Market implied volatilities for defaultable bonds

Typically, implied volatilities for defaultable instruments are not available in the financial market since quotations related to options on defaultable bonds or on credit default swaps are usually not quoted by brokers. However, an estimate of their volatilities is needed for pricing purposes. In this paper, we provide a methodology to infer market implied volatilities for defaultable bonds using equity implied volatilities and CDS spreads quoted by the market in relation to a specific issuer. The theoretical framework we propose is based on the Merton’s model under stochastic interest rates where the short rate is assumed to follow the Hull–White model. A numerical analysis is provided to illustrate the calibration process to be performed starting from financial market data. The market implied volatility calibrated according to the proposed methodology could be used to evaluate options where the underlying is a risky bond, i.e. callable bond or other types of credit-risk sensitive financial instruments.

     

Empirical Evaluation of Hybrid Defaultable Bond Pricing Models

A four‐factor model (the extended model of Schmid and Zagst) is presented for pricing credit risk related instruments such as defaultable bonds or credit derivatives. It is an advancement of an earlier three‐factor model. In addition to a firm‐specific credit risk factor, a new systematic risk factor in the form of GDP growth rate is included. This new model is set in the context of other hybrid defaultable bond pricing models and empirically compared to specific representatives. We find that a model based only on firm‐specific variables is unable to capture changes in credit spreads completely. However, it is shown that in this model, market variables such as GDP growth rates, non‐defaultable interest rates and firm‐specific variables together significantly influence credit spread levels and changes.     

Impact of risk aversion and belief heterogeneity on trading of defaultable claims

This paper studies the problem of pricing and trading of defaultable claims among investors with heterogeneous risk preferences and market views. Based on the utility-indifference pricing methodology, we construct the bid-ask spreads for risk-averse buyers and sellers, and show that the spreads widen as risk aversion or trading volume increases. Moreover, we analyze the buyer’s optimal static trading position under various market settings, including (i) when the market pricing rule is linear, and (ii) when the counterparty—single or multiple sellers—may have different nonlinear pricing rules generated by risk aversion and belief heterogeneity. For defaultable bonds and credit default swaps, we provide explicit formulas for the optimal trading positions, and examine the combined effect of risk aversions and beliefs. In particular, we find that belief heterogeneity, rather than the difference in risk aversion, is crucial to trigger a trade.     

Risk Minimizing Option Pricing for a Class of Exotic Options in a Markov-Modulated Market

We address risk minimizing option pricing in a regime switching market where the floating interest rate depends on a finite state Markov process. The growth rate and the volatility of the stock also depend on the Markov process. Using the minimal martingale measure, we show that the locally risk minimizing prices for certain exotic options satisfy a system of Black-Scholes partial differential equations with appropriate boundary conditions. We find the corresponding hedging strategies and the residual risk. We develop suitable numerical methods to compute option prices.     

多重可违约基金的定价

In this paper a generalized defaultable bond pricing formula is derived by assuming that there exists a defaultable forward rate term structure and that firms in the economy interact when default occurs. Generally,The risk-neutral default intensity χ^Q is not equal to the empirical or actual default intensity λ,. This paper proves that multiple default intensities are invari-ant under equivalent martingale transformation,given a well-diversified portfolio corresponding to the defaultable bond. Thus one can directly apply default intensities and fractional losses empirically estimated to the evaluation of defaultable bonds or contingent claims.     

Hedging costs for two large investors

We consider the cost of hedging contingent claims in a financial market where the trades of two large investors can move market prices. We provide a characterization of the minimal hedging costs in terms of associated stochastic control problems. We also prove that the minimal hedging cost is a viscosity solution of a corresponding dynamic programming equation in the case of a Markov market model.     

Pricing and hedging in the presence of extraneous risks

Given an underlying complete financial market, we study contingent claims whose payoffs may depend on the occurrence of nonmarket events. We first investigate the almost-sure hedging of such claims. In particular, we obtain new representations of the hedging prices and provide necessary and sufficient conditions for a claim to be marketed. The analysis of various examples then leads us to investigate alternative pricing rules. We choose to embed the pricing problem into the agent’s portfolio decision and study reservation prices. We establish the existence and consistency of this pricing rule in a semimartingale model. We characterize the nonlinear dependence of the reservation price with respect to both the agent’s initial capital and the size of her position. The fair price arises as a limiting case.     

Valuation and hedging strategy of currency options under regime-switching jump-diffusion model

The main purpose of this thesis is in analyzing and empirically simulating risk minimizing European foreign exchange option pricing and hedging strategy when the spot foreign exchange rate is governed by a Markov-modulated jump-diffusion model. The domestic and foreign money market interest rates, the drift and the volatility of the exchange rate dynamics all depend on a continuous-time hidden Markov chain which can be interpreted as the states of a macro-economy. In this paper, we will provide a practical lognormal diffusion dynamic of the spot foreign exchange rate for market practitioners. We employing the minimal martingale measure to demonstrate a system of coupled partial-differential-integral equations satisfied by the currency option price and attain the corresponding hedging schemes and the residual risk. Numerical simulations of the double exponential jump diffusion regime-switching model are used to illustrate the different effects of the various parameters on currency option prices.     

A fractional credit model with long range dependent default rate

Motivated by empirical evidence of long range dependence in macroeconomic variables like interest rates we propose a fractional Brownian motion driven model to describe the dynamics of the short and the default rate in a bond market. Aiming at results analogous to those for affine models we start with a bivariate fractional Vasicek model for short and default rate, which allows for fairly explicit calculations. We calculate the prices of corresponding defaultable zero-coupon bonds by invoking Wick calculus. Applying a Girsanov theorem we derive today’s prices of European calls and compare our results to the classical Brownian model.     

Utility-based indifference pricing in regime-switching models

In this paper, we study utility-based indifference pricing and hedging of a contingent claim in a continuous-time, Markov, regime-switching model. The market in this model is incomplete, so there is more than one price kernel. We specify the parametric form of price kernels so that both market risk and economic risk are taken into account. The pricing and hedging problem is formulated as a stochastic optimal control problem and is discussed using the dynamic programming approach. A verification theorem for the Hamilton-Jacobi-Bellman (HJB) solution to the problem is given. An issuer’s price kernel is obtained from a solution of a system of linear programming problems and an optimal hedged portfolio is determined.     

On the simulation of portfolios of interest rate and credit risk sensitive securities

We discuss extensions of reduced-form and structural models for pricing credit risky securities to portfolio simulation and valuation. Stochasticity in interest rates and credit spreads is captured via reduced-form models and is incorporated with a default and migration model based on the structural credit risk modelling approach. Calculated prices are consistent with observed prices and the term structure of default-free and defaultable interest rates. Three applications are discussed: (i) study of the inter-temporal price sensitivity of credit bonds and the sensitivity of future portfolio valuation with respect to changes in interest rates, default probabilities, recovery rates and rating migration, (ii) study of the structure of credit risk by investigating the impact of disparate risk factors on portfolio risk, and (iii) tracking of corporate bond indices via simulation and optimisation models. In particular, we study the effect of uncertainty in credit spreads and interest rates on the overall risk of a credit portfolio, a topic that has been recently discussed by Kiesel et al. [The structure of credit risk: spread volatility and ratings transitions. Technical report, Bank of England, ISSN 1268-5562, 2001], but has been otherwise mostly neglected. We find that spread risk and interest rate risk are important factors that do not diversify away in a large portfolio context, especially when high-quality instruments are considered.     

Using Utility Functions to Model Risky Bonds

This paper prices defaultable bonds by incorporating inherent risks with the use of utility functions. By allowing risk preferences into the valuation of bonds, nonlinearity is introduced in their pricing. The utility‐function approach affords the advantage of yielding exact solutions to the risky bond pricing equation when familiar stochastic models are used for interest rates. This can be achieved even when the default probability parameter is itself a stochastic variable. Valuations are found for the power‐law and log utility functions under the interest‐rate dynamics of the extended Vasicek and CIR models.     

Pricing stock and bond derivatives with a multi-factor Gaussian model

The martingale approach to pricing contingent claims can be applied in a multiple state variable model. The idea is used to derive the prices of derivative securities (futures on stock and bond futures, options on stocks, bonds and futures) given a continuous time Gaussian multi-factor model of the returns of stocks and bonds. The bond market is similar to Langetieg's multi-factor model, which has closed-form solutions. This model is a generalization of Vasicek's model, where the term structure depends on state variables following correlated mean reverting processes. The stock market is affected by systematic and unsystematic risk.     

A Simple Novel Approach to Valuing Risky Zero Coupon Bond in a Markov Regime Switching Economy

We have addressed the problem of pricing risky zero coupon bond in the framework of Longstaff and Schwartz structural type model by pricing it as a Down-and-Out European Barrier Call option on the company’s asset-debt ratio assuming Markov regime switching economy. The growth rate and the volatility of the stochastic asset debt ratio is driven by a continuous time Markov chain which signifies state of the economy. Regime Switching renders market incomplete and selection of a Equivalent martingale measure (EMM) becomes a subtle issue. We price the zero coupon risky bond utilizing the powerful technique of Risk Minimizing hedging of the underlying Barrier option under the so called “Risk Minimal” martingale measure via computing the bond default probability.     

Pricing and hedging of American contingent claims in incomplete markets

0.IntroductionandSummaryThecelebratedpapersof[2]and[3],pavedthewayforpricingoptionsonstocks,onthebasisofthefollowingprinciple:inacompletemarket(suchastheoneinSection1.5),everycontingentclaimcanbeattainedexactlybyinvestinginthemarketandstartingwithala...     

Option Pricing and Hedging under a Markov Switching Lévy Process Model

In this paper,we consider a Markov switching Lévy process model in which the underlying risky assets are driven by the stochastic exponential of Markov switching Lévy process and then apply the model to option pricing and hedging.In this model,the market interest rate,the volatility of the underlying risky assets and the N-state compensator,depend on unobservable states of the economy which are modeled by a continuous-time Hidden Markov process.We use the MEMM(minimal entropy martingale measure) as the equivalent martingale measure.The option price using this model is obtained by the Fourier transform method.We obtain a closed-form solution for the hedge ratio by applying the local risk minimizing hedging.     

Duality and martingales: a stochastic programming perspective on contingent claims

The hedging of contingent claims in the discrete time, discrete state case is analyzed from the perspective of modeling the hedging problem as a stochastic program. Application of conjugate duality leads to the arbitrage pricing theorems of financial mathematics, namely the equivalence of absence of arbitrage and the existence of a probability measure that makes the price process into a martingale. The model easily extends to the analysis of options pricing when modeling risk management concerns and the impact of spreads and margin requirements for writers of contingent claims. However, we find that arbitrage pricing in incomplete markets fails to model incentives to buy or sell options. An extension of the model to incorporate pre-existing liabilities and endowments reveals the reasons why buyers and sellers trade in options. The model also indicates the importance of financial equilibrium analysis for the understanding of options prices in incomplete markets. Received: June 5, 2000 / Accepted: July 12, 2001?Published online December 6, 2001     

Local risk minimization for vulnerable European contingent claims on nontradable assets under regime switching models

This article focuses on an optimal hedging problem of the vulnerable European contingent claims. The underlying asset of the vulnerable European contingent claims is assumed to be nontradable. The interest rate, the appreciation rate and the volatility of risky assets are modulated by a finite-state continuous-time Markov chain. By using the local risk minimization method, we obtain an explicit closed-form solution for the optimal hedging strategies of the vulnerable European contingent claims. Further, we consider a problem of hedging for a vulnerable European call option. Optimal hedging strategies are obtained. Finally, a numerical example for the optimal hedging strategies of the vulnerable European call option in a two-regime case is provided to illustrate the sensitivities of the hedging strategies.     

Pricing and hedging equity-linked life insurance contracts beyond the classical paradigm: The principle of equivalent forward preferences

By applying the principle of equivalent forward preferences, this paper revisits the pricing and hedging problems for equity-linked life insurance contracts. The equity-linked contingent claim depends on, not only the future lifetime of the policyholder, but also the performance of the reference portfolio in the financial market for the segregated account of the policyholder. For both zero volatility and non-zero volatility forward utility preferences, prices and hedging strategies of the contract are represented by solutions of random horizon backward stochastic differential equations. Numerical illustration is provided for the zero volatility case. The derived prices and hedging strategies are also compared with classical results in the literature.     

American Options under Proportional Transaction Costs: Pricing, Hedging and Stopping Algorithms for Long and Short Positions

American options are studied in a general discrete market in the presence of proportional transaction costs, modelled as bid-ask spreads. Pricing algorithms and constructions of hedging strategies, stopping times and martingale representations are presented for short (seller’s) and long (buyer’s) positions in an American option with an arbitrary payoff. This general approach extends the special cases considered in the literature concerned primarily with computing the prices of American puts under transaction costs by relaxing any restrictions on the form of the payoff, the magnitude of the transaction costs or the discrete market model itself. The largely unexplored case of pricing, hedging and stopping for the American option buyer under transaction costs is also covered. The pricing algorithms are computationally efficient, growing only polynomially with the number of time steps in a recombinant tree model. The stopping times realising the ask (seller’s) and bid (buyer’s) option prices can differ from one another. The former is generally a so-called mixed (randomised) stopping time, whereas the latter is always a pure (ordinary) stopping time.