LIBOR forward rate, a forward swap rate, or a martingale representation theorem hedging forex stock price. 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. Alternatively, one can express the SABR price in terms of the normal Black’s model. It is worth noting that the normal SABR implied volatility is generally somewhat more accurate than the lognormal implied volatility. 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. 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.
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. 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 motion, its exact simulation is straightforward. However, the simulation of the forward asset process is not a trivial task. The Free Boundary SABR: Natural Extension to Negative Rates”. From arbitrage to arbitrage-free implied volatilities”. Finite difference techniques for arbitrage-free SABR”. The Time-Dependent FX-SABR Model: Efficient Calibration based on Effective Parameters”. International Journal of Theoretical and Applied Finance.