Put-call Parity Research Articles

Analytically pricing volatility options and capped/floored volatility swaps with nonlinear payoffs in discrete observation case under the Merton jump-diffusion model driven by a nonhomogeneous Poisson process

In this paper, we introduce novel analytical solutions for valuating volatility derivatives, including volatility options and capped/floored volatility swaps, employing discrete sampling within the framework of the Merton jump-diffusion model, which is driven by a nonhomogeneous Poisson process. The absence of a comprehensive understanding of the probability distribution characterizing the realized variance has historically impeded the development of a robust analytical valuation approach for such instruments. Through the application of the cumulative distribution function of the realized variance conditional on Poisson jumps, we have derived explicit expectations for the derivative payoffs articulated as functions of the extremum values of the square root of the realized variance. We delineate precise pricing structures for an array of instruments, encompassing variance and volatility swaps, variance and volatility options, and their respective capped and floored variations, alongside establishing put-call parity and relationships for capped and floored positions. Complementing the theoretical advancements, we substantiate the practical efficacy and precision of our solutions via Monte Carlo simulations, articulated through multiple numerical examples. Conclusively, our analysis extends to the quantification of jump impacts on the fair strike prices of volatility derivatives with nonlinear payoffs, facilitated by our analytic pricing expressions.

METODE BLACK-SCHOLES DALAM PENENTUAN HARGA OPSI

The Black-Scholes Method is one of the option pricing method that introduced by Fischer Black and Myron Scholes in 1973. This study aim to review the determination of the price of an option with stocks as the underlying assets and based on Black and Scholes assumptions. These assumptions lead to construct an equation named Black-Scholes differential equations, which is the equation that must be satisfied for option as the derivative instrument and non-dividend giving stocks as the underlying assets. After the Black-Scholes differential equations formed successfully, the next step is to find the solution of that equation. Consider the option is call option, the solution that will be obtained from solving the equation is the price of call option. Then substitute it to the put-call parity equation, which is the equation that shows the relationship between call and put option prices, so the price of put option can be obtained too. The estimation that obtained from the Black-Scholes method is the highest price for a contract is said to be fair and worth to buy for the holder.

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.

Exploring Implied Certainty Equivalent Rates in Financial Markets: Empirical Analysis and Application to the Electric Vehicle Industry

In this paper, we mainly study the impact of the implied certainty equivalent rate on investment in financial markets. First, we derived the mathematical expression of the implied certainty equivalent rate by using put-call parity, and then we selected some company stocks and options; we considered the best-performing and worst-performing company stocks and options from the beginning of 2023 to the present for empirical research. By visualizing the relationship between the time to maturity, moneyness, and implied certainty equivalent rate of these options, we have obtained a universal conclusion—a positive implied certainty equivalent rate is more suitable for investment than a negative implied certainty equivalent rate, but for a positive implied certainty equivalent rate, a larger value also means a higher investment risk. Next, we applied these results to the electric vehicle industry, and by comparing several well-known US electric vehicle production companies, we further strengthened our conclusions. Finally, we give a warning concerning risk, that is, investment in the financial market should not focus solely on the implied certainty equivalent rate, because investment is not an easy task, and many factors need to be considered, including some factors that are difficult to predict with models.

The Contribution of Transaction Costs to Expected Stock Returns: A Novel Measure

We document that a theoretically founded, real-time, and easy-to-implement option-based measure, termed <italic>synthetic-stock difference</italic> (SSD), accurately estimates the part of stock’s expected return arising from stock’s transaction costs. We calculate SSD for US optionable stocks. SSD can be more than 10% per annum, it can fluctuate significantly over time and its cross-sectional dispersion widens over market crises periods. We confirm the accuracy of SSD by empirically verifying the predictions of a standard asset pricing setting with transaction costs. First, we document its predicted type of connection with various proxies of stocks’ transaction costs. Second, we conduct simple asset pricing tests which render further support. Our setting allows explaining the size of alphas reported by previous literature on the predictive ability of deviations from put-call parity.

Circumventing SEC Rule 201 short sale restrictions with options

This paper provides evidence on the ease and relatively low cost, relative to the potential benefits of circumventing the short selling circuit breaker regulations (SEC Rule 201) using options. Stocks with traded options react more negatively on short sale restriction trigger days relative to their counterparts without traded options. Short sale circuit breaker events are associated with increases in put and call options spreads as well as put-call parity violations. The paper estimates a discount of a synthetic short position to the prevailing stock price when Rule 201 is triggered of about 3%.

Financial Derivatives

Dr Michael Kelly puts Mathematica to work deriving the mathematical and numerical results for option prices by solving PDEs, SDEs using the discounted expectation of stochastic variables and simulations of underlying expiry prices, as well as discovering the risk-neutral value of the rate of return and establishing theorems such as the Put-Call parity formula

Checking the Box on the Efficiency of CME Bitcoin Futures Options

The adoption of new financial instruments is naturally met with skepticism and apprehension. Capital markets as we know them today possess the amazing capability to package any exposure into a digestible instrument for market participants to utilize. However, prudence requires us to verify that these instruments do not suffer from inefficiencies that may prove hazardous to investors. In this article, the authors verify the efficiency of CME Bitcoin Futures Options by testing boundary arbitrage, put-call parity arbitrage, and box spread arbitrage conditions. The results strongly suggest that no reasonable arbitrage opportunities exist and that CME Bitcoin Futures Options are well-suited for institutional scale investing. Hence, we believe their use by institutions to hedge, speculate, and/or facilitate transactions between decentralized markets (“DeFi”) and traditional markets (“TradFi”) will grow significantly in the next few years.

Variation in Put-Call Parity Deviation and Equity Option Returns

Variation in Put-Call Parity Deviation and Equity Option Returns

Pricing and Hedging Bond Power Exchange Options in a Stochastic String Term-Structure Model

We study power exchange options written on zero-coupon bonds under a stochastic string term-structure framework. Closed-form expressions for pricing and hedging bond power exchange options are obtained and, as particular cases, the corresponding expressions for call power options and constant underlying elasticity in strikes (CUES) options. Sufficient conditions for the equivalence of the European and the American versions of bond power exchange options are provided and the put-call parity relation for European bond power exchange options is established. Finally, we consider several applications of our results including duration and convexity measures for bond power exchange options, pricing extendable/accelerable maturity zero-coupon bonds, options to price a zero-coupon bond off of a shifted term-structure, and options on interest rates and rate spreads. In particular, we show that standard formulas for interest rate caplets and floorlets in a LIBOR market model can be obtained as special cases of bond power exchange options under a stochastic string term-structure model.

Arbitrage Opportunities in the Options Market: A Study of the Indian Index Options

The objective of this paper is to find out whether the put-call parity relationship holds in case of index options in the Indian stock market. The index which has been chosen as the underlying asset is NSE Nifty. This paper further aims at finding out different factors responsible for the violation of put-call parity relationship, if any.

The Paradoxical Prices of Options

The synchronized relationship between financial and fundamental prices has been topical for years now. It seems that option pricing theory has not been used to disentangle that relationship between two prices during merger and acquisition (M&amp;A) activities. This paper uses Put-Call parity theorem to explore the divergence of financial and fundamental prices in any firm during the acquisition process. The results illustrate that price differentials are persistent; moreover, the differentials are caused by the exponential factor. Despite the fact that some principles are drawn from the real estate investment trust (REIT) literature, the results have wider implications for industries with similar traits to REITs.

Uncertainty of Put-Call Parity Violation and Option Returns

Uncertainty of Put-Call Parity Violation and Option Returns

Empirical Analysis of Potential Put-Call Parity Arbitrage Opportunities with Particular Focus on the Shanghai Stock Exchange 50 Index

Put-Call-Parity is a major cornerstone of the option pricing theory. The equation provides an answer to the equilibrium of the option market. It tells us what the right call option price should be assuming put price, actual stock price, risk free rate and maturity. The call price depends on these parameters. No arbitrage opportunities are possible if the equilibrium equation is met. In financially well developed countries and regions the put-call-parity holds and allows no arbitrage opportunities except in abnormal market conditions. This paper aims to analyse the put-call-parity in China for a certain period of time. It reviews if arbitrage opportunities can be identified. It shows that the put-call-parity dominates the option market in China as well despite shorter periods in the development of the financial markets and allows no arbitrage opportunities.

Principled pasting: attaching tails to risk-neutral probability density functions recovered from option prices

The popular ‘curve-fitting’ method of using option prices to construct an underlying asset's risk neutral probability density function (RND) first recovers the interior of the density and then attaches left and right tails. Typically, the tails are constructed so that values of the RND and risk neutral cumulative distribution function (RNCDF) from the interior and the tails match at the attachment points. We propose and demonstrate the feasibility of also requiring that the left and right tails accurately price the options with strikes at the attachment points. Our methodology produces a RND that provides superior pricing performance than earlier curve-fitting methods for both those options used in the construction of the RND and those that were not. We also demonstrate that Put-Call Parity complicates the classification of in and out of sample options.

Determinants of Put-Call Disparity: Kospi 200 Index Options

Many studies find that traditional option pricing models fail to work in practice. The implied volatility smile is one example. In this study, we examine deviations of spot prices from prices implied by put-call parity for Korean KOSPI 200 index options, one of the most actively traded derivative products in the world. Deviations are significant and economically meaningful across different moneyness categories spanning deep-in-the-money to deep-out-of-the-money options. Determinants of put-call disparities for the KOSPI 200 index options include past spot return moments, cognitive biases, and prior option trading volume relative to spot trading volume. We show mispricing is more likely to occur after periods of extreme downturns in the stock market, implying demand for put options increases relative to call options when investors become more likely to insure against extreme loss. We also show that put-call disparity rates have predictive power for future spot returns due to overreaction of KOSPI 200 index option traders, rather than to information contained in option prices.

On a strongly continuous semigroup for a Black-Scholes integro-differential operator: European options under jump-diffusion dynamics

We consider a Black-Scholes integro-differential operator associated with a partial integro-differential equation for pricing European options with a jump-diffusion process for the underlying asset. Using the theory of one-parameter semigroups, we prove that the operator is the infinitesimal generator of a strongly continuous semigroup and express the semigroup explicitly as a convolution of a jump function, the Black-Scholes kernel and the payoff function. This is analogous to the Gauss-Weierstrass and Poisson semigroups. Then we investigate the pricing of European options under jump diffusion for two broad classes of payoff functions. A generalised put-call parity relating the functions from both classes is also obtained.

A UNIFIED MARKET MODEL FOR SWAPTIONS AND CONSTANT MATURITY SWAPS

Internal-rate-of-return (IRR) settled swaptions are the main interest rate volatility instruments in the European interest rate markets. Industry practice is to use an approximation formula to price IRR swaptions based on Black model, which is not arbitrage-free. We formulate a unified market model to incorporate both swaptions and constant maturity swaps (CMS) pricing under a single, self-consistent framework. We demonstrate that the model is able to calibrate to market quotes well, and is also able to efficiently price both IRR-settled and swap-settled swaptions, along with CMS products. We use the model to illustrate the difference in implied volatilities for IRR-settled payer and receiver swaptions, the pricing of zero-wide collars and in-the-money (ITM) swaptions, the implication on put-call parity, and the issue of negative vega. These findings offer important insights to the ongoing reform in the European swaption market.

The Performance of Trading Strategies Based on Deviations from Put-Call Parity of Stock Options

According to Cremers and Weinbaum [6], we compute the implied volatility spread by option put-call parity theory. Then, we build strategy based on implied volatility spread and compares it with OS, 52-week high, and contrarian investment strategies to explore whether the investment performance of the implied-volatility-spread based strategy is better than other strategies. Moreover, we combine the implied-volatility-spread based strategy with other strategies to form the two-dimensional investment strategy to explore whether the performance of two-dimensional implied-volatility-spread strategy is better than one-dimensional implied-volatility-spread strategy. The empirical results show that it needs more than one year of investment horizon to get positive abnormal return by implied-volatility-spread based strategy. Otherwise, it will only receive negative abnormal return when the investment horizon is less than one year. In addition, two-dimensional strategy improves bad performance of one-dimensional strategy. After combining the contrarian 52-week high and contrarian investment strategy with implied-volatility-spread strategy, we find that there is the best strategic effect when the holding period is 12 months. Nevertheless, the abnormal returns decrease after the holding period is 24 months. JEL classification numbers: G11, G12. Keywords: Implied-volatility-spread, OS strategy, 52-week highs strategy, Trading volume strategy, Price momentum strategy, Option volume.

The GameStop Short Squeeze: Put-Call Parity and the Effect of Frictions Before, During and After the Squeeze

The short squeeze in GameStop attracted worldwide attention and resulted in congressional hearings. The increase in price from $21 to $483 over a short period of time was not the result of obvious fundamental earnings prospects. Excess demand by investors on a social media site accompanied by the short-covering resulted in the stratospheric ascent of stock price. We use put-call parity to investigate the related issue of the no-arbitrage violations before, during, and after the squeeze. We do not find evidence of abundant free money after accounting for short-selling frictions.