It has useful information of fundamentals of parameter averaging technique for dynamic SABR model calibration
Abstract: We present the Markovian projection method, a method to obtain closed-form approximations to European option prices on various underlyings that, in principle, is applicable to any (diffusive) model. Successful applications of the method have already appeared in the literature, in particular for interest rate models (short rate and forward Libor models with stochastic volatility), and interest rate/FX hybrid models with FX skew. The purpose of this note is thus not to present other instances where the Markovian projection method is applicable (even though more examples are indeed given) but to distill the essence of the method into a conceptually simple plan of attack, a plan that anyone who wants to obtain European option approximations can follow.
Keywords: Local volatility, stochastic volatility, Markovian projection, parameter averaging, Dupire's local volatility, index options, basket options, spread options
by CAROL ALEXANDER and LEONARDO M. NOGUEIRA ABSTRACT There are two unique volatility surfaces associated with any arbitrage-free set of standard European option prices, the implied volatility surface and the local volatility surface. Several papers have discussed the stochastic differential equations for implied volatilities that are consistent with these option prices but the static and dynamic no-arbitrage conditions are complex, mainly due to the large (or even infinite) dimensions of the state probability space. These no-arbitrage conditions are also instrument-specific and have been specified for some simple classes of options. However, the problem is easier to resolve when we specify stochastic differential equations for local volatilities instead. And the option prices and hedge ratios that are obtained by making local volatility stochastic are identical to those obtained by making instantaneous volatility or implied volatility stochastic. After proving that there is a one-to-one correspondence between the stochastic implied volatility and stochastic local volatility approaches, we derive a simple dynamic no-arbitrage condition for the stochastic local volatility model that is modelspecific. The condition is very easy to check in local volatility models having only a few stochastic parameters.
Abstract: The author considers SABR model which is a two factor stochastic volatility model and gives an asymptotic expansion formula of implied volatilities for this model. His approach is based on infinite dimensional analysis on the Malliavin calculus and large deviation.
Furthermore, he applies the approach to a foreign exchange model where interest rates and the FX volatilities are stochastic and gives an asymptotic expansion formula of implied volatilities of foreign exchange options.
by Zaizhi Wang Abstract: This paper tackles the issue of approximated formula for stochastic model with time dependent model parameters, using an averaging principle. The idea lies in finding a similar model but with constant parameters that is the closest to our initial process, along the same lines as results proven by Gy Discuss this paper
Thesis by Besiana Rexhepi Abstract It is well-known that the fair value of options can be determined by using the Black-Scholes model. However, for liquidly traded options, i.e. instruments for which the market price is known, there is clear evidence that the Black-Scholes model is not correct. This is reflected in the existence of the volatility smile phenomenon which is one the most challenging problems in financial economics. Recently, a rigorous analysis of the time evolution of the empirically observed volatility smile, i.e. smile dynamics, has been reported by (CdF02) and (Fen05). However, the quantification of the volatility smile dynamics as implied by smile-consistent models has not been done rigorously, so far. People have addressed this by looking at the evolution of the smile, based on asymptotic analysis and qualitative investigations. In this work, we use similar statistical techniques as employed in the empirical studies, to quantify the smile dynamics that is implied by the following smile-consistent models: Displaced Diffusion, Constant Elasticity of Variance (CEV) and SABR Stochastic Volatility Models. We find that in markets where options exhibit extreme skew, e.g. equity options markets, the displaced diffusion and CEV models should be used with care, since these models have poor fitting capabilities to market prices and impose inaccurate smile dynamics. The SABR model on the other hand, was shown to be able to capture the smile dynamics very closely to the empirically observed dynamics.
by Patrick Hagan and Andrew Lesniewski We propose and study the SABR/LMM model. This is a term structure model of interest rates with stochastic volatility that is a natural extension of both the LIBOR market model and the SABR model. The key result of the paper is a closed form asymptotic formula for swaption volatility in the SABR/LMM model which allows for rapid and accurate valuation of European swaptions.
Original paper by PATRICK S. HAGAN, DEEP KUMAR, ANDREW S. LESNIEWSKI, AND DIANA E. WOODWARD
Abstract. Market smiles and skews are usually managed by using local volatility models a la Dupire. We discover that the dynamics of the market smile predicted by local vol models is opposite of observed market behavior: when the price of the underlying decreases, local vol models predict that the smile shifts to higher prices; when the price increases, these models predict that the smile shifts to lower prices. Due to this contradiction between model and market, delta and vega hedges derived from the model can be unstable and may perform worse than naive Black-Scholes Discuss this paper
by GRAEME WEST ABSTRACT Recently the SABR model has been developed to manage the option smile which is observed in derivatives markets. Typically, calibration of such models is straightforward as there is adequate data available for robust extraction of the parameters required asinputs to the model. The paper considers calibration of the model in situations where input data is very sparse. Although this will require some creative decision making, the algorithms developed here are remarkably robust and can be used confidently for mark to market and hedging of option portfolios. KEY WORDS: SABR model, equity derivatives, volatility skew calibration, illiquid markets Discuss this paper
We apply SABR model (2002) and, in particular, the asymptotic formula for SABR implied volatility to 1) parametrize and fit market implied volatility surface, and 2) produce a smooth local volatility surface. Discuss this paper
ICRA: EC - Early Childhood
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