Option Pricing Formula Research Articles

Finite Mixture Approximation of CARMA(p,q) Models

In this paper we show how to approximate the transition density of a CARMA(p, q) model driven by means of a time changed Brownian Motion based on the Gauss-Laguerre quadrature. We then provide an analytical formula for option prices when the log price follows a CARMA(p, q) model. We also propose an estimation procedure based on the approximated likelihood density.

A pure-jump mean-reverting short rate model

A new multi-factor short rate model is presented which is bounded from below by a real-valued function of time. The mean-reverting short rate process is modeled by a sum of pure-jump Ornstein–Uhlenbeck processes such that the related bond prices possess affine representations. Also the dynamics of the associated instantaneous forward rate is provided and a condition is derived under which the model can be market-consistently calibrated. The analytical tractability of this model is illustrated by the derivation of an explicit plain vanilla option price formula. With view on practical applications, suitable probability distributions are proposed for the driving jump processes. The paper is concluded by presenting a post-crisis extension of the proposed short and forward rate model.

Pricing two-asset alternating barrier options with icicles and their variations

This paper introduces a new class of barrier options and its variations. We call the new class of options as two-asset alternating barrier options, since we consider alternating barrier levels for two underlying assets. The alternating barrier levels are placed in the sub-periods of the option’s lifetime; each being applied to one of the two underlying assets. We also consider vertical branches of the barrier, which are termed as icicles. The alternating barrier with icicles can be often seen as an embedded form in various equity-linked financial products. To price such new options, we obtain the joint distribution of two underlying asset prices at an intermediate time point and the maturity, along with their partial maximums under the Black–Scholes model. This joint distribution plays a critical role in the derivation of the pricing formulas for alternating barrier options and their variants. As in ordinary barrier options, we consider eight types of alternating barrier options and derive their explicit option pricing formulas. To our knowledge, the pricing formulas for these options have never been obtained explicitly in the literature even under the Black–Scholes model. We also examine an autocallable equity-linked investment product to derive its explicit pricing formula. Our results are illustrated with numerical examples, showing the effect of different barrier levels and different values of correlation coefficient between two underlying asset prices.

Constructing Out-of-the-Money Longevity Hedges Using Parametric Mortality Indexes

Proposed by Chan, Li, and Li, parametric mortality indexes (i.e., indexes created using the time-varying parameters in a suitable stochastic mortality model) can be used to develop tradable mortality-linked derivatives such as K-forwards. Compared to existing indexes such as the Life and Longevity Markets Association’s LifeMetrics, parametric mortality indexes are richer in information content, allowing the market to better concentrate liquidity. In this article, we further study this concept in several aspects. First, we consider options written on parametric mortality indexes. Such options enable hedgers to create out-of-the-money longevity hedges, which, compared to at-the-money-hedges created with q-/K-forwards, may better meet hedgers’ needs for protection against downside risk. Second, using the properties of the time series processes for the parametric mortality indexes, we derive analytical risk-neutral pricing formulas for K-forwards and options. In addition to convenience, the analytical pricing formulas remove the need for computationally intensive nested simulations that are entailed in, for example, the calculation of the hedging instruments’ values when a dynamic hedge is adjusted. Finally, we construct static and dynamic Greek hedging strategies using K-forwards and options, and demonstrate empirically the conditions under which an out-of-the-money hedge is more economically justifiable than an at-the-money one.

Marital quality in arranged and couple-initiated marriages: The role of perceived influence over partner selection.

Previous comparisons of relationship quality between individuals in couple-initiated and arranged marriage have yielded inconsistent findings. One factor which may help to explain this inconsistency is variability in the way in which arranged and couple-initiated marriages are practiced. To test this, we recruited 116 Bangladeshi women engaged to be married or within the first 3 years of couple-initiated and arranged marriages. We tested to see which of three models (type of marriage only, perceived influence over partner selection only, and a model that included both relationship type and influence over partner selection) best accounted for variance in self-reported intimacy, passion, commitment as well as positive and negative relationship quality. Results suggest that influence over partner selection is a better predictor of these outcomes than marriage type, with marriage type offering little if any information beyond that provided by perceived influence. Regardless of marriage type, women with greater influence over their partner selection reported higher levels of intimacy, passion, commitment and positive marital quality. Negative marital quality was unrelated. Future research may benefit from reconceptualising the arranged/couple-initiated marriage dichotomy as a continuum.

An Empirical Study of the Option Pricing Formula with the Underlying Banned from Short Sell

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 quantitatively studied until very recently when Guo and Zhu (2017) and He and Zhu (2018) 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 and Zhu (2018). 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.

Valuation and empirical analysis of currency options

In this paper, we study whether the correlated jump risks of interest and exchange rates play an important role in currency option pricing. We augment the model of Jarrow and Yildirim (2003) with correlated jump risks (herein referred to as the CB-CJ model) and derive the pricing formula for currency options under this model. Using the data of the United States, Japan, European Union, and the United Kingdom, we find that CB-CJ outperforms the geometric Brownian model, the original Jarrow and Yildirim model, and the Jarrow and Yildirim model with independent jump risks because it substantially improves the in-sample and out-of-sample pricing errors in most cases. As a result, we conclude that correlated jump risks are important factors when pricing currency options.

A semianalytical formula for European options under a hybrid Heston–Cox–Ingersoll–Ross model with regime switching

AbstractIn this paper, we consider the pricing of European options under a regime‐switching Heston–Cox–Ingersoll–Ross (CIR) hybrid model, where the mean‐reversion levels of both the stochastic volatility and interest rate are assumed to change among different states. Albeit difficult, we have still managed to derive an semianalytical pricing formula for European options after the generalized moment generating function of this particular model is worked out. Numerical experiments are also carried out to demonstrate the accuracy of the newly derived formula as well as the influence of the introduction of the regime‐switching mechanics on option prices. Finally, through a preliminary empirical study, our model is shown to be superior to the Heston‐CIR model, which demonstrates the importance of introducing the regime‐switching mechanics.

How Rational Are the Option Prices of Hong Kong Dollar Exchange Rate?

In this paper, we study the prices of the options on Hong Kong's linked exchange rate. The study was motivated by the apparent contradiction that options with strike prices outside the narrow trading band have positive prices. We developed a simple regime-switching model of the exchange rate and provided a formula for the prices of its options. The option pricing formula allows us to back out the implied probability of the failure of the linked exchange rate regime. With the option price data for the period from June 1, 2005 to July 31, 2018, we find that the market's belief about the likelihoods of the failure, as implied by the option prices, is too high to be justified by the possibility of the failure alone. This is first shown qualitatively, without using any model, with ratio of empirical over implied volatilities and the implied range of strike prices. Then, using the model, we find that, close to 40% of the sample period, the market thinks the failure probability is greater than 10%. When contrasted with the fact that the regime has not failed since 1983, the p-value of that event is less than 0.1%. Our finding suggests that, while the failure of the linked exchange rate regime is a significant risk factor, it cannot explain the risk premium seen in the option prices.

The non-Markovian property of [formula omitted]-Gaussian process

In this paper, the q-Gaussian process based on the non-extensive theory is discussed from a mathematical point of view, which has been widely applied to many anomalous diffusion systems in physics and finance. Firstly, the discussion of non-Markovian property of q-Gaussian process provides a numerical support for the future theoretical research. Secondly, the martingale and self-similarity of this process are obtained by Tsallis distributions. Thirdly, the long dependence is analyzed by simulations and Hurst exponents are compared with those of fractional Brownian motion. At last, the European call option price formula driven by this process is simulated, by which we find that this process can better match anomalous diffusion and the volatility smile.

Analytical valuation of Asian options with counterparty risk under stochastic volatility models

AbstractIn this paper, we consider Asian options with counterparty risk under stochastic volatility models. We propose a simple way to construct stochastic volatility models through the market factor channel. In the proposed framework, we obtain an explicit pricing formula of Asian options with counterparty risk and illustrate the effects of systematic risk on Asian option prices. Specially, the U‐shaped and inverted U‐shaped curves appear when we keep the total risk of the underlying asset and the issuer's assets unchanged, respectively.

A SEMI-ANALYTICAL PRICING FORMULA FOR EUROPEAN OPTIONS UNDER THE ROUGH HESTON-CIR MODEL

We combine the rough Heston model and the CIR (Cox–Ingersoll–Ross) interest rate together to form a rough Heston-CIR model, so that both the rough behaviour of the volatility and the stochastic nature of the interest rate can be captured. Despite the convoluted structure and non-Markovian property of this model, it still admits a semi-analytical pricing formula for European options, the implementation of which involves solving a fractional Riccati equation. The rough Heston-CIR model is more general, taking both the rough Heston model and the Heston-CIR model as special cases. The influence of rough volatility and stochastic interest rate is shown to be significant through numerical experiments.

Representation of exchange option prices under stochastic volatility jump-diffusion dynamics

In this article, we provide representations of European and American exchange option prices under stochastic volatility jump-diffusion (SVJD) dynamics following models by Merton [Option pricing when underlying stock returns are discontinuous. J. Financ. Econ., 1976, 3(1-2), 125–144], Heston [A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud., 1993, 6(2), 327–343], and Bates [Jumps and stochastic volatility: Exchange rate processes implicit in Deutsche Mark options. Rev. Financ. Stud., 1996, 9(1), 69–107]. A Radon–Nikodým derivative process is also introduced to facilitate the shift from the objective market measure to other equivalent probability measures, including the equivalent martingale measure. Under the equivalent martingale measure, we derive the integro-partial differential equation that characterizes the exchange option prices. We also derive representations of the European exchange option price using the change-of-numéraire technique proposed by Geman et al. [Changes of numéraire, changes of probability measure and option pricing. J. Appl. Probab., 1995, 32(2), 443–458] and the Fourier inversion formula derived by Caldana and Fusai [A general closed-form spread option pricing formula. J. Bank. Finance, 2013, 37, 4893–4906], and show that these two representations are comparable. Lastly, we show that the American exchange option price can be decomposed into the price of the European exchange option and an early exercise premium.

Empirical analysis of SH50ETF and SH50ETF option prices under regime-switching jump-diffusion models

In this paper, we use extensive empirical data sets from Shanghai 50ETF (SH50ETF) and SH50ETF options markets in China to study how regime-switching jump-diffusion models improve goodness of fit and option pricing performance. Firstly, the model parameters are estimated by using maximum likelihood estimation and the numerical analysis indicates that the regime-switching jump-diffusion models outperform a range of other models. Secondly, the analytical option pricing formulae are obtained via the fast Fourier transform and the empirical results using the proposed option pricing formulae are presented. Finally, we find that the calculated option prices are fairly consistent with the actual market prices.

Tractable Rare Disaster Probability and Options-Pricing

We derive an option-pricing formula from recursive preference and estimate rare disaster probability. The new options-pricing formula applies to far-out-of-the money put options on the stock market when disaster risk dominates, the size distribution of disasters follows a power law, and the economy has a representative agent with Epstein-Zin utility. The formula conforms with options data on the S&P 500 index from 1983-2018 and for analogous indices for other countries. The disaster probability, inferred from monthly fixed effects, is highly correlated across countries, peaks during the 2008-2009 financial crisis, and forecasts equity index returns and growth vulnerabilities in the economy.

Rare Disaster Probability and Options-Pricing

We derive an option-pricing formula from recursive preferences and estimate rare disaster probability. The new options-pricing formula applies to far-out-of-the money put options on the stock market when disaster risk dominates, the size distribution of disasters follows a power law, and the economy has a representative agent with a constant-relative-risk-aversion utility function. The formula conforms with options data on the S&P 500 index from 1983-2018 and for analogous indices for other countries. The disaster probability, inferred from monthly fixed effects, is highly correlated across countries, peaks during the 2008-2009 financial crisis, and forecasts rates of economic growth.

Research on the American Call Options on the Stocks Paying Multiple Dividends

Cassimon et al. (2007) propose a pricing formula of American call options under the multiple dividends by extending Roll (1977). However, because these studies investigate the option pricing formula under the escrow model, there is inconsistency for the assumption of the stock prices. This paper proposes pricing formulas of American call options under the multiple dividends and piecewise geometric Brownian motion. For the formulas, I approximate the log prices of ex-dividend dates to follow a multivariate normal distribution, and decompose the option price as a function of payoffs and exercise boundaries. Then, I obtain an upper bound of the American call options by substituting approximated log prices into the both of the payoffs and the exercise boundaries. Besides, I obtain a lower bound of the price by substituting approximated price only into the exercise boundaries. These upper and lower bounds are exact prices when the amounts of dividends are linear to the stock prices. According to the numerical study, the lower bound produces relatively small errors. Especially, it produces small errors when the dividends are more sensitive to the stock price changes.

Exponentiation of conditional expectations under stochastic volatility

We use the Itô Decomposition Formula (see Alòs [A decomposition formula for option prices in the Heston model and applications to option pricing approximation. Finance Stoch., 2012, 16(3), 403–422]) to express certain conditional expectations as exponentials of iterated integrals. As one application, we compute an exact formal expression for the leverage swap for any stochastic volatility model expressed in forward variance form. As another, we show how to derive a smile expansion analogous to that of Bergomi and Guyon to all orders. Finally, we compute exact expressions under rough volatility, obtaining in particular the fractional Riccati equation for the rough Heston characteristic function. As a corollary, we compute a closed-form expression for the leverage swap in the rough Heston model which can be used for fast calibration.

An improved Mellin transform approach to BSM formula of ML-payoff function

Panini and Srivastav introduced valuation of the European put option for plain vanilla payoff through Mellin transform. This transform fails to valuing European call option. Fadugba and Nwozo derived formula for plain vanilla put option using improved Mellin transform. This paper contributes to the valuation of European call option for Modified Log-payoff (ML-payoff) using improved Mellin transform. It turns out that the present option pricing formula is quite close to the celebrated plain vanilla option.

Extreme values for solution to uncertain fractional differential equation and application to American option pricing model

Uncertain fractional differential equation plays an important role of describing uncertain dynamic process. This paper focuses on extreme values (including supremum and infimum) for solution to an uncertain fractional differential equation for the Caputo type. Theorems for the inverse uncertain distributions of the extreme values are given based on the definition of α-path. And then, numerical algorithms for them are designed, numerical examples are shown for validating the availability about algorithms. The absolute errors between the numerical and analytical results are also presented to demonstrate the accuracy of the algorithms. Finally, as an application of the extreme values, an uncertain stock model is proposed on the basis of uncertain fractional differential equation of the Caputo type. The American option pricing formulas of such stock model are studied by using the proposed extreme theorems. Besides, numerical calculations are also illustrated with respect to different parameters p.