Pricing vulnerable options in a hybrid credit risk model driven by Heston–Nandi GARCH processes

Review of Derivatives Research - This paper proposes a hybrid credit risk model, in closed form, to price vulnerable options with stochastic volatility. The distinctive features of the model are...     

An economic evaluation of stock–bond return comovements with copula-based GARCH models

Owing to their importance in asset allocation strategies, the comovements between the stock and bond markets have become an increasingly popular issue in financial economics. Moreover, the copula theory can be utilized to construct a flexible joint distribution that allows for skewness in the distribution of asset returns as well as asymmetry in the dependence structure between asset returns. Therefore, this paper proposes three classes of copula-based GARCH models to describe the time-varying dependence structure of stock–bond returns, and then examines the economic value of copula-based GARCH models in the asset allocation strategy. We compare their out-of-sample performance with other models, including the passive, the constant conditional correlation (CCC) GARCH and the dynamic conditional correlation (DCC) GARCH models. From the empirical results, we find that a dynamic strategy based on the GJR-GARCH model with Student-t copula yields larger economic gains than passive and other dynamic strategies. Moreover, a less risk-averse investor will pay higher performance fees to switch from a passive strategy to a dynamic strategy based on copula-based GARCH models.     

Pricing Black–Scholes options with correlated interest rate risk and credit risk: an extension

This article provides a closed-form valuation formula for the Black–Scholes options subject to interest rate risk and credit risk. Not only does our model allow for the possible default of the option issuer prior to the option's maturity, but also considers the correlations among the option issuer's total assets, the underlying stock, and the default-free zero coupon bond. We further tailor-make a specific credit-linked option for hedging the default risk of the option issuer. The numerical results show that the default risk of the option issuer significantly reduces the option values, and the vulnerable option values may be remarkably overestimated in the case where the default can occur only at the maturity of the option.     

On the pricing of forward starting options in Heston’s model on stochastic volatility

We consider the problem of pricing European forward starting options in the presence of stochastic volatility. By performing a change of measure using the asset price at the time of strike determination as a numeraire, we derive a closed-form solution within Hestons stochastic volatility framework applying distribution properties of the volatility process. In this paper we develop a new and more suitable formula for pricing forward starting options. This formula allows to cover the smile effects observed in a Black-Scholes environment, in which the extreme exposure of forward starting options to volatility changes is ignored.Received: July 2004, Mathematics Subject Classification (2000): 91B28, 60G44, 60H30, 60E10JEL Classification: G13It is a pleasure to thank the anonymous referee for his valuable comments and suggestions on this paper. Furthermore, we would like to thank Holger Kraft, University of Kaiserslautern, and Alexander Giese, HypoVereinsbank AG Munich, for fruitful discussions and suggestions.     

Long-term factorization in Heath–Jarrow–Morton models

The long-term factorization decomposes the stochastic discount factor (SDF) into discounting at the rate of return on the long bond and a martingale that defines a long-term forward measure. We establish sufficient conditions for existence of the long-term factorization in HJM models. A condition on the forward rate volatility ensures existence of the long bond volatility. This yields existence of the long bond and convergence of \(T\)-forward measures to the long forward measure. It contrasts with the familiar risk-neutral factorization that decomposes the SDF into discounting at the short rate and a martingale defining the risk-neutral measure.     

On exact pricing of FX options in multivariate time-changed Lévy models

In this paper we discuss foreign-exchange option pricing in conditionally Gaussian models, namely in the variance-gamma and in the normal-inverse Gaussian models. It happens that in the both models closed-form pricing is attainable. The used method developes the one of the work by Madan et al. (Eur Finance Rev 2:79–105, 1998) where the price of the European call is primarily derived. The obtained formulas are based on values of the Gauss and the Appell hypergeometric functions.     

Dealing with risk–risk interdependencies and trade-offs in relation to development and use of new technologies

The paper deals with the recognized need to systematically explore trade-offs in modern among single risks in modern risk management systems. These risk trade-offs have often been downplayed as ‘side effects’ or ‘unintended consequences’ and, even more often, poorly examined, quantified, and managed. The paper proposes how to deal with the real or possible trade-offs and within a framework which covers also the emerging risks: from early notion and indications up to fully developed and recognized (mature) risks. The trade-offs have been quantified primarily by means of indicators key/safety performance indicators. The framework relies very much on the ideas proposed by IRGC, ISO 31000, Solvency directive and other recognized concepts and takes practical examples from the running EU project iNTeg-Risk.     

Differences in the strengths of evidence matters in risk–risk trade-offs

Making decisions between alternatives are challenging when there is weak or unreliable knowledge about the risks and benefits of the alternatives. This requires a trade-off between risks (and benefits). Here, we comment on a recent paper on risk–risk trade-offs and highlight the difficulties of making such trade-offs when the available evidence is of different strength. One current example of a risk–risk trade-off under weak evidence is the restriction and reevaluation of the risks of neonicotinoid insecticides to bees conducted by the European Food Safety Authority (EFSA). We argue that a risk–risk trade-off is essential in this context. Although considerable research efforts have been focused at determining the risks of neonicotinoids to bees, the evidence base is still limited. However, focus on strengthening evidence on impacts of one substance may lead policy-makers and public to believe that its substitutes are less harmful, when in fact evidence is weak on the impacts of these substitutes as well. We argue that a broader management of uncertainty is needed and that the difference in uncertainty underlying evidence of risk for different alternatives needs to be communicated to policy-makers. We suggest that this can be done, for example, using measures of uncertainty, which take into account strength in evidence, and combine these with principles to guide decision-making.     

Parisian options with jumps: a maturity–excursion randomization approach

This paper introduces an analytically tractable method for the pricing of European and American Parisian options in a flexible jump–diffusion model. Our contribution is threefold. First, using a double Laplace–Carson transform with respect to the option maturity and the Parisian (excursion) time, we obtain closed-form solutions for different types of Parisian contracts. Our approach allows us also to analytically disentangle contributions of the jump and diffusion components for Parisian options in the excursion region. Second, we provide numerical examples and quantify the impact of jumps on the option price and the Greeks. Finally, we study the non-monotonic effects of volatility and jump intensity close to the excursion barrier, which are important for shareholders’ investment policy decisions in a levered firm.     

New analytical option pricing models with Weyl–Titchmarsh theory

Abstract

In this paper we consider the valuation of Bermudan callable derivatives with multiple exercise rights. We present in this context a new primal–dual linear Monte Carlo algorithm that allows for efficient simulation of the lower and upper price bounds without using nested simulations (hence the terminology). The algorithm is essentially an extension of the primal–dual Monte Carlo algorithm for standard Bermudan options proposed by Schoenmakers et al. [SIAM J. Finance Math., 2013, 4, 86–116] to the case of multiple exercise rights. In particular, the algorithm constructs upwardly a system of dual martingales to be plugged into the dual representation of Schoenmakers. At each level, the respective martingale is constructed via a backward regression procedure starting at the last exercise date. The thus constructed martingales are finally used to compute an upper price bound. The algorithm also provides approximate continuation functions that may be used to construct a price lower bound. The algorithm is applied to the pricing of flexible caps in a Hull and White model setup. The simple model choice allows for comparison of the computed price bounds with the exact price obtained by means of a trinomial tree implementation. As a result, we obtain tight price bounds for the considered application. Moreover, the algorithm is generically designed for multi-dimensional problems and is tractable to implement.     

The maximum surplus before ruin for dependent risk models through Farlie–Gumbel–Morgenstern copula

We extend the classical compound Poisson risk model to consider the distribution of the maximum surplus before ruin where the claim sizes depend on inter-claim times via the Farlie–Gumbel–Morgenstern copula. We derive an integro-differential equation with certain boundary conditions for this distribution, of which the Laplace transform is provided. We obtain the renewal equation and explicit expressions for this distribution are derived when the claim amounts are exponentially distributed. Finally, we present numerical examples.     

Regime switches in the risk–return trade-off

This paper deals with the estimation of the risk–return trade-off. We use a MIDAS model for the conditional variance and allow for possible switches in the risk–return relation through a Markov-switching specification. We find strong evidence for regime changes in the risk–return relation. This finding is robust to a large range of specifications. In the first regime characterized by low ex-post returns and high volatility, the risk–return relation is reversed, whereas the intuitive positive risk–return trade-off holds in the second regime. The first regime is interpreted as a “flight-to-quality” regime.     

A class of Lévy process models with almost exact calibration to both barrier and vanilla FX options

Vanilla (standard European) options are actively traded on many underlying asset classes, such as equities, commodities and foreign exchange (FX). The market quotes for these options are typically used by exotic options traders to calibrate the parameters of the (risk-neutral) stochastic process for the underlying asset. Barrier options, of many different types, are also widely traded in all these markets but one important feature of the FX options markets is that barrier options, especially double-no-touch (DNT) options, are now so actively traded that they are no longer considered, in any way, exotic options. Instead, traders would, in principle, like to use them as instruments to which they can calibrate their model. The desirability of doing this has been highlighted by talks at practitioner conferences but, to our best knowledge (at least within the realm of the published literature), there have been no models which are specifically designed to cater for this. In this paper, we introduce such a model. It allows for calibration in a two-stage process. The first stage fits to DNT options (or other types of double barrier options). The second stage fits to vanilla options. The key to this is to assume that the dynamics of the spot FX rate are of one type before the first exit time from a ‘corridor’ region but are allowed to be of a different type after the first exit time. The model allows for jumps (either finite activity or infinite activity) and also for stochastic volatility. Hence, not only can it give a good fit to the market prices of options, it can also allow for realistic dynamics of the underlying FX rate and realistic future volatility smiles and skews. En route, we significantly extend existing results in the literature by providing closed-form (up to Laplace inversion) expressions for the prices of several types of barrier options as well as results related to the distribution of first passage times and of the ‘overshoot’.     

Dodd–Frank and risk in the financial services industry

We present evidence that discretionary risk taking by financial institutions has declined following the passage of Dodd–Frank. The largest institutions experience the greatest reduction in risk consistent with the legislation’s objective of reducing systemic risk and an ultimate goal of ending the too-big-to-fail doctrine. Analysis of a sample of banks, the most highly regulated financial institutions, reveals that banks exhibiting characteristics consistent with riskier business strategies prior to Dodd–Frank experience the greatest risk reduction. Further, banks that alter their business practices by increasing their capital ratios and reducing their level of non-performing loans following the law’s passage are shown to experience the greatest reduction in risk. Our results point to the efficacy of Dodd–Frank in reducing risk in the financial system.     

The risk–return relations in the Singapore stock market

This paper examines the risk–return relations in the Singapore stock market for the period April 1986 to December 1998. Though beta is significantly related to realized returns, the explanatory power is low. Adding other stock characteristics such as skewness and kurtosis provides limited incremental benefits. However, when a conditional framework based on up and down markets is introduced, the explanatory power increases more than 100 times and there is a significant positive (negative) relation between beta and returns when the market excess returns are positive (negative). The same relation applies when unsystematic risk, total risk and kurtosis are added separately to the beta–return relation during up and down markets with increased explanatory power. Our results indicate that other stock characteristics in addition to beta are also important in pricing risky assets and investors do not hold diversified portfolios. Our results are also checked and compared with another conditional model with time-varying betas conditional on a set of economic variables.     

On a renewal risk process with dependence under a Farlie–Gumbel–Morgenstern copula

In this article, we consider an extension to the renewal or Sparre Andersen risk process by introducing a dependence structure between the claim sizes and the interclaim times through a Farlie–Gumbel–Morgenstern copula proposed by Cossette et al. (2010) for the classical compound Poisson risk model. We consider that the inter-arrival times follow the Erlang(n) distribution. By studying the roots of the generalised Lundberg equation, the Laplace transform (LT) of the expected discounted penalty function is derived and a detailed analysis of the Gerber–Shiu function is given when the initial surplus is zero. It is proved that this function satisfies a defective renewal equation and its solution is given through the compound geometric tail representation of the LT of the time to ruin. Explicit expressions for the discounted joint and marginal distribution functions of the surplus prior to the time of ruin and the deficit at the time of ruin are derived. Finally, for exponential claim sizes explicit expressions and numerical examples for the ruin probability and the LT of the time to ruin are given.     

Conditional risk–return relationship in a time-varying beta model

We investigate the asymmetric risk–return relationship in a time-varying beta CAPM. A state space model is established and estimated by the Adaptive Least Squares with Kalman foundations proposed by McCulloch. Using S&P 500 daily data from 1987:11–2003:12, we find a positive risk–return relationship in the up market (positive market excess returns) and a negative relationship in the down market (negative market excess returns). This supports the argument of Pettengill et al., who use a constant beta model. However, our model outperforms theirs by eliminating the unexplained returns and improving the accuracy of the estimated risk price.     

On dividends in the phase–type dual risk model

The dual risk model assumes that the surplus of a company decreases at a constant rate over time, and grows by means of upward jumps which occur at random times with random sizes. In the present work, we study the dual risk renewal model when the waiting times are phase-type distributed. Using the roots of the fundamental and the generalized Lundberg’s equations, we get expressions for the ruin probability and the Laplace transform of the time of ruin for an arbitrary single gain distribution. Then, we address the calculation of expected discounted future dividends particularly when the individual common gains follow a phase-type distribution. We further show that the optimal dividend barrier does not depend on the initial reserve. As far as the roots of the Lundberg equations and the time of ruin are concerned, we address the existing formulae in the corresponding Sparre-Andersen insurance risk model for the first hitting time, and we generalize them to cover also the situations where we have multiple roots. We do that working a new approach and technique, approach we also use for working the dividends, unlike others, it can be also applied for every situation.     

Investigating United Kingdom's monetary policy with Macro-Factor Augmented Dynamic Nelson–Siegel models

We employ the Dynamic Nelson–Siegel (DNS) model augmented with macroeconomic factors to investigate interactions of yields, real economic activity, and monetary policy in the United Kingdom. By explicitly accounting for the structural break during the early 90s at the time the UK exited the Exchange Rate Mechanism of the European Monetary System, we document a number of interesting findings. Specifically, there is evidence of a great moderation in the volatility of the term structure post-1992. At the same time, there is a significant reduction of the loading parameter in the DNS model, which suggests a greater influence of monetary policy and economic activity on the UK bond market. We find that the level and slope yield curve factors are related to inflation expectations and monetary policy, respectively, as has been found in related literature. Interestingly, the curvature factor which has been elusive in its relationship to macroeconomic fundamentals is found to be more strongly related to economic activity post-1992.