Pricing and Hedging Options on Defaultable Assets (original) (raw)
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The goal of this chapter is to present a survey of recent developments in the practically important and challenging area of hedging credit risk. In a companion work, Bielecki et al. (2004a), we presented techniques and results related to the valuation of defaultable claims. It should be emphasized that in most existing papers on credit risk, the risk-neutral valuation of defaultable claims is not supported by any other argument than the desire to produce an arbitrage-free model of default-free and defaultable assets. Here, we focus on the possibility of a perfect replication of defaultable claims and, if the latter is not feasible, on various approaches to hedging in an incomplete setting.
Valuation of default-sensitive claims under imperfect information
Finance and Stochastics, 2008
We propose a valuation method for financial assets subject to default risk, where investors cannot observe the state variable triggering the default but observe a correlated price process. The model is sufficiently general to encompass a large class of structural models and can be seen as a generalization of the model of Duffie and Lando (Econometrica 69: 2001). In this setting we prove that the default time is totally inaccessible in the market's filtration and derive the conditional default probabilities and the intensity process. Finally, we provide pricing formulas for default-sensitive claims and illustrate in particular examples the shapes of the credit spreads.
The impact of default risk on the prices of options and other derivative securities
Journal of Banking & Finance, 1995
This paper presents a model for valuing derivative securities when there is default risk. The holder of a security is assumed to recover a proportion of its no-default value in the event of a default by the counterparty. Both the probability of default and the size of the proportional recovery are random. The paper shows how data on bonds issued by the counterparty can be used to provide information about model parameters.
Modelling of default risk: an overview
Mathematical finance: theory and …, 2000
The aim of these notes is to provide a relatively concise - but still self-contained - overview of mathematical notions and results which underpin the valuation of defaultable claims. Though the default risk modelling was extensively studied in numerous recent papers, it seems ...
SSRN Electronic Journal, 2003
We develop a model for pricing derivative and hybrid securities whose value may depend on different sources of risk, namely, equity, interest-rate, and default risks. In addition to valuing such securities the framework is also useful for extracting probabilities of default (PD) functions from market data. Our model is not based on the stochastic process for the value of the firm [which is unobservable], but on the stochastic process for interest rates and the equity price, which are observable. The model comprises a risk-neutral setting in which the joint process of interest rates and equity are modeled together with the default conditions for security payoffs. The model is embedded on a recombining lattice which makes implementation of the pricing scheme feasible with polynomial complexity. We present a simple approach to calibration of the model to market observable data. The framework is shown to nest many familiar models as special cases. The model is extensible to handling correlated default risk and may be used to value distressed convertible bonds, debt-equity swaps, and credit portfolio products such as CDOs. We present several numerical and calibration examples to demonstrate the applicability and implementation of our approach.
Dynamic, nonparametric hedging of European style contingent claims using canonical valuation
Finance Research Letters, 2005
The canonical valuation, proposed by . Journal of Finance 51, 1633-1652, is a nonparametric option pricing approach for valuing European-style contingent claims. This paper derives risk-neutral dynamic hedge formulae for European call and put options under canonical valuation that obey put-call parity. Further, the paper documents the error-metrics of the canonical hedge ratio and analyzes the effectiveness of discrete dynamic hedging in a stochastic volatility environment. The results suggest that the nonparametric hedge formula generates hedges that are substantially unbiased and is capable of producing hedging outcomes that are superior to those produced by Black and Scholes [1973. Journal of Political Economy 81, 637-654] delta hedging. 2004 Elsevier Inc. All rights reserved.
Valuation and Hedging of Credit Derivatives
2007
In this chapter, we present the so-called structural approach to modeling credit risk, which is also known as the value-of-the-firm approach. This methodology refers directly to economic fundamentals, such as the capital structure of a company, in order to model credit events (a default event, in particular). As we shall see in what follows, the two major driving concepts in the structural modeling are: the total value of the firm's assets and the default triggering barrier. It is worth noting that this was historically the first approach used in this area-it goes back to the fundamental papers by Black and Scholes [17] and Merton [76]. 1.1 Basic Assumptions We fix a finite horizon date T * > 0, and we suppose that the underlying probability space (Ω, F, P), endowed with some (reference) filtration F = (F t) 0≤t≤T * , is sufficiently rich to support the following objects: • The short-term interest rate process r, and thus also a default-free term structure model. • The firm's value process V, which is interpreted as a model for the total value of the firm's assets. • The barrier process v, which will be used in the specification of the default time τ. • The promised contingent claim X representing the firm's liabilities to be redeemed at maturity date T ≤ T *. • The process A, which models the promised dividends, i.e., the liabilities stream that is redeemed continuously or discretely over time to the holder of a defaultable claim.
On the hedging strategies for defaultable claims under incomplete information
arXiv (Cornell University), 2016
In this paper we investigate the hedging problem of a defaultable claim with recovery at default time via the local risk-minimization approach when investors have a restricted information on the market. We assume that the stock price process dynamics depends on an exogenous unobservable stochastic factor and that at any time, investors may observe the risky asset price and know if default has occurred or not. We characterize the optimal strategy in terms of the integrand in the Galtchouk-Kunita-Watanabe decomposition of the defaultable claim with respect to the minimal martingale measure and the available information flow. Finally, we provide an explicit formula by means of predictable projection of the corresponding hedging strategy under full information with respect to the natural filtration of the risky asset price and the minimal martingale measure in a Markovian setting via filtering.