By Jim Gatheral
Compliment for The Volatility floor "I'm extremely joyful through the looks of Jim Gatheral's new publication The Volatility floor. The literature on stochastic volatility is colossal, yet tricky to penetrate and use. Gatheral's booklet, in contrast, is obtainable and useful. It effectively charts a center flooring among particular examples and normal models--achieving notable readability with out giving up sophistication, intensity, or breadth." --Robert V. Kohn, Professor of arithmetic and Chair, Mathematical Finance Committee, Courant Institute of Mathematical Sciences, long island college "Concise but complete, both aware of either idea and phenomena, this e-book presents an unsurpassed account of the peculiarities of the implied volatility floor, its outcomes for pricing and hedging, and the theories that fight to provide an explanation for it." --Emanuel Derman, writer of My lifestyles as a Quant "Jim Gatheral is the wiliest practitioner within the company. This very high quality booklet is an outgrowth of the lecture notes ready for essentially the most renowned periods at NYU's esteemed Courant Institute. the subjects coated are on the leading edge of study in mathematical finance and the author's therapy of them is just the simplest on hand during this form." --Peter Carr, PhD, head of Quantitative monetary examine, Bloomberg LP Director of the Masters software in Mathematical Finance, big apple college "Jim Gatheral is an stated grasp of complex modeling for derivatives. within the Volatility floor he unearths the secrets and techniques of facing an important yet so much elusive of economic amounts, volatility." --Paul Wilmott, writer and mathematician "As a instructor within the box of mathematical finance, I welcome Jim Gatheral's ebook as an important improvement. Written through a Wall highway practitioner with broad marketplace and educating event, The Volatility floor provides scholars entry to a degree of information on derivatives which was once no longer formerly on hand. I strongly suggest it." --Marco Avellaneda, Director, department of Mathematical Finance Courant Institute, big apple college "Jim Gatheral couldn't have written a greater book." --Bruno Dupire, winner of the 2006 Wilmott innovative learn Award Quantitative study, Bloomberg LP
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Extra info for The volatility surface: A practitioner's guide
Example text
14 THE VOLATILITY SURFACE Differentiating once with respect to K gives ∂C = −E [θ (ST − K)] ∂K where θ (·) is the Heaviside function. Differentiating again with respect to K gives ∂ 2C = E [δ (ST − K)] ∂K2 where δ(·) is the Dirac δ function. 12) That is, local variance is the risk-neutral expectation of the instantaneous variance conditional on the final stock price ST being equal to the strike price K. CHAPTER 2 The Heston Model n this chapter, we present the most well-known and popular of all stochastic volatility models, the Heston model, and provide a detailed derivation of the Heston European option valuation formula, implementation of which follows straightforwardly from the derivation.
18) We see that in the Heston model, the at-the-money Black-Scholes implied variance σBS (K, T)2 K=FT →v (the instantaneous variance) as the time to expiration T → 0 and as T → ∞, the at-the-money Black-Scholes implied variance reverts to v .
First, we derive an expression for implied volatility in terms of local volatilities. 12) how to express local variance as an expectation of instantaneous variance in a stochastic volatility model. 10) for local volatility in terms of implied volatility. An obvious direct approach might be to invert that expression and express implied volatility in terms of local volatility. 10) in the limit of zero time to expiration. Instead, by exploiting the work of Dupire (1998), we derive a general path-integral representation of Black-Scholes implied variance.
