Underlying Asset Price Research Articles

Fractional Stochastic Volatility Pricing of European Option Based on Self-Adaptive Differential Evolution

The option pricing model can predict the future trend of the financial market. In order to more accurately describe the changing process of the financial market, the Hurst index which can describe the characteristics of long-term memory is introduced into the traditional Heston model. Under the assumption that the underlying asset price follows fractional Brownian motion, the fractional stochastic volatility pricing of European option pricing model (Hurst-Heston model) is constructed, and the closed solution of the model is obtained according to the partial differential equation satisfied by the model. By analyzing the relationship between Hurst index and asset price, it is found that the movement process of asset price under the hypothesis of this model is more consistent with the real market change law, which verifies the rationality of the model. In the process of empirical analysis of SSE 50ETF put option data, Self-adaptive Differential Evolution algorithm is used to estimate parameters. The results showed that the error of Hurst-Heston model is smaller than other models, and the prediction error for consecutive trading days is similar. It showed that the pricing results of Hurst-Heston model are more accurate and stable.

Power Option Pricing Based on Time‐Fractional Model and Triangular Interval Type‐2 Fuzzy Numbers

The problem of generalizing the power option‐pricing model to incorporate more empirical features becomes an urgent and necessary event. A new power option pricing method is designed for the financial market uncertainty that simultaneously involves randomness and fuzziness. The randomness in market uncertainty is modeled by a time‐fractional diffusion model, which describes trend memory in underlying asset prices. The fuzziness in market uncertainty is characterized by a triangular interval type‐2 fuzzy numbers, which better captures the fuzziness of underlying asset prices. Considering the decision‐maker’s subjective judgment, we show the price mean value with the possibility‐necessity weight and pessimistic‐optimistic index under the type‐2 fuzzy environment. We develop the power option pricing model with the time‐fractional diffusion model and the triangular interval type‐2 fuzzy numbers. Furthermore, the analytic solutions of pricing call power option and put power option are obtained and verified by the variational iterative reconstruction method. Our study shows that power option pricing, which adopts the time‐fractional model and the triangular interval type‐2 fuzzy numbers, can better capture the trend memory and double fuzziness of the real market. In addition, a numerical example is provided to illustrate that the power option means the value is decreasing with respect to the pessimistic‐optimistic index and is fluctuating with respect to the possibility‐necessity weight index.

A Valuation Formula for Chained Options with n‐Barriers

This study examines chained options that are connected in the sense that another barrier option becomes active continuously after the underlying asset price crosses a primary barrier. These barrier options have several advantages. First, they preserve the merit of regular barrier options, but demand far lower option premiums, which appeal to option traders. Second, they reduce the higher risk of loss of double barrier options, making option strategies more profitable in certain cases. Third, they have closed‐form pricing formulas, unlike double‐barrier options, and, thus, avoid the complexity of option pricing. Therefore, they help to enlarge the range of trader’s choice according to a variety of demand of buyers. The values of chained options are compared to those of similar single‐ and double‐barrier options. This study extends the chained option with two barriers to a generalized chained option with n‐barriers. In addition, this paper proves the closed formulas of generalized chained options with n‐barriers using mathematical induction.

Option Pricing with a Local Bimodal Distribution of Underlying Asset Price

Option Pricing with a Local Bimodal Distribution of Underlying Asset Price

Pricing Multi-Asset Derivatives by Finite-Difference Method on a Quantum Computer

Following the recent great advance of quantum computing technology, there are growing interests in its applications to industries, including finance. In this article, we focus on derivative pricing based on solving the Black–Scholes partial differential equation by the finite-difference method (FDM), which is a suitable approach for some types of derivatives but suffers from the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">curse of dimensionality</i> , that is, exponential growth of complexity in the case of multiple underlying assets. We propose a quantum algorithm for FDM-based pricing of multi-asset derivative with exponential speedup with respect to dimensionality compared with classical algorithms. The proposed algorithm utilizes the quantum algorithm for solving differential equations, which is based on quantum linear system algorithms. Addressing the specific issue in derivative pricing, that is, extracting the derivative price for the present underlying asset prices from the output state of the quantum algorithm, we present the whole of the calculation process and estimate its complexity. We believe that the proposed method opens the new possibility of accurate and high-speed derivative pricing by quantum computers.

Option Volatility Investment Strategy: The Combination of Neural Network and Classical Volatility Prediction Model

This study focuses on the volatility prediction and option volatility investment. By investigating the traditional Volatility Prediction Model and machine learning algorithms, this study tries to merge these two aspects together. This work setup a bridge of previous financial studies and machine learning studies by proposing an algorithm integrating neural network and three traditional volatility models, called “Quantile based neural network and model integration combination algorithm.” The algorithm effectively lowers the volatility prediction error (measured by root of mean square error, shorted for RMSE: 0.319724) and beat the Wavenet (RMSE: 0.44) which is the benchmark and surpasses integrated model (RMSE: 0.348346) in test set. In terms of option investment strategy, this paper constructs a CSI 300 index option portfolio which hedges the underlying asset price risk and exposes the volatility risk. Then propose the “Option strategy of volatility prediction with dynamic thresholds.” With the new algorithm above, the strategy improves the return‐risk ratio in test set (measured by Sharpe ratio: 1.99–2.07).

Bias reduction in spot volatility estimation from options

We consider the problem of nonparametric spot volatility estimation from options that is robust to time-variation in volatility and presence of jumps in the underlying asset price. Using a higher-order expansion of the characteristic function of the underlying price increment over shrinking time intervals and option-based estimates of the latter over two distinct horizons, we achieve asymptotic bias-reduction in spot volatility estimation, relative to existing methods, that is due to time-variation in volatility and presence of jumps. Further asymptotic improvement is achieved by de-biasing the volatility estimator using an estimate for the bias in it due to the small jumps in the price process. The gains from the newly-developed volatility estimation approach are illustrated on simulated data and in an empirical application.

Computation of powered option prices under a general model for underlying asset dynamics

We derive the Laplace transforms for the prices and deltas of the powered call and put options, as well as for the price and delta of the capped powered call option under a general framework. These Laplace transforms are expressed in terms of the transform of the underlying asset price at maturity. For any model that can derive the transform of the underlying asset price, we can obtain the Laplace transforms for the prices and deltas of the powered options and the capped powered call option. The prices and deltas of the powered options and the capped powered call option can be computed by numerical inversion of the Laplace transforms. Models to which our method can be applied include the geometric Lévy model, the regime-switching model, the Black–Scholes–Vasiček model, and Heston’s stochastic volatility model, which are commonly used for pricing of financial derivatives. In this paper, numerical examples are presented for all four models.

PRICING ASIAN OPTIONS WITH CORRELATORS

We derive a series expansion by Hermite polynomials for the price of an arithmetic Asian option. This requires the computation of moments and correlators of the underlying asset price which for a polynomial jump–diffusion process are given analytically; hence, no numerical simulation is required to evaluate the series. This allows to derive analytical expressions for the option Greeks. The weight function defining the Hermite polynomials is a Gaussian density with scale [Formula: see text]. We find that the rate of convergence of the series depends on [Formula: see text], for which we prove a lower bound to guarantee convergence. Numerical examples show that the series expansion is accurate but unstable for initial values of the underlying process far from zero, mainly due to rounding errors.

SABR equipped with AI wings

This study proposes an artificial neural network (ANN) based option pricing framework under the SABR (stochastic alpha beta rho) and free boundary SABR volatility models. Unlike previous research, we do not directly apply the ANN technique to train and predict option implied volatilities. Instead, we use the technique to train and predict the differences between the implied volatilities calculated by asymptotic approximation and those computed by Monte Carlo simulation. Since the analytical solution for an option written on an underlying asset price in the SABR model is intractable, approximation techniques are used to derive an approximate closed-form solution in practice. However, these approximation formulas worsen as maturity increases and the underlying asset's volatility is high. The accuracy decreases rapidly in wings, which represents deep in-the-money and out-of-the-money cases. By combining the approximation formulas and ANN framework, our new option pricing method offers the following improvements: (1) the training becomes more robust, and the predictions produce more stable and accurate results: (2) it significantly speeds up the training procedure: and (3) the accuracy and efficiency of approximation are improved in the wings without sacrificing performance.

Pricing Vulnerable Options in the Bifractional Brownian Environment with Jumps

In this paper, we study the valuation of European vulnerable options where the underlying asset price and the firm value of the counterparty both follow the bifractional Brownian motion with jumps, respectively. We assume that default event occurs when the firm value of the counterparty is less than the default boundary. By using the actuarial approach, analytic formulae for pricing the European vulnerable options are derived. The proposed pricing model contains many existing models such as Black–Scholes model (1973), Merton jump-diffusion model (1976), Klein model (1996), and Tian et al. model (2014).

Expected Utility Theory on General Affine GARCH Models

ABSTRACT Expected utility theory has produced abundant analytical results in continuous-time finance, but with very little success for discrete-time models. Assuming the underlying asset price follows a general affine GARCH model which allows for non-Gaussian innovations, our work produces an approximate closed-form recursive representation for the optimal strategy under a constant relative risk aversion (CRRA) utility function. We provide conditions for optimality and demonstrate that the optimal wealth is also an affine GARCH. In particular, we fully develop the application to the IG-GARCH model hence accommodating negatively skewed and leptokurtic asset returns. Relying on two popular daily parametric estimations, our numerical analyses give a first window into the impact of the interaction of heteroscedasticity, skewness and kurtosis on optimal portfolio solutions. We find that losses arising from following Gaussian (suboptimal) strategies, or Merton's static solution, can be up to and 5%, respectively, assuming low-risk aversion of the investor and using a five-years time horizon.

Analoging the digital: Designing better binary option contracts

AbstractBinary option pays a fixed dollar amount if it matures in the money and nothing otherwise. While this cash‐or‐nothing payoff structure is very attractive to speculators, it also creates incentives to manipulate the underlying asset price in order to gain extra payoff. In this paper, we propose better designs for binary options that disincentivize market manipulation and highlight two key features of such designs. We demonstrate the effectiveness of the new contract designs using numerical examples.

Lookback Option Pricing Problems of Uncertain Mean-Reverting Stock Model

A lookback option is a path-dependent option, offering a payoff that depends on the maximum or minimum value of the underlying asset price over the life of the option. This paper presents a new mean-reverting uncertain stock model with a floating interest rate to study the lookback option price, in which the processing of the interest rate is assumed to be the uncertain counterpart of the Cox–Ingersoll–Ross (CIR) model. The CIR model can reflect the fluctuations in the interest rate and ensure that such rate is positive. Subsequently, lookback option pricing formulas are derived through the α-path method and some mathematical properties of the uncertain option pricing formulas are discussed. In addition, several numerical examples are given to illustrate the effectiveness of the proposed model.

The closed-form approximation to price basket options under stochastic interest rate

The paper presents closed-form approximation formulas for pricing basket options. We assume that the underlying asset prices follow geometric Brownian motions and the interest rate follows the one-factor Hull-White model. Under a given forward measure, we obtain the analytical lower and upper bounds of basket options. By finding a simple random variable to replace the sum of the lognormal random variables, we combine the conditioning and the moment matching approaches, and derive approximation formulas to price basket options. Numerical results illustrate that our results fall in the sharp lower and upper bounds of basket options and are consistent with Monte Carlo simulation results.

Pricing Product Options and Using Them to Complete Markets for Functions of Two Underlying Asset Prices

Options paying the product of put and/or call option payouts at different strikes on two underlying assets are observed to synthesize joint densities and replicate differentiable functions of two underlying asset prices. The pricing of such options is undertaken from three perspectives. The first perspective uses a geometric two-dimensional Brownian motion model. The second inverts two-dimensional characteristic functions. The third uses a bootstrapped physical measure to propose a risk charge minimizing hedge using options on the two underlying assets. The options are priced at the cost of the hedge plus the risk charge.

Is normal backwardation normal? Valuing financial futures with a local index-rate covariance

Revisiting the two-factor valuation of futures contracts, we propose a new pricing model for financial futures and their derivatives. The linkage between the money market funding rate and the underlying asset price is stochastic and state-dependent, in compliance with investors’ arbitrage strategies. The model explicitly captures the impact of interest rate expectations in the marking-to-market feature of futures, as predicted by Cox, Ingersoll, and Ross (1981) theory. The backwardation vs. contango regime of financial futures depends on a new parameter, the contango factor, which paves the way for future empirical studies. Akin to the implied volatility of option contracts, the contango factor provides market participants with a universal gauge of futures contracts’ level of contango, consistent across futures markets and maturities. Our numerical simulations show significant deviations from the traditional cost-of-carry model of futures prices, with price deviations above 1% even for short-term futures contracts.

Option pricing with illiquidity during a high volatile period

This paper deals with the valuation of options in markets without liquidity and under stress. More precisely, a European option is considered when the dynamic of the underlying asset is governed by a Brownian motion. Following Liu and Young (2005), a term related to the number of invested stocks is embedded into the model. Moreover, the volatility of the asset is augmented by a separate function that models the abnormal increase of the volatility. Under these settings, we deal with the evaluation of European options. Dealing with illiquidity during a financial crisis is an important issue in terms of financial risk management, which has not been tackled with in the literature to the best of our knowledge. It is during these non‐normal periods that volatility in the financial markets is very high and tools for reducing the underlying risk are urgently needed for hedging purposes. In this work, the PDE of the option price within this context is derived, which is a generalization of the PDE of Liu and Young (2005). Moreover, the hedging strategy to replicate the price of the option is obtained. Numerical simulations for the underlying asset price are conducted. The illustrations support several stylized facts of the model that seem to reflect what is operational in the real financial markets.

A numerical approach to pricing exchange options under stochastic volatility and jump-diffusion dynamics

We consider a method of lines (MOL) approach to determine prices of European and American exchange options when underlying asset prices are modeled with stochastic volatility and jump-diffusion dynamics. As with any other numerical scheme for partial differential equations (PDEs), the MOL becomes increasingly complex when higher dimensions are involved, so we first simplify the problem by transforming the exchange option into a call option written on the ratio of the yield processes of the two assets. This is achieved by taking the second asset yield process as the numéraire. Under the equivalent martingale measure induced by this change of numéraire, we derive the exchange option pricing integro-partial differential equations (IPDEs) and investigate the early exercise boundary of the American exchange option. We then discuss a numerical solution of the IPDEs using the MOL, its implementation using computing software and possible alternative boundary conditions at the far limits of the computational domain. Our analytical and numerical investigation shows that the near-maturity behavior of the early exercise boundary of the American exchange option is significantly influenced by the dividend yields and the presence of jumps in the underlying asset prices. Furthermore, with the numerical results generated by the MOL, we are able to show that key jump and stochastic volatility parameters significantly affect the early exercise boundary and exchange option prices. Our numerical analysis also verifies that the MOL performs more efficiently, than other finite difference methods or simulation approaches for American options, since the MOL integrates the computation of option prices, greeks and the early exercise boundary, and does so with the least error.

Callable barrier reverse convertible securities

We study the valuation of callable barrier reverse convertible contracts written on one or two underlying asset prices. We assume the issuer of the contract can call early redemption at any time during a pre-specified time interval. We identify the optimal redemption policy and show, in the single underlying asset case, it is characterized by a time-dependent boundary. The boundary satisfies a nonlinear integral equation of the Volterra type. When there are two underlying assets, the boundary is a surface depending on one price in addition to time. Valuation formulas and associated integral equations are derived. Numerical experiments are performed.