The discussion reveals a man who wears his knowledge and influence lightly, quick to assign credit to his collaborators and, tellingly, one who chose the name for his private vehicle, “Gorilla Science LLP,” because “it was about brute intellect.”

]]>A central concept in risk management, applying the Kelly criterion is in fact more of an art than a science.

]]>All change, or no change at all?

]]>What is the smile problem? How can it be interpreted? Do people really understand it?

]]>Bill's annual look at the Triple Crown races.

]]>Patrick S. Hagan: SABR model, the LGM model, auto-calibration, internal and external adjustors, and MC spline-based stripping methods. Brute intellect in the Western lowlands of quant finance.

]]>Option markets, empirical price data, and theoretical arguments all indicate that asset prices in actively traded markets are driven by Lévy flights and/or tempered Lévy flights, not by Brownian motion. So here we model asset prices in the real world by

- (1)

where *dZ*_{L} is a Lévy or tempered Lévy flight. Derivatives based on such assets cannot be made risk free by dynamic hedging, so these derivatives cannot be priced using the standard Black–Scholes–Merton (BSM) arbitrage-free pricing criterion. Therefore, we develop a pricing theory based on a more general pricing criterion: that the *expected* return of all *diversifiable* portfolios is at the risk-free rate. We show that even though derivatives based on Lévy or tempered Lévy-driven assets cannot be hedged to make risk-free portfolios, they can be hedged to make diversifiable portfolios. This allows us to conclude that these option prices are given by

- (2)

where the expected value uses the “real-world” probability measure, but with the asset price *X(t)* replaced by the alternative “pricing process”

- (3)

which grows at the risk-free rate.

We analyze these models to obtain explicit asymptotic formulas for European option prices. This analysis shows that as the time to expiry decreases, eventually all these Lévy and tempered Lévy-based models reduce to the *same* canonical model, and that European option prices and implied volatilities are given by similarity solutions under this canonical model. These similarity solutions are then examined to assess the mishedging that arises from Brownian-based models in a Lévy world.

Many independent studies on stocks and futures contracts have established that market impact is proportional to the square root of the executed volume. Is market impact quantitatively similar for option markets as well? In order to answer this question, we have analyzed the impact of a large proprietary data set of option trades. We find that the square-root law indeed holds in that case. This finding supports the argument for a universal underlying mechanism.

]]>We present a model for dealing with maturity mismatch in the DRC model. The model is a modified Gaussian copula model which – unlike the Gaussian copula – generates the same default time for assets with correlation 1 that default within the 1-year time horizon.

]]>The hypercar builder from Valencia adds a couple of turbos to its hand-built exotic that's all set to take a fight to Ferrari, McLaren, and Koenigsegg.

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