Discrete-time option pricing with stochastic liquidity

Classical option pricing theories are usually built on the law of one price, neglecting the impact of market liquidity that may contribute to significant bid-ask spreads. Within the framework of conic finance, we develop a stochastic liquidity model, extending the discrete-time constant liquidity model of Madan (2010). With this extension, we can replicate the term and skew structures of bid-ask spreads typically observed in option markets. We show how to implement such a stochastic liquidity model within our framework using multidimensional binomial trees and we calibrate it to call and put options on the S&P 500.     

Mispricing and lasting arbitrage between parallel markets in the Czech Republic

If co-existing parallel markets are efficient, then arbitrage will maintain a correct pricing relationship. A related question is whether two parallel emerging markets offering more or less the same securities but using different institutional designs, can behave as a single, fully integrated market. In this paper an explicit model of price convergence (with transaction costs) is introduced, in which price differences are studied using levels of arbitrage activity. For the empirical analysis two parallel markets in the Czech Republic are used — the Prague Stock Exchange (PSE) and the RMS (over-the-counter system). In particular, the degree of arbitrage activity is studied for different segments of the PSE and the evolution of arbitrage in the early history of these emerging markets. The empirical results provide evidence of market linkage for actively traded stocks. A significant relationship is found between the segment of the market to which a given firm belongs and the estimated level of arbitrage trading. Moreover, the level of arbitrage activity increases over time for all market segments, and as the markets mature, the differences among the segments gradually disappear.     

Futures Options and the Volatility of Futures Prices

Assuming nonstochastic interest rates, European futures options are shown to be European options written on a particular asset referred to as a futures bond. Consequently, standard option pricing results may be invoked and standard option pricing techniques may be employed in the case of European futures options. Additional arbitrage restrictions on American futures options are derived. The efficiency of a number of futures option markets is examined. Assuming that at-the-money American futures options are priced accurately by Black's European futures option pricing model, the relationship between market participants' ex ante assessment of futures price volatility and the term to maturity of the underlying futures contract is also investigated empirically.     

Pricing by hedging and no-arbitrage beyond semimartingales

We show that pricing a big class of relevant options by hedging and no-arbitrage can be extended beyond semimartingale models. To this end we construct a subclass of self-financing portfolios that contains hedges for these options, but does not contain arbitrage opportunities, even if the stock price process is a non-semimartingale of some special type. Moreover, we show that the option prices depend essentially only on a path property of the stock price process, viz. on the quadratic variation. We end the paper by giving no-arbitrage results even with stopping times for our model class.      

Arbitrage-free market models for option prices: the multi-strike case

This paper studies modeling and existence issues for market models of option prices in a continuous-time framework with one stock, one bond and a family of European call options for one fixed maturity and all strikes. After arguing that (classical) implied volatilities are ill-suited for constructing such models, we introduce the new concepts of local implied volatilities and price level. We show that these new quantities provide a natural and simple parametrization of all option price models satisfying the natural static arbitrage bounds across strikes. We next characterize absence of dynamic arbitrage for such models in terms of drift restrictions on the model coefficients. For the resulting infinite system of SDEs for the price level and all local implied volatilities, we then study the question of solvability and provide sufficient conditions for existence and uniqueness of a solution. We give explicit examples of volatility coefficients satisfying the required assumptions, and hence of arbitrage-free multi-strike market models of option prices.      

Option Profit and Loss Attribution and Pricing: A New Framework

This paper develops a new top-down valuation framework that links the pricing of an option investment to its daily profit and loss attribution. The framework uses the Black-Merton-Scholes option pricing formula to attribute the short-term option investment risk to variation in the underlying security price and the option's implied volatility. Taking risk-neutral expectation and demanding no dynamic arbitrage result in a pricing relation that links an option's fair implied volatility level to the underlying volatility level with corrections for the implied volatility's own expected direction of movement, its variance, and its covariance with the underlying security return.     

An analysis of option pricing under systematic consumption risk using GARCH

An issue in the pricing of contingent claims is whether to account for consumption risk. This is relevant for contingent claims on stock indices, such as the FTSE 100 share price index, as investor’s desire for smooth consumption is often used to explain risk premiums on stock market portfolios, but is not used to explain risk premiums on contingent claims themselves. This paper addresses this fundamental question by allowing for consumption in an economy to be correlated with returns. Daily data on the FTSE 100 share price index are used to compare three option pricing models: the Black–Scholes option pricing model, a GARCH (1, 1) model priced under a risk-neutral framework, and a GARCH (1, 1) model priced under systematic consumption risk. The findings are that accounting for systematic consumption risk only provides improved accuracy for in-the-money call options. When the correlation between consumption and returns increases, the model that accounts for consumption risk will produce lower call option prices than observed prices for in-the-money call options. These results combined imply that the potential consumption-related premium in the market for contingent claims is constant in the case of FTSE 100 index options.     

More heat than light: Investor attention and bitcoin price discovery

We investigate how increased attention affects bitcoin's price discovery process. We first decompose bitcoin price into efficient and noise components and then show that the noise element of bitcoin pricing is driven by high levels of attention. This implies that high levels of attention are linked with an increase in uninformed trading activity in the market for bitcoin, while informed trading activity is driven by arbitrage rather than attention.     

Option Pricing on Stocks in Mergers and Acquisitions

We develop an arbitrage‐free and complete framework to price options on the stocks of firms involved in a merger or acquisition deal allowing for the possibility that the deal might be called off at an intermediate time, creating discontinuous impacts on the stock prices. Our model can be a normative tool for market makers to quote prices for options on stocks involved in such deals and also for traders to control risks associated with such deals using traded options. The results of tests indicate that the model performs significantly better than the Black–Scholes model in explaining observed option prices.     

Price Expectation and the Pricing of Stock Index Futures

Capital markets are not perfect or frictionless, and arbitrage mechanism cannot be complete, particularly for index arbitrage. This study constructs a theoretical foundation to explain why the price expectation of the underlying asset should be entered into the pricing formula of stock index futures. The price expectation and incompleteness of arbitrage then are taken into account to develop a pricing model of stock index futures in imperfect markets. This study also presents three approaches for estimating the model parameter. Finally, the concept of the degree of market imperfection is defined and the valuation model is provided.     

沪深300股指期货定价误差及影响因素分析

文章运用持有成本模型、无套利定价原理以及回归分析,分别对日交易数据、日内5分钟数据对我国沪深300股指期货的定价误差及影响定价误差幅度的因素进行了实证研究,研究表明我国沪深300股指期货的价格在大多数时间是偏高的,在考虑套利成本的情况下,股指期货的定价在大多数时间是有效率的,但是在股票市场大幅波动的时段,股指期货的定价在存在较大幅度的定价误差。从影响股指期货定价误差幅度的因素来看,距到期日越远定价误差越大,现货指数波动越剧烈定价误差越大,股指期货持仓量对定价误差没有显著影响,加息对定价误差的影响跟加息日期有关。     

American option pricing under the double Heston model based on asymptotic expansion

This paper focuses on pricing American put options under the double Heston model proposed by Christoffersen et al. By introducing an explicit exercise rule, we obtain the asymptotic expansion of the solution to the partial differential equation for pricing American put options. We calculate American option price by the sum of the European option price and the early exercise premium. The early exercise premium is calculated by the difference between the American and European option prices based on asymptotic expansions. The European option price is obtained by the efficient COS method. Based on the obtained American option price, the double Heston model is calibrated by minimizing the distance between model and market prices, which yields an optimization problem that is solved by a differential evolution algorithm combined with the Matlab function fmincon.m. Numerical results show that the pricing approach is fast and accurate. Empirical results show that the double Heston model has better performance in pricing short-maturity American put options and capturing the volatility term structure of American put options than the Heston model.     

Pricing and Hedging Discount Bond Options in the Presence of Model Risk

This paper focuses on pricing and hedging options on a zero-coupon bond in a Heath–Jarrow–Morton (1992) framework when the value and/or functional form of forward interest rates volatility is unknown, but is assumed to lie between two fixed values. Due to the link existing between the drift and the diffusion coefficients of the forward rates in the Heath, Jarrow and Morton framework, this is equivalent to hedging and pricing the option when the underlying interest rate model is unknown. We show that a continuous rangeof option prices consistent with no arbitrage exist. This range is bounded by the smallest upper-hedging strategy and the largest lower-hedging strategy prices, which are characterized as the solutions of two non-linear partial differential equations. We also discuss several pricing and hedging illustrations.     

Measuring the pricing error of the arbitrage pricing theory

This article provides an exact Bayesian framework for analyzingthe arbitrage pricing theory (APT). Based on the Gibbs sampler,we show how to obtain the exact posterior distributions forfunctions of interest in the factor model. In particular, wepropose a measure of the APT pricing deviations and obtain itsexact posterior distribution. Using monthly portfolio returnsgrouped by industry and market capitalization, we find thatthere is little improvement in reducing the pricing errors byincluding more factors beyond the first one.     

Asset pricing theory for two price economies

We show that nonlinearly discounted nonlinear martingales are related to no arbitrage in two price economies as linearly discounted martingales were related to no arbitrage in economies satisfying the law of one price. Furthermore, assuming risk acceptability requires a positive physical expectation, we demonstrate that expected rates of return on ask prices should be dominated by expected rates of return on bid prices. A preliminary investigation conducted here, supports this hypothesis. In general we observe that asset pricing theory in two price economies leads to asset pricing inequalities. A model incorporating both nonlinear discounting and nonlinear martingales is developed for the valuation of contingent claims in two price economies. Examples illustrate the interactions present between the severity of measure changes and their associated discount rates. As a consequence arbitrage free two price economies can involve unique discount curves and measure changes that are however specific to both the product being priced and the trade direction. Furthermore the developed valuation operators call into question the current practice of Debt Valuation Adjustments.     

Are regime-shift sources of risk priced in the market?

In this paper we develop a discrete-time pricing model for European options where the log-return of the underlying asset is subject to discontinuous regime shifts in its mean and/or volatility which follow a Markov chain. The model allows for multiple regime shifts whose risk cannot be hedge out and thus must be priced in option market. The paper provides estimates of the price of regime-shift risk coefficients based on a joint estimation procedure of the Markov regime-switching process of the underlying stock and the suggested option pricing model. The results of the paper indicate that bull-to-bear and bear-to-crash regime shifts carry substantial prices of risk. Risk averse investors in the markets price these regime shifts by assigning higher transition (switching) probabilities to them under the risk neutral probability measure than under the physical. Ignoring these sources of risk will lead to substantial option pricing errors. In addition, the paper shows that investors also price reverse regime shifts, like the crash-to-bear and bear-to-bull ones, by assigning smaller transition probabilities under the risk neutral measure than the physical. Finally, the paper evaluates the pricing performance of the model and indicates that it can be successfully employed, in practice, to price European options.     

A Note on Option Pricing for the Constant Elasticity of Variance Model

We study the arbitrage free optionpricing problem for the constant elasticity of variance (CEV) model. To treatthestochastic aspect of the CEV model, we direct attention to the relationship between the CEV modeland squared Bessel processes. Then we show the existence of a unique equivalentmartingale measure and derive the Cox's arbitrage free option pricing formulathrough the properties of squared Bessel processes. Finally we show that the CEVmodel admits arbitrage opportunities when it is conditioned to be strictlypositive.     

Price discovery and persistent arbitrage violations in credit markets

This paper investigates price violations in credit markets using a data sample spanning from 2002 to 2016. We find that price violations are highly persistent during the crisis period, particularly for speculative-grade bonds. There is evidence that price distortions and market disintegration are linked to market-wide and firm-level impediments to arbitrage and limited capital provision. Higher firm-level impediments to arbitrage lead to less market integration, and more severe and persistent pricing discrepancies. Moreover, we find that the negative CDS basis persists in the postcrisis period, which is attributable to dealers’ lower capital commitment and deterioration in market-making quality.     

Estimation and pricing under long-memory stochastic volatility

We treat the problem of option pricing under a stochastic volatility model that exhibits long-range dependence. We model the price process as a Geometric Brownian Motion with volatility evolving as a fractional Ornstein–Uhlenbeck process. We assume that the model has long-memory, thus the memory parameter H in the volatility is greater than 0.5. Although the price process evolves in continuous time, the reality is that observations can only be collected in discrete time. Using historical stock price information we adapt an interacting particle stochastic filtering algorithm to estimate the stochastic volatility empirical distribution. In order to deal with the pricing problem we construct a multinomial recombining tree using sampled values of the volatility from the stochastic volatility empirical measure. Moreover, we describe how to estimate the parameters of our model, including the long-memory parameter of the fractional Brownian motion that drives the volatility process using an implied method. Finally, we compute option prices on the S&P 500 index and we compare our estimated prices with the market option prices.