Pricing Of Claims Research Articles

Utility Indifference Pricing: A Time Consistent Approach

This article considers the optimal portfolio selection problem in a dynamic multi-period stochastic framework with regime switching. The risk preferences are of exponential (CARA) type with an absolute coefficient of risk aversion that changes with the regime. The market model is incomplete and there are two risky assets: tradable and non-tradable. In this context, the optimal investment strategies are time inconsistent. Consequently, the subgame perfect equilibrium strategies are considered. The utility indifference ask price of a contingent claim written on the risky assets is computed through an indifference valuation algorithm. By running numerical experiments, we examine how this price varies in response to changes in model parameters.

Financial Derivative Uncertainty Claim Pricing Using Meixner Distribution Skewed Operators

In this paper, consideration is given to the problem of independent/uncertain claim pricing using distribution operators. This method was earlier studied in Insurance Pricing where the original operator was described in relation to normal distribution. We apply the Meixner process to financial pricing since the normal distribution is a very poor model to fit log-returns of financial assets like stocks or indices. For us to realize a better fit, we substitute the normal distribution by Meixner process. Hence, we generalize this approach by using an operator based on the density Meixner distribution. We further show how Meixner operator function can be used to derive the formular for asset pricing of independent/uncertain future returns of a risky asset.

A Path-Independent Humped Volatility Model for Option Pricing

This article presents a path-independent model for evaluating interest-sensitive claims in a Heath–Jarrow–Morton (1992, Bond pricing and the term structure of interest rates: a new methodology for contingent claims valuation, Econometrica, 60, pp. 77–105) framework, when the volatility structure of forward rates shows the deterministic and stationary humped shape analysed by Ritchken and Chuang (2000, Interest rate option pricing with volatility humps, Review of Derivatives Research, 3(3), pp. 237–262). In our analysis, the evolution of the term structure is captured by a one-factor short rate process with drift depending on a three-dimensional state variable Markov process. We develop a lattice to discretize the dynamics of each variable appearing in the short rate process, and establish a three-variate reconnecting tree to compute interest-sensitive claim prices. The proposed approach makes the evaluation problem path-independent, thus overcoming the computational difficulties in managing path-dependent variables as it happens in the Ritchken–Chuang (2000, Interest rate option pricing with volatility humps, Review of Derivatives Research, 3(3), pp. 237–262) model.

Two Processes for Two Prices

Postulating additivity of bid and ask prices for claims comonotone with a long or short stock position, two pricing processes are identified from data on bid and ask prices for options. It is observed that there are two separate put call parity relations in place, with the ask price for call less bid prices for put delivering an ask price for the forward stock. Likewise, the ask for puts less the bid for calls identifies the bid for the forward stock. Two processes are introduced to determine bid and ask prices for claims comonotone with a long or short position in the stock. For a claim comonotone with a long position, one uses the so-called increasing process for the ask price and the so-called decreasing process for the bid price, and vice versa for a claim comonotone with a short position. As candidates for the two processes, one may employ any of the traditional one dimensional Markov processes. We illustrate the theory by using a Sato process, a model known to produce a smile conforming fit over strike and maturity. The two processes are observed to have marginals related by first order stochastic dominance. The increasing process dominates the decreasing process in this sense. These two processes are also used to construct upper and lower bounds for bid and ask prices for claims not comonotone with a long or short stock position. The two processes and their properties are illustrated with data on bid and ask prices for options on the exchange traded fund, SPY.

Pricing of Claims in Discrete Time with Partial Information

We consider the pricing problem of a seller with delayed price information. By using Lagrange duality, a dual problem is derived, and it is proved that there is no duality gap. This gives a characterization of the seller's price of a contingent claim. Finally, we analyze the dual problem, and compare the prices offered by two sellers with delayed and full information respectively.

A Multi Period Equilibrium Pricing Model

In this paper, we propose an equilibrium pricing model in a dynamic multi-period stochastic framework with uncertain income. There are one tradable risky asset (stock/commodity), one non-tradable underlying (temperature), and also a contingent claim (weather derivative) written on the tradable risky asset and the non-tradable underlying in the market. The price of the contingent claim is priced in equilibrium by optimal strategies of representative agent and market clearing condition. The risk preferences are of exponential (CARA) type with a stochastic coefficient of risk aversion. Both Nash subgame perfect strategies and naive strategies are considered. From the numerical result we examine how the equilibrium prices vary in response to changes in model parameters and highlight the importance of Nash equilibrium pricing principle.

Benchmark approach for defaultable claims under partial information

We study the pricing and hedging of defaultable claims under the benchmark approach, introduced by Platen and Heath (2006), with partial information. The model is considered to be driven by an Ornstein–Uhlenbeck (OU) process. We fist convert the partial information model to a complete information model through the innovation process approach by deriving the non-linear filtering equation for the unobservable processes. We then define the total cash flow and use the Schweizer (2008) methodology for hedging the cash flow under the real world probability measure.

Debt Valuation and Chapter 22

Numerous studies have examined the effect on credit spreads of renegotiation. These studies have generally focused on the impact on spread levels in general, and not on how renegotiation influences the relative pricing of senior versus junior debt claims. In this paper, we show that the scope for sequential renegotiation may reduce and even eliminate the premium for debt seniority. Our analysis also explains why companies may engage in repeated Chapter 11 bankruptcy filings (a phenomenon commonly referred to as Chapter 22).

Engineering More Effective Weighted Monte Carlo Option Pricing Models

The Weighted Monte Carlo (WMC) method (Avellaneda (1998) and Avellaneda et al. (2001)) is a practical and robust method for calibrating a contingent claim pricing model that is consistent with the prices of liquid market instruments. We discuss new, tractable instances of the WMC method that make use of underlying asset prices that evolve with Student t (rather than, for example, Gaussian) innovations, relative U-entropy (rather than Kullback-Leibler relative entropy), and constraints that force model prices to lie within the bid-ask bands together with penalties on the discrepancies from the mid-market prices. We demonstrate the effectiveness of these WMC instances via benchmarking exercises on SPX option and LIBOR swaption data. In particular, we find that our WMC instances can be successfully calibrated to more liquid options (in our benchmarking exercises, our WMC instances reduced calibration failure rates – SPX options and LIBOR swaptions priced outside the bid-ask band – by a factors of over 2,000 and 38, respectively), and are more accurate on “out-of-sample” options than previously described WMC instances, particularly short-dated options and highly out-of-the-money options, where our instances are less prone to underpricing.

A STATISTICAL MODEL OF CLAIM PRICES IN PREDICTION MARKETS

This paper finds that claim prices in prediction markets, a new genre of financial markets, follow a Poisson distribution. The significance of this finding is that as soon as a claim in a prediction market is created and thereafter flushes out expert and inside information from around the world regarding that particular claim, claim prices immediately begin forming bell-shaped distributions, implying global agreement regarding the probabilities of claims being realized. This is an interesting finding, implying a surprisingly high degree of global homogeneity of inside information in predictions markets, even though such information is scattered in disconnected and secretive pockets around the world. This finding could also imply that cultural diversities do not significantly affect the interpretation of information in prediction markets.

Pricing Vulnerable Claims in a L�vy Driven Model

We obtain an explicit expression for the price of a vulnerable claim written on a stock whose predefault dynamics follows a Levy-driven SDE. The stock jumps to zero at default with a hazard rate intensity given by a negative power of the stock price. We recover the characteristic function of the terminal log price as the solution of a complex valued infinite dimensional system of first order ordinary differential equations. We provide an explicit eigenfunction expansion representation of the characteristic function in a suitably chosen Banach space, and use it to price defaultable bonds and stock options. We present numerical results to demonstrate the accuracy and efficiency of the method.

EU ETS Futures Spread Analysis and Pricing Contingent Claims Under Different Market Schemes

The European Union Emission Trading Scheme is the largest market mechanism yet implemented to spur emissions reduction. In the scheme emissions certificates are traded within annual periods to compensate for the total emissions of given companies. The rules for how certificates can be passed from one annual period to another is phase dependent. During phase II of the EU ETS, emissions can be borrowed between years falling within phase II, but not from phase III years. We are interested in investigating the way in which this structural change affects the impact of unexpected release of information on futures returns. We propose a continuous-time model that depicts the relationship assimilated into the spread. We assume that two futures that mature at subsequent dates are traded. Their dynamic is driven by Brownian motions augmented by two jump processes. The discontinuous component reflects the impact of any unexpected release of information. We estimate the model parameters and draw economic conclusions. Furthermore our empirical results supports the stance that the market is mature and efficient enough to be comparable to other many futures markets. We present a pricing solution based on the Follmer-Schweizer decomposition. The optimal hedging strategy depends on all traded futures and minimizes the mean conditional square error of the cumulative process. We show how the fair price of any contingent claim can theoretically be computed in this context. Pricing examples under different market schemes are investigated.

ON PRICING CONTINGENT CLAIMS UNDER THE DOUBLE HESTON MODEL

This article presents a lattice based approach for pricing contingent claims when the underlying asset evolves according to the double Heston (dH) stochastic volatility model introduced by Christoffersen et al. (2009). We discretize the continuous evolution of both squared volatilities by a "binomial pyramid", and consider the asset value as an auxiliary state variable for which a subset of possible realizations is attached to each node of the pyramid. The elements of the subset cover the range of asset prices at each time slice, and claim price is computed solving backward through the "binomial pyramid". Numerical experiments confirm the accuracy and efficiency of the proposed model.

PRICING AND SEMIMARTINGALE REPRESENTATIONS OF VULNERABLE CONTINGENT CLAIMS IN REGIME‐SWITCHING MARKETS

Using a suitable change of probability measure, we obtain a Poisson series representation for the arbitrage‐free price process of vulnerable contingent claims in a regime‐switching market driven by an underlying continuous‐time Markov process. As a result of this representation, along with a short‐time asymptotic expansion of the claim's price process, we develop an efficient novel method for pricing claims whose payoffs may depend on the full path of the underlying Markov chain. The proposed approach is applied to price not only simple European claims such as defaultable bonds, but also a new type of path‐dependent claims that we term self‐decomposable, as well as the important class of vulnerable call and put options on a stock. We provide a detailed error analysis and illustrate the accuracy and computational complexity of our method on several market traded instruments, such as defaultable bond prices, barrier options, and vulnerable call options. Using again our Poisson series representation, we show differentiability in time of the predefault price function of European vulnerable claims, which enables us to rigorously deduce Feynman‐Kač representations for the predefault pricing function and new semimartingale representations for the price process of the vulnerable claim under both risk‐neutral and objective probability measures.

NO‐ARBITRAGE PRICING UNDER SYSTEMIC RISK: ACCOUNTING FOR CROSS‐OWNERSHIP

We generalize Merton’s asset valuation approach to systems of multiple financial firms where cross‐ownership of equities and liabilities is present. The liabilities, which may include debts and derivatives, can be of differing seniority. We derive equations for the prices of equities and recovery claims under no‐arbitrage. An existence result and a uniqueness result are proven. Examples and an algorithm for the simultaneous calculation of all no‐arbitrage prices are provided. A result on capital structure irrelevance for groups of firms regarding externally held claims is discussed, as well as financial leverage and systemic risk caused by cross‐ownership.

Time-varying long-run mean of commodity prices and the modeling of futures term structures

The exploration of the mean-reversion of commodity prices is important for inventory management, inflation forecasting and contingent claim pricing. Bessembinder et al. [J. Finance, 1995, 50, 361–375] document the mean-reversion of commodity spot prices using futures term structure data; however, mean-reversion to a constant level is rejected in nearly all studies using historical spot price time series. This indicates that the spot prices revert to a stochastic long-run mean. Recognizing this, I propose a reduced-form model with the stochastic long-run mean as a separate factor. This model fits the futures dynamics better than do classical models such as the Gibson–Schwartz [J. Finance, 1990, 45, 959–976] model and the Casassus–Collin-Dufresne [J. Finance, 2005, 60, 2283–2331] model with a constant interest rate. An application for option pricing is also presented in this paper.

MONTE CARLO DERIVATIVE PRICING WITH PARTIAL INFORMATION IN A CLASS OF DOUBLY STOCHASTIC POISSON PROCESSES WITH MARKS

To model intraday stock price movements we propose a class of marked doubly stochastic Poisson processes, whose intensity process can be interpreted in terms of the effect of information release on market activity. Assuming a partial information setting in which market agents are restricted to observe only the price process, a filtering algorithm is applied to compute, by Monte Carlo approximation, contingent claim prices, when the dynamics of the price process is given under a martingale measure. In particular, conditions for the existence of the minimal martingale measure Q are derived, and properties of the model under Q are studied.

The price of risk and ambiguity in an intertemporal general equilibrium model of asset prices

We consider a version of the intertemporal general equilibrium model of Cox et al. (Econometrica 53:363–384, 1985) with a single production process and two correlated state variables. It is assumed that only one of them, Y 2, has shocks correlated with those of the economy’s output rate and, simultaneously, that the representative agent is ambiguous about its stochastic process. This implies that changes in Y 2 should be hedged and its uncertainty priced, with this price containing risk and ambiguity components. Ambiguity impacts asset pricing through two channels: the price of uncertainty associated with the ambiguous state variable, Y 2, and the interest rate. With ambiguity, the equilibrium price of uncertainty associated with Y 2 and the equilibrium interest rate can increase or decrease, depending on: (i) the correlations between the shocks in Y 2 and those in the output rate and in the other state variable; (ii) the diffusion functions of the stochastic processes for Y 2 and for the output rate; and (iii) the gradient of the value function with respect to Y 2. As applications of our generic setting, we deduct the model of Longstaff and Schwartz (J Financ 47:1259–1282, 1992) for interest-rate-sensitive contingent claim pricing and the variance-risk price specification in the option pricing model of Heston (Rev Financ Stud 6:327–343, 1993). Additionally, it is obtained a variance-uncertainty price specification that can be used to obtain a closed-form solution for option pricing with ambiguity about stochastic variance.

An optimal stopping problem with a reward constraint

This article studies an optimal stopping problem with an endogenous constraint on the set of admissible stopping times. The constraint stipulates that continuation is permitted, at any given date t, only if the endogenous reward achieved exceeds a prespecified threshold. Characterizations of the value function and the optimal stopping time are presented. An application to the pricing of corporate claims, when the capital structure of the firm includes equity-trigger debt, is carried out.

Drawdowns and the Speed of Market Crash

In this paper we examine the probabilistic behavior of two quantities closely related to market crashes. The first is the drawdown of an asset and the second is the duration of time between the last reset of the maximum before the drawdown and the time of the drawdown. The former is the first time the current drop of an investor’s wealth from its historical maximum reaches a pre-specified level and has been used extensively as a path-dependent measure of a market crash in the financial risk management literature. The latter is the speed at which the drawdown occurs and thus provides a measure of how fast a market crash takes place. We call this the speed of market crash. In this work we derive the joint Laplace transform of the last visit time of the maximum of a process preceding the drawdown, the speed of market crash, and the maximum of the process under general diffusion dynamics. We discuss applications of these results in the pricing of insurance claims related to the drawdown and its speed. Our applications are developed under the drifted Brownian motion model and the constant elasticity of variance (CEV) model.