The Relation Between Implied and Realised Probability Density Functions

Recovering risk-neutral probability density functions from options prices using cubic splines and ensuring nonnegativity

“…Among the nonparametric approaches, and following Bondarenko [9], we can point out: implied trees methods [23], kernel methods [2], maximum-entropy methods [12,26], methods applied to the volatility smile [3,11,13,25], methods applied to the risk-neutral probability [19,21]. Other methods share both parametric and nonparametric features [1,9].…”

“…Other authors have also used spline fitting techniques in the context of risk-neutral density estimation, see [3,13]. In contrast to existing techniques, we allow the displacement of spline knots in a superset of the set of points corresponding to option strikes.…”

Closed-form Transformations from Risk-neutral to Real-world Distributions

“…As a result many density specifications have been estimated from options prices. Parametric specifications include a mixture of lognormals [Ritchey (1990), Melick and Thomas (1997)], polynomials multiplied by a lognormal [Madan and Milne (1994)], a generalized beta [Anagnou, Bedendo, Hodges and Tompkins (2002)] and the densities of continuous-time price processes when volatility is stochastic [Bates (2000), Jondeau and Rockinger (2000)]. Other approaches include maximum entropy densities [Buchen and Kelly (1996)], non-parametric estimates [Ait-Sahalia and Lo (1998)], multi-parameter discrete distributions [Jackwerth and Rubinstein (1996)] and densities implied by smile functions, defined by either polynomials [Shimko (1993), Malz (1997)] or spline functions [Bliss and Panigirtzoglou (2002a)].…”

“…One strand of literature investigates methods that transform RNDs into real-world densities [Anagnou et al (2002), Bakshi, Kapadia and Madan (2003), Bliss and Panigirtzoglou (2002b)]. These methods are important for central bankers and other decision takers who wish to infer market beliefs about future distributions from market prices.…”

“…Our paper is most closely related to the contemporaneous research of Anagnou et al (2002) and Bliss and Panigirtzoglou (2002b). These papers only consider the utility transformation and they do not use MLE to estimate their risk aversion Risk-neutral and historical densities provide sufficient information to estimate representative risk aversion functions.…”