Formula For Options Research Articles

Pricing of timer volatility-barrier options under Heston’s stochastic volatility model

Timer options are financial instruments that enable investors to exercise their rights on a random maturity date the realized variance reaches the level of variance budget. These options provide a stable investment opportunity for investors under the unpredictable and complex financial markets, such as global financial crisis or COVID-19 pandemic, which can induce the drastic changes of the volatility for the underlying asset. Meanwhile, in the financial markets, investors who invest in standard timer options may face the problems caused by the postponement of the exercising time for too low volatility, compared to standard vanilla options. In this regard, to overcome such disadvantages, we propose timer volatility-barrier options, which are activated and expired when the volatility arrives at a relatively low barrier level, with the original properties of the standard timer options. In this paper, by making use of the method of images, we derive an analytical formulas for the timer volatility-barrier options so that the volatility process can be driven by Heston stochastic volatility model, and verify the pricing accuracy of the timer options by comparing our solutions with those obtained from Monte Carlo simulations. Finally, we conduct numerical studies on the timer volatility-barrier options to examine the pricing sensitivities with respect to the various model parameters, and implement the discussion for pricing formula of double volatility barrier type of the timer options.

Pricing exchange options under hybrid stochastic volatility and interest rate models

This paper investigates the pricing of exchange options under hybrid models integrating stochastic volatility and stochastic interest rates. It aims to achieve two primary objectives. First, we derive a closed-form pricing formula for exchange options under a two-factor Heston–Hull–White hybrid model, which accounts for long-term volatility and exhibits relatively broad correlations among the dynamics of asset prices, volatilities, and interest rates. Second, we explore the Heston model’s integration with a generalized single-factor stochastic interest rate model, illustrating that the price is not dependent on the specific form of the interest rate process. A closed-form pricing formula for exchange options under this framework is also derived. Our numerical experiments support the proposed formulas and elucidate the effects of various parameters on option prices.

Pricing for perpetual American strangle options under stochastic volatility with fast mean reversion

A perpetual American strangle option refers to an investment strategy combining the features of both call and put options on a single underlying asset, with an infinite time horizon. Investors are known to use this trading strategy when they expect the stock price to fluctuate significantly but cannot predict whether it will rise or fall. In this study, we consider the perpetual American strangle options under a stochastic volatility model and investigate the corrected option values and free boundaries using an asymptotic analysis technique. Further, we examine the pricing accuracy of the approximated formulas for perpetual American strangle options under stochastic volatility by comparing our solutions with the prices that are obtained from Monte Carlo simulations. We also investigate the sensitivities of the option values and free boundaries with respect to several model parameters.

Valuing American options using multi-step rebate options

The determination of optimal exercise boundaries is a critical aspect of pricing American options, which often requires costly numerical methods. This paper proposes a new approach that employs multi-step rebate options to approximate American option prices. Since the rebate options offer payoffs when the multi-step boundaries are touched, the prices of American options are estimated by maximizing the multi-step rebate option prices, and the optimal multi-step barriers replace the true optimal exercise boundaries. To this end, the closed-form pricing formulas for multi-step rebate options are derived and utilized to approximate several American option prices. Through extensive numerical experiments, we demonstrate the validity and performance of our approach.

Counting jumps: does the counting process count?

In this paper, we develop a ‘jump diffusion type’ financial model based on renewal processes for the discontinuous part of the risk driver, and study its ability to price options and accurately reproduce the corresponding implied volatility surfaces. In this model, the log-returns process displays finite mean and variance, and non-vanishing skewness and excess kurtosis over a long period. The proposed construction is parsimonious, and it allows for a simple and intuitive pricing formula for European vanilla options; furthermore, it offers an efficient Monte Carlo sample path generator for the pricing of exotic options. We illustrate the performance of the proposed framework using observed market data, and we study the features of the best fitting model specifications.

Variance and volatility swaps and options under the exponential fractional Ornstein–Uhlenbeck model

Considering the fair strike values of variance and volatility swaps, we use a stochastic volatility model in which the log volatility is given by a fractional Ornstein–Uhlenbeck process with two versions; a stationary version and a version with a deterministic initial value. Under these versions, the fair strike formulas are obtained in exact form for variance swaps and approximated fair strike formulas are derived for volatility swaps based on the fact that an aggregation of log-normal variables is well-approximated by shifted log-normal or log-normal distribution. In addition, we obtain two approximate pricing formulas for European options on the realized variance and volatility. The accuracy and robustness of the approximated fair strike formulas are examined via Monte-Carlo computations. We conduct calibration experiments to show that the Hurst exponent and the mean-reversion property of the fractional Ornstein–Uhlenbeck process are able to produce various shapes resembling the market term-structures of variance swaps when they are put together.

Using a Mix of Finite Difference Methods and Fractional Differential Transformations to Solve Modified Black–Scholes Fractional Equations

This paper discusses finding solutions to the modified Fractional Black–Scholes equation. As is well known, the options theory is beneficial in the stock market. Using call-and-pull options, investors can theoretically decide when to sell, hold, or buy shares for maximum profits. However, the process of forming the Black–Scholes model uses a normal distribution, where, in reality, the call option formula obtained is less realistic in the stock market. Therefore, it is necessary to modify the model to make the option values obtained more realistic. In this paper, the method used to determine the solution to the modified Fractional Black–Scholes equation is a combination of the finite difference method and the fractional differential transformation method. The results show that the combined method of finite difference and fractional differential transformation is a very good approximation for the solution of the Fractional Black–Scholes equation.

Pricing forward-start style exotic options under uncertain stock models with periodic dividends

<p>In this study, we derived pricing formulas for various forward-start style exotic options based on an uncertain stock models with periodic dividends. Specifically, we present valuations for forward-start, Cliquet/Ratchet, and spread options. In addition, we conducted numerical simulations of these formulas and compared them to pricing formulas for the same options based on a dividend-paying stock model driven by standard Brownian motion.</p>

Pricing geometric average Asian options in the mixed sub-fractional Brownian motion environment with Vasicek interest rate model

<p>Considering the characteristics of long-range correlations in financial markets, the issue of valuing geometric average Asian options is examined, assuming that the variations of the underlying asset follow the mixed sub-fractional Brownian motion, and the dynamics of short-term interest rate satisfies the mixed sub-fractional Vasicek model. Based on the principle of no arbitrage, the definite solution of PDE of a zero-coupon bond for geometric average Asian options under the circumstance of the mixed sub-fractional is given by the delta hedging technique. The derivation of the explicit pricing formula for geometric average Asian options with fixed strike price is achieved through the utilization of multiple variable substitutions. Furthermore, we perform numerical calculations to analyze the performance of the model.</p>

Fractal barrier option pricing under sub-mixed fractional Brownian motion with jump processes

<p>In this work, we mainly focused on the pricing formula for fractal barrier options where the underlying asset followed the sub-mixed fractional Brownian motion with jump, including the down-and-out call option, the down-and-out put option, the down-and-in call option, the down-and-in put option, and so on. To start, the fractal Black-Scholes type partial differential equation was established by using the fractal Itô's formula and a self-financing strategy. Then, by transforming the partial differential equation to the Cauchy problem, we obtained the explicit pricing formulae for fractal barrier options. Finally, the effects of barrier price, fractal dimension, Hurst index, jump intensity, and volatility on the value of fractal barrier options were exhibited through numerical experiments.</p>

A closed-form pricing formula for European options under a multi-factor nonlinear stochastic volatility model with regime-switching

A closed-form pricing formula for European options under a multi-factor nonlinear stochastic volatility model with regime-switching

European Option Pricing under Sub-Fractional Brownian Motion Regime in Discrete Time

In this paper, the approximate stationarity of the second-order moment increments of the sub-fractional Brownian motion is given. Based on this, the pricing model for European options under the sub-fractional Brownian regime in discrete time is established. Pricing formulas for European options are given under the delta and mixed hedging strategies, respectively. Furthermore, European call option pricing under delta hedging is shown to be larger than under mixed hedging. The hedging error ratio of mixed hedging is shown to be smaller than that of delta hedging via numerical experiments.

A Simplified Approach to the Pricing of Vulnerable Options with Two Underlying Assets in an Intensity-Based Model

In this paper, we study a simplified approach to determine the pricing formula for vulnerable options involving two correlated underlying assets. We utilize an intensity-based model to describe the credit risk associated with these vulnerable options. Without the change of measure technique, we derive pricing formulas for vulnerable options involving two underlying assets based on the probabilistic approach. We provide closed-form pricing formulas for two specific types of options: the vulnerable exchange option and the vulnerable foreign equity option. Finally, we present numerical results to demonstrate the accuracy of our formulas using the Monte-Carlo method and the effect of various parameters on the price of options.

Pricing European Options under a Fuzzy Mixed Weighted Fractional Brownian Motion Model with Jumps

This study investigates the pricing formula for European options when the underlying asset follows a fuzzy mixed weighted fractional Brownian motion within a jump environment. We construct a pricing model for European options driven by fuzzy mixed weighted fractional Brownian motion with jumps. By converting the partial differential equation (PDE) into a Cauchy problem, we derive explicit solutions for both European call options and European put options. The figures and tables demonstrating the effectiveness of the results highlight the suitability of the fuzzy mixed weighted fractional Brownian motion with jump model for option pricing.

Pricing first-touch digitals with a multi-step double boundary and American barrier options

A first-touch digital option delivers a payoff when an underlying variable first touches a given boundary, and most studies discuss its pricing by assuming a single flat boundary and a fixed payout amount. This paper derives explicit pricing formulas for first-touch digital options with a multi-step double boundary and customizable payoffs. To this end, we establish the probabilities of the first-hitting times of a Brownian motion for the multi-step double boundary. Furthermore, we explain how the first-touch digital option pricing formulas approximate an American knock-out multi-step barrier option.

Pricing European Vulnerable Options with Jumps and Stochastic Default Obstacles Barrier under Regime Switching

In this paper, we propose an enhanced model for pricing vulnerable options. Specifically, our model assumes that parameters such as interest rates, jump intensity, and asset value volatility are governed by an observable continuous-time finite-state Markov chain. We take into account European vulnerable options that are exposed to both default risk and rare shocks from underlying and counterparty assets. We also consider stochastic default barriers driven by a regime-switching model and geometric Brownian motion, thus improving upon the assumption of fixed default barriers. The risky assets follow a related jump-diffusion process, whereas the default barriers are influenced by a geometric Brownian motion correlated with the risky assets. Within the framework of our model, we derive an explicit pricing formula for European vulnerable options. Furthermore, we conduct numerical simulations to examine the effects of default barriers and other related parameters on option prices. Our findings indicate that stochastic default barriers increase credit risk, resulting in a decrease in option prices. By considering the aforementioned factors, our research contributes to a better understanding of pricing vulnerable options in the context of counterparty credit risk in over-the-counter trading.

Pricing quanto options with market liquidity risk.

This paper investigates the pricing problem of quanto options with market liquidity risk using the Bayesian method. The increasing volatility of global financial markets has made liquidity risk a significant factor that should be taken into consideration while evaluating option prices. To address this issue, we first derive the pricing formula for quanto options with liquidity risk. Next, we construct a likelihood function to conduct posterior inference on model parameters. We then propose a numerical algorithm to conduct statistical inferences on the option prices based on the posterior distribution. This proposed method considers the impact of parameter uncertainty on option prices. Finally, we conduct a comparison between the Bayesian method and traditional estimation methods to examine their validity. Empirical results show that our proposed method is feasible for pricing and predicting quanto options with liquidity risk, particularly for parameter estimations with a small sample size.

Funding startups using contingent option of value appreciation: theory and formula

PurposeThis paper provides a structural model to value startup companies and determine the optimal level of research and development (R&D) spending by these companies.Design/methodology/approachThis paper describes a new variant of float-the-money options, which can act as a financial instrument for financing R&D expenses for a specific time horizon or development stage, allowing the investor to share in the startup's value appreciation over that duration. Another innovation of this paper is that it develops a structural model for evaluating optimal level of R&D spending over a given time horizon. The paper deploys the Gompertz-Cox model for the R&D project outcomes, which facilitates investigation of how increased level of R&D input can enhance the company's value growth.FindingsThe author first introduces a time-varying drift term into standard Black-Scholes model to account for the varying growth rates of the startup at different stages, and the author interprets venture capital's investment in the startup as a “float-the-money” option. The author then incorporates the probabilities of startup failures at multiple stages into their financial valuation. The author gets a closed-form pricing formula for the contingent option of value appreciation. Finally, the author utilizes Cox proportional hazards model to analyze the optimal level of R&D input that maximizes the return on investment.Research limitations/implicationsThe integrated contingent claims model links the change in the financial valuation of startups with the incremental R&D spending. The Gompertz-Cox contingency model for R&D success rate is used to quantify the optimal level of R&D input. This model assumption may be simplistic, but nevertheless illustrative.Practical implicationsOnce supplemented with actual transaction data, the model can serve as a reference benchmark valuation of new project deals and previously invested projects seeking exit.Social implicationsThe integrated structural model can potentially have much wider applications beyond valuation of startup companies. For instance, in valuing a company's risk management, the level of R&D spending in the model can be replaced by the company's budget for risk management. As another promising application, in evaluating a country's economic growth rate in the face of rising climate risks, the level of R&D spending in this paper can be replaced by a country's investment in addressing climate risks.Originality/valueThis paper is the first to develop an integrated valuation model for startups by combining the real-world R&D project contingencies with risk-neutral valuation of the potential payoffs.

Valuing of timer path-dependent options

Timer options are financial instruments, first proposed by Société Générale Corporate and Investment Banking in 2007, which allow investors to exercise the options randomly under the level of volatility, unlike a vanilla style option exercised at a fixed maturity date. In this article, we study the problem of valuing the timer path-dependent options where the volatility is governed by a fast-mean reverting process. Specifically, extending and developing the study by Saunders (2010), we derive analytical formulas for path-dependent timer options by using the method of images as shown in Buchen (2001) and the technique of asymptotic expansions as described in Fouque et al. (2011). Moreover, we verify the pricing accuracy of the analytic formulas of path-dependent options by comparing our solutions with the ones from the Monte Carlo simulations. Finally, we experiment with the numerical studies on the timer-path dependent options to demonstrate the pricing sensitivities with respect to the model parameters.

Nonparametric estimates of option prices via Hermite basis functions

We consider approximate pricing formulas for European options based on approximating the logarithmic return’s density of the underlying by a linear combination of rescaled Hermite polynomials. The resulting models, that can be seen as perturbations of the classical Black-Scholes one, are nonpararametric in the sense that the distribution of logarithmic returns at fixed times to maturity is only assumed to have a square-integrable density. We extensively investigate the empirical performance, defined in terms of out-of-sample relative pricing error, of this class of approximating models, depending on their order (that is, roughly speaking, the degree of the polynomial expansion) as well as on several ways to calibrate them to observed data. Empirical results suggest that such approximate pricing formulas, when compared with simple nonparametric estimates based on interpolation and extrapolation on the implied volatility curve, perform reasonably well only for options with strike price not too far apart from the strike prices of the observed sample.