Convergence of numerical methods for valuing path-dependent options using interpolation

One method for valuing path-dependent options is the augmented state space approach described in Hull and White (1993) and Barraquand and Pudet (1996), among others. In certain cases, interpolation is required because the number of possible values of the additional state variable grows exponentially. We provide a detailed analysis of the convergence of these algorithms. We show that it is possible for the algorithm to be non-convergent, or to converge to an incorrect answer, if the interpolation scheme is selected in appropriately. We concentrate on Asian options, due to their popularity and because of some errors in the previous literature. This revised version was published online in June 2006 with corrections to the Cover Date.     

Bounds and prices of currency cross-rate options

This paper derives the pricing bounds of a currency cross-rate option using the option prices of two related dollar rates via a copula theory and presents the analytical properties of the bounds under the Gaussian framework. Our option pricing bounds are useful, because (1) they are general in the sense that they do not rely on the distribution assumptions of the state variables or on the selection of the copula function; (2) they are portfolios of the dollar-rate options and hence are potential hedging instruments for cross-rate options; and (3) they can be applied to generate bounds on deltas. The empirical tests suggest that there are persistent and stable relationships between the market prices and the estimated bounds of the cross-rate options and that our option pricing bounds (obtained from the market prices of options on two dollar rates) and the historical correlation of two dollar rates are highly informative for explaining the prices of the cross-rate options. Moreover, the empirical results are consistent with the predictions of the analytical properties under the Gaussian framework and are robust in various aspects.     

A numerical method to price exotic path-dependent options on an underlying described by the Heston stochastic volatility model

We consider the problem of pricing European exotic path-dependent derivatives on an underlying described by the Heston stochastic volatility model. Lipton has found a closed form integral representation of the joint transition probability density function of underlying price and variance in the Heston model. We give a convenient numerical approximation of this formula and we use the obtained approximated transition probability density function to price discrete path-dependent options as discounted expectations. The expected value of the payoff is calculated evaluating an integral with the Monte Carlo method using a variance reduction technique based on a suitable approximation of the transition probability density function of the Heston model. As a test case, we evaluate the price of a discrete arithmetic average Asian option, when the average over n = 12 prices is considered, that is when the integral to evaluate is a 2n = 24 dimensional integral. We show that the method proposed is computationally efficient and gives accurate results.     

Bounds for Functions of Dependent Risks

The problem of finding the best-possible lower bound on the distribution of a non-decreasing function of n dependent risks is solved when n=2 and a lower bound on the copula of the portfolio is provided. The problem gets much more complicated in arbitrary dimensions. When no information on the structure of dependence of the random vector is available, we provide a bound on the distribution function of the sum of risks which we prove to be better than the one generally used in the literature.     

Arbitrage Free Price Bounds for Property Derivatives

Market frictions inhibit the perfect replication of property derivatives, and define the property spread as a price measure in the incomplete real estate market. We identify transaction costs, transaction time, and short sale constraints as the main frictions in this market. Based on these frictions, we set up a framework of arbitrage free price bounds for property derivatives. In turn, we use observed derivative prices to determine the implied cost of the frictions. Lastly, we verify these values by using other research, which confirms the accuracy of our framework.     

Value-added tax options for India

The pressures of aggregate revenue, the requirement of a reduced role for customs duties for the liberalization of the economy, and the complexity and strains of the current system together point clearly toward the desirability of tax reform in India. Since domestic indirect taxes provide the major source of revenue, they deserve special attention. This paper argues that India would benefit from moving toward a system of value-added taxation (VAT) and focuses on the way in which a VAT (or VATs) can be best introduced into India given the country's federal structure. Three different options are distinguished: a central VAT, dual VAT, and states' VAT. We argue that the first is politically infeasible, that the second represents the best way forward in the short term, and that the third deserves consideration as a long-run option. Special attention is paid to the problems that would arise under either a states' or a dual VAT with regard to taxing interstate trade.     

Bounds for option prices and expected payoffs

This article sharpens Lo's upper bounds for option prices using an alternative approach with the assumption that the underlying asset price is continuously distributed. The increased sharpness is obtained using additional information about the distribution of the underlying assets. It is shown in this article that a large portion of Lo's upper bounds is the maximum of our bounds over all possible distributions.     

Some calculations for Israeli options

Recently Kifer (2000) introduced the concept of an Israeli (or Game) option. That is a general American-type option with the added possibility that the writer may terminate the contract early inducing a payment exceeding the holders claim had they exercised at that moment. Kifer shows that pricing and hedging of these options reduces to evaluating a saddle point problem associated with Dynkin games. In this short text we give two examples of perpetual Israeli options where the solutions are explicit.Received: December 2002, Mathematics Subject Classification: 90A09, 60J40, 90D15JEL Classification: G13, C73I would like to express thanks to Chris Rogers for a valuable conversation.     

On Option Pricing Bounds

The purpose of this article is to compare the Perrakis and Ryan bounds of option prices in a single-period model with option bounds derived using linear programming. It is shown that the upper bounds are identical but that the lower bounds are different. A comparison of these bounds, together with Merton's bounds and the Black-Scholes prices in a lognormal securities market, is presented.     

Two-Sided Bounds for the Finite Time Probability of Ruin

Explicit, two-sided bounds are derived for the probability of ruin of an insurance company, whose premium income is represented by an arbitrary, increasing real function, the claims are dependent, integer valued r.v.s and their inter-occurrence times are exponentially, non-identically distributed. It is shown, that the two bounds coincide when the moments of the claims form a Poisson point process. An expression for the survival probability is further derived in this special case, assuming that the claims are integer valued, i.i.d. r.v.s. This expression is compared with a different formula, obtained recently by Picard & Lefevre (1997) in terms of generalized Appell polynomials. The particular case of constant rate premium income and non-zero initial capital is considered. A connection of the survival probability to multivariate B -splines is also established.     

Non-exponential Bounds for Ruin Probability with Interest Effect Included

In this paper, we consider a discrete time risk model. First we discuss the classical model, both exponential and non-exponential upper bounds for ruin probabilities are obtained by using martingale inequalities. Then similar results are obtained for the model with investment income.     

Bottleneck options

In the spirit of Kyprianou and Ott (in Acta Appl. Math., to appear, 2013) and Ott (in Ann. Appl. Probab. 23:2327–2356, 2013) we consider an option whose payoff corresponds to a capped American lookback option with floating strike and solve the associated pricing problem (an optimal stopping problem) in a financial market whose price process is modelled by an exponential spectrally negative Lévy process. Despite the simple interpretation of the cap as a moderation of the payoff, it turns out that the optimal strategy to exercise the option looks very different compared to the situation without a cap. In fact, we show that the continuation region has a feature that resembles a bottleneck and hence the name “bottleneck option”.     

Hedging options

This paper considers the problem of forming a hedge when there are perceived profit opportunities. We show that the option price obeys a modified Black and Scholes equation. Iterative methods yield the appropriate hedge ratio.     

A closed-form solution for options with stochastic volatility with applications to bond and currency options

I use a new technique to derive a closed-form solution for theprice of a European call option on an asset with stochasticvolatility. The model allows arbitrary correlation between volatilityand spot asset returns. I introduce stochastic interest ratesand show how to apply the model to bond options and foreigncurrency options. Simulations show that correlation betweenvolatility and the spot asset's price is important for explainingreturn skewness and strike-price biases in the Black-Scholes(1973) model. The solution technique is based on characteristicfunctions and can be applied to other problems     

The efficient hedging problem for American options

In this paper, we prove the existence of efficient partial hedging strategies for a trader unable to commit the initial minimal amount of money needed to implement a hedging strategy for an American option. The attitude towards the shortfall is modeled in terms of a decreasing and convex risk functional satisfying a lower semicontinuity property with respect to the Fatou convergence of stochastic processes. Some relevant examples of risk functionals are analyzed. Numerical computations in a discrete-time market model are provided. In a Lévy market, an approximating solution is given assuming discrete-time trading.     

Watermark options

We consider a new family of derivatives whose payoffs become strictly positive when the price of their underlying asset falls relative to its historical maximum. We derive the solution to the discretionary stopping problems arising in the context of pricing their perpetual American versions by means of an explicit construction of their value functions. In particular, we fully characterise the free-boundary functions that provide the optimal stopping times of these genuinely two-dimensional problems as the unique solutions to highly nonlinear first order ODEs that have the characteristics of a separatrix. The asymptotic growth of these free-boundary functions can take qualitatively different forms depending on parameter values, which is an interesting new feature.