Its exact solution for the zero correlation as well as an efficient approximation for a general case are available. Arbitrage problem in the implied volatility formula[ edit ] Although the asymptotic solution is very easy to implement, the density implied by the approximation is not always arbitrage-free, especially not for very low strikes it becomes negative or the density does not integrate to one. One possibility to "fix" the formula is use the stochastic collocation method and to project the corresponding implied, ill-posed, model on a polynomial of an arbitrage-free variables, e. This will guarantee equality in probability at the collocation points while the generated density is arbitrage-free. Another possibility is to rely on a fast and robust PDE solver on an equivalent expansion of the forward PDE, that preserves numerically the zero-th and first moment, thus guaranteeing the absence of arbitrage.

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One can perhaps restrict 1 for indices to highlight the very unlikely probability of a highly liquid index to default. Another rem- edy is to modify the underlying asset process such that default is no longer possible. An example is the CEV by including a minimum asset price level below which the volatility be- comes constant [3, 14]. However, for equities, the CEV becomes an attractive model where bankruptcy is more often than not a recurrent event e.

From a practical point of view we regard the default case as a strength of the models [7, 15, 33]. Also it can play a role in pricing of derivatives. Another possibility is the shifted CEV [1] and SABR models with default occurring when the asset price reaches a small positive value instead of zero.

This has been shown [30] to overcome some of the above difculties and provide a practical alternative to the CEV when default is possible. The popularity of a pricing model is often due to the existence of an exact or approxi- mate pricing formula making it possible to calibrate efciently and rapidly price and hedge 4 R. TIER nancial derivatives in a real time environment, e. Blacks pricing formulas.

The more complex the model the harder it is to derive a practical closed form solution that can be easily implemented for pricing and risk management. To implement such models often requires a numerical solution such as lattice methods, numerical integration routines or numerical methods for partial differential equations, e.

Another robust alternative is a Monte-Carlo simulation for valuation, analysis and risk management. This may be slower to give a price in an intra-day high frequency trading environment for which real time valuation and hedging is required.

Therefore without a useful formula, the practical issues of implementation can become problematic for a trader or portfolio manager. The models lack of practicability hinders its usefulness despite its modeling strengths ver- sus the standard GBM. Analytic formulas would certainly be practical and advisable to look for. Whether they are exact or approximate, they can provide a very fast way to perform real time option hedges obtained by analytically differentiating the pricing formulas.

Moreover it makes it possible to calibrate efciently in a real time environment [44]. The formulas are an improvement over numerical methods such as lattice, numerical integration, PDE or Monte-Carlo methods.

The SABR model [22], despite its lack of an explicit or quasi-closed form solution [23, 25, 32], is popular among practitioners due to the existence of an approximate or asymptotic formula [22, 38] to price European style calls and puts giving good agreement between the theoretical and observed smiles for 1.

The model allows the market price and the market risks to be obtained immediately using the asymptotic formula. It also provides good and sometimes spectacular ts to the implied volatility curves observed in the marketplace. More importantly, the SABR model captures the correct dynamics of the smile, and thus yields stable hedges. In [23] the authors rene the results from [22] by using a more general asymptotic technique [43] as a method to price derivatives using risk-neutral expectation.

The purpose of this paper is to present a general systematic approach to derive approxi- mate analytic formulas for European type of derivatives based on the CEV and SABR models. In particular, we use the ray method that was developed in the theory of wave propagation [31]. It was applied to general diffusion equations in [9, 41]. The method consists of rst con- structing an asymptotic solution or ray solution for 1 valid away from the boundaries.

If necessary a boundary layer solution is constructed in the neighborhood of the boundaries. The ray and boundary layer solutions are then matched to determine any unknown quantities in them.

The present method has a number of virtues. First it is general and can be applied to any diffusion equation in any domain. Thus it does not depend upon separability or any other special property of the equation. Second, it provides very useful and accurate quanti- tative approximations to the probability density function that can be used to price nancial derivatives. The approach taken in this paper differs from that of [4, 22, 23, 24] in several important ways.

First we present the ray method as a systematic approach to derive the asymptotic behavior of the density function for 1 away from and near the boundaries. Next, we present a general asymptotic method [39] to obtain a uniform approximation to the pricing integral for call and put options and its corresponding deltas.

Our approximations provide new analytic formulas for pricing European derivative contracts. The recipe provides a prac- tical set of tools needed for pricing and hedging of European contingent claims undertaken with more sophisticated models in a high frequency trading environment. This paper is organized as follows.

In Section 2 we review the CEV process and derive the ray and boundary layer solutions for the risk-neutral density function.

In Section 3 we consider the SABR model and we derive and the ray solution and boundary layer correction for the density function. In Section 4 we benchmark the asymptotic formulas from Sections 2 and 3 to standard numerical methods as well to special cases for which an exact solution is known.

In addition, a simple calibration is performed using the analytic formulas to illustrate their practicality and speed. Details of the derivations are given in the Appendices and in [29]. The CEV Model.

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## SIAM Journal on Financial Mathematics

The underlying process in the SABR model has the volatility as a stochastic function of the asset price. In such situations, trading desks often resort to numerical methods to solve the pricing and hedging problem. This can be problematic for complex models if real-time valuations, hedging, and calibration are required. A more efficient and practical alternative is to use a formula even if it is only an approximation. A systematic approach is presented, based on the WKB or ray method, to derive asymptotic approximations yielding simple formulas for the pricing problem.

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## Asymptotic Approximations to Cev and Sabr Models

Pages using web citations with no URL. This page was last edited on 3 Novemberat SABR volatility model In mathematical financethe SABR model is a stochastic volatility model, which attempts to capture the volatility smile in derivatives markets. Journal of Computational Finance, Forthcoming. The value of this option is equal to the suitably discounted expected value of the payoff under the probability distribution of the process. In mathematical financethe SABR model is a stochastic volatility model, which attempts to capture the volatility smile in derivatives markets. Another possibility is to rely on a fast and robust PDE solver on an equivalent expansion of the forward PDE, that preserves ceg the zero-th and first moment, thus guaranteeing the absence of arbitrage. Its exact solution for the zero moeels as well as an efficient approximation for a general case are available.

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## SABR volatility model

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## Constant elasticity of variance model

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