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| A '''local volatility''' model, in [[mathematical finance]] and [[financial engineering]], is one which treats [[Volatility (finance)|volatility]] as a function of the current asset level <math> S_t </math> and of time <math> t </math>.
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| ==Formulation==
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| In [[mathematical finance]], the asset ''S''<sub>''t''</sub> which [[Underlying|underlie]] [[financial derivatives]], are typically assumed to follow [[stochastic differential equation]]s of the type
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| :<math> dS_t = (r_t-d_t) S_t\,dt + \sigma_t S_t\,dW_t </math>
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| where <math> W_t </math> is a Brownian motion.
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| ''r'' is the instantaneous [[risk-free interest rate|risk free rate]], giving an average local direction to the dynamics, and ''W'' is a [[Wiener process]], representing the inflow of randomness into the dynamics. The amplitude of this randomness is measured by the instant volatility <math>\sigma_t</math>. In the simplest (naive) model, this instant volatility is assumed to be constant, but in reality realized volatility of an underlier actually rises and falls over time. | |
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| When such volatility has a randomness of its own—often described by a different equation driven by a different ''W''—the model above is called a [[stochastic volatility]] model. And when such volatility is merely a function of the current asset level ''S''<sub>''t''</sub> and of time ''t'', we have a local volatility model. The local volatility model is a useful simplification of the [[stochastic volatility]] model.
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| "Local volatility" is thus a term used in [[quantitative finance]] to denote the set of diffusion coefficients, <math>\sigma_t = \sigma(S_t,t)</math>, that are consistent with market prices for all options on a given underlying. This model is used to calculate [[exotic option]] valuations which are consistent with observed prices of [[vanilla option]]s.
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| ==Development==
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| The concept of a local volatility was developed when [[Bruno Dupire]] <ref name=dupire>{{cite journal | author=Bruno Dupire | title=Pricing with a Smile | publisher=Risk | year= 1994 }}http://www.risk.net/data/risk/pdf/technical/2007/risk20_0707_technical_volatility.pdf</ref> and [[Emanuel Derman]] and [[Iraj Kani]]<ref name=derman>{{cite journal | author=Derman, E., Iraj Kani | title="Riding on a Smile." RISK, 7(2) Feb.1994, pp. 139-145, pp. 32-39. | publisher=Risk | format=PDF | year= 1994 | url=http://www.ederman.com/new/docs/gs-volatility_smile.pdf | accessdate=2007-06-01}}</ref>
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| noted that there is a unique diffusion process consistent with the risk neutral densities derived from the market prices of European options.
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| Derman and Kani described and implemented a local volatility function to model instantaneous volatility. They used this function at each node in a [[binomial options pricing model]]. The tree successfully produced option valuations consistent with all market prices across strikes and expirations.<ref name=derman /> The Derman-Kani model was thus formulated with '''discrete''' time and stock-price steps. The key '''continuous'''-time equations used in local volatility models were developed by [[Bruno Dupire]] in 1994. Dupire's equation states
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| :<math>
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| \frac{\partial C}{\partial T} = \frac{1}{2} \sigma^2(K,T; S_0)K^2 \frac{\partial^2C}{\partial K^2}-(r - q)K \frac{\partial C}{\partial K} - qC
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| </math>
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| There exist few known parametrisation of the volatility surface based on the heston model (Schonbusher, SVI and gSVI) as well as their de-arbitraging methodologies.<ref name=damghani>{{cite journal | author=Babak Mahdavi Damghani and Andrew Kos| title=De-arbitraging with a weak smile | publisher=Wilmott | year = 2013}}http://www.readcube.com/articles/10.1002/wilm.10201?locale=en</ref>
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| ==Use==
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| Local volatility models are useful in any options market in which the underlying's volatility is predominantly a function of the level of the underlying, interest-rate derivatives for example. Time-invariant local volatilities are supposedly inconsistent with the dynamics of the equity index implied volatility surface,<ref name=dumas>{{cite journal | author=Dumas, B., J. Fleming, R. E. Whaley | title=Implied volatility functions: Empirical tests | journal=The Journal of Finance | volume=53 | year=1998}}</ref> but see {{cite journal | author=Crepey, S | title=Delta-hedging Vega Risk | journal=Quantitative Finance | volume=4 | year=2004}}, who claims that such models provide the best average hedge for equity index options. Local volatility models are nonetheless useful in the formulation of [[stochastic volatility]] models.<ref name=gatheral>{{cite book | author=Gatheral, J. | title=The Volatility Surface: A Practitioners's Guide | publisher = Wiley Finance | year=2006 | isbn= 978-0-471-79251-2 }}</ref>
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| Local volatility models have a number of attractive features.<ref name="Derman et al">{{cite journal | author=Derman, E. I Kani & J. Z. Zou | title=The Local Volatility Surface: Unlocking the Information in Index Options Prices | journal = Financial Analysts Journal | volume=(July-Aug 1996) | year=1996}}</ref> Because the only source of randomness is the stock price, local volatility models are easy to calibrate. Also, they lead to complete markets where hedging can be based only on the underlying asset. The general non-parametric approach by Dupire is however problematic, as one needs to arbitrarily pre-interpolate the input implied [[volatility surface]] before applying the method. Alternative parametric approaches have been proposed, notably the highly tractable mixture dynamical local volatility models by [[Damiano Brigo]] and [[Fabio Mercurio]].<ref name=brigomercmixbachelier>{{cite conference | author=Damiano Brigo and Fabio Mercurio|title=Displaced and Mixture Diffusions for Analytically-Tractable Smile Models| booktitle= Mathematical Finance - Bachelier Congress 2000. Proceedings | year=2001|publisher=Springer Verlag }}</ref><ref name=brigomercmixijtaf>{{cite journal | author=Damiano Brigo and Fabio Mercurio|title=Lognormal-mixture dynamics and calibration to market volatility smiles| journal= International Journal of Theoretical and Applied Finance|volume = 5|issue = 4| year=2002 | format=PDF | url=http://www.damianobrigo.it/lognsmile.pdf | accessdate=2011-03-07}}</ref>
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| Since in local volatility models the volatility is a deterministic function of the random stock price, local volatility models are not very well used to price [[cliquet option]]s or [[forward start option]]s, whose values depend specifically on the random nature of volatility itself.
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| ==References==
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| {{Reflist}}
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| #{{cite journal | author=Carol Alexander|title=Normal mixture diffusion with uncertain volatility: Modelling short- and long-term smile effects| journal= Journal of Banking & Finance|volume = 28|issue = 12| year=2004}}
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| {{Volatility}}
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| {{derivatives market|state=collapsed}}
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| [[Category:Derivatives (finance)]]
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