is a martingale that jumps up or down by integer amounts. The spot distribution is discrete, with zero density in between ... 5 The following source contains detailed answers to your questions in a research paper from ETH ZÃ¼rich. van der Weijst, Roel (2017). "Numerical Solutions for the Stochastic Local Volatility Model" http://resolver.tudelft.nl/uuid:029cbbc3-d4d4-4582-8be2-e0979e9f6bc3 5 The LV model is a particular kind of model where the implied volatility of a European vanilla of given strike and maturity emerges a deterministic function of time, spot level and the local volatility function used$\sigma(\cdot, \cdot). $$\hat{\sigma}_{KT} = f(t, S_t; \sigma)$$ such that using ItÃ´ one could write \begin{align} \frac{ dS_t }{S_t } &= \... 5 I'll answer both of your questions in one go: Your ideas are correct. If the Black-Scholes model was true, the implied volatility surface would be flat but it is not in real life. Thus, the geometric Brownian motion as stock price model is misspecified and we need more sophisticated models (sto vol, jumps etc), in particular if we want to price more ... 4 Yes, there is a unique time homogeneous local vol model. This is proven in http://www.sciencedirect.com/science/article/pii/S0304414912002487. There is a slight generalization required that if the option-implied density is zero somewhere, the corresponding local vol is infinite in that region, giving a "gap diffusion". No, there is no nice formula for the ... 4 SinceS_T = S_0 + \sigma W_T, \begin{align*} C &:= E\left((S_T-K)^+ \right)\\ &= E\left((S_0+\sigma W_T-K)^+ \right)\\ &=\int_{\frac{K-S_0}{\sigma \sqrt{T}}}^{\infty}(S_0+\sigma\sqrt{T} x-K) \frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}dx\\ &=(S_0-K)\Phi\left(\frac{S_0-K}{\sigma \sqrt{T}}\right)+\frac{\sigma\sqrt{T}}{\sqrt{2\pi}}e^{-\frac{(S_0-K)... 4 The implied Black-Scholes skew will be downward sloping in the limit on both the left and the right. (I believe @Gordon's derivation claiming upward slope may have a sign error somewhere). Left Side For the left side it is sufficient to note that the lognormal model has no density below zero while the normal model has strictly positive density in that ... 4 Note that \begin{align*} dS_t = S_t\left(\mu dt+\sigma S_t^{\gamma-1} dW_t \right). \end{align*} That is, the volatility function is defined by\sigma S_t^{\gamma-1}$. Then, if$\gamma <1$, the volatility increases as the price falls. 4 The problem with Dupire's formula is that it requires the derivatives of the option prices, where you do not have a continuum of prices. The reason this is a problem is that you now have to come up with some interpolation scheme for your prices (and even if that involves fitting some term vol surface, it's still an interpolation scheme, it's just more ... 4 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 ... 4 In fact, this is a confusion caused by a sloppy notation. The rigorous version of the setup should be $$A(K)\rightarrow \epsilon A(K).$$ Then we let$x:=\frac{f-K}\epsilon\$. The rest is the usual singular perturbation operation.

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Let the risk-neutral dynamics under your LV model be given by $$\frac{d S_t }{S_t } = \mu_t dt + \sigma(t,S_t) dW_t$$ Let's drop the drift contribution (not relevant here) and apply ItÃ´'s lemma to obtain: $$d \ln(S_t) = -\frac{1}{2}\sigma^2(t,S_t) dt + \sigma(t,S_t) dW_t$$ In order to simulate from this SDE, you need to choose a particular discretisation ...

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In the end I found that fitting a SABR smile to each tenor (borrowing a result from this answer) was sufficient to build a local vol surface that was smooth and well-behaved enough to build a variance surface worked nicely. I also fitted a Heston model to it, and the two surfaces do look fairly similar. Here is the final code and the fits generated (the long ...

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