### Displaced Lognormal Volatility Skews: Analysis and Applications to Stochastic Volatility Simulations

"... Abstract We analyze the implied volatility skews generated by displaced lognormal diffusions. In particular, we prove the global monotonicity of implied volatility, and an at-the-money bound on the steepness of downward volatility skews, under displaced lognormal dynamics, which therefore cannot re ..."

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Abstract We analyze the implied volatility skews generated by displaced lognormal diffusions. In particular, we prove the global monotonicity of implied volatility, and an at-the-money bound on the steepness of downward volatility skews, under displaced lognormal dynamics, which therefore cannot reproduce some features observed in equity markets. A variant, the displaced anti-lognormal, overcomes the steepness constraint, but its state space is bounded above and unbounded below. In light of these limitations on what features the displaced (anti-)lognormal (DL) can model, we exploit the DL, not as a model, but as a control variate, to reduce variance in Monte Carlo simulations of the CEV and SABR local/stochastic volatility models. For either use -as model, or as control variate -the DL's parameters require estimation. We find an explicit formula for the DL's short-expiry limiting volatility skew, which allows direct calibration of its parameters to volatility skews implied by market data or by other models.

### THE REGULARIZED IMPLIED LOCAL VOLATILITY EQUATIONS-A NEW MODEL TO RECOVER THE VOLATILITY OF UNDERLYING ASSET FROM OBSERVED MARKET OPTION PRICE

"... Dedicated to Professor Avner Friedman on the occasion of his 80th birthday Abstract. In this paper, we propose a new continuous time model to recover the volatility of underlying asset from observed market European option price. The model is a couple of fully nonlinear parabolic partial differential ..."

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Dedicated to Professor Avner Friedman on the occasion of his 80th birthday Abstract. In this paper, we propose a new continuous time model to recover the volatility of underlying asset from observed market European option price. The model is a couple of fully nonlinear parabolic partial differential equations(see (34), (36)). As an inverse problem, the model is deduced from a Tikhonov regularization framework. Based on our method, the recovering procedure is stable and accurate. It is justified not only in theoretical proofs, but also in the numerical experiments. 1. Introduction. In the Black-Scholes framework one of basic assumptions is that the volatility of the underlying asset is known, and never changes. However in real markets it is well known that the constant-volatility assumption is not valid when Black-Scholes model is applied to option pricing. For example, S & P 500 index options are such

### We model the stock as the exponential of a Lévy process, viz.

, 2008

"... We examine the small expiry behaviour of the price of call options in models of exponential Lévy type. In most cases of interest, it turns out that E ( (Sτ − K) +) − (S0 − K) + ∼ τ ∫ ..."

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We examine the small expiry behaviour of the price of call options in models of exponential Lévy type. In most cases of interest, it turns out that E ( (Sτ − K) +) − (S0 − K) + ∼ τ ∫

### Finance Stoch DOI 10.1007/s00780-008-0078-4 Local volatility dynamic models

, 2007

"... Abstract This paper is concerned with the characterization of arbitrage-free dynamic stochastic models for the equity markets when Itô stochastic differential equations are used to model the dynamics of a set of basic instruments including, but not limited to, the underliers. We study these market m ..."

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Abstract This paper is concerned with the characterization of arbitrage-free dynamic stochastic models for the equity markets when Itô stochastic differential equations are used to model the dynamics of a set of basic instruments including, but not limited to, the underliers. We study these market models in the framework of the HJM philosophy originally articulated for Treasury bond markets. The main thrust of the paper is to characterize absence of arbitrage by a drift condition and a spot consistency condition for the coefficients of the local volatility dynamics.

### Option pricing for stochastic volatility models: Vol-of-Vol expansion

, 2013

"... In this article, we propose an analytical approximation for the pricing of European options for some lognormal stochastic volatility models. This approximation is a second-order Taylor series expansion of the Fourier transform with respect to the "volatility of volatility". We give, using ..."

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In this article, we propose an analytical approximation for the pricing of European options for some lognormal stochastic volatility models. This approximation is a second-order Taylor series expansion of the Fourier transform with respect to the "volatility of volatility". We give, using these formulas, a new method of variance reduction for the Monte-Carlo simulation of the trajectories of the underlying. 1

### MPRA Munich Personal RePEc Archive

, 2007

"... Noname manuscript No. (will be inserted by the editor) An Hilbert space approach for a class of arbitrage free implied volatilities models ..."

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Noname manuscript No. (will be inserted by the editor) An Hilbert space approach for a class of arbitrage free implied volatilities models