Lookback Options Research Articles

Pricing formula of Lookback option in stochastic delay differential equation model

This paper deals with new explicit pricing formulae for Lookback option when underlying asset price processes are represented by stochastic delay differential equation (hereafter “SDDE”). We derive a lemma on the joint distribution of the minimum and itself of a Wiener process in the SDDE model. Using this lemma, we obtain the explicit pricing formulae for the Lookback option. Through some numerical comparison experiment, we assure the correctness of the obtained pricing formula.

On pricing of discrete Asian and Lookback options under the Heston model

We propose a new, data-driven approach for efficient pricing of – fixed- and floating-strike – discrete arithmetic Asian and Lookback options when the underlying process is driven by the Heston model dynamics. The method proposed in this article constitutes an extension of Perotti and Grzelak [Fast sampling from time-integrated bridges using deep learning, J. Comput. Math. Data Sci. 5 (2022)], where the problem of sampling from time-integrated stochastic bridges was addressed. The model relies on the Seven-League scheme [S. Liu et al. The seven-league scheme: Deep learning for large time step Monte Carlo simulations of stochastic differential equations, Risks 10 (2022), p. 47], where artificial neural networks are employed to ‘learn’ the distribution of the random variable of interest utilizing stochastic collocation points [L.A. Grzelak et al. The stochastic collocation Monte Carlo sampler: Highly efficient sampling from expensive distributions, Quant. Finance 19 (2019), pp. 339–356]. The method results in a robust procedure for Monte Carlo pricing. Furthermore, semi-analytic formulae for option pricing are provided in a simplified, yet general, framework. The model guarantees high accuracy and a reduction of the computational time up to thousands of times compared to classical Monte Carlo pricing schemes.

Foreign equity lookback options with partial monitoring

A lookback option on foreign equity reduces the uncertainty of foreign investment by delivering the benefit of hindsight. Despite its high marketability, purchasing the option is stalled due to the failure to meet customized demands. This paper proposes foreign equity lookback options with partial monitoring and floating strike price to control the option premiums. We establish the expected value of the extreme of an underlying process when the specific condition, covering various payoffs of the lookback options, is satisfied. It is exploited to derive the unified closed-form pricing formula. We demonstrate the properties of the option prices through numerical examples.

Pricing vulnerable lookback options using Laplace transforms

This study develops a closed-form solution for pricing vulnerable lookback options, which combines the Black–Scholes model for the underlying asset and a correlated jump–diffusion model for the issuer’s asset to account for default risks. Our approach relies on applying Laplace transforms to establish a closed-form solution and compute them numerically using Laplace inversion algorithms. To determine the price, we derive the joint distribution of the first passage time of a drifted Brownian motion and the value of the correlated jump–diffusion. Through numerical examples, we demonstrate the accuracy of the numerical solutions obtained through our method and the stability of the algorithm. Subsequently, we conduct a numerical analysis to understand how counterparty risk influences option prices based on our formula.

Pricing vanilla, barrier, and lookback options under two-scale stochastic volatility driven by two approximate fractional Brownian motions

<p>In this paper, we proposed a stochastic volatility model in which the volatility was given by stochastic processes representing two characteristic time scales of variation driven by approximate fractional Brownian motions with two Hurst exponents. We obtained an approximate closed-form formula for a European vanilla option price and the corresponding implied volatility formula based on singular and regular perturbations and a Mellin transform. The explicit formula for the implied volatility allowed us to find the slope of the implied volatility skew with respect to the Hurst exponent and time-to-maturity. The proposed model allows the market volatility behavior to be captured uniformly in time-to-maturity. We conducted an empirical analysis to find the validity of the proposed model by comparing it with other models and Monte Carlo simulation. Further, we extended the pricing result for the vanilla option to two path-dependent exotic (barrier and lookback) options and obtained the corresponding price formulas explicitly.</p>

Monte Carlo method for pricing lookback options in Lévy models

В статье построен универсальный метод Монте-Карло для оценивания опционов, функция выплат которых зависит от конечного положения экстремума процесса Леви. Предлагаемый метод позволяет вычислять цены опционов lookback с плавающей и фиксированной ценой исполнения не только в начальный момент времени, но и в течение всего срока, когда текущее положение процесса Леви может не совпадать со своим экстремумом. Предлагаемый алгоритм включает три этапа: аппроксимацию функции распределения процесса экстремума, ее обращение и симуляцию конечного положения экстремума процесса Леви. В работе выведены новые приближенные формулы для вычисления функций распределения процессов супремума и инфимума для моделей Леви, использующие факторизацию Винера-Хопфа. Описаны принципы построения гибридного метода Монте-Карло, комбинирующего классические численные методы построения функции распределения конечного положения процесса экстремума и методы машинного обучения для обращения функции распределения с помощью тензорных нейронных сетей. Проведенные вычислительные эксперименты подтверждают эффективность универсального метода Монте-Карло при вычислении цен опционов lookback.

Comparison of Price Simulation of European Lookback Option and Discrete Asian Option

As a matter of fact, this study focuses on the comparison of European Lookback Option and Discrete Asian option based on Monte Carlo Method and Binary Tree model. In the first step, options and methods are introduced, together with their mathematical principles and expressions. Next, parameters used are provided, including ideal condition and historic data of Unilever. Simulations are made and price gap and price distribution are discussed. Then, sensitive researches are done through changing volatility and strike price. The relationship among price gaps method and the principles of the method has also been taken into consideration. It is found that price trend based on Monte Carlo Method is linear-like while price based on Binary Tree Model is exponential-like. Price gap keeps gradually increasing, which is due to characteristics of the method. Moreover, for a given and fixed volatility, the higher the strike price, the lower the price simulated and vice versa, this is explained mathematically. Finally, limitations, future outlooks and conclusions are made.

A general approach for lookback option pricing under Markov models

We propose a computationally efficient method for pricing various types of lookback options under Markov models. We utilize the model-free representations of lookback option prices as integrals of first passage probabilities. We combine efficient numerical quadrature with continuous-time Markov chain approximation for the first passage problem to price lookbacks. Our method is applicable to a variety of models, including one-dimensional time-homogeneous and time-inhomogeneous Markov processes, regime-switching models and stochastic local volatility models. We demonstrate the efficiency of our method through various numerical examples.

A new options pricing method: semi-stochastic kernel regression method with constraints

This paper presents a unified semi-stochastic kernel regression method for pricing options under general stochastic volatility model. The method combines semi-stochastic sampling for initial asset values with Monte Carlo simulations to construct a least-squares based kernel function regression solution. This approach can not only approximates option prices, but also determines the Greeks of option. The least square problem is augmented with weighted derivative constraints, which enables flexible adjustment of approximate errors for both option prices and Greeks. Numerical results show the efficiency of the proposed method for the Vanilla option and some exotic options: Asian option, Lookback option, discretely monitored Barrier option and the Basket option with several assets under the stochastic volatility model.

No-Transaction Band Network: A Neural Network Architecture for Efficient Deep Hedging

Deep hedging is a versatile framework for computing the optimal hedging strategy of derivatives in incomplete markets. However, it is subject to the action-dependence problem impeding efficient training because the appropriate hedging action at the next step depends on the current action. To overcome this issue, the authors leverage a no-transaction band strategy, an existing technique that provides optimal hedging strategies for European options and exponential utility. The authors theoretically argue this strategy to be optimal for a wider class of utilities and derivatives, including exotics. Based on the result, the authors propose a no-transaction band network, namely, a neural network architecture that facilitates fast training and precise evaluation of the optimal hedging strategy. Moreover, the authors experimentally demonstrate that, for European and lookback options, their architecture rapidly attains a better hedging strategy compared with a standard feed-forward network. The findings thus have important implications for the practical applications of deep hedging.

Valuation of barrier and lookback options under hybrid CEV and stochastic volatility

In this paper, we evaluate down-and-out put option and floating strike lookback option prices when the underlying asset is driven by a hybrid model with constant elasticity of variance and stochastic volatility (SVCEV). Usually, it is difficult to get closed-form solutions for those exotic options under stochastic volatility models. Here, we use an asymptotic expansion approach and the Mellin transform method to obtain explicit closed-form formulae for the zero-order and first-order correction terms. In addition, we perform a sensitivity analysis numerically on the asymptotic terms and compare the option prices corresponding to the Black–Scholes, CEV and SVCEV models with those calculated by Monte-Carlo simulations and the binomial tree method to illustrate the accuracy of our pricing formulae.

Research on The Lookback Option Pricing of Disney Based on BSM Model

The lookback option is a path-dependent option that enables holders to use the most advantageous pricing to execute the underlying asset. This paper adopts the BMS model to price the lookback option and takes risk factors into consideration to calculate the value of an option contract in the BMS model. The main variables in the BMS model are risk-free rate, current spot price, strike price, volatility, and time to maturity. The results are reliable enough for investors to make future decisions. However, the BMS model still has a few drawbacks which can make the prices derived from real future prices. When predicting future prices, using the BMS model can help investors make better decisions. The lookback option allows holders to minimize their regrets and exercise the underlying asset at the most beneficial price.

Application of fuzzy Malliavin calculus in hedging fixed strike lookback option

<abstract><p>In this paper, we develop a Malliavin calculus approach for hedging a fixed strike lookback option in fuzzy space. Due to the uncertainty in financial markets, it is not accurate to describe the problems of option pricing and hedging in terms of randomness alone. We consider a fuzzy pricing model by introducing a fuzzy stochastic differential equation with Skorohod sense. In this way, our model simultaneously involves randomness and fuzziness. A well-known hedging strategy for vanilla options is so-called $ \Delta $-hedging, which is usually derived from the Itô formula and some properties of partial differentiable equations. However, when dealing with some complex path-dependent options (such as lookback options), the major challenge is that the payoff function of these options may not be smooth, resulting in the estimates are computationally expensive. With the help of the Malliavin derivative and the Clark-Ocone formula, the difficulty will be readily solved, and it is also possible to apply this hedging strategy to fuzzy space. To obtain the explicit expression of the fuzzy hedging portfolio for lookback options, we adopt the Esscher transform and reflection principle techniques, which are beneficial to the calculation of the conditional expectation of fuzzy random variables and the payoff function with extremum, respectively. Some numerical examples are performed to analyze the sensitivity of the fuzzy hedging portfolio concerning model parameters and give the permissible range of the expected hedging portfolio of lookback options with uncertainty by a financial investor's subjective judgment.</p></abstract>

An exact and explicit formula for pricing lookback options with regime switching

<p style='text-indent:20px;'>This paper investigates the pricing of European-style lookback options when the price dynamics of the underlying risky asset are assumed to follow a Markov-modulated Geometric Brownian motion; that is, the appreciation rate and the volatility of the underlying risky asset depend on states of the economy described by a continuous-time Markov chain process. We derive an exact, explicit and closed-form solution for European-style lookback options in a two-state regime switching model.</p>

Regularity and higher integrability of weak solutions to a class of non-Newtonian variation-inequality problems arising from American lookback options

<abstract><p>This paper presents the proofs of the higher integrability and regularity of weak solutions to a class of variation-inequality problems that are formulated by a non-Newtonian parabolic operator. After obtaining the gradient estimate, the higher order integrability of the weak solution is analyzed. We also examine the internal regularity estimate of the weak solution by utilizing a test function of the difference type.</p></abstract>

Partial quanto lookback options

Financial instruments for hedging and speculating on the foreign exchange rate and equity risks draw the attention of market participants as financial transactions increase across multiple jurisdictions. Notably, a quanto lookback option has been actively traded because it successfully meets market demands. Although the quanto lookback option provides numerous benefits, a high premium due to the lookback feature is the primary culprit that hinders investors from purchasing it. This paper proposes partial quanto lookback options and provides the closed-form pricing formulas when the lookback feature is applied to the exchange rate or equity value, and the extremes are determined by observing them for a shorter period than the life of the option. Because pricing the options is challenging due to their partial path-dependence, we develop the quanto extreme expectation that facilitates deriving the option prices. Extensive numerical examples demonstrate the efficacy of the partial quanto lookback options in lowering the premiums.

Investigating the difference of option’s real values between Lookback option and European option based on price models

The concept of pricing model is crucial for global financial markets today. Millions of investors tend to seek options to effectively maximize their returns. The lookback option is based on various exotic options in response to this trend. This paper chooses Amazon’s stock as our data to analyze and then simulate its future stock price 1000 times. By calculating the minimum and maximum value of stocks during a given period, we know the returns gained by using the lookback option and then compare the payoffs with those of the European option. We then conclude that the lookback option could lower the uncertainty over the timing of market entry. Also, the lookback option allows buyers to minimize their regret and maximize their profits. The analysis for pricing models in this paper could provide investors with a reference standard when selecting options and help investors avoid a certain degree of risk.

Evaluation of chooser option evidence from Alibaba

Due to the financial market’s globalization, it has resulted in the growingly fierce competition and the international financial system’s volatility in current financial market. This implies that it is growingly necessary to analyze the financial derivatives market’s trend. To assess chooser option, Alibaba is selected. The way of trading a majority of the exotic options is generally determined by the financial organizations based on the customers’ specific demands. Exotic options have three common types which are respectively lookback options, chooser options and binary options. In this proposed study, the Brown Motion Theory can be used to determine the prices of assets through applying Geometric Brownian Motion to simulate the prices of Alibaba’s financial assets in order to calculate the financial assets’ future value. This study offers valuable insights with both financial institutions and investors respectively creating more proper exotic options and making more appropriate investment decisions.

Lookback option pricing under the double Heston model using a deep learning algorithm

Lookback option pricing under the double Heston model using a deep learning algorithm

Pricing Asian Lookback Option based on Monte Carlo simulation

With the emergence and development of exotic options, the diversity of securities has largely increased. However, existing types of popular exotic options seem to be either limited or inflexible to satisfy buyers' personal needs. As for the Lookback option and Asian option, the Lookback option’s application is limited in the OTC area with a high price. The Asian option is cheap but used mostly for hedging, leaving relatively little space for speculating. Thus, we created a new exotic option called the Asian Lookback option (ALB) as a combination and derivative of these two options with a flexible implementing way enabling both the requirements for hedging and speculating. In the process of building the model of ALB, these two options’ models are referenced, together with the widely used option pricing method Monte Carlo Simulations based on the assumptions of the Black-Scholes model. To simulate the practical function and price of ALB, we set a control experiment of the prices of four options: two types of ALB, Asian option, and Lookback option in three different markets (S&P 500, Corn, COMEX Gold). It is shown that the price of ALB can be switched from Asian-like to Lookback-like, realizing the function of customized usage from hedging to speculating. These results shed light on a more flexible and personalized development tendency of derivatives.