Pricing Formula Research Articles

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.

American knock-out options based on floating interest rate in uncertain financial market

The knock-out options are considered as path-dependent barrier options that only expire worthless once the value of the underlying asset reaches a specific threshold. The uncertain differential equations are typically used to describe stock fluctuations in uncertain financial markets. In this study, we build a stock model considering floating interest rate based on uncertainty theory. On this basis, we mainly study the pricing scheme of American call and put options. Based on this model, we mainly research the pricing schemes for call and put options with the American barrier option. Moreover, we develope the parameter estimation for the uncertain stock model and analyze the results of the uncertain hypothesis test. Finally, we design numerical algorithms for the corresponding option pricing formulas. As an application, we verify the validity of the formulas through numerical experiments.

Approximate option pricing under a two-factor Heston–Kou stochastic volatility model

Under a two-factor stochastic volatility jump (2FSVJ) model we obtain an exact decomposition formula for a plain vanilla option price and a second-order approximation of this formula, using Itô calculus techniques. The 2FSVJ model is a generalization of several models described in the literature such as Heston (Rev Financ Stud 6(2):327–343, 1993); Bates (Rev Financ Stud 9(1):69–107, 1996); Kou (Manag Sci 48(8):1086–1101, 2002); Christoffersen et al. (Manag Sci 55(12):1914–1932, 2009) models. Thus, the aim of this study is to extend some approximate pricing formulas described in the literature, like formulas in Alòs (Finance Stoch 16(3):403–422, 2012); Merino et al. (Int J Theor Appl Finance 21(08):1850052, 2018); Gulisashvili et al. (J Comput Finance 24(1), 2020), to pricing under the more general 2FSVJ model. Moreover, we provide numerical illustrations of our pricing method and its accuracy and computational advantage under double exponential and log-normal jumps. Numerically, our pricing method performs very well compared to the Fourier integral method. The performance is ideal for out-of-the-money options as well as for short maturities.

Knock-in options of mean-reverting stock model with floating interest rate in uncertain environment

Knock-in options are a type of barrier option that become active when the value of the underlying asset exceeds a predetermined target. This paper investigates knock-in options based on the mean-reverting stock model within an uncertain environment where the interest rate is dynamic and the stock price follows a geometric process. We subsequently provide the price formulas for the European up-and-in call option and European down-and-in put option. Additionally, we apply real data from the financial market and employ moments estimation to derive the optimal parameters of the stock model under bullish and bearish conditions, respectively. Hypothesis testing is subsequently utilized to assess the fitting effect between the model and data. Finally, the numerical experiments are set to verify the validity of the formulas by observing the influence of parameters on the option price.

Pricing Strategy of Convertible Bond with Memory and Jumps

In this paper, a novel pricing formula for convertible bond is developed under Chinese Regulator’s framework. This formula generalizes existing results by considering the long-memory property of underling assets and rare market events. An empirical analysis is conducted by the end of this paper.

Risk-return tradeoff and serial correlation in the Chinese stock market: A bailout-driven crash feedback hypothesis

This paper explores two puzzling phenomena about aggregate stock market behavior: the weak mean–variance (risk-return) relation and return autocorrelation. Existing literature typically analyzes these two phenomena in isolation, with few attempts to understand their coexistence. To bridge this gap, we integrate two key factors, market-wide bailout guarantees and feedback trading, within the context of the Chinese stock market. We develop joint pricing formulas to theoretically show that feedback traders’ speculation on bailouts can simultaneously produce these anomalies by tying them to features typical of market crashes. Our empirical analysis, based on Chinese market data from 1997 to 2020 and a market crash index, provides consistent support for these theoretical derivations. The results highlight the significant role of speculation driven by bailout expectations in influencing aggregate stock market dynamics. Accordingly, this also illuminates the necessity for further research on the extensive pricing implications of both government intervention and speculative trading.

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 formula for a Barrier call option based on stochastic delay differential equation

We derive new explicit pricing formulae for a type of Barrier call option, down and in call option when underlying asset price processes are represented by a stochastic delay differential equation (hereafter “SDDE”). We note the conditional normality of a stochastic integral with respect to a Wiener process to find the joint distribution of the stochastic integral and their minimum. On the basis of this result, we obtain pricing formulae for the Barrier call option which extends ones in the classical Black-Scholes models without delay. Finally, through Monte-Carlo simulations, we demonstrate that our theoretical prices for a Barrier option are correct.

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.

American Rainbow Option Pricing Formulae in Uncertain Environment

American Rainbow Option Pricing Formulae in Uncertain Environment

Review and Practice of Option Pricing Research

This paper reviews the past research on option pricing, including the Bachelier option pricing formula, Black-Scholes model, and the Cox-Ross-Rubinstein (CRR) binomial option pricing model. Then some practical calculations are performed on the Black-Scholes model and CRR binomial model and the differences between the results of these two models are analyzed for investors to choose the appropriate model for option prices. From the practical calculation results, it is verified that the Black-Scholes model can give out the analytical solutions of option prices and decrease the calculation of solution, while it can only be used for European options with stock prices regarded as a continuous process. For the CRR binomial model, although it can only give numerical solutions and requires lots of calculation because of too many branches, but it can solve the American option prices with early exercises and the European option prices with stock prices regarded as a discrete process.

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.

Hedging Dynamic Fund Protection: A Static Versus Dynamic Comparison

The static nature of the downside protection offered by a standard Put option can motivate investors to use exotic option contracts like dynamic fund protection (DFP), which protects the underlying asset value throughout its life. DFP has a path-dependent payoff structure with its terminal payoff depending on the terminal asset price, the strike price and the minimum price level attained by the asset over the contract tenor. This makes DFP a contract strikingly similar to a Lookback Call option. In this article, we try to borrow the respective strengths from a dynamic and a static hedging strategy, formulate a semi-static hedging procedure for DFP and compare the hedging effectiveness of this procedure with a standard delta hedging strategy. Given the payoff resemblance between a DFP and a Lookback Call, we also seek to verify Tompkins (2002, Journal of Risk Finance, 3(4), 6–34) conclusion, which objectively shows that a dynamic procedure performs better in hedging a Lookback Call option both with and without transaction cost. Using simulated underlying price paths, we perform the delta hedging on DFP based on the closed-form pricing formula given by Gerber and Pafumi (2000, North American Actuarial Journal, 4(2), 28–37) and compare the hedge performance results with its static hedge counterpart.

An analytical approach to the pricing of an exchange option with default risk under a stochastic volatility model

The exchange option, which has two correlated underlying assets, is one of the most popular exotic options in the over-the-counter markets. This paper studies the valuation of exchange options with default risk of option issuer, where default is allowed only at maturity. Moreover, we consider three underlying assets with stochastic volatilities and assume that fast mean-reverting processes determine the stochastic volatilities. Based on the partial differential equation approach, we derive the analytical pricing formula of the exchange option price with default risk using the asymptotic expansion. To verify the accuracy and efficiency of our pricing formula, we compare the results by our pricing formula with those by Monte Carlo simulation, which is considered a benchmark. In addition, we provide several graphs to illustrate the properties of the option for significant parameters.

Rescaling the double-mean-reverting 4/2 stochastic volatility model for derivative pricing

The popular 4/2 stochastic volatility model reveals important features of volatility but a closed-form formula for derivative prices is still lacking. This paper proposes a modified form of the 4/2 model into which two scale double-mean-reverting stochastic volatility is incorporated in order to remedy its shortcomings. We obtain a closed-form formula for the prices of European derivatives. The formula can be explicitly calculated by taking derivatives of the Black–Scholes prices and thus faster calibration of the 4/2 model becomes available. We also show that the rescaled 4/2 model is flexible enough to capture essential features (skew or smile) of market implied volatilities.

A capital asset pricing model based on the value at risk under asymmetric Laplace distribution

PurposeFinancial asset return series usually exhibit nonnormal characteristics such as high peaks, heavy tails and asymmetry. Traditional risk measures like standard deviation or variance are inadequate for nonnormal distributions. Value at Risk (VaR) is consistent with people's psychological perception of risk. The asymmetric Laplace distribution (ALD) captures the heavy-tailed and biased features of the distribution. VaR is therefore used as a risk measure to explore the problem of VaR-based asset pricing. Assuming returns obey ALD, the study explores the impact of high peaks, heavy tails and asymmetric features of financial asset return data on asset pricing.Design/methodology/approachA VaR-based capital asset pricing model (CAPM) was constructed under the ALD that follows the logic of the classical CAPM and derive the corresponding VaR-β coefficients under ALD.FindingsALD-based VaR exhibits a minor tail risk than VaR under normal distribution as the mean increases. The theoretical derivation yields a more complex capital asset pricing formula involving β coefficients compared to the traditional CAPM.The empirical analysis shows that the CAPM under ALD can reflect the β-return relationship, and the results are robust. Finally, comparing the two CAPMs reveals that the β coefficients derived in this paper are smaller than those in the traditional CAPM in 69–80% of cases.Originality/valueThe paper uses VaR as a risk measure for financial time series data following ALD to explore asset pricing problems. The findings complement existing literature on the effects of high peaks, heavy tails and asymmetry on asset pricing, providing valuable insights for investors, policymakers and regulators.

On the design of bail-in-able bonds from the perspective of non-financial firms

To prevent black swan events such as the financial tsunami or COVID-19 from crashing overall economic systems, governments provide bailouts to too-big-to-fail financial institutions and non-financial firms such as airline companies with tax payers’ money. To avoid unfair bailouts, using bail-in-able bonds to absorb an issuing bank’s losses and to fulfill capital requirements such as BASEL III has been widely studied. Without capital adequacy requirements, non-financial firms also issue bail-in-able bonds because the embedded loss-absorbing mechanism can cut debt repayments down to reduce bankruptcy risk and thus increase the overall benefits of these firms’ claim holders. However, bond investors require higher rewards to compensate for their potential losses once this mechanism is triggered. In addition, unlike financial institutions, which are strictly regulated, such a mechanism may induce non-financial issuers to take on excessive risk to benefit themselves at the expense of their debt holders. To investigate how the issuances of various (non)-bail-in-able bonds influence the equity and the existing debt holders’ benefits from a non-financial issuer’s prospective, we develop novel quadrature pricing formulas based on the structural credit risk model. We find that while non-bail-in-able bonds can be wise choices for low-leverage issuers, bail-in-able bonds with benefit-sharing mechanisms allowing investors to share issuers’ upside profits can efficiently suppress interest expenses and circumvent the asset substitution problem for high-leverage issuers. If the value of an issuer (such as an oil shale firm) mainly depends on the value of that asset (such as oil), issuing a bail-in-able bond referring to the asset could be a wise choice. Otherwise, issuing a bond referring to the issuer’s stock price could provide timely fund injections and low interest expenses for a high-leverage public firm.

Overview of Some Recent Results of Energy Market Modeling and Clean Energy Vision in Canada

This paper overviews our recent results of energy market modeling, including The option pricing formula for a mean-reversion asset, variance and volatility swaps on energy markets, applications of weather derivatives on energy markets, pricing crude oil options using the Lévy processes, energy contracts modeling with delayed and jumped volatilities, applications of mean-reverting processes on Alberta energy markets, and alternatives to the Black-76 model for options valuation of futures contracts. We will also consider the clean renewable energy prospective in Canada, and, in particular, in Alberta and Calgary.

Valuation of option price in commodity markets described by a Markov-switching model: A case study of WTI crude oil market

This paper suggests a Markov-switching model to evaluate commodity futures and spot dynamics, such that the diffusion coefficients and jump size parameter are associated with a hidden Markov chain. We improve the current models in the literature of the commodity markets by modeling the sudden jumps in the commodity prices through the hidden Markov chain. From the crude oil spot price in West Texas Intermediate, we estimate the parameters of proposed Markov-switching model based on expectation–maximization algorithm. To perform this task, we apply this estimation algorithm to the model discretized by Euler scheme and provide some convergence analysis for this discretization method. There are options, such as European options, which are written on the commodity futures. In this study, we evaluate them under the regime-switching model with various economic states. In the following, we calibrate the option prices resulting from the proposed commodity model to a set of observed European call options written on crude oil futures. For this purpose, we first apply an inverse Fourier transform and obtain a semi-analytical option pricing formula. Then, we use the fast Fourier transform method to compute option prices. Since the investors need to calculate Greeks in order to understand the risk involved in option investments, the Greek formulas of Delta, Rho, Theta, and Gamma are derived.

A market consistent approach to the valuation of no-negative equity guarantees and equity release mortgages

AbstractThis paper provides a new market consistent approach to the valuation of no negative equity guarantees and equity release mortgages. The paper provides a new approach to the estimation of volatility inputs. The proposed approach to volatility produces a volatility term structure that is dependent on the age and gender of the borrower. Illustrative valuations are provided based on the Black ’76 put pricing formula and mortality projections based on the M5 Cairns–Blake–Dowd mortality model. Results show interesting ramifications for industry practice and prudential regulation.