Forums > Pricing & Modelling > Hedging under Stochastic Volatility

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 lz Banned Total Posts: 13 Joined: Jun 2014
 Posted: 2014-06-14 16:33 "{C(T,K;S0,t0)} is surface of calls in dimensions of strike K and maturity T for FIXED values of current spot price S0 and current time t0"There are two different slangs: in calculus we talk about function C(T,K ) in analytic geometry we interpret function C = C(T,K ) as a surface in 3-dimensional space (T, K, C). Our settings are the same.If p ( t , x ; T , y ) is probabilistic density of the stock S ( T ; t , x ) then p ( t , x ; T , K ) = c_K_K ( t , x ; T , K ) { = C_K_K ( T , K ; t, x ) in your notations}Because function c is known as far as it is BSE solution we know also C. One can calculate by using numerical methods C for different T and K. Using graphic computer applications we can depict correspondent surface C = C ( T , K ). It's o'k. Then what does it follow ?
 lz Banned Total Posts: 13 Joined: Jun 2014
 Posted: 2014-06-14 19:14 Is it clear how does LV surface justifies paradoxes (Derman, Kani, Zou 95)" 1. At any fixed expiration, implied volatilities vary with strike level. Almost always, mplied volatilities increase with decreasing strike – that is, out-of-the-money puts trade at higher implied volatilities than out-of-the-money calls. This feature is often referred to as a “negative” skew.2. For any fixed strike level, implied volatilities vary with time to expiration. Often, long-term implied volatilities exceed short-term implied volatilities."
 sigma Total Posts: 108 Joined: Mar 2009
 Posted: 2014-06-14 20:39 Following some feedback and discussions, I added clarifications for Sections 3 and 4 with new illustrations about interpretation of model parameters and econometric estimation of volatility beta and skew-beta. Please check the updated version of the paper http://ssrn.com/abstract=2387845 To summarize, for analysis of risks and computation of hedges, we need to model the sensitivity of the term structure of ATM volatility, as a function of maturity time, to changes in the spot price. The two key concepts for understanding of the empirical dynamics of the ATM volatility and the underlying spot price are:1) the regression volatility beta, which can be interpreted as sensitivity of ATM volatility to price changes2) the regression volatility skew-beta, which can be interpreted as sensitivity of ATM volatility to price changes normalized by the implied skew.The volatility beta for the continuous-time beta stochastic volatility dynamics measures the sensitivity of the instantaneous driver for the implied volatility V(t) to changes in the spot price.With appropriate specification of the vol beta and the skew premium, the beta SV model can reproduce empirical behavior of implied ATM vols along with empirical skew-beta and skew delta so the model can be directly applied for model based risk hedging (it allows to reduce P&L volatility by about 30% and also reduce transaction costs).If interested, you can start with simple time- and space-homogenous beta SV model. I developed the closed-form analytic solution for valuation of vanillas and calibration of this model with numerical implementation similar to that of affine SV models (see section 6 in my paper). You can start with it and play around to see how you can control the vol skew-beta and the skew delta in this model. You need to adjust the level of ATM vols for this model, as the market moves, and will have some mis-pricing of wings (but very minimal - see illustrations on page 53). If you are happy with it, you can then implement the PDE-based beta SV model to make the calibration automatic.
 sigma Total Posts: 108 Joined: Mar 2009
 Posted: 2014-11-07 09:47 I have updated the paper here http://ssrn.com/abstract=2387845In sections 2.3 and 2.4, I present a quantitative analysis of the model delta as function of local or stochastic volatility components, the question which frequently occurs here and there - see for an example in SABR DeltaIn essence, when we compute the minimum-variance delta with vega adjustment for price-induced change in volatility, all stochastic or local volatility models will produce the same delta if they are all calibrated to the same implied vol surface. So if you compute model hedges using model calibrated to implied vol surface only, it doesn't really matter whether you use local or stoch vol or any combination of them. What matters is how to fit the model to empirical dynamics of underlying and its implied vol surface and re-produce the empirical delta - see section 4 in my paper.
 radikal Total Posts: 263 Joined: Dec 2012
 Posted: 2015-04-18 02:58 Did you do a presentation this year at Global Derivatives? There are no surprising facts, only models that are surprised by facts
 unsmt Total Posts: 196 Joined: Jul 2014
 Posted: 2015-04-19 00:18 sigma, i have some theoretical experience and do not have a sufficient practical one. Stochastic volatility model presents a system of two sdes one for price and second for volatility dynamics. From math point it is o'k. On the other hand when one uses the system for calculations whether we should think that price and volatility should be observed simultaneously?
 sigma Total Posts: 108 Joined: Mar 2009
 Posted: 2015-05-01 08:53 radikal, thank you for your interest.This one is is about the risk-premiums in implied vs realized vols and generating signals for volatility trading strategies with some empirical back-test
 radikal Total Posts: 263 Joined: Dec 2012
 Posted: 2015-05-06 11:54 Haha -- I already stalked your site and read both of those! Huge fan of the work btw. Sadly I'm stuck building out some delta one stuff for next few months, so my interest is currently merely academic. There are no surprising facts, only models that are surprised by facts
 sigma Total Posts: 108 Joined: Mar 2009
 unsmt Total Posts: 196 Joined: Jul 2014
 Posted: 2015-05-26 18:04 In my last message"Posted: 2015-04-18 23:18... Stochastic volatility model presents a system of two sdes one for price and second for volatility dynamics. From math point it is o'k. On the other hand when one uses the system for calculations whether we should think that price and volatility should be observed simultaneously?" I would like to highlight the fact that price is only observable variable and therefore models should bear this fact in mind. It means filtration of the sigma algebras should be generated by asset prices. Correlation between prices and its volatility is an important issue which could not be ignored. In theory we can assume everything we need to get a final result but in the model all details related to implementation should be discussed, ie one need to produce method which calculate correlation between price and volatility.
 amin Total Posts: 279 Joined: Aug 2005
 Posted: 2015-07-02 19:22 In my paper http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2149231I have presented a way of calculating the density of various stochastic volatility integrals from their moments. We can use these synthesized densities to integrate over Black Scholes formulas of volatility greeks to get stochastic volatility greeks in terms of black formula units. You can compare this reasoning to the fact that, without correlation between asset and stoch vol, we can calculate the option value under stochastic volatility by integrating Black Scholes formula over the density of integral of stochastic variance or vol appropriately. Stochastic volatility greeks have similar analogy to option pricing and can be calculated by integrating over density of related integrals in Black Scholes greek equations. And I am sure, we can also deal with the case of correlated volatility greeks with a bit of hack.If someone is actually interested, I would love to write the relevant equations in latex for you.
 tartak Total Posts: 3 Joined: Jan 2014
 Posted: 2015-07-11 05:47 It is often said that the theoretical stickiness ratio SR (sigma calls it skew beta) for stochastic vol models is 2 (same as for local vol). This is true only for very short maturities. In sv theory, SR depends on the roll (rate of decay) of the skew term structure: SR = 2 + dln(Skew)/dlnT. Empirically, the skew decays as about 1/sqrt(T) (and saturates below two weeks), so we get SR=1.5 (and approaching 2 at maturities below two weeks). This SR is commonly observed in indexes, so there is very little skew premium there. To be consistent with this sqrt skew decay, sv models need to be multi-factor though. In practice, the correlation between 2-SR and the skew TS roll seems to be pretty weak though.
 amin Total Posts: 279 Joined: Aug 2005
 Posted: 2015-07-24 13:02 [Quote]In my paper http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2149231I have presented a way of calculating the density of various stochastic volatility integrals from their moments. We can use these synthesized densities to integrate over Black Scholes formulas of volatility greeks to get stochastic volatility greeks in terms of black formula units. You can compare this reasoning to the fact that, without correlation between asset and stoch vol, we can calculate the option value under stochastic volatility by integrating Black Scholes formula over the density of integral of stochastic variance or vol appropriately. Stochastic volatility greeks have similar analogy to option pricing and can be calculated by integrating over density of related integrals in Black Scholes greek equations. And I am sure, we can also deal with the case of correlated volatility greeks with a bit of hack.If someone is actually interested, I would love to write the relevant equations in latex for you. [/Quote]If somebody is interested in calculating stochastic volatility vega as a single statistic by integrating Black-Scholes vega formula over integral of stochastic variance, you do not necessarily have to follow my paper, you can get this statistic through monte carlo as well by calculating the appropriate integrated variance with changes for correlated noise for each monte carlo path and by integrating BS vega formula at the end as average over all monte carlo paths. I think it will be interesting for a lot of people who want robust hedges in SV environment.
 Jurassic Total Posts: 199 Joined: Mar 2018
 Posted: 2018-04-30 23:58 @sigma the link is broken
 sigma Total Posts: 108 Joined: Mar 2009
 Posted: 2018-05-10 17:52 Jurassic,I placed my prior research and new developments to my blog at https://artursepp.com/blog/
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