Pricing stock options in a jump diffusion model

In mathematical financethe SABR model is a stochastic volatility model, which attempts to capture the volatility smile in derivatives markets. The name stands for " stochastic alphabetarho ", 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.

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 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:.

Pricing Stock Options in a Jump-Diffusion Model with Stochastic Volatility and Interest Rates: Applications of Fourier Inversion Methods - Scott - - Mathematical Finance - Wiley Online Library

We have also set. Alternatively, one can express the SABR price in terms of the normal Black's model. Then the implied normal volatility can be asymptotically computed by means of the following expression:. It is worth noting that the normal SABR implied volatility is generally somewhat more accurate than the lognormal implied volatility. A SABR model extension for Negative interest rates that has gained popularity in recent years is the shifted SABR model, where the shifted forward rate is assumed to follow a SABR process.

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. The SABR model can also be modified to cover Negative interest rates by:.

Its exact solution for the zero correlation as well as an efficient approximation for a general case are available. An obvious drawback of this approach is the a priori assumption of potential highly negative interest rates via the free boundary.

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 pricing stock options in a jump diffusion model 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.

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".

As the stochastic volatility process follows a geometric Brownian motionits exact simulation is straightforward. However, the simulation of the forward asset process is not a trivial task. Taylor-based simulation schemes are typically considered, like Euler-Maruyama or Milstein.

Recently, novel methods have been proposed for the almost exact Monte Carlo simulation of the SABR model. From Wikipedia, the free encyclopedia. Natural Extension to Negative Rates January 28, Journal of Computational Finance, Forthcoming. Journal of Computational Finance, August Efficient Calibration based on Effective Parameters". International Journal of Theoretical and Applied Finance.

Pricing Stock Options in a Jump-diffusion Model with Stochastic Volatility - Louis O. Scott - Google Livres

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Monte Carlo methods for option pricing - Wikipedia

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