International Journal of Theoretical and Applied Finance

Markets are modeled as a counterparty accepting at zero cost a set of cash flows that are closed under addition, scaling and contain the nonnegative cash flows. Formulas are then provided for bid and ask prices in terms of this marketed cone. Additionally closed forms are obtained when parametric concave distortions introduced in Cherny and Madan (2009) define the marketed claims. Finally explicit expressions price call and put options at bid and ask. Three applications illustrate. The first estimates the movement of the cone through the financial crisis using data on bid and ask prices for S&P 500 index options. It is observed that the cone contracted significantly in 2008 and slowly opened up thereafter. The second application documents the improvements possible in terms of reduced ask prices by hedging at a flat Black-Scholes volatility even when the underlying assumptions for replication are violated. The third application considers a number of structured products written on daily returns to an underlying asset price and illustrates the use of our closed form expressions for the ask price as an objective function in designing hedges.

This paper develops a general approximation scheme, henceforth called a hybrid asymptotic expansion scheme for valuation of multi-factor European path-independent derivatives. Specifically, we apply it to pricing long-term currency options under a market model of interest rates and a general diffusion stochastic volatility model with jumps of spot exchange rates. Our scheme is very effective for a type of models in which there exist correlations among all the factors whose dynamics are not necessarily affine nor even Markovian so long as the randomness is generated by Brownian motions. It can also handle models that include jump components under an assumption of their independence of the other random variables when the characteristic functions for the jump parts can be analytically obtained. An asymptotic expansion approach provides a closed-form approximation formula for their values, which can be calculated in a moment and thus can be used for calibration or for an explicit approximation of Greeks of options. Moreover, this scheme develops Fourier transform method with an asymptotic expansion as well as with closed-form characteristic functions obtainable in parts of a model, extending the method proposed by Takehara and Takahashi (2008) to be applicable to a general class of models. It also introduces a characteristic-function-based Monte Carlo simulation method with the asymptotic expansion as a control variable in order to make full use of analytical approximations by the asymptotic expansion and of the closed-form characteristic functions. Finally, a series of numerical examples shows the effectiveness of our scheme.

In this paper we propose a new nonparametric approach to interacting failing systems (FS), that is systems whose probability of failure is not negligible in a fixed time horizon, a typical example being firms and financial bonds. The main purpose when studying a FS is to calculate the probability of default and the distribution of the number of failures that may occur during the observation period. A model used to study a failing system is defined default model. In particular, we present a general recursive model constructed by the means of interacting urns. After introducing the theoretical model and its properties we show a first application to credit risk modeling, showing how to assess the idiosyncratic probability of default of an obligor and the joint probability of failure of a set of obligors in a portfolio of risks, that are divided into reliability classes.

We consider a two-factor Heath–Jarrow–Morton (HJM) model under the risk neutral measure and show that it may be decoupled into a particular dynamic Nelson–Siegel (NS) model plus a somewhat counter-intuitive adjustment (lying outside the NS family) which keeps it arbitrage-free. We assess the importance of the adjustment for arbitrage-free pricing by comparing the HJM model with a novel NS model which is selected using projection techniques. We analyze forward curves and derivative prices generated by the HJM and projected NS model and consider two real-world case studies. Our analysis shows that the influence of the adjustment term on arbitrage-free evolution is small.

Asian options incorporate the average stock price in the terminal payoff. Examination of the Asian option partial differential equation (PDE) has resulted in many equations of reduced order that in general can be mapped into each other, although this is not always shown. In the literature these reductions and mappings are typically acquired via inspection or ad hoc methods. In this paper, we evaluate the classical Lie point symmetries of the Asian option PDE. We subsequently use these symmetries with Lie's systematic and algorithmic methods to show that one can obtain the same aforementioned results. In fact we find a familiar analytical solution in terms of a Laplace transform. Thus, when coupled with their methodic virtues, the Lie techniques reduce the amount of intuition usually required when working with differential equations in finance.

It is well known that for highly skewed distributions the standard method of using the t statistic for the confidence interval of the mean does not give robust results. This is an important problem for importance sampling (IS) as its final distribution is often skewed due to a heavy tailed weight distribution. In this paper, we first explain Hall's transformation and its variants to correct the confidence interval of the mean and then evaluate the performance of these methods for two numerical examples from finance which have closed-form solutions. Finally, we assess the performance of these methods for credit risk examples. Our numerical results suggest that Hall's transformation or one of its variants can be safely used in correcting the two-sided confidence intervals of financial simulations.

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.