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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.

The paper presents a financial market model that generates stochastic volatility using a minimal set of factors. These factors, formed by transformations of square root processes, model the dynamics of different denominations of a benchmark portfolio. Benchmarked prices are assumed to be local martingales. Numerical results for the pricing and hedging of basic derivatives on indices are described for the minimal market model. This includes cases where the standard risk neutral pricing methodology fails because of the presence of a strict local martingale measure. However, payoffs can be perfectly hedged using self-financing strategies and a form of arbitrage exists. This is illustrated by hedge simulations. The different term structure of implied volatilities is documented for calls and puts on an index.

We develop a class of models within the pricing kernel framework, i.e., we model the pricing kernel directly, and not a particular interest rate or a set of rates. The construction of the kernel is explicitly linked to the calibrating set of instruments. Thus, once the kernel is constructed it will price correctly the chosen set of instruments, and have a low-dimensional Markov structure. We test our model on yield, at-the-money cap, caplet implied volatility surface, and swaption data. The quality of fit is very good.

The aim of this work is to use a duality approach to study the pricing of derivatives depending on two stocks driven by a bidimensional Lévy process. The main idea is to apply Girsanov's Theorem for Lévy processes, in order to reduce the posed problem to a problem with one Lévy driven stock in an auxiliary market, baptized as "dual market". In this way, we extend the results obtained by Gerber and Shiu [5] for two-dimensional Brownian motion.

A computational technique borrowed from the physical sciences is introduced to obtain accurate closed-form approximations for the transition probability of arbitrary diffusion processes. Within the path integral framework the same technique allows one to obtain remarkably good approximations of the pricing kernels of financial derivatives. Several examples are presented, and the application of these results to increase the efficiency of numerical approaches to derivative pricing is discussed.

We analyze and compare the performance of the Fourier transform method in affine and quadratic term structure models. We explain why the method of the reduction to FFT in dimension 1 is efficient for ATSMs of type A₀(n), but may lead to sizable errors for QTSMs unless computational errors are taken into account properly. We suggest a certain improvement and generalization which make FFT more accurate and, for the same precision, faster than the Leippold and Wu [M. Leippold and L. Wu, Option pricing under the quadratic class, Journal of Financial and Quantitative Analysis37(2) (2002) 271–295] method. We deduce simple general recommendations for the choice of parameters of computational schemes for QTSMs, which ensure a given precision, and an approximate formula for the bias which FFT produces.

We introduce a simple but efficient PDE method that makes use of interpolation wavelets for their advantages in compression and interpolation in order to define a sparse computational domain. It uses finite difference filters for approximate differentiation, which provide us with a simple and sparse stiffness matrix for the discrete system. Since the method only uses a nodal basis, the application of non-constant terms, boundary conditions and free-boundary conditions is straightforward. We give empirical results for financial products from the equity and fixed income markets in 1, 2 and 3 dimensions and show a speed-up factor between 2 and 4 with no significant reduction of precision.

We present an accurate and easy-to-compute approximation of zero-coupon bonds and Arrow–Debreu (AD) prices for the Black–Karasinski (BK) model of interest rates or default intensities. Through this procedure, dubbed exponent expansion, AD prices are obtained as a power series in time to maturity. This provides remarkably accurate results — for time horizons up to several years — even when truncated to the first few terms. For larger time horizons the exponent expansion can be combined with a fast numerical convolution to obtain extremely accurate results.

We derive semi-analytic approximation formulae for bond and swaption prices in a Black–Karasiński (BK) interest rate model. Approximations are obtained using a novel technique based on the Karhunen–Loève expansion. Formulas are easily computable and prove to be very accurate in numerical tests. This makes them useful for numerically efficient calibration of the model.

The problem of fast pricing, hedging, and calibrating of derivatives is considered when the underlying does not follow the standard Black–Scholes–Merton model but rather a mean-reverting and deterministic volatility model. Mean-reverting models are often used for volatility, commodities, and interest-rate derivatives, while the deterministic volatility accounts for the nonconstant implied volatility. Trading desks often use numerical methods for real-time pricing, hedging, and calibration when implementing such models. A more efficient alternative is to use an analytic formula, even if only approximate. A systematic approach is presented, based on the WKB or ray method, to derive asymptotic approximations to the density function that can be used to derive simple formulas for pricing derivatives. Such approximations are usually only valid away from any boundaries, yet for some derivatives the values of the underlying near the boundaries are needed such as when interest rates are very low or for pricing put options. Hence, the ray approximation may not yield acceptable results. A new asymptotic approximation near boundaries is derived, which is shown to be of value for pricing certain derivatives. The results are illustrated by deriving new analytic approximations for European derivatives and their high accuracy is demonstrated numerically.

The paper establishes a convergence result for the Least-Squares Monte Carlo method, applied to the estimation of risk measures (such as VaR) depending on the distribution of the future value of some derivative contract (or portfolio). We extend previous results in the literature by focusing on the specific case where the distribution of the underlying factors is known. In particular, we are able to remove the requirement of bounded conditional variance which is usually not satisfied by common financial models and payoffs at the cost of using a weaker notion of convergence. Our main result shows the convergence of the empirical distribution function when the number of simulations $n$ and of basis variables $m$ are jointly sent to infinity under certain conditions on the payoff and the basis. With a suitable choice of basis functions, we finally prove that convergence can be ensured when $(m^{2}lnm)/n\to 0$.

In this paper, valuation of a derivative partially collateralized in a specific foreign currency defined in its credit support annex traded between default-free counterparties is studied. Two pricing approaches — by hedging and by expectation — are presented to obtain similar valuation formulae which are equivalent under certain conditions. Our findings show that the current marking-to-market value of such a derivative consists of three components: the price of the perfectly collateralized derivative (a.k.a. the price by collateral rate discounting), the value adjustment due to different funding spreads between the payoff currency and the collateral currency, and the value adjustment due to funding requirements of the uncollateralized exposure. These results generalize previous works on discounting for fully collateralized derivatives and on funding value adjustments for partially collateralized or uncollateralized derivatives.

It is possible to base an equity derivatives pricing model on a stochastic driving process for the share price (spot), or for the equity forward. While the former is the classical approach pioneered by Black and Scholes [9], the latter approach separates the modeling of exogenous random price fluctuations from the cost-of-carry modeling of the stock, probably the first and most prominent example of this technique being the paper by Black [8]. While the Black—Scholes spot price approach and Black’s forward approach are equivalent, the present note demonstrates that the introduction of local volatility and/ or level-dependent default intensity into the stochastic driving process destroys this equivalence, if applied carelessly. The advantages and disadvantages of both approaches are discussed.