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Valuation of the American options encountered commonly in finance is quite difficult due to the possibility of early exercise alternatives. Since an exact solution for the American options does not exist, effective numerical methods are needed to understand the behavior of option pricing models. Therefore, in this paper, a new approach based on a high-order difference scheme is proposed to discuss the valuation of an American put option as a free boundary problem. Using a front-fixing approach that transforms the unknown free boundary (optimal stopping) into a fixed one, a sixth-order finite difference scheme (FD6) in space and a third-order strong-stability preserving Runge–Kutta (SSPRK3) in time are applied to the model converted to a nonlinear partial differential equation. The computed results revealed that the combined method is seen to attempt to pull up the capacity of the algorithm to achieve higher accuracy. It is seen that the quantitative and qualitative results produced by the method proposed with minimal computational effort are sufficiently accurate and meaningful. Therefore, this article provides some new insights about the physical characteristics of financial problems and such realistic phenomena.

It has been found that the surface of implied volatility has appeared in financial market embrace volatility “Smile” and volatility “Smirk” through the long-term observation. Compared to the conventional Black-Scholes option pricing models, it has been proved to provide more accurate results by stochastic volatility model in terms of the implied volatility, while the classic stochastic volatility model fails to capture the term structure phenomenon of volatility “Smirk.” More attempts have been made to correct for American put option price with incorporating a fast-scale stochastic volatility and a slow-scale stochastic volatility in this paper. Given that the combination in the process of multiscale volatility may lead to a high-dimensional differential equation, an asymptotic approximation method is employed to reduce the dimension in this paper. The numerical results of finite difference show that the multiscale volatility model can offer accurate explanations of the behavior of American put option price.

Informed traders may prefer the options market to the stock market for reasons including the leverage effect, transaction costs, restrictions on short sale. Many studies try to predict future returns of stocks using informed traders' behavior in the options market. In this study, we examine whether the trading volume ratios of single stock options have the predictive power for future returns of the underlying stock. By analyzing the stock price responses to the “preliminary announcement of performance” of 36 underlying stocks on the Korea Exchange from November 2014 to March 2021 and the trading volume of options written on those stocks, we investigate the relation between the option ratios, which are the call option volume to put option volume ratio (C/P ratio) and the option volume to stock volume ratio (O/S ratio), and the future returns of the underlying stock. We also examine which ratio is better in predicting the future returns. The authors found that both option ratios showed the statistically significant predictability about future returns of the underlying stock and that the return predictability of the O/S ratio is more robust than that of the C/P ratio. This study shows that indicators generated in the options market can be used to predict future underlying stock returns. Further, the findings of this study contributed to a dearth of literature pertaining to single stock options. The results suggest that the single stock options market is efficient and influences the price discovery in the stock market.

A multiperiodic structure always underlies the procurement problem in practice. Option contracts are widely used to make a firm capable of responding flexibly to high demand uncertainty. However, so far, the impact of different option contracts on the multiperiod decisions and performance of a firm has been not discussed. Motivated by this problem, in this paper we analyze the multiperiod procurement problem within the framework of wholesale price, call, put, and bidirectional option contracts. Based on the dynamic programming theory, we characterize the structure of the firm's optimal replenishment policy for each period in four cases. We then develop a heuristic to approximate the corresponding policy parameters. Through system comparisons of four cases, we examine the value of option contracts for the firm and discuss the best option contract type for the firm. The results show that the firm will gain higher profits after the option contracts are introduced. Moreover, if the unit purchase and exercise costs of the bidirectional option are equal to those of the unilateral option, then bidirectional option contracts are preferred over unilateral option contracts. If the unit purchase cost of the bidirectional option is higher than that of the unilateral option, unilateral option contracts may be more profitable than bidirectional option contracts. Furthermore, if the unit backlogging cost is high and the unit inventory holding cost is low, call option contracts are better than put option contracts; if the unit backlogging cost is low and the unit inventory holding cost is high, then put option contracts work better than call option contracts.

Black-Scholes option pricing model is used to decide theoretical price of different Options contracts in many stock markets in the world. In can find many generalizations of BS model by modifying some assumptions of classical BS model. In this paper we compared two such modified Black-Scholes models with classical Black-Scholes model only for Indian option contracts. We have selected stock options form 5 different sectors of Indian stock market. Then we have found call and put option prices for 22 stocks listed on National Stock Exchange by all three option pricing models. Finally, we have compared option prices for all three models and decided the best model for Indian Options. Motivation/Background: In 1973, two economists, Fischer Black, Myron and Robert Merton derived a closed form formula for finding value of financial options. For this discovery, they got a Nobel prize in Economic science in 1997. Afterwards, many researchers have found some limitations of Black-Scholes model. To overcome these limitations, there are many generalizations of Black-Scholes model available in literature. Also, there are very limited study available for comparison of generalized Black-Scholes models in context of Indian stock market. For these reasons we have done this study of comparison of two generalized BS models with classical BS model for Indian Stock market. Method: First, we have selected top 5 sectors of Indian stock market. Then from these sectors, we have picked total 22 stocks for which we want to compare three option pricing models. Then we have collected essential data like, current stock price, strike price, expiration time, rate of interest, etc. for computing the theoretical price of options by using three different option pricing formulas. After finding price of options by using all three models, finally we compared these theoretical option price with market price of respected stock options and decided that which theoretical price has less RMSE error among all three model prices. Result: After going through the method described above, we found that the generalized Black-Scholes model with modified distribution has minimum RMSE errors than other two models, one is classical Black-Scholes model and other is Generalized Black-Scholes model with modified interest rate.

Diplomsko delo opisuje različne možne načine trgovanja z opcijami. Opcija je pogodba med nosilcem in izdajateljem opcije, ki daje lastniku pravico nakupa ali prodaje osnovnega instrumenta po vnaprej dogovorjeni izvršilni ceni. Opcijska pogodba je napisana na določen osnovni instrument, to je lahko delnica, terminska pogodba, surovina, borzni indeks in podobno. Opcije so uporaben osnovni instrument zlasti na nestabilnih trgih, saj omogočajo večje finančne učinke in zavarovanje pred tveganji. Pri trgovanju z opcijami je možno zelo hitro izgubiti večje zneske denarja, vendar so lahko tudi dobički zelo visoki. Kupec opcije zanjo plača premijo, ki investitorju predstavlja zavarovanje za prevzeto tveganje. Višina premije je odvisna od višine dogovorjene izvršilne cene, netvegane obrestne mere, volatilnosti osnovnega instrumenta in časa zapadlosti opcije. Trgovanje z opcijami poteka na organiziranih in prostih trgih. Vlagatelji lahko s pomočjo opcij oblikujejo različne strategije trgovanja. V diplomskem delu najprej opišemo opcijske strategije, ki vključujejo le eno opcijo, te so primerne bolj za izkušene in dobro poučene vlagatelje, ki se zavedajo možnih izgub. Predstavimo pokrite opcijske strategije, pri katerih se vlagatelji pred večjimi izgubami zavarujejo z držanjem osnovnega instrumenta. Strategija, ki se jo poslužujejo veliki institucionalni lastniki, je pokrita nakupna opcijska strategija, izguba te strategije je omejena na razliko med stroškom nakupa osnovnega premoženja in prejeto premijo. Pri pokriti prodajni opcijski strategiji, ki jo investitorji uporabljajo predvsem, ko pričakujejo padec vrednosti osnovega premoženja, je izguba omejena na plačano premijo. Velik del diplomskega dela opisuje opcijske razpone. Ti se delijo v tri skupine: navpični, vodoravni in diagonalni razponi. Največkrat uporabljen razpon je bikov razpon, ta se zaradi enakega datuma zapadlosti opcij in različnih izvršilnih cen uvršča v strategije navpičnega razpona. Zelo pogosto uporabljena strategija navpičnega razpona, je tudi strategija medvedjega razpona. Omenjeni strategiji imata omejeno možnost zaslužka in izgube. Strategija, ki združuje medvedji in bikov razpon se imenuje metuljev razpon. Investitorji, ki pričakujejo znatno gibanje cen delnic, vendar niso prepričani o smeri gibanja cene ponavadi uporabijo strategijo dolgega trikotnika, ki omogoča zaslužek ne glede na smer gibanja cene. Diplomsko delo predstavi tudi strategijo dolgega koledarskega razpona, ki izkorišča hitrejši upad časovne vrednosti opcij s krajšim časom zapadlosti. Podobno izkoriščata to tudi strategiji diagonalnega razpona in strategija dvojne diagonale. Nazadnje omenjene strategije investitorjem prinašajo nekoliko negotovosti o največjem možnem zaslužku in nekatere tudi negotovost o možni izgubi. Ravno zato se te uvrščajo v bolj zahtevne strategije trgovanja z opcijami, ki od vlagatelja zahtevajo dobro poznavanje finančnega trga. In this Bachelor’s thesis we describe different possible ways of trading options. An option is a contract between an option holder and an option writer that gives the owner the right to buy or sell an underlying instrument at a pre-agreed strike price. An option contract is written on a specific underlying instrument, which can be a share, a futures contract, a commodity, a stock index, etc. Options are a useful underlying instrument, particularly in volatile markets, as they allow for greater financial leverage and hedging. When trading options, it is possible to lose large amounts of money very quickly, but the gains can also be very high. The buyer of an option pays a premium for it, which provides the investor with insurance against the risk taken. The amount of the premium depends on the size of the agreed strike price, the risk-free interest rate, the volatility of the underlying instrument and the maturity of the option. Options are traded on organised and over-the-counter markets. Investors can use options to design different trading strategies. In this thesis we first describe option strategies involving only one option, which are more suitable for experienced and knowledgeable investors who are aware of the potential losses. We present covered option strategies, where investors protect themselves against major losses by holding the underlying instrument. A strategy used by large institutional owners is the covered call option strategy, the loss of which is limited to the difference between the cost of buying the underlying asset and the premium received. In the covered put option strategy, which is mainly used by investors when they expect the value of the underlying asset to fall, the loss is limited to the premium paid. In large part of the thesis we describe option spreads. These are divided into three groups: vertical, horizontal and diagonal spreads. The most commonly used spread is the bull spread, which is classified as a vertical spread strategy due to the same maturity date of the options and the different strike prices. A very commonly used vertical spread strategy is also the bear spread strategy. These two strategies have limited profit and loss potential. A strategy that combines bear and bull spreads is called a butterfly spread. Investors who expect a significant movement in share prices but are unsure of the direction of the price movement tend to use the long triangle strategy, which allows earnings to be made regardless of the direction price moves. In thesis we also present a long calendar spread strategy, which takes advantage of the faster decline in the time value of shorter-dated options. In a similar way, the diagonal spread and double diagonal strategies also take advantage of this. The latter strategies provide investors with some uncertainty about the maximum possible gain and also some uncertainty about the possible loss. This is why they are classified as more involved option trading strategies, which require the investor to have a good knowledge of the financial market.

The critical price $$S^{*}\left( t\right) $$ S ∗ t of an American put option is the underlying stock price level that triggers its immediate optimal exercise. We provide a new perspective on the determination of the critical price near the option maturity T when the jump-adjusted dividend yield of the underlying stock is either greater than or weakly smaller than the riskfree rate. Firstly, we prove that $$S^{*}\left( t\right) $$ S ∗ t coincides with the critical price of the covered American put (a portfolio that is long in the put as well as in the stock). Secondly, we show that the stock price that represents the indifference point between exercising the covered put and waiting until T is the European-put critical price, at which the European put is worth its intrinsic value. Finally, we prove that the indifference point’s behavior at T equals $$S^{*}\left( t\right) $$ S ∗ t ’s behavior at T when the stock price is either a geometric Brownian motion or a jump-diffusion. Our results provide a thorough economic analysis of $$S^{*}\left( t\right) $$ S ∗ t and rigorously show the correspondence of an American option problem to an easier European option problem at maturity .

UK equity release actuaries are using a flawed approach to value the no-negative equity guarantees in their equity release mortgages. The approach they use, the Discounted Projection approach, incorrectly uses projected future house prices as the underlying prices in their put option pricing equations. The correct approach uses forward house prices. The Discounted Projection approach entails significant under-valuations of no negative equity guarantees and over-valuations of equity release mortgages and can produce valuations that violate rational pricing principles. The Discounted Projection approach is also inconsistent with both actuarial and accounting standards. Our results have significant ramifications for equity release industry practice and prudential regulation.

The crop insurance in Indonesia (AUTP) is much focused on the area impacted by flood, drought, and pest attack. The complication of the procedure to claim the loss must follow several conditions. The different approaches in the insurance sector, using weather index can be taken into consideration to produce a variety of insurance products. This insurance product used the American put option with the primary asset is the rainfall and the cumulative rainfall to exercise the claim, considering the optimal execution limit. The homotopic analysis is used to determine the valuation of the American put option, which also becomes the insurance premium. The case study is focused on areas experiencing a drought so that insurance claims can be exercise when the rainfall index value is below a predetermined limit. Considering the normality of the rainfall data, the calculation of insurance premium was done for the first growing season. The insurance premium is varies based on the optimal execution limit, while the calculation of profit is based on the optimum limit exercise and the minimum rainfall for the growing season, and its different depended on insurance claim acceptance limits.

Developing a hedging strategy to reduce risk of losses for a given set of stocks in a portfolio is a difficult task due to cost of the hedge. In Vietnam stock market, cross-hedge is involved hedging a long position of a stock because there is no put option for the stock. In addition, only VN30 stock index futures contracts are traded on Hanoi Stock Exchange. Inspired by recently achievement of deep reinforcement learning, we explore feasibility to construct a hedging strategy automatically by leveraging cooperative multi-agent in reinforcement learning techniques without advanced domain knowledge. In this work, we use 10 popular stocks on Ho Chi Minh Stock Exchange, and VN30F1M (VN30 Index Futures contracts within one month settlement) to develop a stock market simulator (including transaction fee, tax, and settlement date of transactions) for reinforcement learning agent training. We use daily return as input data for training process. Results suggest that the agent can learn trading and hedging policy to make profit and reduce losses. Furthermore, we also find that our agent can protect portfolios and make positive profit in case market collapses systematically. In practice, this work can help Vietnam's stock market investors to improve performance and reduce losses in trading, especially when the volatility cannot be controlled.

Recently published research has identified an empirical approximation to the free exercise boundary of an American equity put option. This article examines the accuracy of this simple, readily calculated approximation through linear regression analyses of hundreds of numerically generated free boundaries. The author finds that this Little Ansatz provides an excellent estimate of the optimal boundary for all meaningful parameter values and times to expiration. A second result of this article is the mapping of the two empirical parameters as functions of risk-free rate and volatility. This mapping is necessary to enable fast option pricing and sensitivity calculations for large books of these derivative transactions. Proper calculation of the American put option is a long-standing challenge that has spawned immensely creative and admirable research. The discovery of this empirical approximation has great practical importance. TOPICS:Derivatives, options, security analysis and valuation, quantitative methods, statistical methods Key Findings ▪ The author finds that a recently published empirical expression provides an excellent estimate of the American free exercise boundary for all meaningful parameter values and times to expiration. ▪ The author determines efficient and accurate mapping of the two necessary adjustable parameters of the approximation as functions of risk-free rate and volatility. ▪ The author suggests future research and improvements for this fast, empirical solution to the American free exercise boundary.

Reverse supply chain (RSC) management can be implemented to ameliorate environmental and economic goals simultaneously. In 2020, the devastating influences generated by the COVID-19 global pandemic had established high uncertainty in the manufacturers' capacity, which can hinder the fulfillment of such goals. To address such a problem, in this research, we analyze a two-echelon RSC, including a re-manufacturer who, despite facing the remanufacturing capacity uncertainty, remanufactures eligible obsolete products, then re-enters the marketplace, and a collector who accumulates eligible obsolete products from consumers. We survey centralized and decentralized decisions, and also a condition where the collector, as a Stackelberg game leader, offers a put option contract as a risk-sharing approach and decides on both option and exercise prices; in return, the re-manufacturer determines the order quantity. Contrary to previous studies in which the value of option contracts has been analyzed under demand disruptions, this paper aims to address the performance of a put option contract to mitigate the remanufacturing capacity uncertainty in an RSC. Our results demonstrate that by offering the put option contract and determining the option price as nearly low as the marginal refund cost, not only can the collector motivate the re-manufacturer to augment its order quantity but both parties also attain a win-win profit-sharing outcome. Besides, the customized put option contract can achieve Pareto-improving channel coordination in the condition of remanufacturing capacity uncertainty.

In this study, we propose an implied forward-looking measure for systemic risk that employs the information from put option prices, the Systemic Options Value-at-Risk (SOVaR). This new measure can capture the buildup stage of systemic risk in the financial sector earlier than the standard stock market-based systemic risk measures (SRMs). Non-parametric tests show that our measure exhibits more timely early warning signals (up to one month earlier) regarding the main turbulent events around the global financial crisis of 2007-2009 than the three main stock market-based SRMs. Moreover, this new measure also shows significant predictive power with respect to macroeconomic downturns as well as future recessions. Our results are robust to various specifications, breakdowns of financial sectors, and controlling for the other main risk measures proposed in the literature.

The purpose of this study is to examine the implementation of option contracts using Black Scholes and GARCH on the LQ45 index using the long straddle strategy. This study uses time-series data as a time frame for conducting research, using a sample of closing price data for the LQ 45 daily index for 2009-2018. For the test the model, we used the secondary data of the closing stock price index from February 28, 2009 to March 31, 2019The results of this study are seen by comparing the average percentage value of Average Mean Squared Error (AMSE) of Black Scholes and GARCH with the application of a long straddle strategy, where the smaller the percentage value, the better the model will be. Within one month of option contract due date, Black Scholes is better than GARCH, with an error value on the call option of 2.77% and the put option of 1.56%. Within two months of option contract due date, GARCH is better than Black Scholes, with an error value on the call option of 8.12% and the put option of 4.00%. Within three months of option contract due date, Black Scholes is better than GARCH, with an error value on the call option of 12.38% and on the put option of 5.50%. The long straddle strategy in the LQ45 index only reached a maximum of 60% of possible profits, with an average of around 30% possible profits.

Dividend paying European stock options are modeled using a time-fractional Black–Scholes (tfBS) partial differential equation (PDE). The underlying fractional stochastic dynamics explored in this work are appropriate for capturing market fluctuations in which random fractional white noise has the potential to accurately estimate European put option premiums while providing a good numerical convergence. The aim of this paper is two fold: firstly, to construct a time-fractional (tfBS) PDE for pricing European options on continuous dividend paying stocks, and, secondly, to propose an implicit finite difference method for solving the constructed tfBS PDE. Through rigorous mathematical analysis it is established that the implicit finite difference scheme is unconditionally stable. To support these theoretical observations, two numerical examples are presented under the proposed fractional framework. Results indicate that the tfBS and its proposed numerical method are very effective mathematical tools for pricing European options.