Bounds for the price of a European-style Asian option in a binary tree model

Inspired by the ideas of Rogers and Shi [J. Appl. Prob. 32 (1995) 1077], Chalasani et al. [J. Comput. Finance 1(4) (1998) 11] derived accurate lower and upper bounds for the price of a European-style Asian option with continuous averaging over the full lifetime of the option, using a discrete-time binary tree model. In this paper, we consider arithmetic Asian options with discrete sampling and we generalize their method to the case of forward starting Asian options. In this case with daily time steps, the method of Chalasani et al. is still very accurate but the computation can take a very long time on a PC when the number of steps in the binomial tree is high. We derive analytical lower and upper bounds based on the approach of Kaas et al. [Insurance: Math. Econ. 27 (2000) 151] for bounds for stop-loss premiums of sums of dependent random variables, and by conditioning on the value of underlying asset at the exercise date. The comonotonic upper bound corresponds to an optimal superhedging strategy. By putting in less information than Chalasani et al. the bounds lose some accuracy but are still very good and they are easily computable and moreover the computation on a PC is fast. We illustrate our results by different numerical experiments and compare with bounds for the Black and Scholes model [J. Pol. Econ. 7 (1973) 637] found in another paper [Bounds for the price of discretely sampled arithmetic Asian options, Working paper, Ghent University, 2002]. We notice that the intervals of Chalasani et al. do not always lie within the Black and Scholes intervals. We have proved that our bounds converge to the corresponding bounds in the Black and Scholes model. Our numerical illustrations also show that the hedging error is small if the Asian option is in the money. If the option is out of the money, the price of the superhedging strategy is not as adequate, but still lower than the straightforward hedge of buying one European option with the same exercise price.     

随机利率下亚式期权的定价模型

§1Introduction Asianoptionpayoffdependsontheaverageofassetpricesoverthelifeofoptions.Theirpopularityistoavoidthepossiblepricemanipulationatthematuritydatefor ordinaryoptions.ItturnsouttobedifficulttoderiveBlack-Scholes-likeclosed-form formulaforAsianoptionsbecausethedistributionofarithmetic-averageassetpricesdoes nothavestandardexpression.AlotofworkhasbeendoneonpricingAsianoptionssince KemmaandVorst(1990).Manytreatmentsdealwiththecaseofgeometricaverageforthe firststepeitherasanapproximatio…     

On pricing arithmetic average reset options with multiple reset dates in a lattice framework

We develop a straightforward algorithm to price arithmetic average reset options with multiple reset dates in a Cox et al. (CRR) (1979) [10] framework. The use of a lattice approach is due to its adaptability and flexibility in managing arithmetic average reset options, as already evidenced by Kim et al. (2003) [9]. Their model is based on the Hull and White (1993) [5] bucketing algorithm and uses an exogenous exponential function to manage the averaging feature, but their choice of fictitious values does not guarantee the algorithm’s convergence (cfr., Forsyth et al. (2002) [11]). We propose to overcome this drawback by selecting a limited number of trajectories among the ones reaching each node of the lattice, where we compute effective averages. In this way, the computational cost of the pricing problem is reduced, and the convergence of the discrete time model to the corresponding continuous time one is guaranteed.     

Convergence of the binomial tree method for Asian options in jump-diffusion models

The binomial tree methods (BTM), first proposed by Cox, Ross and Rubinstein [J. Cox, S. Ross, M. Rubinstein, Option pricing: A simplified approach, J. Finan. Econ. 7 (1979) 229-264] in diffusion models and extended by Amin [K.I. Amin, Jump diffusion option valuation in discrete time, J. Finance 48 (1993) 1833-1863] to jump-diffusion models, is one of the most popular approaches to pricing options. In this paper, we present a binomial tree method for Asian options in jump-diffusion models and show its equivalence to certain explicit difference scheme. Employing numerical analysis and the notion of viscosity solution, we prove the uniform convergence of the binomial tree method for European-style and American-style Asian options.     

A mathematical modeling for the lookback option with jump-diffusion using binomial tree method

The binomial tree method (BTM), first proposed by Cox et al. (1979) [4] in diffusion models and extended by Amin (1993) [9] to jump-diffusion models, is one of the most popular approaches to pricing options. In this paper, we present a binomial tree method for lookback options in jump-diffusion models and show its equivalence to certain explicit difference scheme. We also prove the existence and convergence of the optimal exercise boundary in the binomial tree approximation to American lookback options and give the terminal value of the genuine exercise boundary. Further, numerical simulations are performed to illustrate the theoretical results.     

Bounds for Asian basket options

In this paper we propose pricing bounds for European-style discrete arithmetic Asian basket options in a Black and Scholes framework. We start from methods used for basket options and Asian options. First, we use the general approach for deriving upper and lower bounds for stop-loss premia of sums of non-independent random variables as in Kaas et al. [Upper and lower bounds for sums of random variables, Insurance Math. Econom. 27 (2000) 151–168] or Dhaene et al. [The concept of comonotonicity in actuarial science and finance: theory, Insurance Math. Econom. 31(1) (2002) 3–33]. We generalize the methods in Deelstra et al. [Pricing of arithmetic basket options by conditioning, Insurance Math. Econom. 34 (2004) 55–57] and Vanmaele et al. [Bounds for the price of discrete sampled arithmetic Asian options, J. Comput. Appl. Math. 185(1) (2006) 51–90]. Afterwards we show how to derive an analytical closed-form expression for a lower bound in the non-comonotonic case. Finally, we derive upper bounds for Asian basket options by applying techniques as in Thompson [Fast narrow bounds on the value of Asian options, Working Paper, University of Cambridge, 1999] and Lord [Partially exact and bounded approximations for arithmetic Asian options, J. Comput. Finance 10 (2) (2006) 1–52]. Numerical results are included and on the basis of our numerical tests, we explain which method we recommend depending on moneyness and time-to-maturity.     

Pricing catastrophe options in discrete operational time

We employ a doubly-binomial process as in Gerber [Gerber, H.U., 1988. Mathematical fun with the compound binomial process. ASTIN Bull. 18, 161-168] to discretize and generalize the continuous “randomized operational time” model of Chang et al. ([Chang, C.W., Chang, J.S.K., Yu, M.T., 1996. Pricing catastrophe insurance futures call spreads: A randomized operational time approach. J. Risk Insurance 63, 599-616] and CCY hereafter) from a complete-market continuous-time setting to an incomplete-market discrete-time setting, so as to price a richer set of catastrophe (CAT) options. For futures options, we derive the equivalent martingale probability measures by benchmarking to the shadow price of a bond to span arrival uncertainty, and the underlying futures price to span price uncertainty. With a time change from calendar time to the operational transaction-time dimension, we derive CCY as a limiting case under risk-neutrality when both calendar-time and transaction-time intervals shrink to zero. For a cash option with non-traded underlying loss index, we benchmark to the market reinsurance premiums to span claim uncertainty, and with a time change to claim time, we derive the cash option price as a binomial sum of claim-time binomial Asian option prices under the martingale measures.     

NUMERICAL ANALYSIS ON BINOMIAL TREE METHODS FOR AMERICAN LOOKBACK OPTIONS

1　IntroductionLookback options are path-dependent options whose payoffs depend on the maximumor the minimum of the underlying asset price during the life of the options( see[6] [1 0 ][1 4] ) .Here the maximum or minimum realized asset price may be monitored either con-tinuously or discretely.An American lookback call( put) option allows to be exercised atany time prior to expiry and gives the holder the rightto buy( sell) atthe historical mini-mum( maximum) of the underlying asset price on ex…     

Generalised soft binomial American real option pricing model (fuzzy–stochastic approach)

The stochastic discrete binomial models and continuous models are usually applied in option valuation. Valuation of the real American options is solved usually by the numerical procedures. Therefore, binomial model is suitable approach for appraising the options of American type. However, there is not in several situations especially in real option methodology application at to disposal input data of required quality. Two aspects of input data uncertainty should be distinguished; risk (stochastic) and vagueness (fuzzy). Traditionally, input data are in a form of real (crisp) numbers or crisp-stochastic distribution function. Therefore, hybrid models, combination of risk and vagueness could be useful approach in option valuation. Generalised hybrid fuzzy–stochastic binomial American real option model under fuzzy numbers (T-numbers) and Decomposition principle is proposed and described. Input data (up index, down index, growth rate, initial underlying asset price, exercise price and risk-free rate) are in a form of fuzzy numbers and result, possibility-expected option value is also determined vaguely as a fuzzy set. Illustrative example of equity valuation as an American real call option is presented.     

The discontinuous Galerkin method for discretely observed Asian options

Asian options represent an important subclass of the path-dependent contracts that are identified by payoff depending on the average of the underlying asset prices over the prespecified period of option lifetime. Commonly, this average is observed at discrete dates, and also, early exercise features can be admitted. As a result, analytical pricing formulae are not always available. Therefore, some form of a numerical approximation is essential for efficient option valuation. In this paper, we study a PDE model for pricing discretely observed arithmetic Asian options with fixed as well as floating strike for both European and American exercise features. The pricing equation for such options is similar to the Black-Scholes equation with 1 underlying asset, and the corresponding average appears only in the jump conditions across the sampling dates. The objective of the paper is to present the comprehensive methodological concept that forms and improves the valuation process. We employ a robust numerical procedure based on the discontinuous Galerkin approach arising from the piecewise polynomial generally discontinuous approximations. This technique enables a simple treatment of discrete sampling by incorporation of jump conditions at each monitoring date. Moreover, an American early exercise constraint is directly handled as an additional nonlinear source term in the pricing equation. The proposed solving procedure is accompanied by an empirical study with practical results compared to reference values.     

Pricing catastrophe options in discrete operational time

We employ a doubly-binomial process as in Gerber [Gerber, H.U., 1988. Mathematical fun with the compound binomial process. ASTIN Bull. 18, 161–168] to discretize and generalize the continuous “randomized operational time” model of Chang et al. ([Chang, C.W., Chang, J.S.K., Yu, M.T., 1996. Pricing catastrophe insurance futures call spreads: A randomized operational time approach. J. Risk Insurance 63, 599–616] and CCY hereafter) from a complete-market continuous-time setting to an incomplete-market discrete-time setting, so as to price a richer set of catastrophe (CAT) options. For futures options, we derive the equivalent martingale probability measures by benchmarking to the shadow price of a bond to span arrival uncertainty, and the underlying futures price to span price uncertainty. With a time change from calendar time to the operational transaction-time dimension, we derive CCY as a limiting case under risk-neutrality when both calendar-time and transaction-time intervals shrink to zero. For a cash option with non-traded underlying loss index, we benchmark to the market reinsurance premiums to span claim uncertainty, and with a time change to claim time, we derive the cash option price as a binomial sum of claim-time binomial Asian option prices under the martingale measures.     

Market implied volatilities for defaultable bonds

Typically, implied volatilities for defaultable instruments are not available in the financial market since quotations related to options on defaultable bonds or on credit default swaps are usually not quoted by brokers. However, an estimate of their volatilities is needed for pricing purposes. In this paper, we provide a methodology to infer market implied volatilities for defaultable bonds using equity implied volatilities and CDS spreads quoted by the market in relation to a specific issuer. The theoretical framework we propose is based on the Merton’s model under stochastic interest rates where the short rate is assumed to follow the Hull–White model. A numerical analysis is provided to illustrate the calibration process to be performed starting from financial market data. The market implied volatility calibrated according to the proposed methodology could be used to evaluate options where the underlying is a risky bond, i.e. callable bond or other types of credit-risk sensitive financial instruments.

     

Asymptotic approach to the pricing of geometric asian options under the CEV model

This paper studies the pricing of Asian options whose payoffs depend on the average value of an underlying asset during the period to a maturity. Since the Asian option is not so sensitive to the value of underlying asset, the possibility of manipulation is relatively small than the other options such as European vanilla and barrier options. We derive the pricing formula of geometric Asian options under the constant elasticity of variance (CEV) model that is one of local volatility models, and investigate the implication of the CEV model for geometric Asian options.     

Pricing of perpetual American and Bermudan options by binomial tree method

In this paper, we consider the binomial tree method for pricing perpetual American and perpetual Bermudan options. The closed form solutions of these discrete models are solved. Explicit formulas for the optimal exercise boundary of the perpetual American option is obtained. A nonlinear equation that is satisfied by the optimal exercise boundaries of the perpetual Bermudan option is found.      

The pricing of vulnerable options with double Mellin transforms

Many options traded in the over-the-counter markets are subject to default risks resulting from the probability that the option writer could not honor its contractual obligations. There have been growing concerns about financial derivatives subject to default risks, in particular, since the Global Financial Crisis and Eurozone crisis. This paper uses double Mellin transforms to study European vulnerable options under constant as well as stochastic (the Hull–White) interest rates. We obtain explicitly an analytic closed form pricing formula in each interest rate case so that the pricing of the options can be computed both accurately and efficiently.     

跳跃扩散型离散算术平均亚式期权的近似价格公式

在标的资产价格遵循跳跃扩散过程条件下 ,研究没有封闭形式解的离散算术平均亚式期权 ,运用二阶 Edgeworth逼近得到离散算术平均亚式期权的近似价格公式 .     

A volatility decomposition control variate technique for Monte Carlo simulations of Heath Jarrow Morton models

The aim of this work is to develop a simulation approach to the yield curve evolution in the Heath, Jarrow and Morton [Econometrica 60 (1) (1992) 77] framework. The stochastic quantities considered as affecting the forward rate volatility function are the spot rate and the forward rate. A decomposition of the volatility function into a Hull and White [Rev. Financial Stud. 3 (1990) 573] volatility and a remainder allows us to develop an efficient Control Variate Method that makes use of the closed form solution of the Hull and White call option. This technique considerably speeds up the simulation algorithm to approximate call option values with Monte Carlo simulation.     

New insights on testing the efficiency of methods of pricing and hedging American options

With reference to the evaluation of the speed–precision efficiency of pricing and hedging of American Put options, we present and discuss numerical results obtained on the basis of four different large enough random samples according to the relevance of the American quality (relative importance of the early exercise opportunity) of the options. Here we provide a comparison of the best methods (lattice based numerical methods and an approximation of the American Premium analytical procedure) known in literature along with some key methodological remarks.     

Asymptotic behavior of the distribution of the stock price in models with stochastic volatility: the Hull–White model

In the present Note, we study the asymptotic behavior of the distribution density of the stock price process in the Hull–White model. The leading terms in the asymptotic expansions at zero and infinity are found for such a density and the corresponding error estimates are given. Similar problems are solved for time averages of the volatility process, which are also of interest in the study of Asian options. To cite this article: A. Gulisashvili, E.M. Stein, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Numerical valuation of options under Kou's model

Numerical methods are developed for pricing European and American options under Kou's jump-diffusion model which assumes the price of the underlying asset to behave like a geometrical Brownian motion with a drift and jumps whose size is log-double-exponentially distributed. The price of a European option is given by a partial integro-differential equation (PIDE) while American options lead to a linear complementarity problem (LCP) with the same operator. Spatial differential operators are discretized using finite differences on nonuniform grids and time stepping is performed using the implicit Rannacher scheme. For the evaluation of the integral term easy to implement recursion formulas are derived which have optimal computational cost. When pricing European options the resulting dense linear systems are solved using a stationary iteration. Also for pricing American options similar iterations can be employed. A numerical experiment demonstrates that the described method is very efficient as accurate option prices can be computed in a few milliseconds on a PC. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)