Pricing Of Derivative Securities Research Articles

Empirical Martingale Simulation for Asset Prices

This paper proposes a simple modification to the standard Monte Carlo simulation procedure for computing the prices of derivative securities. The modification imposes the martingale property on the simulated sample paths of the underlying asset price. This procedure is referred to as the empirical martingale simulation (EMS). The EMS ensures that the price estimated by simulation satisfies the rational option pricing bounds. The EMS yields a substantial error reduction for the price estimate and can be easily coupled with the standard variance reduction methods. Simulation studies are conducted for European and Asian call options using both the Black and Scholes and GARCH option pricing frameworks. The results indicate that the EMS yields substantial variance reduction particularly for in- and at-the-money or longer-maturity options. The option price estimate based on the EMS is found to exhibit a minor small-sample bias only in few occasions. An analysis of the trade-off between computing time and price accuracy reveals that the EMS dominates the conventional simulation methods.

The use of Markov discontinuous processes in the pricing of derivative securities

The paper deals with one of the most important fields of research in modern financial theory: the valuation of derivatives. The aim of the work is the use of a discontinuous Markov process to model the dynamics of stock prices and, consequently to discover the right price of the derivative instruments, in place of the traditional binomial multiplicative process used frequently in many discrete models. At the end of the work various applications are presented. In them the stochastic process of the stock prices is modelled with the use of a simple and powerful APL program. In the applications APL shows all its capacity of calculation and matrix manipulation for mathematical uses.

Volatility misspecification, option pricing and superreplication via coupling

Consider the performance of an options writer who misspecifies the dynamics of the price process of the underlying asset by overestimating asset price volatility. When does he overprice the option? If he follows the hedging strategy suggested by his model, when does the terminal value of his strategy dominate the option payout? We show that both these events happen if the option payoff is a convex function of the price of the underlying at maturity. The proofs involve the simple, powerful and intuitive techniques of coupling. 1. Robust hedging and superreplication. The standard approach in mathematical finance, and particularly in the pricing of derivative securities, is to begin by writing down a stochastic model, which is assumed, without comment, to correctly and precisely specify the dynamics of the underlying asset. Arbitrage theory, backed by change of measure techniques and martingale representation theorems, then allows options to be priced and hedged. The fairness of the price and the success of the hedge depend crucially on the truth of the underlying model. The purpose of this article is to consider the implications of a misspecification of the dynamics of the asset price process. In particular, if the options writer uses an incorrect model, when does he overcharge for the option? Further, if he attempts to hedge using this incorrect model, can he still replicate (or rather superreplicate) the option payoff? (A superreplicating strategy is a dynamic hedging strategy which generates a terminal wealth which stochastically dominates the option payout.) It is well known that the price of a call option in the Black-Scholes model is an increasing function of the volatility parameter and that this monotonicity property extends to all European options with convex payoff profiles. We generalize this result to prove a price comparison theorem for pairs of models with stochastic volatilities. We have in mind the scenario where one model is used by the options writer to price and hedge options, and the second model represents the (unknown) truth.

Optimality of replication in the CRR model with transaction costs

Recently, there has been a growing interest in optimization problems associated with the arbitrage pricing of derivative securities in imperfect markets (in particular, in models with transaction costs). In this paper, we examine the valuation and hedging of European claims in the multiplicative binomial model proposed by Cox, Ross and Rubinstein [5] (the CRR model), in the presence of proportional transaction costs. We focus on the optimality of replication; in particular, we provide sufficient conditions for the optimality of the replicating strategy in the case of long and short positions in European options. This work can be seen as a continuation of studies by Bensaid et al. [2] and Edirisinghe et al. [13]. We put, however, more emphasis on the martingale approach to the claims valuation in the presence of transaction costs, focusing on call and put options. The problem of optimality of replication in the CRRmodel under proportional transaction costs was recently solved in all generality by Stettner [30].

On the Term Structure of Interest Rates in the Presence of Reflecting and Absorbing Boundaries

To preclude arbitrage, nominal interest rates must remain non-negative. Researchers have captured this feature by modeling the spot rate with volatility structures that vanish as the spot rate tends to zero. However, such volatility structures are in conflict with empirical evidence; volatility can remain relatively large even at low interest rates. In this paper, we investigate the term structure of interest rates for spot rate processes subject to reflecting and absorbing boundaries. Such models preclude negative rates, are consistent with large volatility at low rates, and admit closed form solutions for both bond and European-type derivative security prices. Below, appropriate boundary conditions are derived for spot rates processes that are subject to reflecting and absorbing boundaries. Although it can be demonstrated that spot rate processes with reflecting boundaries can be supported within a general equilibrium framework, we show that reflecting boundaries cannot be placed upon forward rate dynamics without generating arbitrage opportunities. Closed-form bond price solutions for select models are obtained.

Modeling volatility using state space models.

In time series problems, noise can be divided into two categories: dynamic noise which drives the process, and observational noise which is added in the measurement process, but does not influence future values of the system. In this framework, we show that empirical volatilities (the squared relative returns of prices) exhibit a significant amount of observational noise. To model and predict their time evolution adequately, we estimate state space models that explicitly include observational noise. We obtain relaxation times for shocks in the logarithm of volatility ranging from three weeks (for foreign exchange) to three to five months (for stock indices). In most cases, a two-dimensional hidden state is required to yield residuals that are consistent with white noise. We compare these results with ordinary autoregressive models (without a hidden state) and find that autoregressive models underestimate the relaxation times by about two orders of magnitude since they do not distinguish between observational and dynamic noise. This new interpretation of the dynamics of volatility in terms of relaxators in a state space model carries over to stochastic volatility models and to GARCH models, and is useful for several problems in finance, including risk management and the pricing of derivative securities. Data sets used: Olsen & Associates high frequency DEM/USD foreign exchange rates (8 years). Nikkei 225 index (40 years). Dow Jones Industrial Average (25 years).

A technique for calibrating derivative security pricing models: numerical solution of an inverse problem

A technique is presented for calibrating derivative security pricing models with respect to observed market prices. This technique can be applied in a very general multifactor setting where model parameters such as volatilities and correlations are allowed to be functions of the underlying state variables. These functions are estimated from price observations by solving the inverse problem associated with the parabolic partial differential equation governing arbitrage-free derivative security prices. A detailed exposition is given for consistent pricing of equity index options under a stochastic model that treats index volatility as a deterministic function of index level and time.

Arbitrage-Based Pricing When Volatility is Stochastic

Keywords: Mathematical Finance Reference SFI-PB-WORKING-1997-001 Record created on 2008-03-12, modified on 2017-05-12

Nonparametric Pricing of Interest Rate Derivative Securities

We propose a nonparametric estimation procedure for continuous-time stochastic models. Because prices of derivative securities depend crucially on the form of the instantaneous volatility of the underlying process, we leave the volatility function unrestricted and estimate it nonparametrically. Only discrete data are used but the estimation procedure still does not rely on replacing the continuous-time model by some discrete approximation. Instead the drift and volatility functions are forced to match the densities of the process. We estimate the stochastic differential equation followed by the short-term interest rate and compute nonparametric prices for bonds and bond options.

The impact of default risk on the prices of options and other derivative securities

This paper presents a model for valuing derivative securities when there is default risk. The holder of a security is assumed to recover a proportion of its no-default value in the event of a default by the counterparty. Both the probability of default and the size of the proportional recovery are random. The paper shows how data on bonds issued by the counterparty can be used to provide information about model parameters.

Pricing via anticipative stochastic calculus

The paper proposes a general model for pricing of derivative securities. The underlying dynamics follows stochastic equations involving anticipative stochastic integrals. These equations are solved explicitly and structural properties of solutions are studied.

Pricing via anticipative stochastic calculus

The paper proposes a general model for pricing of derivative securities. The underlying dynamics follows stochastic equations involving anticipative stochastic integrals. These equations are solved explicitly and structural properties of solutions are studied.

Forecasting volatility in the Singapore stock market

Volatility forecasting is a major area in the pricing of derivative securities, such as stock and index options. In this paper, we compare three methods of forecasting volatility. These are the naive method based on historical sample variance, the exponentially weighted moving average (EWMA) method, and the generalised autoregressive conditional heteroscedasticity (GARCH) model. Out-of-sample forecasts of monthly return variances generated by these three methods are compared. The results strongly favour the EWMA method.

Valuing Derivative Securities Using the Explicit Finite Difference Method

This paper suggests a modification to the explicit finite difference method for valuing derivative securities. The modification ensures that, as smaller time intervals are considered, the calculated values of the derivative security converge to the solution of the underlying differential equation. It can be used to value any derivative security dependent on a single state variable and can be extended to deal with many derivative security pricing problems where there are several state variables. The paper illustrates the approach by using it to value bonds and bond options under two different interest rate processes.