A Quantum Jump Model of Option Pricing

Electricity markets are becoming a popular field of research amongst academics because of the lack of appropriate models for describing electricity price behavior and pricing derivatives instruments. Models for price dynamics must consider seasonality and spiky behavior of jumps which seem hard to model by standard jump process. Without good models for electricity price dynamics, it is difficult to think about good models for futures, forward, swaps and option pricing. In this paper we attempt to introduce an algorithm for pricing derivatives to intuition from Colombian electricity market. The main ambition of this study is fourfold: 1) First we begin our approach through to simple stochastic models for electricity pricing. 2) Next, we derive analytical formulas for prices of electricity derivatives with different derivatives tools. 3) Then we extent short of the model for price risk in the electricity spot market 4) Finally we construct the model estimation under the physical measures for Colombian electricity market. And this paper end with conclusion.

Canonical Option Pricing and Greeks with Implications for Market Timing

We introduce a canonical representation of call options, and propose a solution to two open problems in option pricing theory. The first problem was posed by (Kassouf, 1969, pg. 694) seeking “theoretical substantiation” for his robust option pricing power law which eschewed assumptions about risk attitudes, rejected risk neutrality, and made no assumptions about stock price distribution. The second problem was posed by (Scott, 1987, pp. 423-424) who could not find a unique solution to the call option price in his option pricing model with stochastic volatility–without appealing to an equilibrium asset pricing model by Hull and White (1987), and concluded: “[w]e cannot determine the price of a call option without knowing the price of another call on the same stock”. First, we show that under certain conditions derivative assets are superstructures of the underlying. Hence any option pricing or derivative pricing model in a given number field, based on an anticipating variable in an extended field, with coefficients in a subfield containing the underlying, is admissible for market timing. For the anticipating variable is an algebraic number that generates the subfield in which it is the root of an equation. Accordingly, any polynomial which satisfies those criteria is admissible for price discovery and or market timing. Therefore, at least for empirical purposes, elaborate models of mathematical physics or otherwise are unnecessary for pricing derivatives because much simpler adaptive polynomials in suitable algebraic numbers are functionally equivalent. Second, we prove, analytically, that Kassouf (1969) power law specification for option pricing is functionally equivalent to Black and Scholes (1973); Merton (1973) in an algebraic number field containing the underlying. In fact, we introduce a canonical polynomial representation theory of call option pricing convex in time to maturity, and algebraic number of the underlying–with coefficients based on observables in a subfield. Thus, paving the way for Wold decomposition of option prices, and subsequently laying a theoretical foundation for a GARCH option pricing model. Third, our canonical representation theory has an inherent regenerative multifactor decomposition of call option price that (1) induces a duality theorem for call option prices, and (2) permits estimation of risk factor exposure for Greeks by standard [polynomial] regression procedures. Thereby providing a theoretical (a) basis for option pricing of Greeks, and (b) solving Scott’s dual call option problem a fortiori with our duality theory in tandem with Riesz representation theory. Fourth, when the Wold decomposition procedure is applied we are able to construct an empirical pricing kernel for call option based on residuals from a model of risk exposure to persistent and transient risk factors. <div><br><div>Keywords: number theory; price discovery; derivatives pricing; asset pricing; canonical representation; Wold decomposition; empirical pricing kernel; option Greeks; dual option pricing</div></div>

We introduce a canonical representation of call options, and propose a solution to two open problems in option pricing theory. The first problem was posed by (Kassouf, 1969, pg. 694) seeking “theoretical substantiation” for his robust option pricing power law which eschewed assumptions about risk attitudes, rejected risk neutrality, and made no assumptions about stock price distribution. The second problem was posed by (Scott, 1987, pp. 423-424) who could not find a unique solution to the call option price in his option pricing model with stochastic volatility–without appealing to an equilibrium asset pricing model by Hull and White (1987), and concluded: “[w]e cannot determine the price of a call option without knowing the price of another call on the same stock”. First, we show that under certain conditions derivative assets are superstructures of the underlying. Hence any option pricing or derivative pricing model in a given number field, based on an anticipating variable in an extended field, with coefficients in a subfield containing the underlying, is admissible for market timing. For the anticipating variable is an algebraic number that generates the subfield in which it is the root of an equation. Accordingly, any polynomial which satisfies those criteria is admissible for price discovery and or market timing. Therefore, at least for empirical purposes, elaborate models of mathematical physics or otherwise are unnecessary for pricing derivatives because much simpler adaptive polynomials in suitable algebraic numbers are functionally equivalent. Second, we prove, analytically, that Kassouf (1969) power law specification for option pricing is functionally equivalent to Black and Scholes (1973); Merton (1973) in an algebraic number field containing the underlying. In fact, we introduce a canonical polynomial representation theory of call option pricing convex in time to maturity, and algebraic number of the underlying–with coefficients based on observables in a subfield. Thus, paving the way for Wold decomposition of option prices, and subsequently laying a theoretical foundation for a GARCH option pricing model. Third, our canonical representation theory has an inherent regenerative multifactor decomposition of call option price that (1) induces a duality theorem for call option prices, and (2) permits estimation of risk factor exposure for Greeks by standard [polynomial] regression procedures. Thereby providing a theoretical (a) basis for option pricing of Greeks, and (b) solving Scott’s dual call option problem a fortiori with our duality theory in tandem with Riesz representation theory. Fourth, when the Wold decomposition procedure is applied we are able to construct an empirical pricing kernel for call option based on residuals from a model of risk exposure to persistent and transient risk factors. <div><br><div>Keywords: number theory; price discovery; derivatives pricing; asset pricing; canonical representation; Wold decomposition; empirical pricing kernel; option Greeks; dual option pricing</div></div>

This article elaborates upon the intuition underlying Doherty and Garven's (1986) option pricing model and extends its basic results to a further consideration of the implications of limited liability and asymmetric taxes for pricing and incentives in property-liability insurance. When compared with CAPM-based models of the insurer, a number of important insights emerge. First, the option pricing framework is shown to encompass the CAPM framework as a special case and may help to explain a number of empirical phenomena. Second, the option pricing framework is used to develop a risk hypothesis which suggests that limited liability and asymmetric taxes provide mutuals with greater disincentives for riskbearing than stock companies, even in the absence of owner/manager conflicts. Although the property-liability insurance industry has been subject to price regulation for many years, researches have only recently derived valuation formulas for property-liability insurance firms. To date, the most promising approaches apply financial theories such as the capital asset pricing model (CAPM) (see Biger and Kahane, 1978; Fairley, 1979; Hill, 1979; Hill and Modigliani, 1987; and Myers and Cohn, 1987) and the option pricing model (see Doherty and Garven, 1986; Cummins, 1988b; and Derrig, 1989). Although details vary, these models are generally organized around the principle that the rate of underwriting profit must be set so as to produce a fair, or competitive rate of return on equity.(1) In spite of their common origins, CAPM and option-based insurance pricing models produce substantially different predictions concerning pricing and incentives for property-liability insurance. It will be shown that these differences are primarily due to the manner in which the effects of insolvency and taxes are modeled. Essentially, CAPM-based models implicitly assume that shareholders have unlimited liability, whereas option-based models assume that shareholders' liability is limited. Similarly, by assuming that losses are rebated at the same rate at which gains are taxed, CAPM-based models effectively assign unlimited liability to the government, whereas option-based models limit the government's liability by assuming that gains and losses are taxed in an asymmetric fashion. This article elaborates upon the intuition underlying Doherty and Garven's (1986) option pricing model and extends its basic results to a further consideration of the implications of limited liability and asymmetric taxes for pricing and incentives in property-liability insurance. This is accomplished by comparing and contrasting option-based with CAPM-based models of the insurance firm. This analysis yields a number of important insights, such as the fact that the option pricing framework encompasses the CAPM framework as a special case. The option pricing model also has several practical advantages over the CAPM. For example, it is not plagued by the CAPM's well-known parameter estimation problems; indeed, it may help to explain the causes of these problems.(2) The option pricing model also provides an explicit linkage between fair return and the risks of insolvency and tax shield underutilization, whereas the CAPM totally ignores these effects. The option pricing model also calls attention to some important incentive effects concerning risk-bearing that are not captured by the CAPM. Under the CAPM, asset and liability is not particularly important so long as these claims are priced to yield appropriate risk-adjusted rates of return. However, under the option pricing model, the extent to which firms will seek to increase or avoid through their investment and underwriting policy choices depends upon the likelihood of being taxed or becoming insolvent. Consequently, the application of the option pricing framework makes it possible to develop a risk hypothesis which predicts that mutual insurers will seek less exposure to than stock companies. …

The Cox-Ross-Rubinstein (CRR) market model is one of the simplest and easiest ways to analyze the option pricing model. CRR has been employed to evaluate a European Option Pricing (call options) model without complex elements, including dividends, stocks, and stock indexes. Instead, it considers only a continuous dividend yield, futures, and currency options. The CRR model is simple but strong enough to describe the general economic intuition behind option pricing and its principal techniques. Also, it gives us basic ideas on how to develop financial products based on current deviations and volatilities. The paper investigates the CRR model using numerical approaches with python code. It provides a practical event using the mathematical model to demonstrate the application of the model in the financial market. First, the paper provides a simple example to figure out the basic concept of the model. Only a two-period binomial model based on the introductory definitions of the call options makes us understand the concept more easily and quickly. Next, we used actual data on Tesla stock fluctuations from the Nasdaq website (See section 3). We developed the python code to make it easier to understand figures, including tables and graphs. The code allows us to visualize and simplify the model and output data. The code analyzes the stock data to evaluate the probability of the stock’s price increasing or decreasing. Then, it used the CRR model to estimate all possible cases for the stock’s prices and investigate the call and put option pricing. The code was based on the introductory code of binomial option pricing, but we improved it to get more information and provide more detailed results from the data. The detailed codes are provided in section 3 of the paper. As a result, we believe the CRR model is a fundamental formula, but the improved python code can suggest a new direction for evaluating the probability and investigating the value of the stocks. Also, we expect to develop the code to extend the Black Scholes Pricing model, increasing the number of periods.

Modelling of pricing and market impacts for water options

INTRODUCTION Since insurance contracts are financial instruments, it seems natural to apply financial models to insurance pricing. Financial pricing models have been developed based on the capital asset pricing model (Biger and Kahane 1978; Fairley 1979), arbitrage pricing theory (Kraus and Ross 1982), capital budgeting principles (Myers and Cohn 1987) and option pricing theory (Merton 1977; Smith 1979; Doherty and Garven 1986; Cummins 1988; and Shimko 1992). Financial models represent a significant advancement over traditional actuarial models because they recognize that insurance prices should be consistent with an asset pricing model or, minimally, avoid the creation of arbitrage opportunities. A limitation of the existing financial pricing models is the implicit or explicit assumption that insurers produce only one type of insurance, even though most insurers produce multiple types of coverage (e.g., automobile insurance, general liability insurance, workers' compensation insurance, etc.). The purpose of this paper is to remedy this deficiency in the existing literature by providing a theoretical and empirical analysis of insurance pricing in a multiple line firm. An option pricing approach is adopted to model the insurer's default risk. The standard Black-Scholes model is generalized to incorporate more than one class of liabilities, and pricing formulae are generated for each liability class. The theoretical predictions of the model are tested using data on an extensive sample of publicly traded U.S. property-liability insurers. Option models of insurance pricing have two primary advantages: First, they explicitly incorporate default risk. This is important given the increase in insurer insolvency rates since the early-1980s (see BarNiv 1990). Second, because of data limitations, the key parameters can be estimated more accurately for option pricing models than for competing models such as the Myers-Cohn (1987) or Kraus-Ross (1982) models.(1) The standard option pricing model of insurance views the liabilities created by issuing insurance policies as analogous to risky corporate debt. The insurer is assumed to issue an insurance policy in return for a premium payment, analogous to the proceeds of a bond issue. In return, it promises to make a payment to the policyholders at the maturity date of the contract. Using this bond analogy, the value of the insurer's promise to policyholders can be thought of as being like the value of a default risk-free loan in the amount of the promised payment less a put option on the value of the insurer. In reality, however, most insurers issue more than one type of insurance and in this case the analogy with a single debt issue is no longer exact. The problem of pricing multiple classes of debt has been considered by Black and Cox (1976). In their analysis senior debt has priority over junior debt in the event of bankruptcy. However, with multiple lines of insurance, each line has equal priority in the event of bankruptcy (see National Association of Insurance Commissioners 1993), and this is the case investigated in our paper. In a multiple line insurance company, equity capital is held in a common pool. If one or more lines incur deficits of losses over premiums, the lines in difficulty can draw upon the full amount of the firm's equity capital, including earnings from the solvent lines. Given this sharing of resources, it is not obvious how to allocate the cost of equity capital to each line. There have only been a few prior papers on insurance pricing in a multiple line firm, mostly in the actuarial literature. Nearly all have approached the problem by assuming that the insurer's equity capital is allocated among lines of business, usually in proportion to each line's share of the insurer's liabilities (see Knuer 1987; Derrig 1989; and D'Arcy and Garven 1990). Prices for a given line of insurance then incorporate an aggregate profit charge equal to the assumed cost of capital for the line multiplied by its assigned equity. …

Option pricing, a core part of options trading, has been fruitfully researched over the years. This article reviews the history of the emergence and advancement of option pricing in terms of a thorough classification of the widely used option pricing models and their empirical studies that follow. The three option pricing models are summarized, including the Black-Scholes pricing model, the tree diagram model, and the Monte Carlo simulation techniques, which have all represented significant progress in the field of option pricing theory. Moreover, the differences between various pricing models are analyzed and compared to show their applications and to provide an outlook on future work in this theory. It is vital to carry out some research on option pricing in order to better suit the preferences of investors. In order to meet the continuous development of financial markets, valuation of financial derivative securities is the key to effective investment in risky assets.

PurposeThe purpose of this paper is to ascertain the effectiveness of major deterministic and stochastic volatility-based option pricing models in pricing and hedging exchange-traded dollar–rupee options over a five-year period since the launch of these options in India.Design/methodology/approachThe paper examines the pricing and hedging performance of five different models, namely, the Black–Scholes–Merton model (BSM), skewness- and kurtosis-adjusted BSM, NGARCH model of Duan, Heston’s stochastic volatility model and an ad hoc Black–Scholes (AHBS) model. Risk-neutral structural parameters are extracted by calibrating each model to the prices of traded dollar–rupee call options. These parameters are used to generate out-of-sample model option prices and to construct a delta-neutral hedge for a short option position. Out-of-sample pricing errors and hedging errors are compared to identify the best-performing model. Robustness is tested by comparing the performance of all models separately over turbulent and tranquil periods.FindingsThe study finds that relatively simpler models fare better than more mathematically complex models in pricing and hedging dollar–rupee options during the sample period. This superior performance is observed to persist even when comparisons are made separately over volatile periods and tranquil periods. However the more sophisticated models reveal a lower moneyness-maturity bias as compared to the BSM model.Practical implicationsThe study concludes that incorporation of skewness and kurtosis in the BSM model as well as the practitioners’ approach of using a moneyness-maturity-based volatility within the BSM model (AHBS model) results in better pricing and hedging effectiveness for dollar–rupee options. This conclusion has strong practical implications for market practitioners, hedgers and regulators in the light of increased volatility in the dollar–rupee pair.Originality/valueExisting literature on this topic has largely centered around either US equity index options or options on major liquid currencies. While many studies have solely focused on the pricing performance of option pricing models, this paper examines both the pricing and hedging performance of competing models in the context of Indian currency options. Robustness of findings is tested by comparing model performance across periods of stress and tranquility. To the best of the author’s knowledge, this paper is one of the first comprehensive studies to focus on an emerging market currency pair such as the dollar–rupee.

In today’s financial world there is a great need to predict the value of the assets, using which strategic decisions can be made to make short term or long term capital gains. Due to the dynamic and uncertain nature of the financial markets, the prediction of the asset prices are really difficult. Many models have been developed to predict the option prices in the financial market. The certainity of these models to predict the option prices to the most accurate level or to the level of minimum deviation is questionable. This study is aimed at analyzing the feasibility of Black - Scholes – Merton differential equation model for stock option pricing in Indian stock exchanges. The result of this study can be used to predict the suitability of using Black - Scholes – Merton differential equation model to predict stock option prices in Indian market. Further the regression analysis has been used to see the impact of time to expiry over the option price and anova test has been used to check whether the mean difference between expected price as computed by Black - Scholes – Merton differential equation model and actual price have any significant difference. The result of analysis found that Black - Scholes – Merton model is more usefull in call option pricing than the put option pricing and also impact of timing is more relevenat for put option pricing than for call option pricing.

This paper is focused on research into options pricing models. The most popular ones are variancegamma and Heston models. They are powerful and flexible to some extent, but they also have drawbacks. Among their shortcomings are instability and long-time calibration. The model proposed in the paper combines neural network (autoencoder) and relatively simple option pricing (mixture normal) model. The autoencoder gives flexibility to the model and reduces the number of parameters. The mixture normal model gives a certain logic to neural network and minimizes calibration time of the new model. So the resulting model eliminates drawbacks of variance-gamma and Heston models and also keeps their advantages. It has fast calibration and shows good enough precision on S&P futures options market.

<sec>Under the background that stock index options urgently need launching in China, the research on stock option pricing model has important theoretical and practical significance. In the traditional B-S-M model it is assumed that the volatility remains unchanged, which differs tremendously from the market’s reality. When the market fluctuates drastically, it is difficult to realize the risk management function of stock index options. Although in the Heston model, as one of the traditional stochastic volatility option pricing models, the correlation risk between the volatility and underlying price is taken into consideration, its pricing accuracy is still to be improved. From the quantum finance perspective, in this paper we use the Feynman path integral method to explore a more practical stock index option pricing model.</sec><sec>In this paper, we construct a Feynman path integral pricing model of stock index options with stochastic volatility by taking Hang Seng index option as the research object and Heston model as the control group. It is found that the Feynman path integral pricing model is significantly superior to the Heston model either at different strike prices on the same expiration date or at different expiration dates for the same strike price. The stock index option pricing model constructed in this paper can give the numerical solution of Feynman's pricing kernel, and directly realizes the forecast of stock index option price. The pricing accuracy is significantly improved compared with the pricing accuracy given by the Heston model through using the characteristic function method.</sec><sec>The remarkable advantages of Feynman path integral stock index option pricing model are as follows. Firstly, the path integral has advantages in solving multivariate problems: the Feynman pricing kernel represents all the information about pricing and can be easily expanded from one-dimensional to multidimensional case, so the change of closing price of stock index and volatility of underlying index can be taken into account simultaneously. Secondly, based on the relationship between the Feynman path generation principle and the law of large number, the mean values of pricing kernel obtained by MATLAB not only optimizes the calculation process, but also significantly improves the pricing accuracy. Feynman path integral is the main method in quantum finance, and the research in this paper will provide reference for its further application in the pricing of financial derivatives.</sec>

Contemporarily, various models have been developed and used to price the returns of an asset or an option for decades. This paper will discuss and compare two option pricing models (the BSM model and the Monte Carlo Simulation) and three financial pricing models (the CAPM, the FF3 Model, and the FF5 Model). It will start with discussing the history, assumptions, and formulae of each model. Then, it will lead into how each model has changed over time, whether that be changes from the researchers themselves or other researchers within the community. Additionally, specifics of the change will be addressed, and the paper will outline what the new model has done to improve upon those concerns. Eventually, an empirical comparison will contrast the effectiveness and accuracy of each model. Ultimately, it will be seen that the BSM model and the MCS are useful in different time frames and the FF5 model is superior to the original financial pricing models as it builds on the original models’ limitations. Even so, the BSM model, MCS, and the FF5 model still have their own limitations that vary from reality and can be improved on in future research. These results shed light on guiding further exploration of newer and more refined models, methodologies, and approaches aimed to enhance our ability to make informed decisions and manage risks with greater precision.

Using GMM to flatten the option volatility smile