Option Bounds with Finite Revision Opportunities

This article generalizes the single-period linear-programming bounds on option prices by allowing for a finite number of revision opportunities. It is shown that, in an incomplete market, the bounds on option prices can be derived using a modified binomial option-pricing model. Tighter bounds are developed under more restrictive assumptions on probabilities and risk aversion. For this case the upper bounds are shown to coincide with the upper bounds derived by Perrakis, while the lower bounds are shown to be tighter.     

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.     

Option Bounds and the Pricing of the Volatility Smile

This paper presents a new approach forthe estimation of the risk-neutral probability distribution impliedby observed option prices in the presence of a non-horizontalvolatility smile. This approach is based on theoretical considerationsderived from option pricing in incomplete markets. Instead ofa single distribution, a pair of risk-neutral distributions areestimated, that bracket the option prices defined by the volatilitybid/ask midpoint. These distributions define upper and lowerbounds on option prices that are consistent with the observableoption parameters and are the tightest ones possible, in thesense of minimizing the distance between the option upper andlower bounds. The application of the new approach to a sampleof observations on the S&P 500 option market showsthat the bounds produces are quite tight, and also that theirderivation is robust to the presence of violations of arbitragerelations in option quotes, which cause many other methods tofail.     

Shedding light on a dark matter: Jump diffusion and option-implied investor preferences

We compare equilibrium jump diffusion option prices with endogenously determined stochastic dominance (SD) option bounds. We use model parameters from earlier studies and find that most equilibrium model prices consistent with SD bounds yield economically meaningless results. Further, the implied distributions of the SD bounds exhibit a tail risk comparable to that of the underlying return data, thus shedding light on the dark matter of the inconsistency of physical and risk-neutral tail probabilities. Since the SD bound assumptions are weaker, we conclude that these bounds should either replace or be used to verify the equilibrium model results.     

Minimum option prices under decreasing absolute risk aversion

We establish bounds on option prices in an economy where the representative investor has an unknown utility function that is constrained to belong to the family of nonincreasing absolute risk averse functions. For any distribution of terminal consumption, we identify a procedure that establishes the lower bound of option prices. We prove that the lower bound derives from a particular negative exponential utility function. We also identify lower bounds of option prices in a decreasing relative risk averse economy. For this case, we find that the lower bound is determined by a power utility function. Similar to other recent findings, for the latter case, we confirm that under lognormality of consumption, the Black Scholes price is a lower bound. The main advantage of our bounding methodology is that it can be applied to any arbitrary marginal distribution for consumption. This revised version was published online in June 2006 with corrections to the Cover Date.     

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.     

A simple approach for pricing barrier options with time-dependent parameters

Abstract

In this paper we present a simple and easy-to-use method for computing accurate estimates (in closed form) of Black-Scholes barrier option prices with time-dependent parameters. This new approach is also able to provide tight upper and lower bounds (in closed form) for the exact barrier option prices.     

Option Prices and the Underlying Asset's Return Distribution

This work examines the relation between option prices and the true, as opposed to risk-neutral, distribution of the underlying asset. If the underlying asset follows a diffusion with an instantaneous expected return at least as large as the instantaneous risk-free rate, observed option prices can be used to place bounds on the moments of the true distribution. An illustration of the paper's results is provided by the analysis of the information concerning the mean and standard deviation of market returns contained in the prices of S&P 100 Index Options.     

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.     

Option Pricing Bounds: Synthesis And Extension

The main option pricing bounds in the literature were originally obtained through various disparate methods. I show that those bounds can be derived from a single analytical framework. The key to this synthesis lies in the use of a general expression for the price of a call option depending on the corresponding put option's discount factor. Although the put's discount factor is unknown, it can be bounded from below. I use this lower bound on the put's discount factor to derive traditional lower bounds for call prices. In addition, I extend the literature by finding a new tighter lower bound.     

Option Pricing and Replication with Transactions Costs

Transactions costs invalidate the Black-Scholes arbitrage argument for option pricing, since continuous revision implies infinite trading. Discrete revision using Black-Scholes deltas generates errors which are correlated with the market, and do not approach zero with more frequent revision when transactions costs are included. This paper develops a modified option replicating strategy which depends on the size of transactions costs and the frequency of revision. Hedging errors are uncorrelated with the market and approach zero with more frequent revision. The technique permits calculation of the transactions costs of option replication and provides bounds on option prices.     

Non-parametric method for European option bounds

There is much research whose efforts have been devoted to discovering the distributional defects in the Black–Scholes model, which are known to cause severe biases. However, with a free specification for the distribution, one can only find upper and lower bounds for option prices. In this paper, we derive a new non-parametric lower bound and provide an alternative interpretation of Ritchken’s (J Finance 40:1219–1233, 1985) upper bound to the price of the European option. In a series of numerical examples, our new lower bound is substantially tighter than previous lower bounds. This is prevalent especially for out of the money options where the previous lower bounds perform badly. Moreover, we present how our bounds can be derived from histograms which are completely non-parametric in an empirical study. We discover violations in our lower bound and show that those violations present arbitrage profits. In particular, our empirical results show that out of the money calls are substantially overpriced (violate the lower bound).     

Retrieving risk neutral moments and expected quadratic variation from option prices

This paper derives exact formulas for retrieving risk neutral moments of future payoffs of any order from generic European-style option prices. It also provides an exact formula for retrieving the expected quadratic variation of the stock market implied by European option prices, which nowadays is used as an estimate of the implied volatility, and a formula approximating the jump component of this measure of variation. To implement the above formulas to discrete sets of option prices, the paper suggests a numerical procedure and provides upper bounds of its approximation errors. The performance of this procedure is evaluated through a simulation and an empirical exercise. Both of these exercises clearly indicate that the suggested numerical procedure can provide accurate estimates of the risk neutral moments, over different horizons ahead. These can be in turn employed to obtain accurate estimates of risk neutral densities and calculate option prices, efficiently, in a model-free manner. The paper also shows that, in contrast to the prevailing view, ignoring the jump component of the underlying asset can lead to seriously biased estimates of the new volatility index suggested by the Chicago Board Options Exchange.     

Tight bounds on American option prices

In contrast to the constant exercise boundary assumed by Broadie and Detemple (1996) [Broadie, M., Detemple, J., 1996. American option valuation: New bounds, approximations, and comparison of existing methods. Review of Financial Studies 9, 1211–1250], we use an exponential function to approximate the early exercise boundary. Then, we obtain lower bounds for American option prices and the optimal exercise boundary which improve the bounds of Broadie and Detemple (1996). With the tight lower bound for the optimal exercise boundary, we further derive a tight upper bound for the American option price using the early exercise premium integral of Kim (1990) [Kim, I.J., 1990. The analytic valuation of American options. Review of Financial Studies 3, 547–572]. The numerical results show that our lower and upper bounds are very tight and can improve the pricing errors of the lower bound and upper bound of Broadie and Detemple (1996) by 83.0% and 87.5%, respectively. The tightness of our upper bounds is comparable to some best accurate/efficient methods in the literature for pricing American options. Moreover, the results also indicate that the hedge ratios (deltas and gammas) of our bounds are close to the accurate values of American options.     

Efficient estimation of lower and upper bounds for pricing higher-dimensional American arithmetic average options by approximating their payoff functions

In this paper, we develop an efficient payoff function approximation approach to estimating lower and upper bounds for pricing American arithmetic average options with a large number of underlying assets. The crucial step in the approach is to find a geometric mean which is more tractable than and highly correlated with a given arithmetic mean. Then the optimal exercise strategy for the resultant American geometric average option is used to obtain a low-biased estimator for the corresponding American arithmetic average option. This method is particularly efficient for asset prices modeled by jump-diffusion processes with deterministic volatilities because the geometric mean is always a one-dimensional Markov process regardless of the number of underlying assets and thus is free from the curse of dimensionality. Another appealing feature of our method is that it provides an extremely efficient way to obtain tight upper bounds with no nested simulation involved as opposed to some existing duality approaches. Various numerical examples with up to 50 underlying stocks suggest that our algorithm is able to produce computationally efficient results.     

American option valuation: new bounds, approximations, and a comparison of existing methods

We develop lower and upper bounds on the prices of Americancall and put options written on a dividend-paying asset. Weprovide two option price approximations one based on the lowerbound (termed LBA) and one based on both bounds (termed LUBA).The LUBA approximation has an average accuracy comparable toa l,000-step binomial tree. We introduce a modification of thebinomial method (termed BBSR) that is very simple to implementand performs remarkably well. We also conduct a careful large-scaleevaluation of many recent methods for computing American optionprices.     

Arbitrage pricing,transaction costs and taxation of capital gains: A study of government bonds with the same maturity date

This paper examines one of the few cases of seemingly redundant securities: sets of three government bonds with the same maturity date. Within the bounds on relative bond prices established by tax-exempt investors in a market with proportional transaction costs, the taxation of capital gains on the basis of realization has a significant impact on relative prices. The empirical evidence supports the tax option effect discussed by Constantinides and Ingersoll, but does not generally support the segmented tax-clientele equilibrium discussed by Schaefer.     

Analysis of the Nonlinear Option Pricing Model Under Variable Transaction Costs

In this paper we analyze a nonlinear Black–Scholes model for option pricing under variable transaction costs. The diffusion coefficient of the nonlinear parabolic equation for the price V is assumed to be a function of the underlying asset price and the Gamma of the option. We show that the generalizations of the classical Black–Scholes model can be analyzed by means of transformation of the fully nonlinear parabolic equation into a quasilinear parabolic equation for the second derivative of the option price. We show existence of a classical smooth solution and prove useful bounds on the option prices. Furthermore, we construct an effective numerical scheme for approximation of the solution. The solutions are obtained by means of the efficient numerical discretization scheme of the Gamma equation. Several computational examples are presented.