Stochastic Implied Volatility: A Factor-Based Model by Reinhold Hafner PDF

By Reinhold Hafner

ISBN-10: 3540221832

ISBN-13: 9783540221838

ISBN-10: 3642171176

ISBN-13: 9783642171178

This publication offers a factor-based version of the stochastic evolution of the implied volatility floor. The version enables the built-in and constant pricing and hedging, possibility administration, and buying and selling of fairness index derivatives in addition to volatility derivatives. within the first half, the e-book develops a unifying idea for the research of contingent claims below either the real-world degree and the risk-neutral degree in an atmosphere of stochastic implied volatility. at the foundation of transaction info, the second one a part of the publication presents broad statistical analyses at the dynamics of the implied volatility floor of German DAX recommendations and proposes a four-factor version to explain its evolution. The version is demonstrated and verified on marketplace info. the ultimate half offers with power functions of the version within the fields of unique choice pricing, worth in danger, and volatility buying and selling.

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Additional resources for Stochastic Implied Volatility: A Factor-Based Model

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3i Possible explanations for the existence of these patterns are given in the next section. 1 Volatility Smiles The most often quoted phenomenon testifying to the limitations of the BlackScholes model is the smile effect: that implied volatilities vary with the strike price of the option contract. 8 (Volatility Smile). For any fixed maturity date T, T ~ T*, the junction O"t(K,ยท) oj implied volatility against strike price K, K > 0, is called the (implied) volatility smile or just smile (for maturity T) at date tE [0, T).

It provides the implied volatility of an option in closed-form across a wide range of underlying prices. In the absence of arbitrage and market frictions, the implied volatility of a European put option and a European call option with the same strike price and the same maturity date must coincide if the underlying stock pays no dividends. 16 To see this, suppose that, for a particular value of v, CBS(t, St, K, T, r, v) and PBS(t, St, K, T, r, v) are the time t Black-Scholes values of European call and put options with strike price K and maturity date T.

More specifically, if H = (8T ) is a path-independent contingent claim, then the arbitrage price process is also given by IIt(H) = f(t, 8 t ), where f solves the (parabolic) Black-Scholes partial differential equation (PDE):6 of(t,s) +rsof(t,s) ot os +~v2s202f(t,s) -rf(t s) 2 with terminal condition os2 ' f (T, s) = (s). 25) for (s) = max {s - K; O} yields the BlackScholes call option pricing formula CBS(t, s). e. for path-independent claims, is established by the Feynman-Kac stochastic representation formula, which basically expresses the solution of a parabolic PDE as the expected value of a certain functional of a Brownian motion.

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Stochastic Implied Volatility: A Factor-Based Model by Reinhold Hafner

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