Narrow Results

We show that the frequent claim that the implied tree prices exotics consistently with an arbitrage-free market is untrue if the local volatilities are stochastic. This is a consequence of the market incompleteness under stochastic volatility. We also show that the problem cannot be mitigated by conveniently defining some 'weakly stochastic' local volatility, as this would violate the no-arbitrage condition. In the process of constructing the proof, we analyse — in the most general context — the impact of stochastic variables on the P&L of a hedged portfolio. We find that any stochastic tradeable either has quadratic variation — and therefore a Γ-like P&L on instruments with non-linear exposure to that asset — or it introduces arbitrage opportunities.

We propose a new method for evaluating fixed strike Asian options using moments. In particular we show that the density of the logarithm of the arithmetic average is uniquely determined from its moments. Resorting to the maximum entropy density, we show that the first four moments are sufficient to recover with great accuracy the true density of the average. Then the Asian option price is estimated with high accuracy. We compare the proposed method with others based on the computation of moments.

Recently, stock price models based on Lévy processes with stochastic volatility were introduced. The resulting vanilla option prices can be calibrated almost perfectly to empirical prices. Under this model, we will price exotic options, like barrier, lookback and cliquet options, by Monte–Carlo simulation. The sampling of paths is based on a compound Poisson approximation of the Lévy process involved. The precise choice of the terms in the approximation is crucial and investigated in detail. In order to reduce the standard error of the Monte–Carlo simulation, we make use of the technique of control variates. It turns out that there are significant differences with the classical Black–Scholes prices.

A new simulation based algorithm to approximate prices of path dependent European options is introduced. The algorithm is defined for tree-like approximations to the underlying process and makes extensive use of structural properties of the discrete approximation. We indicate the advantages of the new algorithm in comparison to standard Monte Carlo algorithms. In particular, we prove a probabilistic error bound that compares the quality of both approximations. The algorithm is of general applicability and, for a large class of options, it has the same computational complexity as Monte Carlo.

We discuss how implied volatilities for OTC traded Asian options can be computed by combining Monte Carlo techniques with the Newton method in order to solve nonlinear equations. The method relies on accurate and fast computation of the corresponding vegas of the option. In order to achieve this we propose the use of logarithmic derivatives instead of the classical approach. Our simulations document that the proposed method shows far better results than the classical approach. Furthermore we demonstrate how numerical results can be improved by localization.

We suggest a modification of an American option such that the option holder can exercise the option early before the expiration and can revert later this decision to exercise; it can be repeated a number of times. This feature gives additional flexibility and risk protection for the option holder. A classification of these options and pricing rules are given. We found that the price of some call options with this feature is the same as for the European call. This means that the additional flexibility costs nothing, similarly to the situation with American and European call options. For the market model with zero interest rate, the price of put options with this feature is also the same as for the standard European put options. Therefore, these options can be more competitive than the standard American options.

We study Vanna-Volga methods which are used to price first generation exotic options in the Foreign Exchange market. They are based on a rescaling of the correction to the Black–Scholes price through the so-called "probability of survival" and the "expected first exit time". Since the methods rely heavily on the appropriate treatment of market data we also provide a summary of the relevant conventions. We offer a justification of the core technique for the case of vanilla options and show how to adapt it to the pricing of exotic options. Our results are compared to a large collection of indicative market prices and to more sophisticated models. Finally we propose a simple calibration method based on one-touch prices that allows the Vanna-Volga results to be in line with our pool of market data.

It is known that the market in a Markovian regime-switching model is, in general, incomplete, so not all contingent claims can be perfectly hedged. We show, in this paper, how certain contingent claims are attainable in the regime-switching market using a money market account, a share and a zero-coupon bond. General contingent claims with payoffs depending on both the share price and the state of the regime-switching process are considered. We apply a martingale representation result to show the attainability of a European-style contingent claim. We also extend our analysis to Asian-style and American-style contingent claims.

The paper introduces special options such that the holder selects dynamically a continuous time process controlling the distribution of the payments (benefits) over time. For instance, the holder can select dynamically the quantity of a commodity purchased or sold by a fixed price given constraints on the cumulative quantity. In a modification of the Asian option, the control process can represent the averaging kernel describing the distribution of the purchases. The pricing of these options requires to solve special stochastic control problems with constraints for the cumulative control similar to a knapsack problem. Some existence results and pricing rules are obtained via modifications of parabolic Bellman equations.

Swing options are a type of exotic financial derivative which generalize American options to allow for multiple early-exercise actions during the contract period. These contracts are widely traded in commodity and energy markets, but are often difficult to value using standard techniques due to their complexity and strong path-dependency. There are numerous interesting varieties of swing options, which differ in terms of their intermediate cash flows, and the constraints (both local and global) which they impose on early-exercise (swing) decisions. We introduce an efficient and general purpose transform-based method for pricing discrete and continuously monitored swing options under exponential Lévy models, which applies to contracts with fixed rights clauses, as well as recovery time delays between exercise. The approach combines dynamic programming with an efficient method for calculating the continuation value between monitoring dates, and applies generally to multiple early-exercise contracts, providing a unified framework for pricing a large class of exotic derivatives. Efficiency and accuracy of the method are supported by a series of numerical experiments which further provide benchmark prices for future research.

We consider the problem of finding a consistent upper price bound for exotic options whose payoff depends on the stock price at two different predetermined time points (e.g. Asian option), given a finite number of observed call prices for these maturities. A model-free approach is used, only taking into account that the (discounted) stock price process is a martingale under the no-arbitrage condition. In case the payoff is directionally convex we obtain the worst case marginal pricing measures. The speed of convergence of the upper price bound is determined when the number of observed stock prices increases. We illustrate our findings with some numerical computations.

There is a vast literature on numerical valuation of exotic options using Monte Carlo (MC), binomial and trinomial trees, and finite difference methods. When transition density of the underlying asset or its moments are known in closed form, it can be convenient and more efficient to utilize direct integration methods to calculate the required option price expectations in a backward time-stepping algorithm. This paper presents a simple, robust and efficient algorithm that can be applied for pricing many exotic options by computing the expectations using Gauss–Hermite integration quadrature applied on a cubic spline interpolation. The algorithm is fully explicit but does not suffer the inherent instability of the explicit finite difference counterpart. A "free" bonus of the algorithm is that it already contains the function for fast and accurate interpolation of multiple solutions required by many discretely monitored path dependent options. For illustrations, we present examples of pricing a series of American options with either Bermudan or continuous exercise features, and a series of exotic path-dependent options of target accumulation redemption note (TARN). Results of the new method are compared with MC and finite difference methods, including some of the most advanced or best known finite difference algorithms in the literature. The comparison shows that, despite its simplicity, the new method can rival with some of the best finite difference algorithms in accuracy and at the same time it is significantly faster. Virtually the same algorithm can be applied to price other path-dependent financial contracts such as Asian options and variable annuities.

We present a generic formalism for option pricing which does not require specifying the stochastic process of the underlying asset price, undergoing a Markovian stochastic behavior. We first derive a consistency condition that the risk neutral density function to maturity must satisfy in order to guarantee no arbitrage. As an example, we show that when the underlying asset price undergoes a continuous stochastic process with deterministic time dependent standard deviation the formalism produces the Black–Scholes–Merton pricing formula. We provide data from the market to prove that the price of European options is independent of the term structure of the volatility prior to maturity. Based on this observation we offer a method to calculate the density function to maturity that satisfies the consistency condition we derived. In the general case the price of European options depends only on the moments of the price return of the underlying asset. When the underlying asset undergoes a continuous time process then only moments up to second order contribute to the European option price. In this case any set of option prices on three strikes with the same maturity contains the information to determine the whole volatility smile for this maturity. Using a great amount of data from the option markets we show that our formalism generates European option prices that match the markets prices very accurately in all asset classes: currencies, equities, interest rates and commodities. Finally, using bootstrapping method with market data of the whole term structure we determine the probability transfer density function from inception to maturity, thus allowing the calculation of path dependent options. Comparison of the results of the model to the market shows a very high level of accuracy.