Nick Constantinou

Pricing Extendible Options Using the Fast Fourier Transform

This paper applies the fast Fourier transform (FFT) approach, within the Black-Scholes framework, to the valuation of options whose time to maturity can be extended to a future date (extendible options). We determine the valuation of the extendible options as sums of expectations of indicator functions, leading to a semianalytic expression for the value of the options over a range of strikes. Compared to Monte Carlo simulation, numerical examples demonstrate that the FFT is both computationally more efficient and higher in accuracy.

Risk-neutral valuation of power barrier options

Abstract Barrier options are standard exotic options traded in the financial market. These instruments are different from the vanilla options as the payoff of the option depends on whether the underlying asset price reaches a predetermined barrier level, during the life of the option. In this work, we extend the vanilla call barrier options to power call barrier options where the underlying asset price is raised to a constant power, within the standard Black–Scholes framework. It is demonstrated that the pricing of the power barrier options can be obtained from standard barrier options by a transformation which involves the power contract and a adjusted barrier. Numerical results are considered.

A Framework for Extracting the Probability of Default from Stock Option Prices

This paper develops a framework to estimate the probability of default (PD) implied in listed stock options. The underlying option pricing model measures PD as the intensity of a jump diffusion process, in which the underlying stock price jumps to zero at default. We adopt a two-stage calibration algorithm to obtain the precise estimator of PD. In the calibration procedure, we improve the fitness of the option pricing model via the implementation of the time inhomogeneous term structure model in the option pricing model. Since the term structure model perfectly fits the actual term structure, we resolve the estimation bias caused by the poor fitness of the time homogeneous term structure model. It is demonstrated that the PD estimator from listed stock options can provide meaningful insights on the pricing of credit derivatives like credit default swap.

Power option pricing via Fast Fourier Transform

The basis of the option universe has been the European option, and much literature has been devoted to the extension of this option to create many new exotic options, including some with nonlinear payoffs. In this work, we study a European-style power option pricing, under a constant volatility dynamics, using the risk-neutral valuation within the Black-Scholes framework. Apart from applying the closed-form solution, we price the power option using the Fast Fourier Transform (FFT) technique which requires an analytical characteristic function of the power option. The resulting approximations are then compared with other numerical methods such as the Monte Carlo simulations, which show promising results and demonstrate the efficiency of the FFT technique as it can compute option prices for a whole range of strike prices. Besides, we show that there exists a relationship between the power call option and the power put option that is similar to the put-call parity relationship of vanilla options. We also find a transformation between the underlying asset and the power contract which enables us to obtain the pricing formulas of the power options from the vanilla options, as well as simplify the Greeks for power options. In addition to the Greeks derived from the closed-form solution, we present the Greeks using the pricing formula obtained from a characteristic function.

Testing Adaptive Expectations Models of a Continuous Double Auction Market against Empirical Facts

It is well known that empirical financial time series data exhibit long memory phenomena: the behaviour of the market at various times in the past continues to exert an influence in the present. One explanation for these phenomena is that they result from a process of social learning in which poorly performing agents switch their strategy to that of other agents who appear to be more successful. We test this explanation using an agent-based model and we find that the stability of the model is directly related to the dynamics of the learning process; models in which learning converges to a stationary steady state fail to produce realistic time series data. In contrast, models in which learning leads to dynamic switching behaviour in the steady state are able to reproduce the long memory phenomena. We demonstrate that a model which incorporates contrarian trading strategies results in more dynamic behaviour in steady state, and hence is able to produce more realistic results.

Learning is neither sufficient nor necessary: a dynamic agent-based model of long memory in financial markets

Financial time series data exhibits long memory phenomena, where certain behaviours in the market have a persistent influence on the market over time. It has been suggested that imitation of successful trader strategies by other less successful traders is an important factor in contributing to this persistence. We test this explanation by using an existing adaptive agent-based model and we find that the robustness of the model is directly related to the dynamics of learning; models in which learning converges to a stationary steady state fail to produce realistic time series data. In contrast, models in which learning leads to dynamic strategy switching behaviour in the steady state are able to reproduce the long memory phenomena. We demonstrate that a model which incorporates contrarian trading strategies results in more dynamic behaviour in steady state, and hence is able to produce more realistic results. We also demonstrate that a non-learning contrarian model that performs dynamic strategy switching produces long memory phenomena and therefore that learning is not necessary. Models that can be validated against properties of empirical high frequency financial data should allow exploration of the robustness and reliability qualities of market mechanism modifications.

A path-independent approach to integrated variance under the CEV model

In this paper, a closed form path-independent approximation of the fair variance strike for a variance swap under the constant elasticity of variance (CEV) model is obtained by applying the small disturbance asymptotic expansion. The realized variance is sampled continuously in a risk-neutral market environment. With the application of a Brownian bridge, we derive a theorem for the conditionally expected product of a Brownian motion at two different times for arbitrary powers. This theorem enables us to provide a conditional Monte-Carlo scheme for simulating the fair variance strike. Compared with results in the recent literature, the method outlined in our paper leads to a simplified approach for pricing variance swaps. The method may also be applied to other more sophisticated volatility derivatives. An empirical comparison of this model with the Heston model and a conditional Monte Carlo scheme is also presented using option data on the S&P 500.

The Relationship between Credit Default Swap Spreads and Equity Prices

To meet new Basel III capital requirements, banks have to proxy unobserved credit default swap (CDS) time series for their over-the-counter derivative counterparties to determine the credit valuation adjustment (CVA) value-at-risk. This paper establishes a link between returns on CDSs and corresponding equity prices that allows CDS proxy series to be generated that retain important idiosyncratic risk, which is lost in more typical mapping methods. In so doing, the multifactor model developed shows a consistent pricing of credit risk in both the CDS market and the equity market. The model is intuitive, requires few assumptions and relies on readily available data. The model also performs well when compared with the Credit Grades model for proxy CDS spreads. Therefore, it tackles the modelling challenges set by Basel III for proxying single-name CDS spreads.

Learning is neither sufficient nor necessary: An agent-based model of long memory in financial markets

Financial markets exhibit long memory phenomena; certain actions in the market have a persistent influence on market behaviour over time. It has been conjectured that this persistence is caused by social learning; traders imitate successful strategies and discard poorly performing ones. We test this conjecture with an existing adaptive agent-based model, and we note that the robustness of the model is directly related to the dynamics of learning. Models in which learning converges to a stationary steady state fail to produce realistic time series data. In contrast, models in which learning leads to continuous dynamic strategy switching behaviour in the steady state are able to reproduce the long memory phenomena over time. We demonstrate that a model which incorporates contrarian trading strategies results in more dynamic behaviour in steady state, and hence is able to produce more realistic results. We also demonstrate that a non-learning contrarian model that performs dynamic strategy switching produces long memory phenomena and therefore that learning is not necessary.

Periodicities of foreign exchange markets and the directional change power law

This paper utilizes advanced methods from Fourier analysis in order to describe periodicities in financial ultrahigh frequency foreign exchange data. The Lomb�Scargle Fourier transform is used to take into account the irregularity in spacing in the time domain. It provides a natural framework for the power spectra of different inhomogeneous time-series processes to be easily and quickly estimated. Furthermore, an event-based approach in intrinsic time based on a power-law relationship is employed using different event thresholds to filter the foreign exchange tick-data. The calculated spectral density demonstrates that the price process in intrinsic time contains different periodic components for directional changes, especially in the medium�long term, implying the existence of stylized facts of ultrahigh frequency data in the frequency domain. Copyright