Pricing Formula Research Articles

Pricing of insurance-linked securities: a multi-peril approach

In this paper we build a methodology for pricing of insurance-linked securities which are tied to multiple natural catastrophe perils. As a representative example, we construct a multi-peril catastrophe (CAT) bond which can be linked to the industry loss indices or actual losses incurred by an insurer. We provide pricing formulas for such CAT bonds. We illustrate the introduced methodology on the US natural catastrophe data obtained from Property Claim Services (PCS). Within this dataset, we specifically examine two types of risks: losses associated with wind and thunderstorm events, and those linked to winter storm events. Then, we fit and validate the underlying compound non-homogeneous Poisson processes taking into account the fact that the data are left-truncated. The best fitted loss distributions appear to be Burr and Generalised Extreme Value and for the first peril and log-normal for the second. Finally, we visualise the zero-coupon CAT bond prices for the selected best-fitted models.

Numerical methods base on trinomial trees for option pricing

This paper investigates the approach to pricing European options, starting with one-step binomial tree pricing (a relatively simple way to calculate option value). In the next step, an additional possible rate of change of stock price is added to make the model more realistic, resulting in the one-step trinomial tree model. The model bounds the option price under the no-arbitrary principle. The paper then analyzes the circumstances under which options have a xed price by completing the market and giving the solution formula of option price through the model. Last, put-call parity is used to prove the rationality of one-step trinomial model so that the model eectively prevents the occurrence of risk-free arbitrage in the market. This helps traders to price options reasonably in the market and maintains the stability of the options market.

Variance dynamics and term structure of the natural gas market

In this paper, we develop a stochastic model to investigate the variance dynamics and term structure of the US natural gas market. The model features two variance factors, the Samuelson effect, seasonality and linearly analytical pricing formula. With variance swap rates synthesized from options prices spanning more than 20 years, we show that the two-factor variance model has better performance and is preferred to describe the variance term structure. Moreover, we find that the two variance factors are particularly needed to capture humps in variance term curves implied from the market. Finally, we apply the two-factor model to hedging natural gas straddles and obtain improved performance. These results highlight the importance of introducing multiple variance factors to the natural gas market, and may be applied to other similar sectors.

Pricing of discretely sampled arithmetic Asian options, under the Hull–White interest rate model

This paper studies the pricing of discrete arithmetic Asian options (AAOs) with fixed strikes under the Hull–White interest rate model. For the pricing of AAOs, we first investigate the stochastic dynamics of the price of the underlying asset under the T-forward measure, and then study the distribution of the discrete arithmetic average of the underlying asset price. Specifically, we provide the first three moments of the discrete arithmetic average under the T-forward measure. Then, we derive approximate pricing formulas for AAOs using the three-moment matching method. Furthermore, we calculate the first three conditional moments of the discrete arithmetic average, given the final value of the underlying asset, under the T-forward measure. These conditional moments can be used to improve the accuracy of the approximation of the AAO prices. The numerical results show that our three-moment matching approximations are very accurate. Additionally, the accuracy can be further improved by combining the conditioning approach with the three-moment matching method. Our procedure is also applied to the computation of deltas of AAOs.

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.

Introducing and testing the Carr model of default

The Merton (On the pricing of corporate debt: The risk structure of interest rates. J. Finance, 1974, 29, 449–470) model is often considered the simplest structural model of default. Its famous modification developed by KMV is also widely used and well understood by both practitioners and academics. However, as the underlying asset process is driven by a diffusion process, the Merton model suffers from the well-known issue of the vanishing of credit spreads for progressively shorter maturities. To overcome such shortfall, several modifications have been proposed in the literature; however, none of these, despite succeeding at better pricing spreads and explaining default dynamics, are able to retain the simplicity of the Merton model. In this paper, we propose a new structural model of default driven by an additive process whose pricing formulas are as simple as – if not even simpler than – those of the Merton model. We named such model the Carr model of default after Peter Carr, given his recent passing and the fact that the underlying asset distribution is the one introduced in Carr and Torricelli (Additive logistic processes in option pricing. Finance Stoch., 2021, 25, 689–724) and Carr and Maglione (Compound option pricing and the Roll-Geske-Whaley formula under the conjugate-power Dagum distribution. J. Deriv., 2022, 31, 1–32). We first provide pricing formulas for the firm's claims and credit spreads, and then discuss a general theory of change of measure for such processes in order to introduce a distance-to-default version of this newly-introduced model. Finally, we empirically test the Carr versus the Merton model and document the far better ability of the Carr model to reproduce CDS spreads as well as explain their cross-sectional and time-series variation.

A rational finance explanation of the stock predictability puzzle

AbstractWe address the stock predictability puzzle, a challenge in the stock market often discussed in behavioral finance. Our approach formulates a statistical model within rational finance, avoiding reliance on behavioral finance assumptions, and integrates stock return predictability into the Black–Scholes option pricing framework. Empirical analysis focuses on the predictability of stock prices by option and spot traders, introducing a forward‐looking measure we term “implied excess predictability.” Results show that option traders' predictability of stock returns positively correlates with moneyness, whereas for spot traders, this relationship is inverse. These findings suggest a potential asymmetry in stock price predictability between spot and option traders. Additionally, we demonstrate the importance of incorporating stock return predictability into option pricing formulas, particularly for options with strike prices significantly different from the stock price. Conversely, when moneyness is close to unity, predictability is not integrated into option pricing, indicating equal information among spot and option traders. Comparison of volatility measures reveals the difference between implied and realized variances or variance risk premia as potential predictors of stock returns.

Pricing airbag option via first passage time approach

Airbag option is a new structured product that has been highly favored by investors in the over-the-counter derivatives market. It provides protections against the downside loss to investors like the ‘airbag’ in a car crash. In contrast with its popularity, there is little literature on the pricing of airbag options, especially from the mathematical finance point of view. In this paper, we study the pricing of airbag options using the first passage time approach. We propose two innovative new types of airbag options, namely airbag-TB and airbag-ER, which enhance the downside loss protections by integrating a time-varying barrier and early redemption features respectively in the payoffs. Explicit and closed-form pricing formulas for the original and the two new types of airbag options are derived under the mixed-exponential jump diffusion (MEJD) model, double exponential jump diffusion (DEJD) model and Black-Scholes (BS) model. In doing so, we significantly improve the Hybrid model proposed by Jaimungal and Sigloch (Incorporating risk and ambiguity aversion into a hybrid model of default. Math. Finance, 2012, 22, 57–81) by incorporating a stochastic redemption intensity model and more general underlying asset dynamics. We further demonstrate the performance of our pricing formulas and the two new types of airbag options with numerical examples based on Monte Carlo simulation, which verifies the accuracy and efficiency of our pricing formulas.

Pricing CDS index tranches under thinning-dependence structure with regime switching

In this paper, we consider a portfolio credit risk model with the thinning-dependence structure under the case with regime switching and the one with no regime switching. We obtain the explicit expressions for the copulas of the default times. Based on the quotation convention of CDS index tranches, we derive some closed-form expressions for the pricing formulas for the CDS index tranche via the copulas and the marginal distribution functions of the default times. As an empirical example, we consider the pricing of the CDX NA IG Series 25. We demonstrate how the model parameter can be calibrated based on market data and how the numerical results can be calculated via the proposed pricing formulas. Compared to the empirical results under the pricing model with some commonly used copulas, such as the Clayton copula, the Gumbel copula, and the Frank copula, we find that the proposed pricing model with regime switching can provide a valuable reference for the market quotes for the CDX NA IG Series 25.

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.

Market consistent bid-ask option pricing under Dempster-Shafer uncertainty

We refer to the discrete-time market model under ambiguity introduced in [Cinfrignini, A., Petturiti, D. and Vantaggi, B., Dynamic bid–ask pricing under Dempster-Shafer uncertainty. J. Math. Econ., 2023a, 107, 102871], formed by a frictionless risk-free bond and a non-dividend paying stock with bid-ask spread. For a European derivative, we generalize the classical binomial pricing formula by allowing for bid-ask prices and investigate the properties of the ensuing replicating strategies. Next, for an American derivative, we propose a backward bid-ask pricing procedure and prove that the resulting discounted price processes are the bid-ask Choquet-Snell envelopes of the discounted payoff process, respectively. Moreover, for an American call option, we prove a generalization of the well-known Merton's theorem [Merton, R.C., Theory of rational option pricing. Bell J. Econ. Manage. Sci., 1973, 4, 141–183] holding for both the bid and the ask price processes. Finally, we introduce a market consistent calibration procedure and show the use of the calibrated model in bid-ask option pricing.

Foreign equity lookback options with partial monitoring

A lookback option on foreign equity reduces the uncertainty of foreign investment by delivering the benefit of hindsight. Despite its high marketability, purchasing the option is stalled due to the failure to meet customized demands. This paper proposes foreign equity lookback options with partial monitoring and floating strike price to control the option premiums. We establish the expected value of the extreme of an underlying process when the specific condition, covering various payoffs of the lookback options, is satisfied. It is exploited to derive the unified closed-form pricing formula. We demonstrate the properties of the option prices through numerical examples.

Valuations of generalized variance swaps under the jump–diffusion model with stochastic liquidity risk

This paper focuses on the pricing problem of generalized variance swaps with jump risk in the underlying asset price under a stochastic liquidity model. We obtain a pricing formula of generalized variance swap in a jump–diffusion model with stochastic liquidity risk by the joint moment generating function (MGF) generated by solving the partial integral differential equation (PIDE). Using asymptotic analysis, we also demonstrate that as the sampling interval approaches zero, the pricing formula of discretely sampled generalized variance swap tends to be that of continuously sampled generalized variance swap. Finally, to verify the feasibility of the pricing formula of the generalized variance swap presented in this paper, we conduct some numerical experiments, including a comparison with the results of Monte Carlo (MC) simulation, the impact of various model parameters on the delivery prices of generalized variance swaps, and empirical research using actual market data.

Explicit formulae for the valuation of European options with price impacts

In this work, we examine the consequences of trading a large position in vanilla European options within a multi-period binomial model framework for the underlying asset price, S. Given the significant size of the transaction, we expect both the derivative's price and the underlying asset's price to be affected by market impacts. Consequently, derivative valuation should incorporate these effects. To address this, we not only utilize a multi-period binomial model to represent the price process S but also incorporate trading impacts in a multiplicative manner.Moreover, we conduct our analysis in discrete time to better capture the influence of price impacts. Our findings suggest, for instance, that the strike price should be determined by both the trade's magnitude and parameterized market impacts. We present explicit formulas for European option prices under market impacts and offer numerical examples to elucidate our findings. Upon request, we can provide code implemented in the statistical package R.

Pricing VIX options based on mean-reverting models driven by information

Financial time series are dynamic and influenced by different types of information from the market. In this study, we propose new models for SPX and VIX options using the Hawkes process, jump process with stochastic intensity, and tempered stable process to capture these changes in financial time series based on three distinct characteristics of market information. We calculate the VIX option pricing formula using these models and find that the simplified VIX model based on VIX characteristics has significantly less pricing error than the consistent VIX model derived from the SPX model. Additionally, our findings suggest that the tempered stable process effectively models the volatility of VIX, sparse large jumps, and infinitesimal jumps. It also shows potential as an alternative to Brownian motion for representing volatility. Conversely, jump processes with stochastic jump intensities adeptly describe asymmetric jumps, and their integration with Brownian motion provides a more accurate depiction of the VIX’s volatility and jump dynamics. Finally, the introduction of jump processes into mean-reverting models for the VIX indicates a relatively low correlation between volatility magnitudes and the current VIX levels. This research contributes to the theory of SPX and VIX options and offers guidance for the development of other information-driven economic models.

Reduced interest option pricing for green bonds

PurposeThis paper originally proposed the fuzzy option pricing method for green bonds. Based on the requirements of arbitrage equilibrium, this paper draws on Merton's corporate bond option pricing model.Design/methodology/approachDescribing the asset value behavior of green bond issuing enterprises through diffusion-jump processes to reflect the uncertainty brought by carbon emission reduction policies and technologies, using approximation methods to get the analytical pricing formula and then, using a fuzzification technique of Choquet expectation under λ-additive fuzzy measures after considering fuzzy factors, the paper provides fuzzy intervals for the parity coupon rates of green bonds with different subjective levels for investors.FindingsThe paper proposes and argues the classical and fuzzy option pricing methods in turn for both corporate ordinary bonds and green bonds, considering carbon risk or climate risk. It implements the scenario analysis varying with industry emission standards and discusses the sensitiveness of the related key parameters of the option.Practical implicationsThe fuzzy option pricing for the green bonds provides the scope of the variable equilibrium values, operational theoretical supports and some policy implications of carbon reduction and promoting green funding.Originality/valueThe logic of introducing the fuzziness of the option pricing for the green bonds lies with considering the existence of fuzzy information about the project supported by the green bond and the subjectivity of investors and it also responds to changes in technological uncertainty and policy uncertainty in the process of “carbon peaking and carbon neutrality.”

Carbon option pricing based on uncertain fractional differential equation: A binomial tree approach

In this paper, we use a Caputo–Hadamard uncertain fractional differential equation (UFDE) to describe the change of carbon emission rights price. Based on a binomial tree approach, a portfolio is constructed by buying options and shorting the underlying assets in a certain proportion. The pricing formulas for carbon finance European option are presented, and the algorithms are designed to formulate the price of carbon finance American option without explicit pricing formulas. Besides, a moment estimation method is used to obtain the parameter values in Caputo–Hadamard UFDE, and a method is provided to simulate the observation data of solution of Caputo–Hadamard UFDE. Finally, the specific Caputo–Hadamard UFDE with mean-reverting process is applied to simulating the change of carbon emission rights price in Chinese carbon market, and the feasibility of the proposed carbon option pricing method is illustrated by some numerical examples.

Research on the Optimal Pricing for Life Insurance

Term life insurance is a policy that guarantees the buyer of the policy insurance benefit upon their death for a period of time in exchange for a premium paid by the insurance buyer to the seller yearly for a set number of years. Purchasing life insurance has become an important, if not necessary, financial investment in everyones life, as it protects their family from a potentially devastating financial loss if they unexpectedly pass away. This paper combined multiple variables in order to compute a pricing formula for life insurance. The final result is that as the age of purchase increases, the premium, or, in other words, the price of the life insurance using my pricing formula, increases exponentially, while the profit received decreases exponentially.

Option pricing under jump diffusion model

We provide an European option pricing formula written in the form of an infinite series of Black–Scholes-type terms under double Lévy jumps model, where both the interest rate and underlying price are driven by Lévy processes. The series solution converges with a radius of convergence, and it is complemented by some numerical experiments to demonstrate its speed of convergence.

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.