By Jim Gatheral

Compliment for The Volatility floor "I'm delighted by way of the looks of Jim Gatheral's new e-book The Volatility floor. The literature on stochastic volatility is titanic, yet tricky to penetrate and use. Gatheral's booklet, in contrast, is on the market and sensible. It effectively charts a center floor among particular examples and normal models--achieving extraordinary 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, ny college "Concise but complete, both responsive to either concept and phenomena, this booklet offers an unsurpassed account of the peculiarities of the implied volatility floor, its results for pricing and hedging, and the theories that fight to give an explanation for it." --Emanuel Derman, writer of My lifestyles as a Quant "Jim Gatheral is the wiliest practitioner within the enterprise. This very superb ebook is an outgrowth of the lecture notes ready for the most well known periods at NYU's esteemed Courant Institute. the subjects lined are on the vanguard of study in mathematical finance and the author's therapy of them is just the easiest on hand during this form." --Peter Carr, PhD, head of Quantitative monetary learn, Bloomberg LP Director of the Masters application in Mathematical Finance, manhattan collage "Jim Gatheral is an said grasp of complicated modeling for derivatives. within the Volatility floor he finds the secrets and techniques of facing an important yet such a lot elusive of monetary amounts, volatility." --Paul Wilmott, writer and mathematician "As a instructor within the box of mathematical finance, I welcome Jim Gatheral's booklet as an important improvement. Written by way of a Wall road practitioner with wide industry and instructing adventure, The Volatility floor offers scholars entry to a degree of information on derivatives which used to be now not formerly on hand. I strongly suggest it." --Marco Avellaneda, Director, department of Mathematical Finance Courant Institute, manhattan college "Jim Gatheral couldn't have written a greater book." --Bruno Dupire, winner of the 2006 Wilmott innovative examine Award Quantitative examine, Bloomberg LP

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Example text

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.

5) is that to compute the BlackScholes implied volatility of an option, we need to average the possible realized volatilities over all possible scenarios, in particular over all possible paths of the underlying stock. Each such scenario is weighted by the gamma of the option; the profitability of the delta hedger in any time interval is directly proportional to the gamma and the difference between ‘‘expected instantaneous variance’’ (or local variance) and realized instantaneous variance. In particular, at inception of the delta hedge, there is only one possible stock price (the then stock price) and only paths that end at the strike price need be included in the average because gamma elsewhere is precisely zero.

We see that q (St , t; S0 , K, T) looks like a Brownian Bridge density for the stock price: p (St , t; S0 ) has a delta function peak at S0 at time 0 and BS (St ) has a delta function peak at K at expiration T. 3 with a flat 20% volatility. We see that q (xt , t; xT , T) peaks on a line, which we will denote by x˜ t , joining the stock price today with the strike price at expiration. Moreover, the density looks roughly symmetric around the peak. This suggests an expansion around the peak x˜ t , at which the derivative of q (xt , t; xt , T) with respect to xt is zero.

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