Pricing Formula Research Articles

Option pricing under mixed hedging strategy in time-changed mixed fractional Brownian model

In this paper, we obtain a mixed hedging strategy and a pricing formula in a discrete time setting for a European call option in the time-changed mixed fractional Brownian model. In this manner, we generalize the mixed hedging and the pricing formula in Brownian motion to the time-changed mixed fractional Brownian motion. Finally, through some numerical experiments and empirical analysis, we show that our mixed hedging is better than delta hedging on the hedging error ratio in some cases.

The Pricing Model of Pension Benefit Guaranty Corporation Insurance with Regime-Switching Processes

This paper aims to evaluate Pension Benefit Guaranty Corporation (PBGC) insurance values through regime-switching models. We separate periods of the economy with faster growth from those with slower growth to observe long-term trends in the economy. We derive a fair PBGC insurance pricing formula under distress termination and intervention termination using regime-switching processes. We set parameters by estimating the S&P 500 index and one-year treasury bills via expectation maximization particle swarm optimization (EM-PSO)-Gradient, which is an extension of the EM-Gradient method. Then, we conduct sensitivity analysis to investigate the impact of model parameters on insurance values. According to the maximum likelihood estimation results, the Akaike information criterion (AIC) and Bayesian information criterion (BIC) estimators show that the regime-switching process has better goodness of fit than the geometric Brownian motion. Scenario analysis also supports the adequacy of our pricing formula.

VALUATION OF GENERAL CONTINGENT CLAIMS WITH SHORT SELLING BANS: AN EQUAL-RISK PRICING APPROACH

This paper studies the valuation of general contingent claims with short selling bans under the equal-risk pricing (ERP) framework proposed in I. Guo & S.-P. Zhu (2017) [Journal of Economic Dynamics and Control 76, 136–151]. In existing literature, analytical pricing formulae were derived in the special case, where the payoff function is monotonic under risk-neutral measures. In this paper, we establish a unified framework for this new pricing approach so that its range of application can be significantly expanded. The results of I. Guo & S.-P. Zhu (2017) [Journal of Economic Dynamics and Control 76, 136–151] are extended to the case of non-monotonic payoffs (such as a butterfly spread option) under risk-neutral measures. We also provide numerical schemes for computing equal-risk prices under other measures such as the original physical measure. Furthermore, we demonstrate how short selling bans can affect the valuation of contingent claims by comparing equal-risk prices with Black–Scholes prices.

An Informational Theory of the Dynamic Value of the Firm

This paper formulates an informational theory of the evolving value of the firm under uncertainties and unknowns in the future payoff structures. In general, the horizon of business firms would last for an indefinitely long period of time, and events in the considerably far future are intrinsically unknown. The existing study of indefinite horizon firms often relies on the assumption of time-invariant structures for the derivation of an optimal solution. In this paper, information about the firm’s future payoffs will be revealed as time goes by. The firm will revise its strategies accordingly, and the process will continue indefinitely. This new approach for the analysis of infinite horizon firms via information updating provides a more realistic and practical alternative to the study of the dynamic value of the firm. Finally, information-based option pricing formulae and non-random walks and cycles in asset price can also be generated with this theory.

A semi-analytic valuation of two-asset barrier options and autocallable products using Brownian bridge

Barrier options based upon the extremum of more than one underlying prices do not allow for closed-form pricing formulas, and thus require numerical methods to evaluate. One example is the autocallable structured product with knock-in feature, which has gained a great deal of popularity in the recent decades. In order to increase numerical efficiency for pricing such products, this paper develops a semi-analytic valuation algorithm which is free from the computational burden and the monitoring bias of the crude Monte Carlo simulation. The basic idea is to combine the simulation of the underlying prices at certain time points and the exit (or non-exit) probability of the Brownian bridge. In the literature, the algorithm was developed to deal with a single-asset barrier option under the Black–Scholes model. Now we extend the framework to cover two-asset barrier options and autocallable product. For the purpose, we explore the non-exit probability of the two-dimensional Brownian bridge, which has not been researched before. Meanwhile, we employ the actuarial method of Esscher transform to simplify our calculation and improve our algorithm via importance sampling. We illustrate our algorithm with numerical examples.

Barrier swaption pricing formulae of mean-reverting model in uncertain environment

Barrier swaption is an exotic option, in which the option purchaser has the right to decide whether the swaption will come into effect within a period and it becomes effective (invalid) only when the underlying rises (falls) to the barrier price. This paper studies four kinds of barrier swaptions based on the mean-reverting model, which are up-and-in payer swaption, down-and-in receiver swaption, down-and-out payer swaption, up-and-out receiver swaption, and the price calculation formulae are given. Then, the related parameters are calculated by the minimum cover estimation method. Finally, the examples are given.

Option pricing with state‐dependent pricing kernel

AbstractWe introduce a new volatility model for option pricing that combines Markov switching with the realized generalized autoregressive conditional heteroskedasticity (GARCH) framework. This leads to a novel pricing kernel with a state‐dependent variance risk premium and a pricing formula for European options, which is derived with an analytical approximation method. We apply the Markov‐switching Realized GARCH model to Standard and Poor's 500 index options from 1990 to 2019 and find that investors' aversion to volatility‐specific risk is time‐varying. The proposed framework outperforms competing models and reduces (in‐sample and out‐of‐sample) option‐pricing errors by 15% or more.

Foreign equity lookback options with guarantees

A foreign equity lookback option plays a vital role in hedging foreign exchange rate and asset price risks. Despite its importance, valuing the foreign equity lookback option is problematic because its path dependence and stochastic exchange rate complicate calculating the expected payoff. This paper delivers a unified closed-form pricing formula for the foreign equity lookback call (or put) with fixed (or floating) strike by relying on the extreme-or-nothing formulas that facilitate computing expectations. In addition, it admits valuing the options systematically with the guarantees of exchange rate and equity extremes. Numerical experiments validate the prices obtained from the analytical pricing formula.

An integral equation representation for American better-of option on two underlying assets

In this paper, we study the problem for pricing of American better-of option on two assets. Due to two correlated underlying assets and early-exercise feature which requires two free boundaries to be determined for the option price, this problem is a complex. We propose a new and efficient approach to solve this problem. Mellin transform methods are mainly used to find the pricing formula, and explicit formula for the option price is derived as an integral equation representation. The formula has two free boundaries which are represented by the coupled integral equations. We propose the numerical scheme based on recursive integration method to implement the integral equations and show that our approach with the proposed numerical scheme is accurate and efficient in computing the prices. In addition, we illustrate significant movements on the option prices and two free boundaries with respect to the selected parameters.

Pricing catastrophe equity puts with counterparty risks under Markov-modulated, default-intensity processes

According to the observation of the catastrophic events with regime-switching phenomena and default rate varying with economic condition, we extend the results of Chang et al. (2011) and also take the default rate varying with economic condition into consideration by using the Markov-modulated reduced-form model. In order to value options under stochastic interest rates and a default intensity environment, we employ Girsanov’s theorem to obtain two different forward measures and to derive a pricing formula. We also conduct numerical analyses using Monte Carlo simulations to illustrate the influence of the recovery rate, the time to maturity, the frequency of catastrophic events, and the effect of counterparties’ default intensity on the catastrophe equity put price.

Barrier Option Pricing in the Sub-Mixed Fractional Brownian Motion with Jump Environment

This paper investigates the pricing formula for barrier options where the underlying asset is driven by the sub-mixed fractional Brownian motion with jump. By applying the corresponding Ito^’s formula, the B-S type PDE is derived by a self-financing strategy. Furthermore, the explicit pricing formula for barrier options is obtained through converting the PDE to the Cauchy problem. Numerical experiments are conducted to test the impact of the barrier price, the Hurst index, the jump intensity and the volatility on the value of barrier option, respectively.

Analytical cyclical price–dividend ratios

How non-linear are exact log price–dividend ratios in the fundamental state variables? We work out in continuous time a novel exact formula for the log price–dividend ratio to study how much non-linearity is generated by the persistence of an exogenous mean-reverting and homoskedastic state variable. Our pricing formula highlights that persistence fosters endogenous conditional heteroskedasticity of the stock returns by fueling the non-linearity of the log price–dividend ratio.

Skew-Brownian motion and pricing European exchange options

This article derives a closed-form pricing formula for European exchange options under a non-Gaussian framework for the underlying assets, intending to resolve mispricing associated with a geometric Brownian motion. The dynamics of each of the two correlated underlying assets are assumed to be governed by the exponential of a skew-Brownian motion, which is specified as a sum of a standard Brownian motion and an independent reflected Brownian motion. The proposed pricing formula does not incur additional computational costs than the standard Black–Scholes framework, which one can quickly recover as a particular case of the proposed framework. Finally, we present some numerical experiments followed by a valuable discussion on the results.

A closed-form pricing formula for European options in an illiquid asset market

This article addresses the problem of pricing European options when the underlying asset is not perfectly liquid. A liquidity discounting factor as a function of market-wide liquidity governed by a mean-reverting stochastic process and the sensitivity of the underlying price to market-wide liquidity is firstly introduced, so that the impact of liquidity on the underlying asset can be captured by the option pricing model. The characteristic function is analytically worked out using the Feynman–Kac theorem and a closed-form pricing formula for European options is successfully derived thereafter. Through numerical experiments, the accuracy of the newly derived formula is verified, and the significance of incorporating liquidity risk into option pricing is demonstrated.

AN EMPIRICAL ANALYSIS OF OPTION PRICING WITH SHORT SELL BANS

Short sell bans are often imposed during a financial crisis as a desperate measure to stabilize financial markets. Yet, the impact of short sell bans on option pricing and hedging is not well studied, at least quantitatively, until very recently when Guo & Zhu [(2017) Equal risk pricing under convex trading constraints, Journal of Economic Dynamics and Control 76, 136–151] and He & Zhu [(2020) A revised option pricing formula with the underlying being banned from short selling, Quantitative Finance 20 (6), 935–948] formulated a new pricing framework with the underlying being either completely or partially banned from short selling. However, no empirical results were provided to substantiate the usefulness of the formulae, as well as to deepen our understanding on the effects of short sell bans. This paper provides a comprehensive empirical study on the effects of short sell bans to the standard option pricing theory by carrying out both cross-sectional and options time series model calibration of the model devised by He & Zhu (2020) [A revised option pricing formula with the underlying being banned from short selling, Quantitative Finance 20 (6), 935–948]. Overall, our empirical results indicate that the alternative option pricing formula considering short sell restrictions has the ability to capture highly-quoted implied volatility, with an evident improvement of 39% out-of-sample performance compared to the benchmark Black–Scholes model during the period of short sell ban.

Effective Markovian projection: application to CMS spread options and mid-curve swaptions

Pricing of interest rate derivatives, such as CMS spread or mid-curve options, depends on the modelling of the underlying single rates. For flexibility and realism, these rates are often described in the framework of stochastic volatility models. In this paper we allow rates to be modelled within a class of general stochastic volatility models, which includes the SABR, ZABR, free SABR and Heston models. We provide a versatile technique called Effective Markovian Projection, which allows a tractable model to be found that mimics the distribution of the more complex models used to price multi-rate derivatives. Three different numerical approaches are outlined and applied to relevant examples from practice. Finally, a new method that involves moment-matching of Johnson distributions is applied to facilitate closed-form pricing formulas.

HEDGE TRANSACTIONS, TAXATION, AND ACCOUNTING IN THE NATURAL GAS TRADE IN THE TURKISH ENERGY SECTOR

Due to its significant advantages, natural gas, which forms an increasing share of world energy resources, has become one of the important energy resources worldwide. When natural gas prices are analyzed around the world, it is seen that the price formulas depend on petroleum products. Natural gas prices are increasing in proportion to the price increase in petroleum products. There is a simultaneous change in the same direction between oil prices and exchange rates. Natural gas prices are also determined depending on oil prices substantially in Turkey. Therefore, a hedging mechanism should be implemented against fluctuations in oil prices. Thus, the effect of prices on natural gas cost is reduced and the costs are controlled within a certain band interval or fixed by a single price. In this study, firstly the hedging of natural gas prices is explained with an example application. Then taxation processes of the natural gas trade and accounting records including taxation processes are presented. Herein, a guiding resource has been created for both real sector actors in the natural gas business and academics who plan to design an academic study on this subject.

APPROXIMATE OPTION PRICING FORMULA FOR BARNDORFF-NIELSEN AND SHEPHARD MODEL

For the Barndorf-Nielsen and Shephard model, we present approximate expressions of call option prices based on the decomposition formula developed by [T. Arai (2021) Alos type decomposition formula for Barndor-Nielsen and Shephard model, Journal of Stochastic Analysis 2 (2), 3]. Besides, some numerical experiments are also implemented to make sure how effective our approximations are.

Pricing VXX options by modeling VIX directly

AbstractIn this paper, we first develop a theoretical and model‐free VXX formula in terms of Volatility Index (VIX) futures in both discrete and continuous forms. The discrete form of VXX can quantify the roll yield of VXX, which can be used to explain VXX's underperformance. Using the log‐normal Ornstein–Uhlenbeck (LOU) diffusion model, we show how the number of rolls of VIX futures affects the VXX option pricing formula and its implied volatility (IV). To further verify the nonflat IV of VXX, the VXX option pricing formula under the LOU with stochastic volatility model is also derived. Finally, we analyze their pricing performance, and the ability to forecast implied volatilities.

Pricing defaultable bonds under Hawkes jump-diffusion processes

In this paper, we propose a reduced-form model for the embedded credit risk in corporate bonds. We specify the default hazard rate as an affine function of a series of influential variables. To capture the clustering property in some extreme situations, Hawkes jump-diffusion processes are adopted to model the variables. We derive the semi-analytical pricing formula for defaultable bonds. The empirical results from U.S. bond market illustrate the significance of jump clustering when pricing low credit-rating bonds.