Convergence of the Critical Price In the Approximation of American Options

We consider the American put option in the Black-Scholes model. When the value of the option is computed through numerical methods (such as the binomial method and the finite difference method) the approximation yields an approximate critical price. We prove the convergence of this approximate critical price towards the exact critical price.     

CRITICAL PRICE NEAR MATURITY FOR AN AMERICAN OPTION ON A DIVIDEND-PAYING STOCK IN A LOCAL VOLATILITY MODEL

We consider an American put option on a dividend-paying stock whose volatility is a function of the stock value. Near the maturity of this option, an expansion of the critical stock price is given. If the stock dividend rate is greater than the market interest rate, the payoff function is smooth near the limit of the critical price. We deduce an expansion of the critical price near maturity from an expansion of the value function of an optimal stopping problem. It turns out that the behavior of the critical price is parabolic. In the other case, we are in a less regular situation and an extra logarithmic factor appears. To prove this result, we show that the American and European critical prices have the same first-order behavior near maturity. Finally, in order to get an expansion of the European critical price, we use a parity formula for exchanging the strike price and the spot price in the value functions of European puts.     

Knock‐in American options

A knock‐in American option under a trigger clause is an option contract in which the option holder receives an American option conditional on the underlying stock price breaching a certain trigger level (also called barrier level). We present analytic valuation formulas for knock‐in American options under the Black‐Scholes pricing framework. The price formulas possess different analytic representations, depending on the relation between the trigger stock price level and the critical stock price of the underlying American option. We also performed numerical valuation of several knock‐in American options to illustrate the efficacy of the price formulas. © 2004 Wiley Periodicals, Inc. Jrl Fut Mark 24:179–192, 2004     

Optimal Stopping and the American Put

We show that the problem of pricing the American put is equivalent to solving an optimal stopping problem. the optimal stopping problem gives rise to a parabolic free-boundary problem. We show there is a unique solution to this problem which has a lower boundary. We identify an integral equation solved by the boundary and show that it is the unique solution to this equation satisfying certain natural additional conditions. the proofs also give a natural decomposition of the price of the American option as the sum of the price of the European option and an "American premium."     

Arbitrage Values Generally Depend On A Parametric Rate of Return

Let X denote a positive Markov stochastic integral, and let S ( t , μ) = exp(μ t ) X ( t ) represent the price of a security at time t with infinitesimal rate of return μ. Contingent claim (option) pricing formulas typically do not depend on μ. We show that if a contingent claim is not equivalent to a call option having exercise price equal to zero, then security prices having this property—option prices do not depend on μ—must satisfy: for some V (0, T ), In( S ( t , μ) X ( V )) is Gaussian on a time interval [ V, T ], and hence S ( t , μ) has independent observed returns. With more assumptions, V = 0, and there exist equivalent martingale measures.     

ANALYTIC APPROXIMATIONS FOR MULTI‐ASSET OPTION PRICING

We derive general analytic approximations for pricing European basket and rainbow options on N assets. The key idea is to express the option’s price as a sum of prices of various compound exchange options, each with different pairs of subordinate multi‐ or single‐asset options. The underlying asset prices are assumed to follow lognormal processes, although our results can be extended to certain other price processes for the underlying. For some multi‐asset options a strong condition holds, whereby each compound exchange option is equivalent to a standard single‐asset option under a modified measure, and in such cases an almost exact analytic price exists. More generally, approximate analytic prices for multi‐asset options are derived using a weak lognormality condition, where the approximation stems from making constant volatility assumptions on the price processes that drive the prices of the subordinate basket options. The analytic formulae for multi‐asset option prices, and their Greeks, are defined in a recursive framework. For instance, the option delta is defined in terms of the delta relative to subordinate multi‐asset options, and the deltas of these subordinate options with respect to the underlying assets. Simulations test the accuracy of our approximations, given some assumed values for the asset volatilities and correlations. Finally, a calibration algorithm is proposed and illustrated.     

Early exercise of American put options: Investor rationality on the Swedish equity options market

Using Swedish equity option data, this study investigates how well the actual exercise behavior of American put options corresponds to the early exercise rules. The optimal exercise strategy is established in two ways. First, the critical exercise price, above which a put option should be exercised early, is computed and compared to the actual exercise price. Second, the exercise value of the option is compared to its market bid price. The results show that most early exercise decisions conform to rational exercise behavior, even though a large number of failures to exercise are found. Most of the faulty exercises can also be discarded after a sensitivity analysis, although several failures to exercise are considered irrational, even after taking transaction costs into account. © 2000 John Wiley & Sons, Inc. Jrl Fut Mark 20:167–188, 2000     

A generalization of Rubinstein's “Pay now,choose later”

This article provides quasi‐analytic pricing formulae for forward‐start options under stochastic volatility, double jumps, and stochastic interest rates. Our methodology is a generalization of the Rubinstein approach and can be applied to several existing option models. Properties of a forward‐start option may be very different from those of a plain vanilla option because the entire uncertainty of evolution of its price is cut off by the strike price at the time of determination. For instance, in contrast to the plain vanilla option, the value of a forward‐start option may not always increase as the maturity increases. It depends on the current term structure of interest rates. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28:488–515, 2008     

Strategic Role of Retailer Bundling in a Distribution Channel

We study retailer bundling in a distribution channel when the manufacturer for one bundled product can strategically set the wholesale price. We show that the retailer can use a bundling option as a strategic leverage to extract concessions from the manufacturer in form of a lower wholesale price. This finding contributes a novel rationale for retailer bundling to the bundling literature. Whenever the bundling option causes this concession-extraction effect, the retailer always benefits from the lower wholesale price. The manufacturer, nevertheless, does not necessarily suffer because bundling can lead to a higher consumer demand. We also show that the manufacturer's marginal production cost plays a critical role in driving the retailer's bundling decision, concession extraction behavior and consequently the total channel profit.     

EXACT SOLUTION OF A MARTINGALE STOCHASTIC VOLATILITY OPTION PROBLEM AND ITS EMPIRICAL EVALUATION

Exact explicit solution of the log-normal stochastic volatility (SV) option model has remained an open problem for two decades. In this paper, I consider the case where the risk-neutral measure induces a martingale volatility process, and derive an exact explicit solution to this unsolved problem which is also free from any inverse transforms. A representation of the asset price shows that its distribution depends on that of two random variables, the terminal SV as well as the time average of future stochastic variances. Probabilistic methods, using the author's previous results on stochastic time changes, and a Laplace–Girsanov Transform technique are applied to produce exact explicit probability distributions and option price formula. The formulae reveal interesting interplay of forces between the two random variables through the correlation coefficient. When the correlation is set to zero, the first random variable is eliminated and the option formula gives the exact formula for the limit of the Taylor series in Hull and White's (1987) approximation. The SV futures option model, comparative statics, price comparisons, the Greeks and practical and empirical implementation and evaluation results are also presented. A PC application was developed to fit the SV models to current market prices, and calculate other option prices, and their Greeks and implied volatilities (IVs) based on the results of this paper. This paper also provides a solution to the option implied volatility problem, as the empirical studies show that, the SV model can reproduce market prices, better than Black–Scholes and Black-76 by up to 2918%, and its IV curve can reproduce that of market prices very closely, by up to within its 0.37%.     

ALTERNATIVE CHARACTERIZATIONS OF AMERICAN PUT OPTIONS

We derive alternative representations of the McKean equation for the value of the American put option. Our main result decomposes the value of an American put option into the corresponding European put price and the early exercise premium. We then represent the European put price in a new manner. This representation allows us to alternatively decompose the price of an American put option into its intrinsic value and time value, and to demonstrate the equivalence of our results to the McKean equation.     

A note on the derivation of Black‐Scholes hedge ratios

An option hedge ratio is the sensitivity of an option price with respect to price changes in the underlying stock. It measures the number of shares of stocks to hedge an option position. This article presents a simple derivation of the hedge ratios under the Black‐Scholes option‐pricing framework. The proof is succinct and easy to follow. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:1119–1122, 2003     

Valuation and management of money-back guarantee options

In this article, we model money-back guarantees (MBGs) as put options. This use of option theory provides retailers with a framework to optimize the price and the return option independently and under various market conditions. This separation of product price and option value enables retailers to offer an unbundled MBG policy, that is, to allow the customer to choose whether to purchase an MBG option with the product or to buy the product without the MBG but at a lower price. The option value of having an MBG is negatively correlated with the likelihood of product fit and with the opportunity to test the product before purchase, and positively correlated with price and contract duration. Simulation of our model reveals that when customers are highly heterogeneous in their product valuation and probability of need-fit, and if return costs are low, an unbundled MBG policy is optimal. When customers have high likelihood of fit or return costs are excessive, no MBG is the best policy. When customers have small variance in product valuation, but vary greatly in likelihood of product fit, the retailer may prefer to offer a bundled MBG contract, extracting consumer surplus by charging a price close to the valuation level.     

Hedging under the influence of transaction costs: An empirical investigation on FTSE 100 index options

The Black–Scholes (BS; F. Black & M. Scholes, 1973) option pricing model, and modern parametric option pricing models in general, assume that a single unique price for the underlying instrument exists, and that it is the mid‐ (the average of the ask and the bid) price. In this article the authors consider the Financial Times and London Stock Exchange (FTSE) 100 Index Options for the time period 1992–1997. They estimate the ask and bid prices for the index, and show that, when substituted for the mid‐price in the BS formula, they provide superior option price predictors, for call and put options, respectively. This result is reinforced further when they .t a non‐parametric neural network model to market prices of liquid options. The empirical .ndings in this article suggest that the ask and bid prices of the underlying asset provide a superior fit to the mid/closing price because they include market maker's, compensation for providing liquidity in the market for constituent stocks of the FTSE 100 index. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:471–494, 2007     

Robustness of the Black and Scholes Formula

Consider an option on a stock whose volatility is unknown and stochastic. An agent assumes this volatility to be a specific function of time and the stock price, knowing that this assumption may result in a misspecification of the volatility. However, if the misspecified volatility dominates the true volatility, then the misspecified price of the option dominates its true price. Moreover, the option hedging strategy computed under the assumption of the misspecified volatility provides an almost sure one-sided hedge for the option under the true volatility. Analogous results hold if the true volatility dominates the misspecified volatility. These comparisons can fail, however, if the misspecified volatility is not assumed to be a function of time and the stock price. The positive results, which apply to both European and American options, are used to obtain a bound and hedge for Asian options.     

Price Clustering in Individual Equity Options: Moneyness,Maturity, and Price Level

Equity options have a significant influence on the price discovery process. This study presents unique evidence of substantial price clustering in individual equity options contracts. A particular contribution arises from investigating competing hypotheses on the roles of moneyness and maturity as determinants of option price clustering. We assert that options price clustering can be decomposed to price level, moneyness, and maturity effects. After controlling for other factors, price clustering has an inverse relation with time‐to‐maturity. This supports the negotiation hypothesis, but not the price resolution hypothesis. Price clustering also tends to be inversely related to moneyness. This effect is linked to the intrinsic value component of option price. Both the maturity and moneyness effects act in an opposite direction to what would be anticipated on the basis of price level alone; hence, these two effects are identified as additional influences on option price clustering. It is also found that the designated market maker scheme at NYSE Euronext London International Financial Futures Exchange (LIFFE) has little influence on trade price clustering. © 2012 Wiley Periodicals, Inc. Jrl Fut Mark 33:55–76, 2013     

The Statistical Properties of the Black–Scholes Option Price

This paper investigates the statistical properties of the Black–Scholes option price, considered as a random variable. The option is conditioned on the current price and/or the estimated volatility of the underlying security. In both cases, some exact results for the distribution functions of the true option price and the predicted option price are derived. Extensions to puts and American contracts are considered. Numerical results are presented for option prices based on parameters appropriate for the FTSE 100 Index.     

Anatomy of option features in convertible bonds

Several earlier theoretical studies on the optimal issuer's calling policy of a convertible bond suggest that the issuer should call the bond as soon as the conversion value exceeds the call price. However, empirical studies on actual cases of calling by convertible bond issuers reveal that firms “delayed” calling their convertible bonds until the conversion value well exceeded the call price. In this paper, we construct valuation algorithms that price risky convertible bonds with embedded option features. In particular, we examine the impact of the soft call and hard call constraints, notice period requirement and other factors on the optimal issuer's calling policy. Our results show that the critical stock price at which the issuer should optimally call the convertible bond depends quite sensibly on these constraints and requirements. The so‐called “delayed call phenomena” may be largely attributed to the underestimation of the critical call price due to inaccurate modeling of the contractual provisions. © 2004 Wiley Periodicals, Inc. Jrl Fut Mark 24:513–532, 2004     

DISUTILITY, OPTIMAL RETIREMENT, AND PORTFOLIO SELECTION

We study the optimal retirement and consumption/investment choice of an infinitely-lived economic agent with a time-separable von Neumann–Morgenstern utility. A particular aspect of our problem is that the agent has a retirement option. Before retirement the agent receives labor income but suffers a utility loss from labor. By retiring, he avoids the utility loss but gives up labor income. We show that the agent retires optimally if his wealth exceeds a certain critical level. We also show that the agent consumes less and invests more in risky assets when he has an option to retire than he would in the absence of such an option.
An explicit solution can be provided by solving a free boundary value problem. In particular, the critical wealth level and the optimal consumption and portfolio policy are provided in explicit forms.     

Improving lattice schemes through bias reduction

Lattice schemes for option pricing, such as tree or grid/partial differential equation (p.d.e.) methods, are usually designed as a discrete version of an underlying continuous model of stock prices. The parameters of such schemes are chosen so that the discrete version “best” matches the continuous one. Only in the limit does the lattice option price model converge to the continuous one. Otherwise, a discretization bias remains. A simple modification of lattice schemes which reduces the discretization bias is proposed. The modification can, in theory, be applied to any lattice scheme. The main idea is to adjust the lattice parameters in such a way that the option price bias, not the stock price bias, is minimized. European options are used, for which the option price bias can be evaluated precisely, as a template to modify and improve American option methods. A numerical study is provided. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:733–757, 2006