Lookback Options Research Articles

The pricing of lookback options and binomial approximation

Refining a discrete model of Cheuk and Vorst, we obtain a closed formula for the price of a European lookback option at any time between emission and maturity. We derive an asymptotic expansion of the price as the number of periods tends to infinity, thereby solving a problem posed by Lin and Palmer. We prove, in particular, that the price in the discrete model tends to the price in the continuous Black–Scholes model. Our results are based on an asymptotic expansion of the binomial cumulative distribution function that improves several recent results in the literature.

The maximum maximum of a martingale with given $\mathbf{n}$ marginals

We obtain bounds on the distribution of the maximum of a martingale with fixed marginals at finitely many intermediate times. The bounds are sharp and attained by a solution to $n$-marginal Skorokhod embedding problem in Ob{\l}\'oj and Spoida [An iterated Az\'ema-Yor type embedding for finitely many marginals (2013) Preprint]. It follows that their embedding maximizes the maximum among all other embeddings. Our motivating problem is superhedging lookback options under volatility uncertainty for an investor allowed to dynamically trade the underlying asset and statically trade European call options for all possible strikes and finitely-many maturities. We derive a pathwise inequality which induces the cheapest superhedging value, which extends the two-marginals pathwise inequality of Brown, Hobson and Rogers [Probab. Theory Related Fields 119 (2001) 558-578]. This inequality, proved by elementary arguments, is derived by following the stochastic control approach of Galichon, Henry-Labord\`ere and Touzi [Ann. Appl. Probab. 24 (2014) 312-336].

Transform-based evaluation of prices and Greeks of lookback options driven by Lévy processes

In this paper, we develop a technique, based on numerical inversion, to compute the prices and Greeks of lookback options driven by Lévy processes. In this setup, the risk neutral evolution of the stock price, say St , is given by S0eΧt , with S0 the initial price and Χt a Lévy process. Lookback option prices are functions of the stock prices ST at maturity time T and the running maximum ST := sup0≤t≤T St. As a consequence, the Wiener-Hopf decomposition provides us with all the probabilistic information needed to evaluate these prices. To overcome the complication that, in general, only an implicit form of the Wiener-Hopf factor is available, we approximate the Lévy process under consideration by an appropriately chosen other Lévy process, for which the double transform ?e−αXt(q)is known, where t(q) is an exponentially distributed random variable with mean q−1. The second step is to write the transform of the lookback option prices in terms of this double transform. Finally, we use state-of-the-art numerical inversion techniques to compute the prices and Greeks (ie, sensitivities with respect to initial price S0 and maturity time T). We test our procedure for a broad range of relevant Lévy processes, including a number of "traditional" models (Black-Scholes, Merton) and more recently proposed models (Carr-Geman-Madan-Yor (CGMY) and Beta processes), and show excellent performance in terms of speed and accuracy.

Lattice Boltzmann methods for solving partial differential equations of exotic option pricing

This paper establishes a lattice Boltzmann method (LBM) with two amending functions for solving partial differential equations (PDEs) arising in Asian and lookback options pricing. The time evolution of stock prices can be regarded as the movement of randomizing particles in different directions, and the discrete scheme of LBM can be interpreted as the binomial models. With the Chapman-Enskog multi-scale expansion, the PDEs are recovered correctly from the continuous Boltzmann equation and the computational complexity is O(N), where N is the number of space nodes. Compared to the traditional LBM, the coefficients of equilibrium distribution and amending functions are taken as polynomials instead of constants. The stability of LBM is studied via numerical examples and numerical comparisons show that the LBM is as accurate as the existing numerical methods for pricing the exotic options and takes much less CPU time.

RETRACTED ARTICLE: The distribution of the maximum of a variance gamma process and path-dependent option pricing

Although numerical procedures often supply a required accuracy, closed-form expressions allow one to escape any accumulation of errors. In this paper, we discuss the possibility of obtaining explicit results for a variance gamma process. We derive the exact distribution of the maximum of the variance gamma process over a finite interval of time and establish the prices of path-dependent options including digital barrier, fixed-strike lookback, and lookback options. The obtained formulas are based on values of hypergeometric functions.

Geometric stopping of a random walk and its applications to valuing equity-linked death benefits

We study discrete-time models in which death benefits can depend on a stock price index, the logarithm of which is modeled as a random walk. Examples of such benefit payments include put and call options, barrier options, and lookback options. Because the distribution of the curtate-future-lifetime can be approximated by a linear combination of geometric distributions, it suffices to consider curtate-future-lifetimes with a geometric distribution. In binomial and trinomial tree models, closed-form expressions for the expectations of the discounted benefit payment are obtained for a series of options. They are based on results concerning geometric stopping of a random walk, in particular also on a version of the Wiener–Hopf factorization.

Local risk-minimization for Lévy markets

In this paper, we aim to obtain explicit representations of locally risk-minimizing by using Malliavin calculus for Lévy processes. For incomplete market models whose asset price is described by a solution to a stochastic differential equation driven by a Lévy process, we derive general formulas of locally risk-minimizing including Malliavin derivatives; and calculate its concrete expressions for call options, Asian options and lookback options.

The Value of Being Lucky: Option Backdating and Non-Diversifiable Risk

Empirical evidence shows that backdating of executive stock option grants was prevalent, particularly at firms with highly volatile stock prices. Yet extant Black-Scholes style models used for empirical estimates of gains to executives from backdating suggest that this benefit is not only modest but is greatest for low volatility firms. In this paper we take the stance that, ex ante, executives who have the opportunity to backdate should take this into account in their valuation and we quantify the value to a risk averse executive of a "lucky" option grant with strike chosen to coincide with the lowest stock price of the month. This enables us to reconcile theory with empirics - we show the ex ante gain to risk averse executives is significant and increases with volatility. Our explanation hinges on valuing the embedded lookback option, and key features of risk aversion, inability to diversify and American early exercise.

A fast numerical method for the valuation of American lookback put options

A fast and efficient numerical method is proposed and analyzed for the valuation of American lookback options. American lookback option pricing problem is essentially a two-dimensional unbounded nonlinear parabolic problem. We reformulate it into a two-dimensional parabolic linear complementary problem (LCP) on an unbounded domain. The numeraire transformation and domain truncation technique are employed to convert the two-dimensional unbounded LCP into a one-dimensional bounded one. Furthermore, the variational inequality (VI) form corresponding to the one-dimensional bounded LCP is obtained skillfully by some discussions. The resulting bounded VI is discretized by a finite element method. Meanwhile, the stability of the semi-discrete solution and the symmetric positive definiteness of the full-discrete matrix are established for the bounded VI. The discretized VI related to options is solved by a projection and contraction method. Numerical experiments are conducted to test the performance of the proposed method.

Lookback option pricing for regime-switching jump diffusion models

In this paper, we will introduce a numerical method to price theEuropean lookback floating strike put options where the underlyingasset price is modeled by a generalized regime-switching jumpdiffusion process. In the Markov regime-switching model, the optionvalue is a solution of a coupled system of nonlinearintegro-differential partial differential equations. Due to thecomplexity of regime-switching model, the jump process involved, andthe nonlinearity, closed-form solutions are virtually impossible toobtain. We use Markov chain approximation techniques to construct adiscrete-time Markov chain to approximate the option value.Convergence of the approximation algorithms is proved. Examples arepresented to demonstrate the applicability of the numerical methods.

The British Lookback Option with Fixed Strike

We continue research of the new type of options called ‘British’ that was introduced recently by presenting the British lookback option with fixed strike. This article generalizes the work about the British Russian option and provides financial analysis of lookback options with fixed non-zero strike. The British holder enjoys the early exercise feature of American options whereupon his pay-off (deliverable immediately) is the ‘best prediction’ of the European lookback pay-off under the hypothesis that the true drift of the stock price equals a contract drift. We derive a closed-form expression for the arbitrage-free price in terms of the optimal stopping boundary of two-dimensional optimal stopping problem with a scaling strike and show that the rational exercise boundary of the option can be characterized via the unique solution to a nonlinear integral equation. We also show the remarkable numerical example where the rational exercise boundary exhibits a discontinuity. Using these results, we perform a financial analysis of the British lookback option with fixed strike, which shows that with the contract drift properly selected this instrument not only provides an effective protection mechanism, but becomes a very attractive alternative to the classic European/American lookback option from speculator’s point of view and gives high returns when stock movements are favourable.

VALUING EQUITY-LINKED DEATH BENEFITS IN A REGIME-SWITCHING FRAMEWORK

AbstractIn this article, we consider the problem of computing the expected discounted value of a death benefit, e.g. in Gerber et al. (2012, 2013), in a regime-switching economy. Contrary to their proposed discounted density approach, we adopt the Laplace transform to value the contingent options. By this alternative approach, closed-form expressions for the Laplace transforms of the values of various contingent options, such as call/put options, lookback options, barrier options, dynamic fund protection and the dynamic withdrawal benefits, have been obtained. The value of each contingent option can then be recovered by the numerical Laplace inversion algorithm, and this efficient approach is documented by several numerical illustrations. The strength of our methodology becomes apparent when we tackle the valuations of exotic contingent options in the cases when (1) the contracts have a finite expiry date; (2) when the time-until-death variable is uniformly distributed in accordance with De Moivre's law.

A finite volume method for pricing the American lookback option

Due to the characteristic of risk aversion, option has become one of the most fashionable derivatives in the financial field. More and more investigators are attracted to devote themselves to exploring the option pricing problem. In this paper, we are concerned with the valuation of American lookback options in terms of the Black-Scholes model. It is well known that the American lookback option satisfies a two-dimensional nonlinear partial differential equation in an unbounded domain, which couldn't be numerically solved directly. Based on the analysis of the issues for solving this problem, this paper introduces an approach to settle it. First, we transform the problem into a one-dimensional form by the numeraire transformation. And then, the Landau's transformation is applied to normalize the defined domain. For the nonlinear feature of the resulting problem, we propose a finite volume method coupled with Newton iterative method to obtain the optional value and the optimal exercise boundary simultaneously. We also give a proof on the nonnegativity of the numerical solutions under some appropriate assumptions. Finally, some numerical simulations are presented using the proposed method in this paper. Comparing with the binomial method, we can conclude that the proposed method is an effective one, which provides a theoretical basis for practical applications.

Lookback Option Pricing with Fixed Proportional Transaction Costs under Fractional Brownian Motion.

The pricing problem of lookback option with a fixed proportion of transaction costs is investigated when the underlying asset price follows a fractional Brownian motion process. Firstly, using Leland's hedging method a partial differential equation satisfied by the value of the lookback option is derived. Then we obtain its numerical solution by constructing a Crank-Nicolson format. Finally, the effectiveness of the proposed form is verified through a numerical example. Meanwhile, the impact of transaction cost rate and volatility on lookback option value is discussed.

Pricing Path-Dependent Options with Discrete Monitoring under Time-Changed Lévy Processes

This paper proposes a pricing method for path-dependent derivatives with discrete monitoring when an underlying asset price is driven by a time-changed Lévy process. The key to our method is to derive a backward recurrence relation for computing the multivariate characteristic function of the intertemporal joint distribution of the time-changed Lévy process. Using the derived representation of the characteristic function, we obtain semi-analytical pricing formulas for geometric Asian, forward start, barrier, fader and lookback options, all of which are discretely monitored.

Generalized Barndorff-Nielsen and Shephard Model and Discretely Monitored Option Pricing

This paper proposes a generalization of the Generalized Barndorff-Nielsen and Shephard model, in which the log return on an asset price is governed by a Levy process with stochastic volatility modeled by a non-Gaussian Ornstein-Uhlenbeck process. Under the generalized model, we derive a closed-form expression for the multivariate characteristic function of the intertemporal joint distribution of the underlying log return. Then, we also investigate asymptotic behavior of the log return and its variance. Moreover, we evaluate discretely monitored path-dependent derivatives such as geometric Asian, forward start, barrier, fade-in, and lookback options as well as European options.

PERFORMANCE MEASURE OF LAPLACE TRANSFORMS FOR PRICING PATH DEPENDENT OPTIONS

This paper presents a performance measure of Laplace transforms for pricing path dependent options. We obtain a simple expression for the double transform by means of Fourier and Laplace transforms, (with respect to the logarithm of the strike and time to maturity) of the price of continu- ously monitored Asian options. The double transform is expressed in terms of Gamma functions only. The computation of the price requires a multivariate numerical inversion. Under jump-diffusion model, we show that the Laplace transforms of lookback options can be obtained through a recursion involving only analytical formulae for standard European call and put options. We also show that the numerical inversion can be performed with great accuracy and low computational cost.

Consistent Pricing of Spot Options and Forward Start Options

The idea of this paper is to present how we can use a specific form of local volatility, namely a forward start local vol, that has already been evoked in different sources, and use it in order to fit a specific Forward Start Smile. A local formula, i.e. a formula similar to a Dupire formula, is available for these models and will be used to fit the Forward Smile Market. These models are therefore particularly adapted for pricing Cliquet Products or generally Forward Starting Products with given frequencies. In particular, it is possible to fit different options like Vanilla options and Forward Start Options using them. We will also consider an extension of these models adapted for the pricing of lookback options.

CONVERGENCE OF EUROPEAN LOOKBACK OPTIONS WITH FLOATING STRIKE IN THE BINOMIAL MODEL

In this paper, we study the convergence of a European lookback option with floating strike evaluated with the binomial model of Cox–Ross–Rubinstein to its evaluation with the Black–Scholes model. We do the same for its delta. We confirm that these convergences are of order [Formula: see text]. For this, we use the binomial model of Cheuk–Vorst which allows us to write the price of the option using a double sum. Based on an improvement of a lemma of Lin–Palmer, we are able to give the precise value of the term in [Formula: see text] in the expansion of the error; we also obtain the value of the term in 1/n if the risk free interest rate is nonzero. This modelization will also allow us to determine the first term in the expansion of the delta.

Dynamic Programming and Error Estimates for Stochastic Control Problems with Maximum Cost

This work is concerned with stochastic optimal control for a running maximum cost. A direct approach based on dynamic programming techniques is studied leading to the characterization of the value function as the unique viscosity solution of a second order Hamilton–Jacobi–Bellman (HJB) equation with an oblique derivative boundary condition. A general numerical scheme is proposed and a convergence result is provided. Error estimates are obtained for the semi-Lagrangian scheme. These results can apply to the case of lookback options in finance. Moreover, optimal control problems with maximum cost arise in the characterization of the reachable sets for a system of controlled stochastic differential equations. Some numerical simulations on examples of reachable analysis are included to illustrate our approach.