Bermudan Swaptions Research Articles

Fast drift-approximated pricing in the BGM model

step together with a separability assumption on the volatility function allows for representation by a low-dimensional Markov process. This in turn leads to efficient pricing by, for example, finite differences. We then develop a discretization based on the Brownian bridge that is especially designed to have high accuracy for single time stepping. The scheme is proven to converge weakly with order one. We compare the single time step method for pricing on a grid with multi-step Monte Carlo simulation for a Bermudan swaption, reporting a computational speed increase by a factor 10, yet maintaining sufficiently accurate pricing.

A PDE based Implementation of the Hull&White Model for Cashflow Derivatives

A new implementation for the one-dimensional Hull&White model is developed. It is motivated by a geometrical approach to construct an invariant manifold for the future dynamics of forward zero coupon bond prices under a forward martingale measure. This reduces the option-pricing problem for cashflow derivatives to the solution of a series of heat equations. The heat equation is solved by a standard Crank-Nicolson scheme. The new method avoids the calibration used in traditional solution approaches. The computation of prices for European and Bermudan swaptions shows the convergence behavior of our new implementation. We also demonstrate the efficiency of our new approach resulting in a speed improvement by one order of magnitude compared to traditional trinomial tree implementations.

Risk Managing Bermudan Swaptions in the Libor BGM Model

This article presents a novel approach for calculating swap vega per bucket in the Libor BGM model. We show that for some forms of the volatility an approach based on re-calibration may lead to a large uncertainty in estimated swap vega, as the instantaneous volatility structure may be distorted by re-calibration. This does not happen in the case of constant swap rate volatility. We then derive an alternative approach, not based on re-calibration, by comparison with the swap market model. The strength of the method is that it accurately estimates vegas for any volatility function and at a low number of simulation paths. The key to the method is that the perturbation in the Libor volatility is distributed in a clear, stable and well understood fashion, whereas in the re-calibration method the change in volatility is hidden and potentially unstable.

Efficient Control Variates and Strategies for Bermudan Swaptions in a Libor Market Model

This paper concerns the problem of valuing Bermudan swaptions in a Libor market model. In particular we consider various efficiency improvement techniques for a Monte Carlo based valuation method. We suggest a simplification of the Andersen (2000) exercise strategy replacing a maximum over several option values by a carefully chosen single option and find it to be much more efficient. Furthermore, we test a range of control variates for Bermudan swaptions sampling the control variate at the exercise time of the Bermudan rather than at a fixed point in time. In addition, we examine the variance-bias trade-off between the numbers of outer and inner paths for the Primal-Dual simulation algorithm of Andersen (2000). We show numerically that application of these efficiency improvements in the Primal-Dual simulation algorithm - in combination with more standard antithetic variate techniques - improves both upper and lower bounds for the price estimates. Finally, we demonstrate that the presence of stochastic volatility increases the expected losses from using the simple strategy in Andersen (2000).

Bounding Bermudan swaptions in a swap-rate market model

We develop a new method for finding upper bounds for Bermudan swaptions in a swap-rate market model. By comparing with lower bounds found by exercise boundary parametrization, we find that the bounds are well within bid-offer spread. As an application, we study the dependence of Bermudan swaption prices on the number of instantaneous factors used in the model. We also establish an equivalence with LIBOR market models and show that virtually identical lower bounds for Bermudan swaptions are obtained.

A PATH INTEGRAL APPROACH TO DERIVATIVE SECURITY PRICING II: NUMERICAL METHODS

We discuss two numerical techniques, based on the path integral approach described in a previous paper, for solving the stochastic equations underlying the financial markets: the path integral Monte Carlo, and the path integral deterministic evaluation. In particular, we apply the latter to some specific financial problems: the pricing of a European option, a zero-coupon bond, a caplet, an American option, and a Bermudan swaption.

Factor dependence of Bermudan swaptions: fact or fiction?

This paper investigates the effect of interest rate correlation in pricing and exercise of Bermudan swaptions. Investigating both Gaussian Markov models and Libor market models, we find that Bermudan swaption prices change only moderately (and in fact typically decrease slightly) when the number of factors in the underlying interest rate model is increased from one to two. We explain the rationale behind these results, and also demonstrate that exercise information generated within a best-fit one-factor model only leads to insignificant losses when properly applied in a two-factor model. Our results provide support for the standard Wall Street practice of using continuously re-calibrated one-factor models to price Bermudan swaptions, as long as calibration procedures are sufficiently comprehensive.

MARKOV MARKET MODEL CONSISTENT WITH CAP SMILE

New interest rate models have emerged recently in which distributional assumptions are made directly on financial observables. In these "Market Models" the Libor rates have a log-normal distribution in the corresponding forward measure, and caps are priced according to the Black–Scholes formula. These models present two disadvantages. First, Libor rates do not in reality have a log-normal distribution since the implied volatility of a cap depends typically on the strike. Second, these models are difficult to use for pricing derivatives other than caps. In this paper, we extend these models to allow for a broader class of Libor rate distributions. In particular, we construct multi-factor Market Models that are consistent with an initial cap smile surface, and have the useful feature of exhibiting Markovian Libor rates. We show that these Markov Market Models can be used relatively easily to price complex Libor derivatives, such as Bermudan swaptions, captions or flexi-caps, by construction of a tree of Libor rates.

Factor Dependence of Bermudan Swaption Prices: Fact or Fiction?

This paper investigates the effect of interest rate correlation in the pricing of Bermudan swaptions. Investigating both Gaussian Markov models and Libor Market models, we find that Bermudan swaption prices depend only weakly on the number of factors in the underlying interest rate model. Moreover, we find that prices of standard Bermudan swaptions typically decrease slightly in the number of factors, primarily a consequence of effects on the time evolution of volatility induced by calibration of the model dynamics. Our findings are markedly different from those of Longstaff, Schwarz, and Santa-Clara (1999) who conclude that single-factor interest rate models significantly undervalue Bermudan swaptions. We argue that the conclusions of Longstaff, Schwarz, and Santa-Clara are due to non-standard choices of model dynamics and calibration methodology. Our study highlights the importance of using a reasonable set of calibration instruments when applying and comparing interest rate models.

A simple approach to the pricing of Bermudan swaptions in the multifactor LIBOR market model

This paper considers the pricing of Bermuda-style swaptions in the LIBOR market model and its extensions. Application of lattice methods to this model class is generally not feasible because of the large number of state variables, so instead a simple technique to incorporate early exercise features into the Monte Carlo method is considered. The approach here involves a direct search for an early exercise boundary parametrized in intrinsic value and the values of still-alive swaptions. The results of the proposed algorithm are compared against prices obtained from Markov chain approximations, nonrecombining trees, and finite difference methods. The proposed algorithm is fast and robust, and produces a lower bound on Bermudan swaption prices that appears to be very tight for many realistic structures. The paper contains several numerical results against which other methods can be tested.

Bermudan Swaptions in the LIBOR Market Model

Bermudan swaptions have until recently been valued using only one-factor models such as the Black-Derman-Toy (BDT) or Black-Karasinski (BK) models. The LIBOR Market (LM) model which is a more general multi-factor model is becoming increasingly popular as a benchmark model. Whereas the BDT and BK models can be approximated using a lattice facilitating easy valuation of Bermudan swaption, the LM model doesn't conform to the lattice framework and as such the valuation seems very difficult. Monte-Carlo simulation is a popular alternative to the lattice framework for derivatives valution. In order to facilitate valuation of Bermudan swaptions the Monte-Carlo simulation technique must be extended. A few methods doing this are presently available, eg [And98]. A common feature of these methods is that the estimated option premia are only lower bounds on the true premia. The Stochastic Mesh method proposed by [BG97b] for valuation of Bermudan (equity) options with applications to equity options provides a lower and an upper bound. We have applied this method to the LM model and use this to verify the premia found by Andersen. We will also apply the approach suggested in [LS98] to the LM model and verify the premia found using that approach. As it turns out this approach is a special case of the [And98] approach. Furthermore we also examine the impact on the Bermudan swaption premia when moving from a LM model with only one factor to a LM model with multiple factors and do indeed find a significant---but not dramatic---impact. We find the [And98] and [LS98] approaches to be mutually consistent and in line with results obtained from low-biased Stochastic Mesh estimates.

A Simple Approach to the Pricing of Bermudan Swaptions in the Multi-Factor Libor Market Model

This paper considers the pricing of Bermuda-style swaptions in the Libor market model (Brace et al (1997), Jamshidian (1997), Miltersen et al (1997)) and its extensions (Andersen and Andreasen (1998)). Due to its large number of state variables, application of lattice methods to this model class is generally not feasible, and we instead focus on a simple technique to incorporate early exercise features into the Monte Carlo method. Our approach involves a direct search for an early exercise boundary parametrized in intrinsic value and the values of still-alive swaptions. We compare results of the proposed algorithm against prices obtained from Markov Chain approximations and finite difference methods. The proposed algorithm is fast and robust, and produces a lower bound on Bermuda swaption prices that appears to be very tight for many realistic structures. The paper contains several numerical results against which other methods can be tested.

Does Correlation Matter in Pricing Caps and Swaptions?

This paper studies the effect of forward rate correlations on caplet and swaption prices. A two-factor HJM lognormal model of forward rates that implies a realistic covariance matrix of forward rates is constructed. A one-factor lognormal model, with the same forward rate volatilities as the two-factor one, is employed for comparison purposes. The one- and two-factor models price European caplets identically. The one-factor model overprices European swaptions as expected. But the magnitude of overpricing is surprisingly small, less than three percent for at-the-money swaptions on five-year semi-annual swaps. The overpricing is less for shorter swap lengths. The surprising result is that the one-factor model underprices both American caplets and American-type swaptions. Five-year at-the-money American caplets on six-month rates are underpriced by as much as twelve percent and three-year at-the-money constant maturity Bermudan swaptions on two-year semi-annual swaps by as much as ten percent. The underpricing is relatively low for six-month and one-year options but increases with option maturity and forward rate decorrelation. Unlike constant maturity Bermudan swaptions, regular Bermudan swaptions are overpriced by the one-factor model by more than four percent in the case of three-year maturity swaptions. An intuitive explanation for the underpricing of American options under the one-factor model is offered. This explanation implies that American options on any type of interest rate security would be underpriced if perfect correlation across the forward rate term structure is assumed.