Black-Scholes in a CEV Random Environment: A New Approach to Smile Modelling

Abstract

Abstract. Classical (Itô diffusions) stochastic volatility models are not able to capture the steepness of smallmaturity implied volatility smiles. Jumps, in particular exponential Lévy and affine models, which exhibit small-maturity exploding smiles, have historically been proposed to remedy this (see [60] for an overview). A recent breakthrough was made by Gatheral, Jaisson and Rosenbaum [30], who proposed to replace the Brownian driver of the instantaneous volatility by a short-memory fractional Brownian motion, which is able to capture the short-maturity steepness while preserving path continuity. We suggest here a different route, randomising the Black-Scholes variance by a CEV-generated distribution, which allows us to modulate the rate of explosion (through the CEV exponent) of the implied volatility for small maturities. The range of rates includes behaviours similar to exponential Lévy models and fractional stochastic volatility models.

Asymptotic behaviour of randomised fractional volatility models

“…We quantify how the tail behaviour (parametrised by a scaling coefficient b > 0) of the random starting point modulates the rate of explosion in the implied volatility in the presence of rough fractional volatility. Finally, in a specific simplified setting we highlight how our model blends naturally into the setting of forward-start options in stochastic volatility models, whose asymptotic properties have been studied in [38]. In proving our results, we improve the large deviations literature on both SDEs with random starting points and fractional SDEs.…”

“…Motivated by financial applications, we derive here asymptotic small-time and tail behaviours of the solution to a generalised version of the Stein-Stein stochastic volatility model, originally proposed in [50,51]. We in particular consider two important (in light of the recent trends in the literature-proposed models) extensions: (i) the stochastic differential equation (SDE) driving the instantaneous volatility process is started from a random distribution; this so-called 'randomised' type of model was recently proposed in [38,39,41], in particular to understand the behaviour of the so-called 'forward volatility'; (ii) the volatility process is driven by a fractional Brownian motion. Fractional stochastic volatility models, originally proposed by Comte and Renault [14,15] with H ∈ (1/2, 1), have recently been extended to the case H ∈ (0, 1/2), and a recent flourishing activity in this area [3,7,26,30,35] has established these models as the go-to standards for estimation and calibration.…”

Asymptotic Behaviour of Randomised Fractional Volatility Models

“…We quantify how the tail behaviour (parametrised by a scaling coefficient b > 0) of the random starting point modulates the rate of explosion in the implied volatility in the presence of rough fractional volatility. Finally, in a specific simplified setting we highlight how our model blends naturally into the setting of forward-start options in stochastic volatility models, whose asymptotic properties have been studied in [46]. In proving our results, we improve the large deviations literature on both SDEs with random starting points and fractional SDEs.…”

“…These asymptotics have overall provided a deeper understanding of the behaviour of models, and, ultimately, allow for better calibration of real data; a general overview can be found in [55].Motivated by financial applications, we derive here asymptotic small-time and tail behaviours of the solution to a generalised version of the Stein-Stein stochastic volatility model, originally proposed in [59,60]. We in particular consider two important (in light of the recent trends in the literature proposed models) extensions:(i) the SDE driving the instantaneous volatility process is started from a random distribution; this so-called 'randomised' type of models was recently proposed in [46,47,51], in particular to understand the behaviour of the so-called 'forward volatility'; (ii) the volatility process is driven by a fractional Brownian motion. Fractional stochastic volatility models, originally proposed by Comte and Renault [18,19] with H ∈ (1/2, 1), have recently been extended to the case H ∈ (0, 1/2), and a recent flourishing activity in this area [2,8,33,37,42] has established these models as the go-to standards for estimation and calibration.…”

The asymptotic smile of a multiscaling stochastic volatility model

“…In [JR15], Jacquier and Roome suggest a simple generalization of the Black&Scholes model, obtained by plugging in a random initial volatility instead of a deterministic one. In the special case when the initial volatility is distributed as the solution, at some time τ > 0, of the CEV stochastic differential equqtion dY u = ξ Y p u dB u , they obtain the explicit asymptotic behavior of the implied volatility close to maturity, which displays steepness of the smile.…”