A Markov Chain Method for Pricing Contingent Claims

We provide a framework for learning risk-neutral measures (Martingale measures) for pricing options from high frequency financial data. In a simple geometric Brownian motion model, a price volatility, a fixed interest rate and a no-arbitrage condition suffice to determine a unique risk-neutral measure. On the other hand, in our framework, we relax some of these assumptions to obtain a class of allowable risk-neutral measures. We then propose a framework for learning the appropriate risk-neural measure. Since the riskneutral measure prices all options simultaneously, we can use all the option contracts on a particular underlying stock for learning. We demonstrate the performance of these models on historical data. In particular, we show that both learning without a no-arbitrage condition and a no-arbitrage condition without learning are worse than our framework; however the combination of learning with a no-arbitrage condition has the best result. These results indicate the potential to learn Martingale measures with a no-arbitrage condition providing just the right constraint. We also compare our approach to standard Binomial models with volatility estimates (historical volatility and GARCH volatility predictors). Finally, we illustrate the power of such a framework by developing a real time trading system based upon these pricing methods.

Option Pricing Under Dividend Barrier Strategies

In the years following the publication of Black and Scholes (J Political Econ, 81(3), 637–654, 1973), numerous alternative models have been proposed for pricing and hedging equity derivatives. Prominent examples include stochastic volatility models, jump-diffusion models, and models based on Levy processes. These all have their own shortcomings, and evidence suggests that none is up to the task of satisfactorily pricing and hedging extremely long-dated claims. Since they all fall within the ambit of risk-neutral valuation, it is natural to speculate that the deficiencies of these models are (at least in part) attributable to the constraints imposed by the risk-neutral approach itself. To investigate this idea, we present a simple two-parameter model for a diversified equity accumulation index. Although our model does not admit an equivalent risk-neutral probability measure, it nevertheless fulfils a minimal no-arbitrage condition for an economically viable financial market. Furthermore, we demonstrate that contingent claims can be priced and hedged, without the need for an equivalent change of probability measure. Convenient formulae for the prices and hedge ratios of a number of standard European claims are derived, and a series of hedge experiments for extremely long-dated claims on the S&P 500 total return index are conducted. Our model serves also as a convenient medium for illustrating and clarifying several points on asset price bubbles and the economics of arbitrage.

Pricing VIX Futures: Evidence from Integrated Physical and Risk-Neutral Probability Measures

This paper provides a unifying framework for the modeling of various types of credit risks such as contagion effects. We argue that Markov chains can efficiently be used to tackle these problems. However, our approach is not limited to pricing problems with contagion. On the theoretical side, we derive pricing formulas for three building blocks that are generalizations of contingent claims studied in Lando (1998). These claims can be thought of as atoms forming the basis for all credit risk payments. Furthermore, we demonstrate that, in general, all contingent claims exposed to credit risk satisfy a system of partial differential equations. This is the key result to calculate prices of credit risk claims explicitly and efficiently.

Downside Risk Neutral Probabilities

Models of the term structure of interest rates have become increasingly important in financial modeling. However, the understanding of these models by practitioners has not always kept pace with the breadth of the application of these models. In particular, misinterpretation of the proper uses of a particular model can lead to significant errors. The confusion regarding these models has arisen because of the overuse and misuse of the term “arbitrage-free.” Keywords: term structure of interest rates; look; act; market risk; risk-neutral valuation; risk-adjusted short rate; risk-neutral model; risk-neutral probability measure; real; artificial; realistic; arbitrage-free model; equilibrium model; arbitrage-free; realistic model

We consider the valuation of credit default swaps (CDSs) under an extended version of Merton’s structural model for a firm’s corporate liabilities. In particular, the interest rate process of a money market account, the appreciation rate, and the volatility of the firm’s value have switching dynamics governed by a finite-state Markov chain in continuous time. The states of the Markov chain are deemed to represent the states of an economy. The shift from one economic state to another may be attributed to certain factors that affect the profits or earnings of a firm; examples of such factors include changes in business conditions, corporate decisions, company operations, management strategies, macroeconomic conditions, and business cycles. In this article, the Esscher transform, which is a well-known tool in actuarial science, is employed to determine an equivalent martingale measure for the valuation problem in the incomplete market setting. Systems of coupled partial differential equations (PDEs) satisfied by the real-world and risk-neutral default probabilities are derived. The consequences for the swap rate of a CDS brought about by the regimeswitching effect of the firm’s value are investigated via a numerical example for the case of a two-state Markov chain. We perform sensitivity analyses for the real-world default probability and the swap rate when different model parameters vary. We also investigate the accuracy and efficiency of the PDE approach by comparing the numerical results from the PDE approach to those from the Monte Carlo simulation.

‘Forward and Futures Prices of Contingent Claims’ applies the multi-period rational expectations model to analyse the pricing of forward and futures contracts and derives an expression for the difference between the prices of the two, which is commonly known as the convexity adjustment. These are illustrated using an example where the asset price, the pricing kernel and the bond prices are joint lognormal variables. The authors then consider the futures and forward prices of contingent claims, such as call options. Futures prices of options can be significantly different from the forward prices of the contingent claims, especially when the options are on interest-rate related assets such as bonds or swaps. Forward prices of long-term bond can be obtained by taking appropriate expected values of future spot pries under the risk-neutral measure.

Efficient Monte Carlo resampling for probability measure changes from Bayesian updating

Downside Risk Neutral Probabilities

For complex multidimensional systems, Monte Carlo methods are useful for sampling probable regions of a configuration space and, in the context of annealing, for determining “low energy” or “high scoring” configurations. Such methods have been used in protein design as means to identify amino acid sequences that are energetically compatible with a particular backbone structure. As with many other applications of Monte Carlo methods, such searches can be inefficient if trial configurations (protein sequences) in the Markov chain are chosen randomly. Here a mean-field biased Monte Carlo method (MFBMC) is presented and applied to designing and sampling protein sequences. The MFBMC method uses predetermined sequence identity probabilities wi(α) to bias the sequence selection. The wi(α) are calculated using a self-consistent, mean-field theory that can estimate the number and composition of sequences having predetermined values of energetically related foldability criteria. The MFBMC method is applied to both a simple protein model, the 27-mer lattice model, and an all-atom protein model. Compared to conventional Monte Carlo (MC) and configurational bias Monte Carlo (BMC), the MFBMC method converges faster to low energy sequences and samples such sequences more efficiently. The MFBMC method also tolerates faster cooling rates than the MC and BMC methods. The MFBMC method can be applied not only to protein sequence search, but also to a wide variety of polymeric and condensed phase systems.

Monte Carlo methods (based on iid sampling or Markov chains) for estimating integrals with respect to a proper target distribution (that is, a probability distribution) are well known in the statistics literature. If the target distribution $\pi$ happens to be improper then it is shown here that the standard time average estimator based on Markov chains with $\pi$ as its stationary distribution will converge to zero with probability 1, and hence is not appropriate. In this paper, we present some limit theorems for regenerative sequences and use these to develop some algorithms to produce strongly consistent estimators (called regeneration and ratio estimators) that work whether $\pi$ is proper or improper. These methods may be referred to as regenerative sequence Monte Carlo (RSMC) methods. The regenerative sequences include Markov chains as a special case. We also present an algorithm that uses the domination of the given target $\pi$ by a probability distribution $\pi_{0}$. Examples are given to illustrate the use and limitations of our algorithms.

This paper presents a unifying theory for valuing contingent claims under a stochastic term of interest rates. The methodology, based on the equivalent martingale measure technique, takes as given an initial forward rate curve and a family of potential stochastic processeE for its subsequent movements. A no arbitrage condition restricts this family of processes yielding valuation formulae for interest rate sensitive contingent claims which do not explicitly depend on the market prices of risk. Examples are provided to illustrate the key results. IN RELATION TO the term of interest rates, arbitrage pricing theory has two purposes. The first, is to price all zero coupon (default free) bonds of varying maturities from a finite number of economic fundamentals, called state variables. The second, is to price all interest rate sensitive contingent claims, taking as given the prices of the zero coupon bonds. This paper presents a general theory and a unifying framework for understanding arbitrage pricing theory in this context, of which all existing arbitrage pricing models are special cases (in particular, Vasicek (1977), Brennan and Schwartz (1979), Langetieg (1980), Ball and Torous (1983), Ho and Lee (1986), Schaefer and Schwartz (1987), and Artzner and Delbaen (1988)). The primary contribution of this paper, however, is a new methodology for solving the second problem, i.e., the pricing of interest rate sensitive contingent claims given the prices of all zero coupon bonds. The methodology is new because (i) it imposes its stochastic directly on the evolution of the forward rate curve, (ii) it does not require an inversion of the term structure to eliminate the market prices of risk from contingent claim values, and (iii) it has a stochastic spot rate process with multiple stochastic factors influencing the term structure. The model can be used to consistently price (and hedge) all contingent claims (American or European) on the term structure, and it is derived from necessary and (more importantly) sufficient conditions for the absence of arbitrage. The arbitrage pricing models of Vasicek (1977), Brennan and Schwartz (1979), Langetieg (1980), and Artzner and Delbaen (1988) all require an IFormerly titled Bond Pricing and the Term Structure of Interest Rates: A New Methodology.

Considering the diversity of travel habits of users, spatiotemporal distribution model of charging load for highly random electric vehicles(EV), based on Markov chain, is proposed. According to the residents’ travel habits of the 2017 National Household Travel Survey (NHTS) in the United States, the destinations are divided into homes, work places, and other locations. Based on Markov Chain and Monte Carlo method, a highly random and complex process chain with unlimited processes is constructed. The distribution of start times and end times in one-day journey is fitted by the parameterless distribution with normal distribution as the kernel. The distribution of travel time, travel distance and dwell time is fitted by lognormal distribution. Then, a spatiotemporal distribution model of one-day vehicle travel is established. Considering the influence of EVs, the dual-input fuzzy algorithm of travel time and travel distance is used to calculate the power consumption of the trip. According to the crowd’s travel anxiety and different travel needs, different charging fredquencies and different charging power levels are used for vehicle charging behavior. Finally, the Monte Carlo method is used to calculate and analyse the charging load of EVs in different classification scenarios, such as household income, home address, weekdays and weekends. The one-day electric vehicle load distribution model was successfully established in three locations. The results show that the load of electric vehicle charging stations is not only affected by factors such as holidays, but also by the composition of the urban population.