Explicit Pricing Formula Research Articles

Pricing formula of Lookback option in stochastic delay differential equation model

This paper deals with new explicit pricing formulae for Lookback option when underlying asset price processes are represented by stochastic delay differential equation (hereafter “SDDE”). We derive a lemma on the joint distribution of the minimum and itself of a Wiener process in the SDDE model. Using this lemma, we obtain the explicit pricing formulae for the Lookback option. Through some numerical comparison experiment, we assure the correctness of the obtained pricing formula.

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.

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.

Pricing geometric average Asian options in the mixed sub-fractional Brownian motion environment with Vasicek interest rate model

<p>Considering the characteristics of long-range correlations in financial markets, the issue of valuing geometric average Asian options is examined, assuming that the variations of the underlying asset follow the mixed sub-fractional Brownian motion, and the dynamics of short-term interest rate satisfies the mixed sub-fractional Vasicek model. Based on the principle of no arbitrage, the definite solution of PDE of a zero-coupon bond for geometric average Asian options under the circumstance of the mixed sub-fractional is given by the delta hedging technique. The derivation of the explicit pricing formula for geometric average Asian options with fixed strike price is achieved through the utilization of multiple variable substitutions. Furthermore, we perform numerical calculations to analyze the performance of the model.</p>

Extreme values of solution of Caputo–Hadamard uncertain fractional differential equation and applications

Uncertain fractional differential equation (UFDE) is a useful tool for studying complex systems in uncertain environments. The mathematical characteristics of solution of an UFDE are also widely used in various fields. In this paper, we give the extreme value theorems of solution of Caputo–Hadamard UFDE and applications. A numerical algorithm for obtaining the inverse uncertainty distributions (IUDs) for extreme values of solution of Caputo–Hadamard UFDE is presented; the stability and feasibility of the proposed algorithm are validated by numerical experiments. As an application of extreme value theorems in uncertain financial market, the pricing formulas of the American option based on the new uncertain stock model are given. Besides, the algorithms for computing the price of the American option without explicit pricing formulas based on the Simpson's rule are designed. Finally, the price fluctuation of the American option is illustrated by numerical experiments.

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.

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.

Min–max multi-step barrier options and their variants

This paper studies a new type of barrier option, min–max multi-step barrier options with diverse multiple up or down barrier levels placed in the sub-periods of the option’s lifetime. We develop the explicit pricing formula of this type of option under the Black–Scholes model and explore its applications and possible extensions. In particular, the min–max multi-step barrier option pricing formula can be used to approximate double barrier option prices and compute prices of complex barrier options such as discrete geometric Asian barrier options. As a practical example of directly applying the pricing formula, we introduce and evaluate a re-bouncing equity-linked security. The main theorem of this work is capable of handling the general payoff function, from which we obtain the pricing formulas of various min–max multi-step barrier options. The min–max multi-step reflection principle, the boundary-crossing probability of min–max multi-step barriers with icicles, is also derived.

Pricing vulnerable fader options under stochastic volatility models

<p style='text-indent:20px;'>In this paper, we incorporate default risk into Heston's stochastic volatility model and focus on the valuation of vulnerable fader options. Fader options are path-dependent derivatives, depending on the time the underlying asset price stays inside a given range. We obtain an explicit pricing formula of vulnerable fader options, including fader options and (vulnerable) European options as special cases. Finally, we illustrate the effect of stochastic volatility and default risk on fader option prices. Specially, an inverted U-shaped curve is observed when we keep initial levels of the default intensity constant, but change the relative proportions of two factors in the default intensity.</p>

Pricing formulas of binary options in uncertain financial markets

<abstract><p>Binary options have a payoff that is either a fixed value or nothing at all. In this paper, the generalized pricing formulas of binary options, including European binary call options, European binary put options, American binary call options and American binary put options, are investigated in uncertain financial markets. By applying the Liu's stock model to describe the stock price, the explicit pricing formulas of binary options are derived successfully. Besides, the corresponding numerical examples for the above four kinds of binary options are discussed in this paper.</p></abstract>

Explicit pricing formulas for vulnerable path-dependent options with early counterparty credit risk

Explicit pricing formulas for vulnerable path-dependent options with early counterparty credit risk

Equilibrium pricing for carbon emission in response to the target of carbon emission peaking

The climate warming effect on global economic output and social sustainability should be paid adequate attention. Nowadays, the best way is thought as the market-based pricing policy for carbon emission, which can reflect the emission cost by directly considering the economic efficiency. In response to the target of peak carbon emission before 2030 by China, this paper formulates an equilibrium carbon price by setting up a stochastic equilibrium model, depicting the interplay between the national economic output, carbon emission and climate change. An explicit pricing formula for carbon emission is presented, which clearly discloses the intrinsic link of carbon price to other considered variables and parameters. In addition to this, some interesting results are achieved. First, we calculate the drift and the approximate volatility of the carbon price. It is found that the drift is completely determined by the climate elements, while the approximate volatility shows a three-dimensional surface structure, not only time- and price-dependent, but also dependent on the carbon emission. Second, we find that the carbon price happens a jump at the deadline of the abatement period, in response to the failure of carbon abatement. Third, via sensitivity analysis, we capture the bifurcation phenomena in the carbon price induced by the economic growth rate and negative effect of climate warming. In conclusion, our equilibrium carbon price plays the role of benchmark in the national carbon emission trading market, leading to the realization of the abatement target.

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.

Pricing some life-contingent lookback options under regime-switching Lévy models

In this paper, we study the valuation problem of life-contingent lookback options embedded in variable annuity with guaranteed minimum death benefit (GMDB). Specifically, the underlying asset price process is assumed to be an exponential regime-switching Lévy process, which is observed periodically. The Fourier cosine series expansion method is applied to compute exponential moments of the discretely monitored maximum and minimum of the regime-switching Lévy process. Furthermore, some explicit pricing formulas for the life-contingent lookback options embedded in GMDB products are derived. Finally, numerical experiments confirm the accuracy and efficiency of our method.

The Variance Gamma++ process and applications to energy markets

AbstractThe purpose of this article is to introduce a new Lévy process, termed the Variance Gamma++ process, to model the dynamics of assets in illiquid markets. Such a process has the mathematical tractability of the Variance Gamma process and is obtained by applying the self‐decomposability of the gamma law. Compared to the Variance Gamma model, it has an additional parameter representing the measure of the trading activity. We give a full characterization of the Variance Gamma++ process in terms of its characteristic triplet, characteristic function, and transition density. In addition, we provide efficient path simulation algorithms, both forward and backward in time. We also obtain an efficient “integral‐free” explicit pricing formula for European options. These results are instrumental to apply Fourier‐based option pricing and maximum likelihood techniques for the parameter estimation. Finally, we apply our model to illiquid markets, namely to the calibration of European power futures market data. We accordingly evaluate exotic derivatives using the Monte Carlo method and compare these values to those obtained using the Variance Gamma process and give an economic interpretation. Finally, we illustrate an extension to the multivariate framework.

Pricing catastrophe reinsurance under the standard deviation premium principle

<abstract><p>Catastrophe reinsurance is an important way to prevent and resolve catastrophe risks. As a consequence, the pricing of catastrophe reinsurance becomes a core problem in catastrophic risk management field. Due to the severity of catastrophe loss, the Peak Over Threshold (POT) model in extreme value theory (EVT) is extensively applied to capture the tail characteristics of catastrophic loss distribution. However, there is little research available on the pricing formula of catastrophe excess of loss (Cat XL) reinsurance when the catastrophe loss is modeled by POT. In the context of POT model, we distinguish three different relations between retention and threshold, and then prove the explicit pricing formula respectively under the standard deviation premium principle. Furthermore, we fit POT model to the earthquake loss data in China during 1990–2016. Finally, we give the prices of earthquake reinsurance for different retention cases. The computational results illustrate that the pricing formulas obtained in this paper are valid and can provide basis for the pricing of Cat XL reinsurance contracts.</p></abstract>

Pricing equity warrants under the sub-mixed fractional Brownian motion regime with stochastic interest rate

<abstract> <p>This paper proposes a pricing model for equity warrants under the sub-mixed fractional Brownian motion regime with the interest rate following the Merton short rate model. By using the delta hedging strategy, the corresponding partial differential equations for equity warrants are obtained. Moreover, the explicit pricing formula for equity warrants and some numerical results are given.</p> </abstract>

Power Exchange Option with a Hybrid Credit Risk under Jump-Diffusion Model

In this paper, we study the valuation of power exchange options with a correlated hybrid credit risk when the underlying assets follow the jump-diffusion processes. The hybrid credit risk model is constructed using two credit risk models (the reduced-form model and the structural model), and the jump-diffusion processes are proposed based on the assumptions of Merton. We assume that the dynamics of underlying assets have correlated continuous terms as well as idiosyncratic and common jump terms. Under the proposed model, we derive the explicit pricing formula of the power exchange option using the measure change technique with multidimensional Girsanov’s theorem. Finally, the formula is presented as the normal cumulative functions and the infinite sums.

Piecewise linear double barrier options

AbstractA piecewise linear double barrier option generalizes classical double barrier options because of its versatility in designing various double boundaries. This paper discusses how to price piecewise linear double barrier options. To this purpose, we derive the probability that an underlying process does not cross a given piecewise linear double barrier, where the underlying process follows the Brownian motion of piecewise constant drift. Using the established non‐crossing probability, we provide the explicit pricing formulas of piecewise linear double barrier options and show how the shape of a double barrier affects the option prices through extensive numerical experiments.