José Fajardo
In this paper we present a very simple way to price a class of barrier options when the underlying process is driven by a huge class of Lévy processes. To achieve our goal we assume that our market satisfies a symmetry property. In case of not satisfying that property some approximations can be obtained.
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We determine the price of digital double barrier options with an arbitrary number of barrier periods in the Black-Scholes model. This means that the barriers are active during some time intervals, but are switched off in between. As an application, we calculate the value of a structure floor for structured notes whose individual coupons are digital double barrier options. This value can also be approximated by the price of a corridor put.
It turns out that in the bivariate Black-Scholes economy Margrabe type options exhibit symmetry properties leading to semi-static hedges of rather general barrier options. Some of the results are extended to variants obtained by means of Brownian subordination. In order to increase the liquidity of the hedging instruments for certain currency options, the duality principle can be applied to set up hedges in a foreign market by using only European vanilla options sometimes along with a risk-less bond. Since the semi-static hedges in the Black-Scholes economy are exact, closed form valuation for…
Local time and the pricing of time-dependent barrier options
时间相关双重障碍期权是一种衍生证券，如果在时间间隔$[0,T]$内没有连续时间相关障碍$b_\pm:[0,T]\to \RR_+$被击中，则在到期$T$时交付终端值$ϕ(S_T)$。使用概率的方法，我们得到了一个分解的障碍期权价格到相应的欧式期权价格减去障碍溢价支付函数$ϕ$，障碍函数$b_\pm$和线性扩散$(S_t)_{t\in[0,T]}$的广泛的一类。我们表明，障碍溢价可以表示为沿障碍$b_\pm$的期权的三角洲$Δ_\pm:[0,T]\to\RR$在障碍的积分的总和，并对功能$(Δ_+,Δ_-)$解决了Volterra积分方程的第一类系统。我们发现这个系统的半解析解的情况下，常数的双障碍和简要讨论的数值算法的时间依赖的情况下。
Brownian excursions an Parisian barrier options: a note
本说明重新讨论了Yor及其合作者提出的巴黎障碍备选办法及其使用拉普拉斯变换方法的估值。巴黎障碍期权的概念，基于游览理论和使用布朗曲流，是扩展，使他们的估值是现在可能在任何点在其寿命。必要时修改相关的拉普拉斯变换。
Robust hedging of double touch barrier options
我们考虑数字期权的无模型定价，如果标的资产越过了上限和下限障碍，则会支付。我们只对基本过程（通常是连续性）做了较弱的假设，但假设具有相同到期日和所有执行价格的看涨期权的初始价格是已知的。在这种情况下，我们能够给出相关期权的无套利价格的上界和下界，并进一步使用Skorokhod嵌入理论的技术来证明这些界是紧的。此外，鞅不等式推导，这提供了交易策略，我们能够实现任何潜在的套利。我们发现，根据投资者的风险厌恶，由此产生的对冲策略可以显着优于标准的Delta/Vega对冲存在的市场摩擦和/或模型误设定。
This paper examines a class of barrier options-multi-step barrier options, which can have any finite number of barriers of any level. We obtain a general, explicit expression of option prices of this type under the Black-Scholes model. Multi-step barrier options are not only useful in that they can handle barriers of different levels and time steps, but can also approximate options with arbitrary barriers. Moreover, they can be embedded in financial products such as deposit insurances based on jump models with simple barriers. Along the way, we derive multi-step reflection principle, which gen…
Path integral method in quantum mechanics provides a new thinking for barrier option pricing. For proportional step options, the option price changing process is similar to the one dimensional trapezoid potential barrier scattering problem in quantum mechanics; for double-barrier step options, the option price changing process is analogous to a particle moving in a finite symmetric square potential well. Using path integral method, the analytical expressions of pricing kernel and option price could be derived. Numerical results of option price as a function of underlying price, potential and e…
Pricing of barrier options by marginal functional quantization
本文提出利用最优二次函数量化方法对障碍期权进行定价，并在Black-Scholes和局部波动率模型中进行了数值验证，同时与布朗桥方法进行了对比。
A Hamiltonian Approach to Floating Barrier Option Pricing
本文利用量子力学中的哈密顿方法，将浮动障碍阶梯期权定价问题映射为势垒散射和方势阱问题，推导出定价核与期权价格的解析表达式，并通过数值结果验证了方法的正确性。
In this paper, we will discuss an approximation of the characteristic function of the first passage time for a Levy process using the martingale approach. The characteristic function of the first passage time of the tempered stable process is provided explicitly or by an indirect numerical method. This will be applied to the perpetual American option pricing and the barrier option pricing. Numerical illustrations are provided for the calibrated parameters using the market call and put prices.
The presence of discrete dividends complicates the derivation and form of pricing formulas even for vanilla options. Existing analytic, numerical, and theoretical approximations provide results of varying quality and performance. Here, we compare the analytic approach, developed and effective for European puts and calls, of Buryak and Guo with the formulas, designed in the context of barrier option pricing, of Dai and Chiu.
We show that prices and shortfall risks of game (Israeli) barrier options in a sequence of binomial approximations of the Black--Scholes (BS) market converge to the corresponding quantities for similar game barrier options in the BS market with path dependent payoffs and the speed of convergence is estimated, as well. The results are new also for usual American style options and they are interesting from the computational point of view, as well, since in binomial markets these quantities can be obtained via dynamical programming algorithms. The paper continues the study of [11]and [7] but requ…
High-order accurate implicit methods for the pricing of barrier options
本文提出一种高阶精确隐式有限差分方法，通过概率最优边界条件处理连续和离散监测的各类障碍期权，显著提高定价速度和精度。
We investigate the pricing of financial options under the 2-hypergeometric stochastic volatility model. This is an analytically tractable model that reproduces the volatility smile and skew effects observed in empirical market data. Using a regular perturbation method from asymptotic analysis of partial differential equations, we derive an explicit and easily computable approximate formula for the pricing of barrier options under the 2-hypergeometric stochastic volatility model. The asymptotic convergence of the method is proved under appropriate regularity conditions, and a multi-stage method…
Two novel closed-form formulas for the price of barrier options in stochastic volatility models with zero interest rate and dividend yield but nonzero correlation between the asset and its instantaneous volatility are derived. The first is a Hull and White type formula, and the second is a decomposition formula similar in form to the Alòs decomposition for vanilla options. A model-free approximation is also given.
In this paper, we provide a unified treatment of the Vanna-Volga pricing technique. We derive the value of single and double barriers FX options, as well as closed formulas for the Delta, Vega, Vanna and Volga of those contracts.
Path integral method in quantum mechanics provides a new thinking for barrier option pricing. For proportional double-barrier step (PDBS) options, the option price changing process is analogous to a particle moving in a finite symmetric square potential well. We have derived the pricing kernel of PDBS options with time dependent interest rate and volatility. Numerical results of option price as a function of underlying asset price are shown as well. Path integral method can be easily generalized to the pricing of PDBS options with curved boundaries.
We provided an analytical representation of the price of a barrier option with one type of special moving barrier. We consider the case that risk free rate, dividend rate and stock volatility are time dependent. We get a pricing formula and put call parity for barrier option when the moving barrier has a special relation with risk free rate, dividend rate and stock volatility.
We continue a series of papers devoted to construction of semi-analytic solutions for barrier options. These options are written on underlying following some simple one-factor diffusion model, but all the parameters of the model as well as the barriers are time-dependent. We managed to show that these solutions are systematically more efficient for pricing and calibration than, eg., the corresponding finite-difference solvers. In this paper we extend this technique to pricing double barrier options and present two approaches to solving it: the General Integral transform method and the Heat Pot…
Barrier options are one of the most widely traded exotic options on stock exchanges. In this paper, we develop a new stochastic simulation method for pricing barrier options and estimating the corresponding execution probabilities. We show that the proposed method always outperforms the standard Monte Carlo approach and becomes substantially more efficient when the underlying asset has high volatility, while it performs better than multilevel Monte Carlo for special cases of barrier options and underlying assets. These theoretical findings are confirmed by numerous simulation results.
Most models for barrier pricing are designed to let a market maker tune the model-implied covariance between moves in the asset spot price and moves in the implied volatility skew. This is often implemented with a local volatility/stochastic volatility mixture model, where the mixture parameter tunes that covariance. This paper defines an alternate model where the spot/volatility correlation is a separate mean-reverting stochastic variable which is itself correlated with spot. We also develop an efficient approximation for barrier option and one touch pricing in the model based on semi-static …
Recent work of Dupire and Carr and Lee has highlighted the importance of understanding the Skorokhod embedding originally proposed by Root for the model-independent hedging of variance options. Root's work shows that there exists a barrier from which one may define a stopping time which solves the Skorokhod embedding problem. This construction has the remarkable property, proved by Rost, that it minimizes the variance of the stopping time among all solutions. In this work, we prove a characterization of Root's barrier in terms of the solution to a variational inequality, and we give an alterna…
Application of simplest random walk algorithms for pricing barrier options
本文展示了一阶算法在障碍期权定价中的有效性，该算法基于弱欧拉近似和线性插值，易于实现且通用，无需触发概率知识。
In this paper we develop an algorithm to calculate the prices and Greeks of barrier options in a hyper-exponential additive model with piecewise constant parameters. We obtain an explicit semi-analytical expression for the first-passage probability. The solution rests on a randomization and an explicit matrix Wiener-Hopf factorization. Employing this result we derive explicit expressions for the Laplace-Fourier transforms of the prices and Greeks of barrier options. As a numerical illustration, the prices and Greeks of down-and-in digital and down-and-in call options are calculated for a set o…
Classification of barrier options
针对给定期权定价精度，研究当定价参数变化时障碍期权何时退化为更简单期权类型的问题，通过概率方法找到初始股价的临界值。
In this paper we present an algorithm for pricing barrier options in one-dimensional Markov models. The approach rests on the construction of an approximating continuous-time Markov chain that closely follows the dynamics of the given Markov model. We illustrate the method by implementing it for a range of models, including a local Levy process and a local volatility jump-diffusion. We also provide a convergence proof and error estimates for this algorithm.
In this paper we analyse financial implications of exchangeability and similar properties of finite dimensional random vectors. We show how these properties are reflected in prices of some basket options in view of the well-known put-call symmetry property and the duality principle in option pricing. A particular attention is devoted to the case of asset prices driven by Levy processes. Based on this, concrete semi-static hedging techniques for multi-asset barrier options, such as certain weighted barrier spread options, weighted barrier swap options or weighted barrier quanto-swap options are…
A transform approach to compute prices and greeks of barrier options driven by a class of Levy processes
本文提出了一种计算由一类Levy过程驱动的障碍期权的价格和收益的变换方法。在具有超指数跳的指数Levy模型中，我们得到了单障碍期权价格和灵敏度随时间的Laplace变换的解析表达式。反演这些单一的拉普拉斯变换产生快速，准确的结果。这些结果被用来构建指数广义超指数（GHE）Levy模型中障碍期权的价格和灵敏度的近似。后一类包括许多在量化金融中使用的Levy模型，如方差伽马（VG）、KoBoL、广义双曲和正态逆高斯（NIG）模型。证明了近似价格和灵敏度的收敛性。为了提供一个数值例子，在驱动过程是VG和NIG Levy过程的情况下，将这种变换方法与蒙特卡罗模拟进行了比较。参数校准为Stoxx50 E看涨期权。
A Hamiltonian Approach to Barrier Option Pricing Under Vasicek Model
提出用哈密顿方法处理Vasicek模型下的障碍期权定价，通过时间无限细分、量子力学矩阵元与完备性条件导出定价核和价格积分式，并给出数值结果。
Short-term barrier option price expansion
本文推导了短期到期障碍期权价格的渐近展开，首项为Black-Scholes价格，修正项由波动率波动阶数决定，数值实验验证其改进效果。
In this work, the Fourier-cosine series (COS) method has been combined with the Boundary Element Method (BEM) for a fast evaluation of barrier option prices. After a description of its use in the Black and Scholes (BS) model, the focus of the paper is on the application of the proposed methodology to the barrier option evaluation in the Heston model, where its contribution is fundamental to improve computational efficiency and to make BEM appealing among Finance practitioners as a valid alternative to Monte Carlo (MC) or other more traditional approaches. An error analysis is provided on the n…
As is known, an option price is a solution to a certain partial differential equation (PDE) with terminal conditions (payoff functions). There is a close association between the solution of PDE and the solution of a backward stochastic differential equation (BSDE). We can either solve the PDE to obtain option prices or solve its associated BSDE. Recently a deep learning technique has been applied to solve option prices using the BSDE approach. In this approach, deep learning is used to learn some deterministic functions, which are used in solving the BSDE with terminal conditions. In this pape…
In this paper we develop a semi-closed form solutions for the barrier (perhaps, time-dependent) and American options written on the underlying stock which follows a time-dependent OU process with a log-normal drift. This model is equivalent to the familiar Hull-White model in FI, or a time dependent OU model in FX. Semi-closed form means that given the time-dependent interest rate, continuous dividend and volatility functions, one need to solve numerically a linear (for the barrier option) or nonlinear (for the American option) Fredholm equation of the first kind. After that the option prices …
Alternative models for FX: pricing double barrier options in regime-switching Lévy models with memory
本文提出带记忆的状态切换 Lévy 模型以改进外汇双重障碍期权定价，并修改原有数值方法，使其在保持与马尔可夫状态切换模型相同主要计算模块数量的同时，兼具灵活性和机器学习适用性。
In this paper we derive semi-closed form prices of barrier (perhaps, time-dependent) options for the Hull-White model, ie., where the underlying follows a time-dependent OU process with a mean-reverting drift. Our approach is similar to that in (Carr and Itkin, 2020) where the method of generalized integral transform is applied to pricing barrier options in the time-dependent OU model, but extends it to an infinite domain (which is an unsolved problem yet). Alternatively, we use the method of heat potentials for solving the same problems. By semi-closed solution we mean that first, we need to …
The aim of this paper is twofold. First, we extend the results of [33] concerning the existence and uniqueness of second-order reflected 2BSDEs to the case of two obstacles. Under some regularity assumptions on one of the barriers, similar to the ones in [10], and when the two barriers are completely separated, we provide a complete wellposedness theory for doubly reflected second-order BSDEs. We also show that these objects are related to non-standard optimal stopping games, thus generalizing the connection between DRBSDEs and Dynkin games first proved by Cvitanic and Karatzas [11]. More prec…
This paper considers the pricing of long-term options on assets such as housing, where either government intervention or the economic nature of the asset is assumed to limit large falls in prices. The observed asset price is modelled by a geometric Brownian motion (the 'notional price') reflected at a lower barrier. The resulting observed price has standard dynamics but with localised intervention at the barrier, which allows arbitrage with interim losses; this is funded by the government's unlimited powers of intervention, and its exploitation is subject to credit constraints. Despite the lac…
Addressing the bias in Monte Carlo pricing of multi-asset options with multiple barriers through discrete sampling
一个有效的条件技术，所谓的布朗桥模拟，以前已被应用于消除定价偏差中出现的应用程序的标准离散时间蒙特卡罗方法来评估期权写在连续时间极值的基础资产。它是基于简单和易于实现的一维布朗桥极值分布的解析公式。本文将该方法推广到具有全部或部分标的资产淘汰障碍的多资产期权的定价。基于多维布朗桥相依极值的联合分布，推导了无偏期权价格估计公式。由于一般情况下联合分布没有解析公式，我们基于独立极值的分布和未知分布的Fréchet上下界，建立了上、下有偏的期权价格估计。所有估计器都很简单，易于实现。它们总是可以用于通过置信区间来绑定真值。数值试验表明，我们的有偏估计快速收敛到真正的期权价值的资产路径模拟的时间步长的数量相比，基于标准的离散时间方法的估计增加。收敛速度取决于标的资产的相关性和障碍结构。
We extend the approach of Carr, Itkin and Muravey, 2021 for getting semi-analytical prices of barrier options for the time-dependent Heston model with time-dependent barriers by applying it to the so-called $λ$-SABR stochastic volatility model. In doing so we modify the general integral transform method (see Itkin, Lipton, Muravey, Generalized integral transforms in mathematical finance, World Scientific, 2021) and deliver solution of this problem in the form of Fourier-Bessel series. The weights of this series solve a linear mixed Volterra-Fredholm equation (LMVF) of the second kind also deri…
Pricing Barrier Options with DeepBSDEs
本文针对现有深度BSDE方法无法直接处理障碍期权边界条件的问题，通过在计算图中新增监控与状态保存节点拓展前向深度BSDE，实现了高低维场景下任意障碍条件与Dirichlet边界条件的障碍期权定价。
There exist several methods how more general options can be priced with call prices. In this article, we extend these results to cover a wider class of options and market models. In particular, we introduce a new pricing formula which can be used to price more general options if prices for call options and digital options are known for every strike price. Moreover, we derive similar results for barrier type options. As a consequence, we obtain a static hedging for general options in the general class of models. Our result can be utilised in several significant applications. As a simple example…
Extrema, Barrier Options, and Semi-Analytic Leverage Corrections in Stochastic-Clock Volatility Models
本文研究了随机时钟波动率模型中障碍期权的定价问题，通过半解析方法和小ρ展开，提出快速且稳定的定价公式，并展示了如何在实际校准中应用这些方法以提高准确性。
We develop the general integral transforms (GIT) method for pricing barrier options in the time-dependent Heston model (also with a time-dependent barrier) where the option price is represented in a semi-analytical form as a two-dimensional integral. This integral depends on yet unknown function $Φ(t,v)$ which is the gradient of the solution at the moving boundary $S = L(t)$ and solves a linear mixed Volterra-Fredholm equation of the second kind also derived in the paper. Thus, we generalize the one-dimensional GIT method, developed in (Itkin, Lipton, Muravey, Generalized integral transforms i…
We continue a series of papers where prices of the barrier options written on the underlying, which dynamics follows some one factor stochastic model with time-dependent coefficients and the barrier, are obtained in semi-closed form, see (Carr and Itkin, 2020, Itkin and Muravey, 2020). This paper extends this methodology to the CIR model for zero-coupon bonds, and to the CEV model for stocks which are used as the corresponding underlying for the barrier options. We describe two approaches. One is generalization of the method of heat potentials for the heat equation to the Bessel process, so we…
When the underlying stock price is a strict local martingale process under an equivalent local martingale measure, Black-Scholes PDE associated with an European option may have multiple solutions. In this paper, we study an approximation for the smallest hedging price of such an European option. Our results show that a class of rebate barrier options can be used for this approximation. Among of them, a specific rebate option is also provided with a continuous rebate function, which corresponds to the unique classical solution of the associated parabolic PDE. Such a construction makes existing …
Henstock--Kurzweil Path Integral in Financial Mathematics: A Machine-Verified Pricing of European and Barrier Options
该研究将Henstock–Kurzweil gauge积分应用于Black–Scholes期权定价模型，在Lean 4 / Mathlib中机器验证了欧式期权等定价公式，证明其与经典Ito微积分兼容。
The Heston stochastic volatility model is a standard model for valuing financial derivatives, since it can be calibrated using semi-analytical formulas and captures the most basic structure of the market for financial derivatives with simple structure in time-direction. However, extending the model to the case of time-dependent parameters, which would allow for a parametrization of the market at multiple timepoints, proves more challenging. We present a simple and numerically efficient approach to the calibration of the Heston stochastic volatility model with piecewise constant parameters. We …
In the present paper, we introduce a numerical scheme for the price of a barrier option when the price of the underlying follows a diffusion process. The numerical scheme is based on an extension of a static hedging formula of barrier options. For getting the static hedging formula, the underlying process needs to have a symmetry. We introduce a way to "symmetrize" a given diffusion process. Then the pricing of a barrier option is reduced to that of plain options under the symmetrized process. To show how our symmetrization scheme works, we will present some numerical results applying (path-in…
In this paper, we derive closed-form formulas of first-order approximation for down-and-out barrier and floating strike lookback put option prices under a stochastic volatility model, by using an asymptotic approach. To find the explicit closed-form formulas for the zero-order term and the first-order correction term, we use Mellin transform. We also conduct a sensitivity analysis on these formulas, and compare the option prices calculated by them with those generated by Monte-Carlo simulation.
In this paper, we study the asymptotic behavior of Asian option prices in the worst case scenario under an uncertain volatility model. We give a procedure to approximate the Asian option prices with a small volatility interval. By imposing additional conditions on the boundary condition and cutting the obtained Black-Scholes-Barenblatt equation into two Black-Scholes-like equations, we obtain an approximation method to solve the fully nonlinear PDE.
Using spectral decomposition techniques and singular perturbation theory, we develop a systematic method to approximate the prices of a variety of options in a fast mean-reverting stochastic volatility setting. Four examples are provided in order to demonstrate the versatility of our method. These include: European options, up-and-out options, double-barrier knock-out options, and options which pay a rebate upon hitting a boundary. For European options, our method is shown to produce option price approximations which are equivalent to those developed in [5]. [5] Jean-Pierre Fouque, George Papa…
Keeping Up with the Correlations: Stochastic Spot/Volatility Correlation and Exotic Pricing
本文提出了一种包含随机即期/波动率相关性的双Heston模型，用于更准确地定价外汇市场中的障碍衍生品和波动率交换。
Pricing Exotic Options in a Path Integral Approach
在Black-Scholes-Merton金融衍生工具模型的框架下，提出了期权定价的路径积分方法。给出了多维资产上欧式路径相关期权定价的一般公式，并利用各种灵活有效的算法实现了该公式。作为一个例子，我们详细的情况下，亚洲，障碍淘汰，反向集团和一篮子看涨期权，评估价格和希腊。数值结果与其他程序中使用的定量金融，并发现是在良好的协议。特别是，当定价在的钱和钱外的选择，路径积分的方法表现出竞争力的性能。
We study pricing and (super)hedging for American options in an imperfect market model with default, where the imperfections are taken into account via the nonlinearity of the wealth dynamics. The payoff is given by an RCLL adapted process $(ξ_t)$. We define the {\em seller's superhedging price} of the American option as the minimum of the initial capitals which allow the seller to build up a superhedging portfolio. We prove that this price coincides with the value function of an optimal stopping problem with nonlinear expectations induced by BSDEs with default jump, which corresponds to the so…
We derive a closed-form solution for the price of an average price as well as an average strike geometric Asian option, by making use of the path integral formulation. Our results are compared to a numerical Monte Carlo simulation. We also develop a pricing formula for an Asian option with a barrier on a control process, combining the method of images with a partitioning of the set of paths according to the average along the path. This formula is exact when the correlation is zero, and is approximate when the correlation increases.
Optimal exercise of American options under time-dependent Ornstein-Uhlenbeck processes
本文将时变 Ornstein–Uhlenbeck 过程下的美式期权行权建模为最优停时问题，推导行权边界的 Volterra 积分方程、界与 Lipschitz 连续性，并用 Picard 迭代数值求解。
This paper starts by defining the criteria where the early-exercise of an American option is never optimal, under positive, or negative rates. It follows with a short analysis of the various shapes of the exercise region under negative interest rates. It then presents a new integral equation, which establishes the option price, and the two early exercise boundaries, under negative rates. It shows how to solve this new equation, through modifications of the modern and efficient algorithm of Andersen and Lake, from the initial guess of the two boundaries to more subtle changes required in their …
Convolution-FFT for option pricing in the Heston model
本文提出了一种基于卷积-FFT的方法，用于在Heston模型中高精度定价欧式期权，并提供了显式的误差界分析。
This paper extends the Singular Fourier--Padé (SFP) method proposed by Chan (2018) to pricing/hedging early-exercise options--Bermudan, American and discrete-monitored barrier options--under a Lévy process. The current SFP method is incorporated with the Filon--Clenshaw--Curtis (FCC) rules invented by Domínguez et al. (2011), and we call the new method SFP--FCC. The main purpose of using the SFP--FCC method is to require a small number of terms to yield fast error convergence and to formulate option pricing and option Greek curves rather than individual prices/Greek values. We also numerically…
The present article provides an efficient and accurate hybrid method to price American standard options in certain jump-diffusion models as well as American barrier-type options under the Black & Scholes framework. Our method generalizes the quadratic approximation scheme of Barone-Adesi & Whaley (1987) and several of its extensions. Using perturbative arguments, we decompose the early exercise pricing problem into sub-problems of different orders and solve these sub-problems successively. The obtained solutions are combined to recover approximations to the original pricing problem of multiple…
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