22

The reason for put and call volatilities to appear different is that the implied vol has been calculated using different drift parameters than those implied by the market. Let's take everything in the model as given except the interest rate $r$ and the volatility $\sigma$. For European options we have the Black-Scholes formula for put and call values $V_{P,...


18

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 ...


11

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 ...


9

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) ...


8

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

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 ...


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

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 ...


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

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

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) = \...


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

For pricing, there are a few products whose prices are sensitive to the forward smile and when you compute that with just local vol, it is not realistic. So if you are a seller, you go to the next church and find something that looks kindof reasonable, and that kind of can reconstruct a reasonnable forward smile structure. The game in pricing is to not ...


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 ...


4

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 ...


4

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 ...


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

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

The following source contains detailed answers to your questions in a research paper from ETH Zürich. van der Weijst, Roel (2017). "Numerical Solutions for the Stochastic Local Volatility Model" http://resolver.tudelft.nl/uuid:029cbbc3-d4d4-4582-8be2-e0979e9f6bc3


4

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 Itô one could write \begin{align} \frac{ dS_t }{S_t } &= \...


3

I'll address your questions in order: 1a) For TSRV constructed using high frequency returns from NYSE market open to market close on a single day, the output should be numbers on the order of magnitude of 1e-4 to 1e-5. In other words, your numbers look about right. I got these number from calculating TSRV for IBM data myself using Kevin Sheppard's MatLab ...


3

For short maturity SPX option chain, the analytic form of the V-shape volatility smile has been fully worked out in my latest paper on SSRN. You can take a look.


3

The local volatility is just a $\mathbb{R}_+\times[0,T]\mapsto \mathbb{R}_+$ function where $T$ is some time horizon. It is the solution of a simple equation so it expression is written as $\sigma(K,t)$ but here $K$ is essentially a notation to denote a strike value as the Dupire equation relates the function $\sigma$ to vanilla market prices at a given ...


3

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 ...


3

Loosely speaking: Local volatility is the instantaneous volatility after time T if the spot is S at that time. Implied volatility is the expected integrated volatility from today up to time T if the spot ends up at S at that time.


3

First, please make sure that when you resimulate sample paths, you are keeping your underlying random samples constant, as in this answer. For your delta, vega and rho there is some ambiguity in the definition of the greeks. Consider the simple case of delta in the presence of a skew $\sigma(K/S)$, and say that the underlying price right now is $S_0$. We ...


3

The OpenGamma Analytics Library definitely does have a Local Volatility model available. In addition, in our Quantitative Papers page there's a link to the full mathematics and basis for our Local Volatility implementation. I'd be interested to know why you decided to write your own rather than using one of the above.


3

Since $S_T = S_0 + \sigma W_T$, \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)...


3

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 ...


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