Binomial option pricing with stochastic parameters: A beta distribution approach

This research extends the binomial option-pricing model of Cox, Ross, and Rubinstein (1979) and Rendleman and Barter (1979) to the case where the up and down percentage changes of stock prices are stochastic. Assuming stochastic parameters in the discrete-time binomial option pricing is analogous to assuming stochastic volatility in the continuous-time option pricing. By assuming that the up and down parameters are independent random variables following beta distributions, we are able to derive a closed-form solution to this stochastic discrete-time option pricing. We also derive an upper and a lower bounds of the option price.     

Arbitrage violations and implied valuations: the option market

The ideas presented in this paper are those of the authors and not necessarily reflect the views of the National bank of Canada. Both authors thank the National Bank of Canada and the SSHRC of Canada for their help. Thanks are also due to Professor Y. Tian for his comments, and for participating, together with students of the Financial Engineering program at York University, in the data preparation and the execution of the Matlab programs. In this paper, we propose a necessary and sufficient condition for bid and ask prices of European options to be free of arbitrage, and derive from it an efficient numerical methodology to determine its satisfaction by a given set of prices. If the bid and ask prices satisfy the no-arbitrage (NA) condition, our methodology produces a vector of NA prices that lie between the bid and ask prices. Otherwise, our methodology generates a vector of arbitrage-free prices that is as close as possible, in some sense, to the bid–ask strip. The arbitrage-free prices detected by our methodology render the commonly used practice of using mid-points and then ‘cleaning’ arbitrage from them as unnecessary. Moreover, a vector of ‘cleaned’ prices obtained from mid-point prices may be outside the bid–ask spread even in an arbitrage-free market and, hence, in this case will not be representative of the current market. A new procedure of estimating implied valuation operators is also suggested here. This procedure is rooted in the economic properties of put and call prices and is based on Phillips and Taylor's approximation of a convex function. This approach is superior to common estimation techniques in that it produces an analytical expression for the implied valuation operator and is not data intensive as some other studies. Empirical findings for the new methods are documented and their economic implications are discussed.     

Binomial Option Pricing Biases and Inconsistent Implied Volatilities

We evaluate the binomial option pricing methodology (OPM) by examining simulated portfolio strategies. A key aspect of our study involves sampling from the empirical distribution of observed equity returns. Using a Monte Carlo simulation, we generate equity prices under known volatility and return parameters. We price American–style put options on the equity and evaluate the risk–adjusted performance of various strategies that require writing put options with different maturities and moneyness characteristics. The performance of these strategies is compared to an alternative strategy of investing in the underlying equity. The relative performance of the strategies allows us to identify biases in the binomial OPM leading to the well–known volatility smile . By adjusting option prices so as to rule out dominated option strategies in a mean–variance context, we are able to reduce the pricing errors of the OPM with respect to option prices obtained from the LIFFE. Our results suggest that a simple recalibration of inputs may improve binomial OPM performance.     

Local volatility dynamic models

This paper is concerned with the characterization of arbitrage-free dynamic stochastic models for the equity markets when Itô stochastic differential equations are used to model the dynamics of a set of basic instruments including, but not limited to, the underliers. We study these market models in the framework of the HJM philosophy originally articulated for Treasury bond markets. The main thrust of the paper is to characterize absence of arbitrage by a drift condition and a spot consistency condition for the coefficients of the local volatility dynamics.     

Equilibrium positive interest rates: a unified view

This article develops precise connections among two generalapproaches to building interest rate models: a general equilibriumapproach using a pricing kernel and the Heath, Jarrow, and Mortonframework based on specifying forward rate volatilities andthe market price of risk. The connections exploit the observationthat a pricing kernel is uniquely determined by its drift. Throughthese connections we provide, for any arbitrage-free term structuremodel, a representative-consumer real production economy supportingthat term structure model in equilibrium. We put particularemphasis on models in which interest rates remain positive.By modeling the dynamics of the drift of the pricing kernel,we construct a new family of Markovian-positive interest ratemodels.     

Lean Trees—A General Approach for Improving Performance of Lattice Models for Option Pricing

The well-known binomial and trinomial tree models for option pricing are examined from the point of view of numerical efficiency. Common lattices use a large part of time resources for calculations which are almost irrelevant for the solution. To avoid this waste of resources, the tree is reduced to a lean form which yields the same order of convergence, but with a reduction of numerical effort. In numerical tests it is shown that the proposed method leads to a significant improvement in real calculation time without loss of accuracy for a broad class of derivatives.     

A sufficient and necessary condition for arbitrage-free pricing

This paper derives a sufficient and necessary condition for arbitrage-free pricing, by the mathematical definition of linear dependency. It states that any pricing function that can be expressed as a linear combination of some of its partial derivatives inherently possesses the arbitrage-free property. This condition can serve as a quick ‘reality check’ to help search for arbitrage-free asset pricing.     

Generic market models

Currently, there are two market models for valuation and risk management of interest rate derivatives: the LIBOR and swap market models. We introduce arbitrage-free constant maturity swap (CMS) market models and generic market models featuring forward rates that span periods other than the classical LIBOR and swap periods. We develop generic expressions for the drift terms occurring in the stochastic differential equation driving the forward rates under a single pricing measure. The generic market model is particularly apt for pricing of, e.g., Bermudan CMS swaptions and fixed-maturity Bermudan swaptions.     

Hedging Strategies of Financial Intermediaries: Pricing Options with a Bid-Ask Spread

This paper uses a model similar to the Boyle-Vorst and Ritchken-Kuo arbitrage-free models for the valuation of options with transactions costs to determine the maximum price to be charged by the financial intermediary writing an option in a non-auction market. Earlier models are extended by recognizing that, in the presence of transactions costs, the price-taking intermediary devising a hedging portfolio faces a tradeoff: to choose a short trading interval with small hedging errors and high transactions costs, or a long trading interval with large hedging errors and low transactions costs. The model presented here also recognizes that when transactions costs induce less frequent portfolio adjustments, investors are faced with a multinomial distribution of asset returns rather than a binomial one. The price upper bound is determined by selecting the trading frequency that will equalize the marginal gain from decreasing hedging errors and the marginal cost of transactions.     

Momentum and reversion in risk neutral martingale probabilities

A time homogeneous, purely discontinuous, parsimonous Markov martingale model is proposed for the risk neutral dynamics of equity forward prices. Transition probabilities are in the variance gamma class with spot dependent parameters. Markov chain approximations give access to option prices. The model is estimated on option prices across strike and maturity for five days at a time. Properties of the estimated processes are described via an analysis of return quantiles, momentum functions that measure the response of tail probabilities to such moves. Momentum and reversion are also addressed via the construction of reverse conditional expectations. Term structures for the moments of marginal distributions support a decay in skewness and excess kurtosis with maturity at rates slower than those implied by Lévy processes. Out of sample performance is additionally reported. It is observed that risk neutral dynamics by and large reflect the presence of momentum in numerous probabilities. However, there is some reversion in the upper quantiles of risk neutral return distributions.     

Fast accurate binomial pricing

We discuss here an alternative interpretation of the familiar binomial lattice approach to option pricing, illustrating it with reference to pricing of barrier options, one- and two-sided, with fixed, moving or partial barriers, and also the pricing of American put options. It has often been observed that if one tries to price a barrier option using a binomial lattice, then one can find slow convergence to the true price unless care is taken over the placing of the grid points in the lattice; see, for example, the work of Boyle & Lau [2]. The placing of grid points is critical whether one uses a dynamic programming approach, or a Monte Carlo approach, and this can make it difficult to compute hedge ratios, for example. The problems arise from translating a crossing of the barrier for the continuous diffusion process into an event for the binomial approximation. In this article, we show that it is not necessary to make clever choices of the grid positioning, and by interpreting the nature of the binomial approximation appropriately, we are able to derive very quick and accurate pricings of barrier options. The interpretation we give here is applicable much more widely, and helps to smooth out the ‘odd-even’ ripples in the option price as a function of time-to-go which are a common feature of binomial lattice pricing.     

Pricing Contingent Claims on a Risky Asset and Stochastic Interest Rates: A New Discrete Time Approach

This paper presents a new discrete time approach to pricing contingent claims on a risky asset and stochastic interest rates. The term structure of interest rates is modeled so that arbitrage-free bond prices depend on an observable initial forward rate curve rather than an exogenously specified market price of risk. A restricted binomial process is employed to model both interest rates and an asset price. As a result, a complete market valuation formula obtains. By choosing the parameters of the discrete joint distribution such that, in the limit, the discrete model converges to the continuous one, a model is obtained that requires the estimation of only three parameters. The approach is parsimonious with respect to alternative models in the literature and can be used to price contingent claims on any two state variables. The procedure is used to numerically analyze the effects of the volatility of interest rates on the determination of mortgage contract rates.     

A hyperbolic diffusion model for stock prices

In the present paper we consider a model for stock prices which is a generalization of the model behind the Black–Scholes formula for pricing European call options. We model the log-price as a deterministic linear trend plus a diffusion process with drift zero and with a diffusion coefficient (volatility) which depends in a particular way on the instantaneous stock price. It is shown that the model possesses a number of properties encountered in empirical studies of stock prices. In particular the distribution of the adjusted log-price is hyperbolic rather than normal. The model is rather successfully fitted to two different stock price data sets. Finally, the question of option pricing based on our model is discussed and comparison to the Black–Scholes formula is made. The paper also introduces a simple general way of constructing a zero-drift diffusion with a given marginal distribution, by which other models that are potentially useful in mathematical finance can be developed.     

A new approach for option pricing under stochastic volatility

We develop a new approach for pricing European-style contingent claims written on the time T spot price of an underlying asset whose volatility is stochastic. Like most of the stochastic volatility literature, we assume continuous dynamics for the price of the underlying asset. In contrast to most of the stochastic volatility literature, we do not directly model the dynamics of the instantaneous volatility. Instead, taking advantage of the recent rise of the variance swap market, we directly assume continuous dynamics for the time T variance swap rate. The initial value of this variance swap rate can either be directly observed, or inferred from option prices. We make no assumption concerning the real world drift of this process. We assume that the ratio of the volatility of the variance swap rate to the instantaneous volatility of the underlying asset just depends on the variance swap rate and on the variance swap maturity. Since this ratio is assumed to be independent of calendar time, we term this key assumption the stationary volatility ratio hypothesis (SVRH). The instantaneous volatility of the futures follows an unspecified stochastic process, so both the underlying futures price and the variance swap rate have unspecified stochastic volatility. Despite this, we show that the payoff to a path-independent contingent claim can be perfectly replicated by dynamic trading in futures contracts and variance swaps of the same maturity. As a result, the contingent claim is uniquely valued relative to its underlying’s futures price and the assumed observable variance swap rate. In contrast to standard models of stochastic volatility, our approach does not require specifying the market price of volatility risk or observing the initial level of instantaneous volatility. As a consequence of our SVRH, the partial differential equation (PDE) governing the arbitrage-free value of the contingent claim just depends on two state variables rather than the usual three. We then focus on the consistency of our SVRH with the standard assumption that the risk-neutral process for the instantaneous variance is a diffusion whose coefficients are independent of the variance swap maturity. We show that the combination of this maturity independent diffusion hypothesis (MIDH) and our SVRH implies a very special form of the risk-neutral diffusion process for the instantaneous variance. Fortunately, this process is tractable, well-behaved, and enjoys empirical support. Finally, we show that our model can also be used to robustly price and hedge volatility derivatives.     

Smooth convergence in the binomial model

In this article, we consider a general class of binomial models with an additional parameter λ. We show that in the case of a European call option the binomial price converges to the Black–Scholes price at the rate 1/n and, more importantly, give a formula for the coefficient of 1/n in the expansion of the error. This enables us, by making special choices for λ, to prove that convergence is smooth in Tian’s flexible binomial model and also in a new center binomial model which we propose. Ken Palmer was supported by NSC grant 93-2118-M-002-002.     

Warrant pricing: a review of empirical research

Recently, several warrant pricing studies have become available for different models as well as for different countries. The most important conclusions that can be drawn from reviewing these studies are: (1) it is not necessary to make a correction on option valuation models for the dilution effect; (2) the only model that systematically outperforms the Black-Scholes (1973) type models is the Square Root model; (3) US and German warrants seem to be priced correctly, while deviations are found for English and Japanese warrants (underpriced by the market) and Swiss and Dutch warrants (overpriced by the market).     

Cross-sectional tests of deterministic volatility functions

We study the cross-sectional performance of option pricing models in which the volatility of the underlying stock is a deterministic function of the stock price and time. For each date in our sample of FTSE 100 index option prices, we fit an implied binomial tree to the panel of all European style options with different strike prices and maturities and then examine how well this model prices a corresponding panel of American style options. We find that the implied binomial tree model performs no better than an ad-hoc procedure of smoothing Black–Scholes implied volatilities across strike prices and maturities. Our cross-sectional results complement the time-series findings of Dumas et al. [J. Finance 53 (1998) 2059].     

A Fuzzy Set Approach for Generalized CRR Model: An Empirical Analysis of S&;P 500 Index Options

This paper applies fuzzy set theory to the Cox, Ross and Rubinstein (CRR) model to set up the fuzzy binomial option pricing model (OPM). The model can provide reasonable ranges of option prices, which many investors can use it for arbitrage or hedge. Because of the CRR model can provide only theoretical reference values for a generalized CRR model in this article we use fuzzy volatility and fuzzy riskless interest rate to replace the corresponding crisp values. In the fuzzy binomial OPM, investors can correct their portfolio strategy according to the right and left value of triangular fuzzy number and they can interpret the optimal difference, according to their individual risk preferences. Finally, in this study an empirical analysis of S&P 500 index options is used to find that the fuzzy binomial OPM is much closer to the reality than the generalized CRR model.This project has been supported by NSC 93-2416-H-009-024.JEL Classification:     

On a general class of one-factor models for the term structure of interest rates

We propose a general one-factor model for the term structure of interest rates which based upon a model for the short rate. The dynamics of the short rate is described by an appropriate function of a time-changed Wiener process. The model allows for perfect fitting of given term structure of interest rates and volatilities, as well as for mean reversion. Moreover, every type of distribution of the short rate can be achieved, in particular, the distribution can be concentrated on an interval. The model includes several popular models such as the generalized Vasicek (or Hull-White) model, the Black-Derman-Toy, Black-Karasinski model, and others. There is a unified numerical approach to the general model based on a simple lattice approximation which, in particular, can be chosen as a binomial or -nomial lattice with branching probabilities .