Closed form option pricing under generalized hermite expansions

We propose a new method to estimate the empirical pricing kernel based on option data. We estimate the pricing kernel nonparametrically by using the ratio of the risk-neutral density estimator and the subjective density estimator. The risk-neutral density is approximated by a weighted kernel density estimator with varying unknown weights for different observations, and the subjective density is approximated by a kernel density estimator with equal weights. We represent the European call option price function by the second order integration of the risk-neutral density, so that the unknown weights are obtained through one-step penalized least squares estimation with the Kullback-Leibler divergence as the penalty function. Asymptotic results of the resulting estimators are established. The performance of the proposed method is illustrated empirically by simulation and real data application studies.

From Continuous to Discrete: Studies on Continuity Corrections and Monte Carlo Simulation with Applications to Barrier Options and American Options Menghui Cao This dissertation 1) shows continuity corrections for first passage probabilities of Brownian bridge and barrier joint probabilities, which are applied to the pricing of two-dimensional barrier and partial barrier options, and 2) introduces new variance reduction techniques and computational improvements to Monte Carlo methods for pricing American options. The joint distribution of Brownian motion and its first passage time has found applications in many areas, including sequential analysis, pricing of barrier options, and credit risk modeling. There are, however, no simple closed-form solutions for these joint probabilities in a discrete-time setting. Chapter 2 shows that, discrete two-dimensional barrier and partial barrier joint probabilities can be approximated by their continuous-time probabilities with remarkable accuracy after shifting the barrier away from the underlying by a factor of βσ √ ∆t. We achieve this through a uniform continuity correction theorem on the first passage probabilities for Brownian bridge, extending relevant results in Siegmund (1985a). The continuity corrections are applied to the pricing of two-dimensional barrier and partial barrier options, extending the results in Broadie, Glasserman & Kou (1997) on one-dimensional barrier options. One interesting aspect is that for type B partial barrier options, the barrier correction cannot be applied throughout one pricing formula, but only to some barrier values and leaving the other unchanged, the direction of correction may also vary within one

Inventory models play an important role in determining the optimal ordering and pricing policies. Much work has been reported in literature regarding inventory models with finite or infinite replenishment. But in many practical situations the replenishment is governed by random factors like procurement, transportation, environmental condition, availability of raw material etc., hence, it is needed to develop inventory models with random replenishment. In this paper, an EPQ model for deteriorating items is developed and analyzed with the assumption that the replenishment is random and follows a Weibull distribution. It is further assumed that the life time of a commodity is random and follows a generalized Pareto distribution and demand is a function of on hand inventory. Using the differential equations, the instantaneous state of inventory is derived. With suitable cost considerations, the total cost function is obtained. By minimizing the total cost function, the optimal ordering policies are derived. Through numerical illustrations, the sensitivity analysis is carried. The sensitivity analysis of the model reveals that the random replenishment has significant influence on the ordering and pricing policies of the model. This model also includes some of the earlier models as particular cases for specific values of the parameters.

In this paper we show that in the lognormal discrete-time stochastic volatility model with predictable conditional expected returns, the conditional expected value of the discounted payo of a European call option is infinite. Our empirical illustration shows that the characteristics of the predictive distributions of the discounted payos, obtained using Monte Carlo methods, do not indicate directly that the expected discounted payos are infinite.

Option prices can be used to extract the implied risk-neutral density functions of the future underlying asset prices and returns. These market expectations provide valuable information that can be helpful to policy makers and investors. We tested the accuracy and stability of four nonstructural models in estimating the “true” risk-neutral density functions from option prices: the density functional based on the confluent hypergeometric function, the mixture of lognormal distributions, the smoothed implied volatility smile and Edgeworth expansions. The “true” risk-neutral density is unknown, so it was generated using the Carr–Geman–Madan–Yor (CGMY) Gamma-Ornstein–Uhlenbeck (Gamma-OU) model, a structural model able to generate flexible “true” risk-neutral densities. We observed that the density functional based on the confluent hypergeometric function and mixture of lognormal distributions outperformed the smoothed implied volatility smile and the Edgeworth expansion models in capturing the “true” risk-neutral density. The smoothed implied volatility smile had the best performance in terms of stability. This work aims to be more exhaustive than previous studies, by testing a wider variety of nonstructural models. In addition, we tested the effectiveness of these models in more demanding conditions that are closer to reality. In fact, the use of the CGMY Gamma-OU model to build the “true” risk-neutral density allows densities with a higher probability of extreme events and more leptokurtosis, and also generates sudden, discontinuous moves in prices, which has advantages over the diffusion models used in previous studies.

Derivatives are alternative financial instruments which extend traders opportunities to achieve some financial goals. They are risk management instruments that are related to a data in the future, and also they react to uncertain prices. Study on pricing futures can provide useful tools to understand the stochastic behavior of prices to manage the risk of price volatility. Thus, this study evaluates commodity futures contracts by considering Ross (1995) one-factor future pricing model as a function of spot price, Gibson and Schwartz (1990) two-factor futures pricing model as a function of spot price and convenience yield and finally Schwartz (1997) three-factor futures pricing model as a function of spot price, convenience yield and instantaneous interest rate by adding jump to stochastic behavior of commodity spot price. For this purpose, it is assumed that spot price follows Jump-diffusion stochastic process with exponential probability distribution of jump domain. Finally, commodity pricing future relations in three basic models are presented as a function of above factor(s) and jump parameters by using Duffy-Pan-Singleton approach. JEL Classification: G12, G13

Essays on Price Adjustment and Imperfect Information Luminita Stevens Understanding how firms set prices is a key step towards settling classic debates in economics regarding the sources of nominal price rigidities, the mechanisms through which disturbances are transmitted within and across countries, and the effectiveness of monetary policy in dampening business cycle fluctuations. This dissertation examines patterns of price adjustment at the firm level, both empirically and theoretically. The first chapter studies pricing patterns in US grocery store data. Using a novel empirical method, I identify changes in the distribution of product-level prices over time. These changes typically occur every seven months and mark the transition to new pricing regimes. Inside regimes, prices alternate among a small set of prices with high frequency. This evidence motivates a theory of price setting in which firms respond to shocks using multiple-price policies that are simple enough to only specify a small number of prices, and that are updated only on discrete occasions. The second chapter presents a theory of costly information that generates such simple, sticky policies. In order to economize on the costs of acquiring information, the firm designs a pricing policy that is a noisy, coarse representation of market conditions. Moreover, it updates this policy infrequently, based on imprecise signals about the state of the economy. Despite the high volatility of observed prices, the firm responds imperfectly to changes in market conditions. The third chapter, co-authored with Ryan Chahrour, addresses the patterns of adjustment in international relative prices. We develop a two-country model in which retailers have imperfect information and search for producers operating in different regions in the two countries. We demonstrate that frictions at the regional level within countries generate dispersion in international relative prices in the absence of additional frictions at the national border.

June 12, 2020 John Cochrane: COVID-19 Interview Policy@McCombs on COVID-19 LISTEN LISTEN ON: Apple Spotify Stitcher John H. Cochrane is the Rose-Marie and Jack Anderson Senior Fellow at the Hoover Institution. He is also a research associate of the National Bureau of Economic Research and an adjunct scholar of the CATO Institute. Before joining Hoover, Cochrane was a Professor of Finance at the University of Chicago’s Booth School of Business, and earlier at its Economics Department. Cochrane earned a bachelor’s degree in physics at MIT and his PhD in economics at the University of California at Berkeley. He was a junior staff economist on the Council of Economic Advisers (1982–83). Cochrane’s recent publications include the book Asset Pricing and articles on dynamics in stock and bond markets, the volatility of exchange rates, the term structure of interest rates, the returns to venture capital, liquidity premiums in stock prices, the relation between stock prices and business cycles, and option pricing when investors can’t perfectly hedge. His monetary economics publications include articles on the relationship between deficits and inflation, the effects of monetary policy, and the fiscal theory of the price level. He has also written articles on macroeconomics, health insurance, time-series econometrics, financial regulation, and other topics. He was a coauthor of The Squam Lake Report. His Asset Pricing PhD class is available online via Coursera. Cochrane frequently contributes editorial opinion essays to the Wall Street Journal, Bloomberg.com, and other publications. He maintains the Grumpy Economist blog. Listen to the Policy@McCombs podcast to learn more about the data shaping public policy and the data that should be. Subscribe to our YouTube Channel to watch video versions of the podcasts.

This article proposes a closed pricing formula for European options when the return of the underlying asset follows extended normal distribution, that is, any different degrees of skewness and kurtosis relative to the normal distribution induced by the Black-Scholes model. The moment restriction is suggested, so that the pricing model under any arbitrary distribution for an underlying asset must satisfy the arbitrage-free condition. Numerical experiments and comparison of empirical performance of the proposed model with the Black-Scholes, ad hoc Black-Scholes, and Gram-Charlier distribution models are carried out. In particular, an estimation of implied parameters such as standard deviation, skewness, and kurtosis of the return on the underlying asset from the market prices of the KOSPI 200 index options is made, and in-sample and out-of-sample tests are performed. These results not only support the previous finding that the actual density of the underlying asset shows skewness to the left and high peaks, but also demonstrate that the present model has good explanatory power for option prices. © 2005 Wiley Periodicals, Inc. Jrl Fut Mark 25:845–871, 2005

Pricing of American options in discrete time by means of regression-based Monte Carlo methods is considered in this thesis. Thereby, the options are allowed to be based on several underlyings (so-called basket options). It is assumed that the underlyings price processes satisfy the Markov property and consequently are Markov processes. In this dissertation, Monte Carlo methods are used to generate artificial sample paths of these price processes, and subsequently adaptive nonparametric least squares regression estimates are used to estimate the so-called continuation values from these data. The continuation values describe the mean values of the American options for given values of the underlyings at time t, subject to the constraint that the options are not exercised at time t, but optimally exercised in the future. Adaptive least squares neural networks as well as adaptive least squares spline estimates are used as nonparametric regression estimates. In conjunction with the pricing of American options, these estimates are theoretically analysed, and results with respect to consistency and rate of convergence are derived. Finally, the pricing of American options on simulated data by means of the estimates is illustrated.

This article presents a multi-GPU adaptation of a specific Monte Carlo and classification based method for pricing American basket options, due to Picazo. The first part relates how to combine fine and coarse-grained parallelization to price American basket options. A dynamic strategy of kernel calibration is proposed. Doing so, our implementation on a reasonable size (18) GPU cluster achieves the pricing of a high dimensional (40) option in less than one hour against almost 8 as observed for runs we conducted in the past, using a 64-core cluster (composed of quad-core AMD Opteron 2356). In order to benefit from different GPU device types, we detail the dynamic strategy we have used to load balance GPU calculus which greatly improves the overall pricing time we obtained. An analysis of possible bottleneck effects demonstrates that there is a sequential bottleneck due to the training phase that relies upon the AdaBoost classification method, which prevents the implementation to be fully scalable, and so prevents to envision further decreasing pricing time down to handful of minutes. For this we propose to consider using Random Forests classification method: it is naturally dividable over a cluster, and available like AdaBoost as a black box from the popular Weka machine learning library. However our experimental tests will show that its use is costly.

This paper obtains an analytic pricing formula for vulnerable European options. This formula allows the option writer to default at any time during the life of the option, and the recovery value of the option in default to be an amount equal to the writerfs asset-to-debt ratio multiplied by the economic equivalent of the option right, rather than its default-free value. Numerical results indicate that the economic equivalent of the option right, assumed as its default-free value, is inappropriate, and a default, restricted to occur only at the exercise date, seriously understates the impact of credit risk on option prices. Key words: vulnerable option, option pricing, credit risk

Power systems have been undergoing radical changes in recent years, and their planning and operation will be surely undertaken according to the Smart Grid (SG) vision in the near future. The SG initiatives aim at introducing new technologies and services in power systems, to make the electrical networks more reliable, efficient, secure and environmentally-friendly. In particular, it is expected that communication technologies, computational intelligence and distributed energy sources will be widely used for the whole power system in an integrated fashion. In particular, nowadays, unprecedented challenges like as stringent regulations, environmental concerns, growing demand for high quality, reliable electricity and rising customer expectations are forcing utilities to rethink about electricity generation and delivery from the bottom up. Moreover, the availability of low cost computing and telecommunications technologies, new generation options, and scalable, modular automation systems push utilities to be dynamic, innovative and ambitious enough to take advantage of them. Driven by the dynamics of the new energy environment, leading utilities, technology vendors and government organizations have created a vision of the next generation of energy delivery systems: the Smart Grid. Operational changes of the grid, caused by restructuring of the electric utility industry and electricity storage technology advancements, have created an opportunity for storage systems to provide unique services to the evolving grid. Especially Battery Energy Storage Systems (BESSs), thanks to the large number and variety of services they can provide, are powerful tools for the solution of some challenges that future grids will face. This consideration makes BESSs critical components of the future grids. The BESS can be applied for different services into the different levels of power system chain to satisfy technical challenges and provide financial benefits. In the context of the application of BESSs in SGs, there are two main problems that need to be addressed in a way that exploits the BESS potential, that are linked to their operation and sizing. This thesis focuses on both these aspects, proposing new strategies that allow optimizing the BESS adoption. When dealing with BESSs, sizing and operation are strictly linked. The correct sizing of a BESS, in fact, needs to take into account its operation which in turn will be effected with the aim of optimizing the whole system where it is included. In the first part of this research study, advanced optimal operating strategies were proposed for BESSs by considering both the distribution system operator perspective and the end user. Thus, the proposed operating strategies were performed with the aim of (i) leveling the active power requested by the loads connected to a distribution system (distribution system operator service), (ii) reducing the electricity costs sustained by an end-use costumer that provides demand response (DR) (end user service) and (iii) scheduling a microgrid (µG) with DR resources such as Plug-in Electric Vehicles (PEVs) and Data Centers (DCs) (both the two section service). The proposed strategies also satisfied technical constraints of BESSs and other components of the µG. The second part of the thesis presented the optimal sizing of BESSs aimed at maximizing the benefits related to their use. In the thesis, the sizing, which is performed by considering the end user point of view with reference to both the industrial and residential customers, is effected by adopting both deterministic and probabilistic approaches. With reference to the deterministic approach, a simple and quick closed form procedure for the sizing of BESSs in residential and industrial applications was proposed. In case of probabilistic approach, the case of a BESS installed in an industrial facility was considered and the sizing was performed based on the decision theory. Technical improvements and economic benefits of optimal operation and optimal sizing of BESSs in SG are demonstrated by the obtained results which are reported in the numerical applications. More specifically, it was clearly determined that BESSs can offer technical supports into the distribution operator section of the grid in terms of load management and security challenges. Moreover optimal integration of BESSs into the grid was also appealing for end users thanks to valuable amounts of electricity bill cost reduction. Regarding the original contribution of the thesis, the following considerations can be done. With reference to the load leveling service, an innovative two-step procedure (day-ahead scheduling and very short time predictive control) was proposed which optimally controls a BESS connected to a distribution substation in order to perform load leveling. In case of DR, a proper control of the BESS was proposed in order to perform DR under different price schemes, such as Real Time Pricing (RTP) and Time of Use (TOU) without modifying the daily work cycle of the industrial loads. The control procedure allows achieving contemporaneously two important goals that are the reduction of the bill costs and the prolonging the battery's lifetime so further reducing the costs sustained by the customer. With reference to the scheduling of microgrids, the original contribution of the thesis is focused on the proposal of optimization strategies aimed at managing and coordinating, simultaneously, batteries on board of vehicles or equipping data centers' Uninterruptable Power Supply (UPS) and Distributed Generation (DG) units. Also comparisons among different single-objective based strategies are made in order to highlight the most convenient. With reference to the sizing based on deterministic approach, unlike the other relating literature, the innovative contribution is that the closed form procedure takes into account both the technical constraints of the battery and contractual agreements between the customer and the utility. Moreover, in the economical analysis performed for the sizing, which is applied with reference to both residential and small industrial customers and is based on actual TOU tariffs, a wide sensitivity analysis to consider different perspectives in terms of life span and future costs was performed. Some aspects that affect the profitability of the battery, such as technological limitations (e.g. the battery and converter efficiency), economic barriers (e.g. capital cost and the rate of change of the cost) and variation of the load profile along the years were deeply analyzed. In case of sizing based on probabilistic approach, the original contributions of the thesis are mainly referred to the proposal of a new method that uses a decision theory-based process to obtain the best sizing alternative considering the various uncertainties affecting the sizing procedure. The thesis is organized in three chapters which are dealing with integration of BESSs in SGs. The first chapter reports basic concepts and characteristics of BESSs, fundamental components and features of SGs and different services that BESSs can provide. The optimal operation strategies of BESS are considered in second chapter which includes their problem formulation, solving procedures and results. The third chapter deals with the optimal sizing problem of BESSs for which the problem formulation, solving procedures and results are reported. Finally, the conclusions are presented in the last part of thesis.

Multiscale stochastic elasticity of variance for options and equity linked annuity; A Mellin transform approach

The SABR stochastic volatility model with β-volatility β ? (0,1) and an absorbing barrier in zero imposed to the forward prices/rates stochastic process is studied. The presence of (possibly) nonzero correlation between the stochastic differentials that appear on the right hand side of the model equations is considered. A series expansion of the transition probability density function of the model in powers of the correlation coefficient of these stochastic differentials is presented. Explicit formulae for the first three terms of this expansion are derived. These formulae are integrals of known integrands. The zero-th order term of the expansion is a new integral formula containing only elementary functions of the transition probability density function of the SABR model when the correlation coefficient is zero. The expansion is deduced from the final value problem for the backward Kolmogorov equation satisfied by the transition probability density function. Each term of the expansion is defined as the solution of a final value problem for a partial differential equation. The integral formulae that give the solutions of these final value problems are based on the Hankel and on the Kontorovich-Lebedev transforms. From the series expansion of the probability density function we deduce the corresponding expansions of the European call and put option prices. Moreover we deduce closed form formulae for the moments of the forward prices/rates variable. The moment formulae obtained do not involve integrals or series expansions and are expressed using only elementary functions. The option pricing formulae are used to study synthetic and real data. In particular we study a time series (of real data) of futures prices of the EUR/USD currency's exchange rate and of the corresponding option prices. The website: http://www.econ.univpm.it/recchioni/finance/w18 contains material including animations, an interactive application and an app that helps the understanding of the paper. A more general reference to the work of the authors and of their coauthors in mathematical finance is the website:http://www.econ.univpm.it/recchioni/finance.