A non‐lattice pricing model of American options under stochastic volatility

In this article, an analytical approach to American option pricing under stochastic volatility is provided. Under stochastic volatility, the American option value can be computed as the sum of a corresponding European option price and an early exercise premium. By considering the analytical property of the optimal exercise boundary, the formula allows for recursive computation of the American option value. Simulation results show that a nonlattice method performs better than the lattice‐based interpolation methods. The stochastic volatility model is also empirically tested using S&P 500 futures options intraday transactions data. Incorporating stochastic volatility is shown to improve pricing, hedging, and profitability in actual trading. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:417–448, 2006     

Futures price limit moves as options

This note demonstrates that an asset's price in an environment with price limit rules can be replicated by the price of a portfolio consisting of a riskless asset and two synthetic options. A procedure is developed to unbundle the unobservable option values imbedded in the actual futures price and impute a theoretical true futures price. Using this framework, evidence from the Treasury Bond futures market suggests that theoretical true futures prices diverge from actual futures prices, on average, 3 h prior to the activation of price limit rules, indicating that price limit moves might be predictable. The reversal of both the actual futures prices and the theoretical futures prices back within the limit range after a limit move provides support for the possibility that traders tend to overreact when market prices are near price limits. © 2002 Wiley Periodicals, Inc. Jrl Fut Mark 22:901–913, 2002     

The Chinese equity index options market

Using Bakshi et al. (2000), and Bakshi and Kapadia's (2003) methodology, this paper studies the Chinese equity index options market that has been developing since 2015. Empirical evidence shows that the market price of call (put) option is generally lower (higher) than their Black-Scholes prices with historical volatility. The prices of the options do not support the one-dimensional diffusion model properties. We find 61.79% (63.25%) of delta-hedged gains in call (put) options to be negative. The analysis of the non-zero delta-hedged gain suggests that the investors are mainly trading on additional volatility risk in the options market in China.     

Pricing arithmetic Asian and Amerasian options: A diffusion operator integral expansion approach

In this paper, we propose a new explicit series expansion formula for the price of an arithmetic Asian option under the Black–Scholes model and Merton's jump-diffusion model. The method is based on an equivalence in law relation together with the diffusion operator integral method proposed by Heath and Platen. The method yields explicit series expansion formula for the Asian options' prices. The theoretical convergence of the expansion to the true value is established. We also consider the American Asian option (i.e., Amerasian option) and derive the corresponding expansion formula through the early exercise premium representation. Numerical results illustrate the accuracy and efficiency of the method as compared with benchmarks in the literature.     

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.     

Empirical performance of alternative pricing models of currency options

This article examines the out‐of‐sample pricing performance and biases of the Heston’s stochastic volatility and modified Black‐Scholes option pricing models in valuing European currency call options written on British pound. The modified Black‐Scholes model with daily‐revised implied volatilities performs as well as the stochastic volatility model in the aggregate sample. Both models provide close and similar correspondence to actual prices for options trading near‐ or at‐the‐money. The prices generated from the stochastic volatility model are subject to fewer and weaker aggregate pricing biases than are the prices from the modified Black‐Scholes model. Thus, the stochastic volatility model may provide improved estimates of the measures of option price sensitivities to key option parameters that may lead to more effective hedging and speculative strategies using currency options. © 2000 John Wiley & Sons, Inc. Jrl Fut Mark 20:265–291, 2000     

Step‐reset options: Design and valuation

This study proposes a new design of reset options in which the option's exercise price adjusts gradually, based on the amount of time the underlying spent beyond prespecified reset levels. Relative to standard reset options, a step‐reset design offers several desirable properties. First of all, it demands a lower option premium but preserves the same desirable reset attribute that appeals to market investors. Second, it overcomes the disturbing problem of delta jump as exhibited in standard reset option, and thus greatly reduces the difficulties in risk management for reset option sellers who hedge dynamically. Moreover, the step‐reset feature makes the option more robust against short‐term price movements of the underlying and removes the pressure of price manipulation often associated with standard reset options. To value this innovative option product, we develop a tree‐based valuation algorithm in this study. Specifically, we parameterize the trinomial tree model to correctly account for the discrete nature of reset monitoring. The use of lattice model gives us the flexibility to price step‐reset options with American exercise right. Finally, to accommodate the path‐dependent exercise price, we introduce a state‐to‐state recursive pricing procedure to properly capture the path‐dependent step‐reset effect and enhance computational efficiency. © 2002 John Wiley & Sons, Inc. Jrl Fut Mark 22:155–171, 2002     

Monte Carlo valuation of American options

This paper introduces a dual way to price American options, based on simulating the paths of the option payoff, and of a judiciously chosen Lagrangian martingale. Taking the pathwise maximum of the payoff less the martingale provides an upper bound for the price of the option, and this bound is sharp for the optimal choice of Lagrangian martingale. As a first exploration of this method, four examples are investigated numerically; the accuracy achieved with even very simple choices of Lagrangian martingale is surprising. The method also leads naturally to candidate hedging policies for the option, and estimates of the risk involved in using them.     

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.     

The promoter's role in ticket pricing: Implications of real options for optimal posted prices and rationing

We consider the problem of pricing event tickets for initial sale when demand is uncertain. It is a standard industry practice for a performer to contract with a promoter who underwrites the event and offers the tickets for sale at a posted price that is sticky in that it is either fixed or costly to adjust once sales begin. Promoters, therefore, bear price risk, and we show that bearing the risk associated with posting a sticky offer price amounts to writing a put option on the ticket revenue. Further, we show that optimal posted-offer prices can be expected to result in rationing (surpluses) if price uncertainty and price elasticity of demand are material (immaterial), even when the demand forecast is accurate. Our results have implications for a more general set of pricing problems in which items are offered for sale at sticky posted prices.     

The optimal method for pricing Bermudan options by simulation

Least‐squares methods enable us to price Bermudan‐style options by Monte Carlo simulation. They are based on estimating the option continuation value by least‐squares. We show that the Bermudan price is maximized when this continuation value is estimated near the exercise boundary, which is equivalent to implicitly estimating the optimal exercise boundary by using the value‐matching condition. Localization is the key difference with respect to global regression methods, but is fundamental for optimal exercise decisions and requires estimation of the continuation value by iterating local least‐squares (because we estimate and localize the exercise boundary at the same time). In the numerical example, in agreement with this optimality, the new prices or lower bounds (i) improve upon the prices reported by other methods and (ii) are very close to the associated dual upper bounds. We also study the method's convergence.     

The valuation of American options in a multidimensional exponential Lévy model

We consider the problem of valuation of American options written on dividend‐paying assets whose price dynamics follow a multidimensional exponential Lévy model. We carefully examine the relation between the option prices, related partial integro‐differential variational inequalities, and reflected backward stochastic differential equations. In particular, we prove regularity results for the value function and obtain the early exercise premium formula for a broad class of payoff functions.     

Lower‐boundary violations and market efficiency: Evidence from the German DAX‐index options market

The informational efficiency of the market for options on the German stock index DAX is examined using intraday transactions data. Problems of previous studies on options‐market efficiency, arising from dividend estimation and the early‐exercise effect, are avoided, because the DAX is a performance index and DAX options are European options. Ex‐post and ex‐ante tests are carried out to simulate trading strategies that exploit irrational lower‐boundary violations of observed option prices. Because the lower‐boundary conditions are solely based on arbitrage considerations, the test results do not depend on the assumption that investors use a particular option‐pricing model. The investigation shows that ex‐post profits are, in general, dramatically reduced when the execution of arbitrage strategies is delayed and/or transaction costs are accounted for. However, arbitrage restrictions, which rely on short selling of the component stocks of the index, tend to be violated more often and with higher persistence. An analysis of consecutive subsamples suggests that, over time, traders have been subjected to a learning process when pricing this relatively new instrument. © 2000 John Wiley & Sons, Inc. Jrl Fut Mark 20: 405–424, 2000     

The Price of Being a Systemically Important Financial Institution (SIFI)

After reviewing the notion of Systemically Important Financial Institution, we propose a first principles way to compute the price of the implicit put option that the State gives to such an institution. Our method is based on important results from extreme value theory, one for the aggregation of heavy‐tailed distributions and the other one for the tail behavior of the value at risk versus the tail value at risk. We show that the value of the put option is proportional to the value at risk of the institution and thus would provide the wrong incentive to banks who are qualified as Systemically Important Financial Institutions. This wrong incentive exists even if the guarantee is not explicitly granted. We conclude with a proposal to make the institution pay the price of this option to a fund, whose task would be to guarantee the orderly bankruptcy of such an institution. This fund would function like an insurance selling a cover to clients.     

Pricing American exchange options in a jump‐diffusion model

A way to estimate the value of an American exchange option when the underlying assets follow jump‐diffusion processes is presented. The estimate is based on combining a European exchange option and a Bermudan exchange option with two exercise dates by using Richardson extrapolation as proposed by R. Geske and H. Johnson (1984). Closed‐form solutions for the values of European and Bermudan exchange options are derived. Several numerical examples are presented, illustrating that the early exercise feature may have a significant economic value. The results presented should have potential for pricing over‐the‐counter options and in particular for pricing real options. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:257–273, 2007     

Price behavior in the light of balance of payments theories

The price predictions of the elasticity and monetary theories of balance of payments adjustment are compared with actual price behavior. Price behavior differs more from the relatively demanding monetary approach in that price levels and price movements for GDP as a whole and for specific types of export goods varied substantially even among major industrial countries. As for the elasticity approach, price levels tended to rise with appreciations and fall with depreciations, as expected.     

Pricing American options by canonical least‐squares Monte Carlo

Options pricing and hedging under canonical valuation have recently been demonstrated to be quite effective, but unfortunately are only applicable to European options. This study proposes an approach called canonical least‐squares Monte Carlo (CLM) to price American options. CLM proceeds in three stages. First, given a set of historical gross returns (or price ratios) of the underlying asset for a chosen time interval, a discrete risk‐neutral distribution is obtained via the canonical approach. Second, from this canonical distribution independent random samples of gross returns are taken to simulate future price paths for the underlying. Third, to those paths the least‐squares Monte Carlo algorithm is then applied to obtain early exercise strategies for American options. Numerical results from simulation‐generated gross returns under geometric Brownian motions show that the proposed method yields reasonably accurate prices for American puts. The CLM method turns out to be quite similar to the nonparametric approach of Alcock and Carmichael and simulations done with CLM provide additional support for their recent findings. CLM can therefore be viewed as an alternative for pricing American options, and perhaps could even be utilized in cases when the nature of the underlying process is not known. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:175–187, 2010     

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     

CRITICAL STOCK PRICE NEAR EXPIRATION

We study the critical price of an American put option near expiration in the Black-Scholes model. Our main result is an estimate for the difference ( t )- K between the critical price at time t and the exercise price as t approaches the maturity of the option.     

Pricing American options with stochastic volatility: Evidence from S&P 500 futures options

This article is the first attempt to test empirically a numerical solution to price American options under stochastic volatility. The model allows for a mean‐reverting stochastic‐volatility process with non‐zero risk premium for the volatility risk and correlation with the underlying process. A general solution of risk‐neutral probabilities and price movements is derived, which avoids the common negative‐probability problem in numerical‐option pricing with stochastic volatility. The empirical test shows clear evidence supporting the occurrence of stochastic volatility. The stochastic‐volatility model outperforms the constant‐volatility model by producing smaller bias and better goodness of fit in both the in‐sample and out‐of‐sample test. It not only eliminates systematic moneyness bias produced by the constant‐volatility model, but also has better prediction power. In addition, both models perform well in the dynamic intraday hedging test. However, the constant‐volatility model seems to have a slightly better hedging effectiveness. The profitability test shows that the stochastic volatility is able to capture statistically significant profits while the constant volatility model produces losses. © 2000 John Wiley & Sons, Inc. Jrl Fut Mark 20:625–659, 2000