Arbitrage-free Sabr

[Niobe Hennigan]

Exactfit of model to market prices. Neither Heston nor SABR can claim to fit exactly market prices. There are two reasons: both have a limited number of parameters (5 for Heston, 4 for SABR), and their corresponding models further restrict the shapes attainable with those parameters. Even for a small number (let's say 3) quotes at a given maturity, they are not guaranteed to match those exactly.

Arbitrages in the parameterization: both stochastic volatility models do not allow arbitrage. But, for SABR, the practice is to use an approximation formula, which is not always arbitrage-free (it is only an approximation of the actual SABR stochastic volatility model).

Dupire local vol: it is true that in presence of arbitrage the local vol is not defined (the local variance is negative). But by definition, the "equivalent local vol model" is the Dupire local vol model. There is no Dupire local vol model without a some continuous representation of the option prices, which may be realized via some specific interpolation, or via a stoch vol model. There is just one local vol model and it is arbitrage-free also by definition. There is also no need to go through Dupire to find out if the fit of a some model is perfect.

For a perfect fit, you typically need at least as many parameters as there are quotes (an approach sometimes called non-parametric). You could use a spline but this will not be arbitrage free in general. In terms of arbitrage-free methods, there is:

In mathematical finance, the SABR model is a stochastic volatility model, which attempts to capture the volatility smile in derivatives markets. The name stands for "stochastic alpha, beta, rho", referring to the parameters of the model. The SABR model is widely used by practitioners in the financial industry, especially in the interest rate derivative markets. It was developed by Patrick S. Hagan, Deep Kumar, Andrew Lesniewski, and Diana Woodward.[1]

The SABR model describes a single forward F \displaystyle F , such as a LIBOR forward rate, a forward swap rate, or a forward stock price. This is one of the standards in market used by market participants to quote volatilities. The volatility of the forward F \displaystyle F is described by a parameter σ \displaystyle \sigma . SABR is a dynamic model in which both F \displaystyle F and σ \displaystyle \sigma are represented by stochastic state variables whose time evolution is given by the following system of stochastic differential equations:

Except for the special cases of β = 0 \displaystyle \beta =0 and β = 1 \displaystyle \beta =1 , no closed form expression for this probability distribution is known. The general case can be solved approximately by means of an asymptotic expansion in the parameter ε = T α 2 \displaystyle \varepsilon =T\alpha ^2 . Under typical market conditions, this parameter is small and the approximate solution is actually quite accurate. Also significantly, this solution has a rather simple functional form, is very easy to implement in computer code, and lends itself well to risk management of large portfolios of options in real time.

It is convenient to express the solution in terms of the implied volatility σ impl \displaystyle \sigma _\textrm impl of the option. Namely, we force the SABR model price of the option into the form of the Black model valuation formula. Then the implied volatility, which is the value of the lognormal volatility parameter in Black's model that forces it to match the SABR price, is approximately given by:

for some positive shift s \displaystyle s .Since shifts are included in a market quotes, and there is an intuitive soft boundary for how negative rates can become, shifted SABR has become market best practice to accommodate negative rates.

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.g. normal. This will guarantee equality in probability at the collocation points while the generated density is arbitrage-free.[4] Using the projection method analytic European option prices are available and the implied volatilities stay very close to those initially obtained by the asymptotic formula.

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.[5]

The SABR model can be extended by assuming its parameters to be time-dependent. This however complicates the calibration procedure. An advanced calibration method of the time-dependent SABR model is based on so-called "effective parameters".[6]

Alternatively, Guerrero and Orlando[7] show that a time-dependent local stochastic volatility (SLV) model can be reduced to a system of autonomous PDEs that can be solved using the heat kernel, by means of the Wei-Norman factorization method and Lie algebraic techniques. Explicit solutions obtained by said techniques are comparable to traditional Monte Carlo simulations allowing for shorter time in numerical computations.

While the model has proven to be a consistent solution for pricing options in an arbitrage-free way for positive rates, it has weakness as a tool for arbitrage-free pricing for very low or negative rates. Market participants would therefore have to resort to other means, such as extrapolation techniques or financial engineering tricks to try and fix this weakness. The authors, as is proven in the book, have discovered a natural way for SABR to accurately handle pricing in a low or negative rates environment.

Another solution presented in the book is an extension to SABR the authors came up with that they call Mixture SABR model, which makes SABR more flexible by making it possible to calibrate to swaption quotes and CMS quotes at the same time. Prior to this extension, SABR could not reproduce CMS quotes. This and other illuminating insights and formulas are included throughout the book, which is based on but not limited to the three papers that appeared in Risk Magazine:

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In the current low rates environment, the classic stochastic alpha beta rho (SABR) formula used to compute option-implied volatilities leads to arbitrages. In "Arbitrage free SABR", Hagan et al proposed a new arbitrage-free SABR solution based on a finite difference discretization of an expansion of the probability density. They rely on a Crank-Nicolson discretization, which can lead to undesirable oscillations in the option price. This paper applies a variety of second-order finite difference schemes to the SABR arbitrage-free density problem and explores alternative formulations. It is found that the trapezoidal rule with the second-order backward difference formula (TR-BDF2) and Lawson-Swayne schemes stand out for this problem in terms of stability and speed. The probability density formulation is the most stable and benefits greatly from a variable transformation. A partial differential equation is also derived for the so-called free-boundary SABR model, which allows for negative interest rates without any additional shift parameter, leading to a new arbitrage-free solution for this model. Finally, the free-boundary model behavior is analyzed.

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This paper explains how to calibrate a stochastic collocation polynomial against market option prices directly. The method is first applied to the interpolation of short-maturity equity option prices in a fully arbitrage-free manner and then to the joint calibration of the constant maturity swap convexity adjustments with the interest rate swaptions smile. To conclude, we explore some limitations of the stochastic collocation technique.

The market provides option prices for a discrete set of strikes and maturities. In order to price over-the-counter vanilla options with different strikes, or to hedge more complex derivatives with vanilla options, it is useful to have a continuous arbitrage-free representation of the option prices, or equivalently of their implied volatilities. For example, the variance swap replication of Carr and Madan consists in integrating a specific function over a continuum of vanilla put and call option prices (Carr et al 1998; Carr and Lee 2008). An arbitrage-free representation is also particularly important for the Dupire local volatility model (Dupire 1994), where arbitrages will translate to a negative implied variance.

More recently, Andreasen and Huge (2011) have proposed to calibrate the discrete piecewise constant local volatility of the single-step finite difference representation for the forward Dupire equation. In their representation, the authors use as many constants as the number of market option strikes for an optimal fit. It works well but often yields a noisy probability density estimate, as the prices are overfitted.

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