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## Hot answers tagged local-volatility

17

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,... 11 There is another reason why Stoc Vol Models should be usually preferred to Local Vol Models, this reason is explained in the Hagan et al. paper "Managing Smile Risk" about SABR process and is in simple terms the fact that "smile dynamics" is poorly predicted by local vol models leading to bad Hedging of exotic options. Anyway Local Vol models have the good ... 7 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 random processes. In the Black-Scholes model both$r$and$\sigma$are deterministic functions (actually constant). This produces a flat smile for any $$C(t,S;T,K) = BS(S,T-t,K;\Sigma(... 6 For pricing and hedging a portfolio of vanilla options, stochastic volatility is almost always preferable to local volatility since empirically it more accurately captures the evolution of the smile. 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 Jump volatility is a term sometimes used to describe randomly varying jump sizes in a model with asset value jumps. So strictly speaking it is merely a parameter in generic jump diffusion. Both local volatility models and jump diffusions end up resulting in skew and kurtosis (of Black-Scholes volatilities). However, they are complementary in practice, at ... 3 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 ... 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 ... 2 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 ... 2 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. 2 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) = \... 2 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. 2 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 ... 2 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 ... 2 The only problem I see with this approach, which remains completely valid from a theoretical perspective, is the embedded (and probably not accounted for) calibration risk: what if your LV surface does not allow you to correctly reproduce the observed vanilla option prices in the first place? In that case, you'll have lost information in the process and ... 2 It is valid to do that, but if your local volatility surface is calibrated to the same OTM options, then your price will converge to the same answer. A local volatility surface is mainly a way of treating path-dependent options consistently with the option volatility surface. Variance swaps are path dependent on the face of it, but as you note the math ... 2 One does not estimate the local volatility at a given$T$and$K$. Instead, Dupire's formula actually gives$\sigma(T,K)$for all$T$and$K$. $$\sigma^2(t_0,S_0;T,K)= \frac{\frac{\partial C}{\partial T} + (r - q)K \frac{\partial C}{\partial K} + qC}{\frac{1}{2} K^2 \frac{\partial^2C}{\partial K^2}}$$ where$C(t_0,S_0;T,K)$are the call prices for ... 1 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)... 1 This is a result of Ledoit et al Lemma proof in Appendix of http://www.ledoit.net/9-98.pdf 1 Local volatility trats volatility as a function of price (St) and time (t), considering volatility constant. It means that you are building a volatility surface depending on price and time. What you are doing is given a certain value of volatility, what will be the price in Xdays long, or viceversa. If you are dong market making with plain vanilla ... 1 Instead of just considering a parallel shift of the whole volatility surface, you can decompose the surface into maturities/strikes domains, so called buckets and consider Vega buckets which are sensitivities wrt to bumps of each of these domains. The vol smile is often inter/extra-polated using a model calibrated to market prices, e.g. the SABR model or ... 1 I know one article (download) that explaining how to calculate local vol surface from IV surface and also chapter 18 of this book is very good In this context. However you know that Dupire’s (1994) formula for local volatility is \begin{align} \sigma_L(k,T)=\sqrt\frac{\frac{\partial C}{\partial T}}{\frac{1}{2}K^2\frac{\partial^2 C}{\partial K^2}} \end{align} ... 1 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. ... 1 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 ... 1 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. 1 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$to vanilla market prices at a given ... 1 Dupire model is just one way of generating a local volatility surface from an implied volatility surface. There are many other ways to generate a local volatility surface. One critical aspect of Dupire model is that the input implied volatility (IV) surface should be arbitrage free. If not, you will negative instantaneous variance when generating the local ... 1 No, if you are referring to the famous Dupire Model (there are others), then they are the same. It is usually referred to as the Local Volatility Model and the Dupire Equation. I would disentagle those with the concept of Local Volatility, which is model independent and a fairly deep result. 1 There is no difference in information, though the fitting algorithm may increase in complexity. First note that in practice you never have an entire curve or surface of prices$C(K,T)\$ of any kind of option. You only have a finite number of observations and even those typically have a bid and an offer. I would therefore argue that the correct picture of ...

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I ended up writing my own library - not easy, but doable.

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