The Hull-White one factor model is used to price interest rate options. The parameters of the model are often calibrated to simple liquid instruments, in particular European swaptions. It is therefore very important to have very efficient pricing formula for simple instruments. Such a formula is proposed here for European swaption. Based on a very efficient corrector type approximation the approximation is efficient both in term of precision and in term of spped. In our implementation the approximation is more than ten time faster than the direct pricing formula and more than twenty time faster than the Jamshidian trick.
Closed form formulas for swaption prices in HJM model are derived. These formulas are used for nonparametric fit of deterministic forward volatility. It is demonstrated that this formula and non-parametric fit works very well and can be used to identify arbitrage opportunities
We propose an elementary model to price European physical delivery swaptions in multicurve setting with a simple exact closed formula. The proposed model is very parsimonious: it is a three-parameter multicurve extension of the two-parameter Hull-White (1990) model. The model allows also to obtain simple formulas for all other plain vanilla Interest Rate derivatives. Calibration issues are discussed in detail.
American and Bermudan-type financial instruments are often priced with specific Monte Carlo techniques whose efficiency critically depends on the effective dimensionality of the problem and the available computational power. In our work we focus on Bermudan Swaptions, well-known interest rate derivatives embedded in callable debt instruments or traded in the OTC market for hedging or speculation purposes, and we adopt an original pricing approach based on Supervised Learning (SL) algorithms. In particular, we link the price of a Bermudan Swaption to its natural hedges, i.e. the underlying Euro…
We develop and test a fast and accurate semi-analytical formula for single-name default swaptions in the context of a shifted square root jump diffusion (SSRJD) default intensity model. The model can be calibrated to the CDS term structure and a few default swaptions, to price and hedge other credit derivatives consistently. We show with numerical experiments that the model implies plausible volatility smiles.
We propose a fast and accurate numerical method for pricing European swaptions in multi-factor Gaussian term structure models. Our method can be used to accelerate the calibration of such models to the volatility surface. The pricing of an interest rate option in such a model involves evaluating a multi-dimensional integral of the payoff of the claim on a domain where the payoff is positive. In our method, we approximate the exercise boundary of the state space by a hyperplane tangent to the maximum probability point on the boundary and simplify the multi-dimensional integration into an analyt…
We derive measure change formulae required to price midcurve swaptions in the forward swap annuity measure with stochastic annuities' ratios. We construct the corresponding linear and exponential terminal swap rate pricing models and show how they capture the midcurve swaption correlation skew.
In this paper we derive semi-closed form prices of barrier (perhaps, time-dependent) options for the Hull-White model, ie., where the underlying follows a time-dependent OU process with a mean-reverting drift. Our approach is similar to that in (Carr and Itkin, 2020) where the method of generalized integral transform is applied to pricing barrier options in the time-dependent OU model, but extends it to an infinite domain (which is an unsolved problem yet). Alternatively, we use the method of heat potentials for solving the same problems. By semi-closed solution we mean that first, we need to …
We introduce a multiple curve framework that combines tractable dynamics and semi-analytic pricing formulas with positive interest rates and basis spreads. Negatives rates and positive spreads can also be accommodated in this framework. The dynamics of OIS and LIBOR rates are specified following the methodology of the affine LIBOR models and are driven by the wide and flexible class of affine processes. The affine property is preserved under forward measures, which allows us to derive Fourier pricing formulas for caps, swaptions and basis swaptions. A model specification with dependent LIBOR r…
In this paper we consider the pricing of options on interest rates such as caplets and swaptions in the Lévy Libor model developed by Eberlein and Özkan (2005). This model is an extension to Lévy driving processes of the classical log-normal Libor market model (LMM) driven by a Brownian motion. Option pricing is significantly less tractable in this model than in the LMM due to the appearance of stochastic terms in the jump part of the driving process when performing the measure changes which are standard in pricing of interest rate derivatives. To obtain explicit approximation for option price…
Two novel closed-form formulas for the price of barrier options in stochastic volatility models with zero interest rate and dividend yield but nonzero correlation between the asset and its instantaneous volatility are derived. The first is a Hull and White type formula, and the second is a decomposition formula similar in form to the Alòs decomposition for vanilla options. A model-free approximation is also given.
We revisit the problem of pricing and hedging plain vanilla single-currency interest rate derivatives using multiple distinct yield curves for market coherent estimation of discount factors and forward rates with different underlying rate tenors. Within such double-curve-single-currency framework, adopted by the market after the credit-crunch crisis started in summer 2007, standard single-curve no-arbitrage relations are no longer valid, and can be recovered by taking properly into account the forward basis bootstrapped from market basis swaps. Numerical results show that the resulting forward…
A heat kernel approach is proposed for the development of a general, flexible, and mathematically tractable asset pricing framework in finite time. The pricing kernel, giving rise to the price system in an incomplete market, is modelled by weighted heat kernels which are driven by multivariate Markov processes and which provide enough degrees of freedom in order to calibrate to relevant data, e.g. to the term structure of bond prices. It is shown how, for a class of models, the prices of bonds, caplets, and swaptions can be computed in closed form. The dynamical equations for the price process…
We construct models for the pricing and risk management of inflation-linked derivatives. The models are rational in the sense that linear payoffs written on the consumer price index have prices that are rational functions of the state variables. The nominal pricing kernel is constructed in a multiplicative manner that allows for closed-form pricing of vanilla inflation products suchlike zero-coupon swaps, year-on-year swaps, caps and floors, and the exotic limited-price-index swap. We study the conditions necessary for the multiplicative nominal pricing kernel to give rise to short rate models…
We derive analytic series representations for European option prices in polynomial stochastic volatility models. This includes the Jacobi, Heston, Stein-Stein, and Hull-White models, for which we provide numerical case studies. We find that our polynomial option price series expansion performs as efficiently and accurately as the Fourier transform based method in the nested affine cases. We also derive and numerically validate series representations for option Greeks. We depict an extension of our approach to exotic options whose payoffs depend on a finite number of prices.
In the context of stochastic volatility models, we study representation formulas in terms of expectations for the power series' coefficients associated to the call price-function. As in a recent paper by Antonelli and Scarlatti the expansion is done w.r.t. the correlation between the noises driving the underlying asset price process and the volatility process. We first obtain expressions for the power series' coefficients from the generalized Hull and White formula obtained by Elisa Alòs. Afterwards, we provide representations turning out from the approach for the sensitivity problem tackled b…
In this work we want to provide a general principle to evaluate the CVA (Credit Value Adjustment) for a vulnerable option, that is an option subject to some default event, concerning the solvability of the issuer. CVA is needed to evaluate correctly the contract and it is particularly important in presence of WWR (Wrong Way Risk), when a credit deterioration determines an increase of the claim's price. In particular, we are interested in evaluating the CVA in stochastic volatility models for the underlying's price (which often fit quite well the market's prices) when admitting correlation with…
The aim of this work is to provide fast and accurate approximation schemes for the Monte Carlo pricing of derivatives in LIBOR market models. Standard methods can be applied to solve the stochastic differential equations of the successive LIBOR rates but the methods are generally slow. Our contribution is twofold. Firstly, we propose an alternative approximation scheme based on Picard iterations. This approach is similar in accuracy to the Euler discretization, but with the feature that each rate is evolved independently of the other rates in the term structure. This enables simultaneous calcu…
In this paper, we study the price of Variable Annuity Guarantees, especially of Guaranteed Annuity Options (GAO) and Guaranteed Minimum Income Benefit (GMIB), and this in the settings of a derivative pricing model where the underlying spot (the fund) is locally governed by a geometric Brownian motion with local volatility, while interest rates follow a Hull-White one-factor Gaussian model. Notwithstanding the fact that in this framework, the local volatility depends on a particularly complicated expectation where no closed-form expression exists and it is neither directly related to European c…
In this paper, we obtain asymptotic formulas with error estimates for the implied volatility associated with a European call pricing function. We show that these formulas imply Lee's moment formulas for the implied volatility and the tail-wing formulas due to Benaim and Friz. In addition, we analyze Pareto-type tails of stock price distributions in uncorrelated Hull-White, Stein-Stein, and Heston models and find asymptotic formulas with error estimates for call pricing functions in these models.
