The information content of option-implied volatility for credit default swap valuation

Credit default swaps (CDS) are similar to out-of-the-money put options in that both offer a low cost and effective protection against downside risk. This study investigates whether put option-implied volatility is an important determinant of CDS spreads. Using a large sample of firms with both CDS and options data, we find that individual firms’ put option-implied volatility dominates historical volatility in explaining the time-series variation in CDS spreads. To understand this result, we show that implied volatility is a more efficient forecast for future realized volatility than historical volatility. More importantly, the volatility risk premium embedded in option prices covaries with the CDS spread. These findings complement existing empirical evidence based on market-level data.     

Effects of market default risk on index option risk-neutral moments

We investigate the relative importance of market default risk in explaining the time variation of the S&P 500 Index option-implied risk-neutral moments. The results demonstrate that market default risk is positively (negatively) related to the index risk-neutral volatility and skewness (kurtosis). These relations are robust in the presence of other factors relevant to the dynamics and microstructure nature of the spot and option markets. Overall, this study sheds light on a set of economic determinants which help to understand the daily evolution of the S&P 500 Index option-implied risk-neutral distributions. Our findings offer explanations of why theoretical predictions of option pricing models are not consistent with what is observed in practice and provide support that market default risk is important to asset pricing.     

Valuation of collateralized debt obligations with hierarchical Archimedean copulae

Modelling portfolio credit risk is one of the crucial challenges faced by financial services industry in the last few years. We propose the valuation model of collateralized debt obligations (CDO) based on hierarchical Archimedean copulae (HAC) with up to three parameters, with default intensities calibrated to market data and with random loss given defaults that are correlated with default times. The methods presented are used to reproduce the spreads of the iTraxx Europe tranches. Our approach describes the market prices better than the standard pricing procedure based on the Gaussian distribution. We also obtain a flat correlation smile across tranches thereby solving the implied correlation puzzle.     

Copula dynamics in CDOs

Values of tranche spreads of collateralized debt obligations (CDOs) are driven by the joint default performance of the assets in the collateral pool. The dependence between the entities in the portfolio mainly depends on current economic conditions. Therefore, a correlation implied from tranches can be seen as a measure of the general situation of the credit market. We analyse the European market of standardized CDOs using tranches of the iTraxx index in the periods before and during the global financial crisis. We investigate the evolution of the correlations using different copula models: the standard Gaussian, the NIG, the double-t, and the Gumbel copula model. After calibration of these models, one obtains a time varying vector of parameters. We analyse the dynamic pattern of these coefficients. That enables us to forecast future parameters and consequently calculate Value-at-Risk measures for iTraxx Europe tranches.     

Implied volatility skews and stock return skewness and kurtosis implied by stock option prices

The Black-Scholes* option pricing model is commonly applied to value a wide range of option contracts. However, the model often inconsistently prices deep in-the-money and deep out-of-the-money options. Options professionals refer to this well-known phenomenon as a volatility ‘skew’ or ‘smile’. In this paper, we examine an extension of the Black-Scholes model developed by Corrado and Su that suggests skewness and kurtosis in the option-implied distributions of stock returns as the source of volatility skews. Adapting their methodology, we estimate option-implied coefficients of skewness and kurtosis for four actively traded stock options. We find significantly nonnormal skewness and kurtosis in the option-implied distributions of stock returns.     

Pricing distressed CDOs with stochastic recovery

In this article, a framework for the joint modelling of default and recovery risk in a portfolio of credit risky assets is presented. The model especially accounts for the correlation of defaults on the one hand and correlation of default rates and recovery rates on the other hand. Nested Archimedean copulas are used to model different dependence structures. For the recovery rates a very flexible continuous distribution with bounded support is applied, which allows for an efficient sampling of the loss process. Due to the relaxation of the constant 40% recovery assumption and the negative correlation of default rates and recovery rates, the model is especially suited for distressed market situations and the pricing of super senior tranches. A calibration to CDO tranche spreads of the European iTraxx portfolio is performed to demonstrate the fitting capability of the model. Applications to delta hedging as well as base correlations are presented.     

Regime dependent determinants of credit default swap spreads

Credit default swap (CDS) spreads display pronounced regime specific behaviour. A Markov switching model of the determinants of changes in the iTraxx Europe indices demonstrates that they are extremely sensitive to stock volatility during periods of CDS market turbulence. But in ordinary market circumstances CDS spreads are more sensitive to stock returns than they are to stock volatility. Equity hedge ratios are three or four times larger during the turbulent period, which explains why previous research on single-regime models finds stock positions to be ineffective hedges for default swaps. Interest rate movements do not affect the financial sector iTraxx indices and they only have a significant effect on the other indices when the spreads are not excessively volatile. Raising interest rates may decrease the probability of credit spreads entering a volatile period.     

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.     

Implied integrated variance and hedging

If the volatility is stochastic, stock price returns and European option prices depend on the time average of the variance, i.e. the integrated variance, not on the path of the volatility. Applying a Bayesian statistical approach, we compute a forward-looking estimate of this variance, an option-implied integrated variance. Simultaneously, we obtain estimates of the correlation coefficient between stock price and volatility shocks, and of the parameters of the volatility process. Due to the convexity of the Black–Scholes formula with respect to the volatility, pricing and hedging with Black–Scholes-type formulas and the implied volatility often lead to inaccuracies if the volatility is stochastic. Theoretically, this problem can be avoided by using Hull–White-type option pricing and hedging formulas and the integrated variance. We use the implied integrated variance and Hull–White-type formulas to hedge European options and certain volatility derivatives.     

The information content of CDS implied volatility and associated trading strategies

Using the theoretical link between put options and credit default swaps (CDS) in a very general setting, we develop a robust measure of CDS implied volatility (CIV) that captures the information content of CDS markets. Specifically, we use the unit recovery claim to bridge CDS and deep out-of-the-money put options of the same firm and then back out CIV via the binomial tree. Our CIV measure strongly co-moves with the option implied volatility (OIV), with a correlation coefficient of 0.8. Based on the standardized difference between CIV and OIV, we construct CDS and option trading strategies. Without taking transaction costs into account, the long–short CDS trading strategy achieves an annualized return of 58.29% and a Sharpe ratio of 2.97, which cannot be explained by non-parametric skewness and volatility risk.     

Pricing vulnerable European options when the option’s payoff can increase the risk of financial distress

We extend the results of Johnson and Stulz (Johnson, H., Stulz, R., 1987. Journal of Finance 42, 267–280) and Klein (Klein, P.C., 1996. Journal of Banking and Finance 20, 1211–1229) for valuing European options subject to the risk of financial distress on the part of the option writer. Our model incorporates a default boundary which depends on the potential liability of the written option as in Johnson and Stulz (1987), and also on the option writer’s other liabilities as in Klein (1996). As in both of these papers, the pay-out ratio in the event of financial distress is linked to the assets of the option writer, and the correlation between the assets of the option writer and the asset underlying the option is explicitly modeled. Although no analytical solution is available, we illustrate the importance of this approach through examples, which are evaluated numerically. We also develop an approximate analytical solution, which works well in most situations.     

Managing Employee Stock Option Expense: A Fair Value Approach

Owing to special characteristics, classic option pricing models are not well suited to the valuation of employee stock options (ESOs). This paper attempts to conduct a more general fair value estimation based on attaching performance targets to option vesting. Considering a setting that includes factors such as options that may be exercised early at employee discretion, employee exit rates and firm default risk, this paper presents a sensitivity analysis and empirical tests of option value. The results highlight the importance of considering the characteristics of ESOs in the design of performance‐vested option plans so as to provide the most attractive incentives for employees.     

Risk-adjusted option-implied moments

This paper provides a new way of converting risk-neutral moments into the corresponding physical moments, which are required for many applications. The main theoretical result is a new analytical representation of the expected payoffs of put and call options under the physical measure in terms of current option prices and a representative investor’s preferences. This representation is then used to derive analytical expressions for a variety of ex-ante physical return moments, showing explicitly how moment premiums depend on current option prices and preferences. As an empirical application of our theoretical results, we provide option-implied estimates of the representative stock market investor’s disappointment aversion using S&P 500 index option prices. We find that disappointment aversion has a procyclical pattern. It is high in times of high index levels and declines when the index falls. We confirm the view that investors with high risk aversion and disappointment aversion leave the stock market during times of turbulence and reenter it after a period of high returns.     

The valuation and behavior of black-scholes options subject to intertemporal default risk

This paper addresses the valuation and behavior of European options subject to intertemporal writer default risk. The framework allows the timing of default and recovery value to be uncertain. Default is said to occur if the writer's creditworthiness violates a specified critical level-both stochastic. Various recovery scenarios are considered including linking recovery to the moneyness of the option at the time of default. In an application of the model, it is estimated that current customer margin requirements for exchange-traded options are set far in excess of the fair market value.     

Do U.S. Insurance Firms Offer the “Wrong” Incentives to Their Executives?

We examine the relation between executive compensation and market‐implied default risk for listed insurance firms from 1992 to 2007. Shareholders are expected to encourage managerial risk sharing through equity‐based incentive compensation. We find that long‐term incentives and other share‐based plans do not affect the default risk faced by firms. However, the extensive use of stock options leads to higher future default risk for insurance firms. We argue that this is because option‐based incentives induce managerial risk‐taking behavior, which seeks to maximize managerial payoff through equity volatility. This could be detrimental to the interests of shareholders, especially during a financial crisis.     

Option strategies: Good deals and margin calls

We provide evidence that trading frictions have an economically important impact on the execution and the profitability of option strategies that involve writing out-of-the-money put options. Margin requirements, in particular, limit the notional amount of capital that can be invested in the strategies and force investors to close down positions and realize losses. The economic effect of frictions is stronger when the investor seeks to write options more aggressively. Although margins are effective in reducing counterparty default risk, they also impose a friction that limits investors from supplying liquidity to the option market.     

Pricing Black-Scholes options with correlated credit risk

This paper presents an improved method of pricing vulnerable Black-Scholes options under assumptions which are appropriate in many business situations. An analytic pricing formula is derived which allows not only for correlation between the option's underlying asset and the credit risk of the counterparty, but also for the option writer to have other liabilities. Further, the proportion of nominal claims paid out in default is endogenous to the model and is based on the terminal value of the assets of the counterparty and the amount of other equally ranking claims. Numerical examples compare the results of this model with those of other pricing formulas based on alternative assumptions, and illustrate how the model can be calibrated using market data.     

Default Risk and Equity Returns: Evidence from the Taiwan Equities Market

Studies of risk and return characteristics of different portfolios have recently gained enormous attention. Differing from past studies, this paper uses a compound option model to build the proxy of default risk and evaluate the relationship between default risk effect and equity returns. The primary goal of this paper is to evaluate the relationship among default risk, size, book-to-market, and equity returns, using data drawn from the Taiwan equities market, and to also examine whether size and book-to-market are proxies for default risk. The results show that the effects of size and book-to-market exist in different default portfolios when default risks are controlled. If size or book-to-market is controlled, there are no default effects. In the regression analysis, when default risk is included in Fama and French’s Three Factor Model, it shows that size, book-to-market and default risk have significant influence on equity returns and default risk is a systematic risk. Default risk is also more powerful in explaining returns when the compound option model is adopted for estimating default risks.     

The appropriateness of default investment options in defined contribution plans: Australian evidence

For participants in defined contribution (DC) plans who refrain from exercising investment choice, plan contributions are invested following the default investment option of their respective plans. Since default investment options of different plans vary widely in terms of their benchmark asset allocation, the most important determinant of investment performance, participants enrolled in these options face significantly different wealth outcomes at retirement. This paper simulates the terminal wealth outcomes under different static asset allocation strategies to evaluate their relative appeal as default investment choice in DC plans. We find that strategies with low or moderate allocation to stocks are consistently outperformed in terms of upside potential of exceeding the participant's wealth accumulation target at retirement as well as downside risk of falling below that target outcome by aggressive strategies whose allocation to stocks approach 100%. The risk of extremely adverse wealth outcomes for plan participants also does not appear to be very sensitive to asset allocation. Our evidence suggests the appropriateness of strategies heavily tilted towards stocks to be nominated as default investment options in DC plans unless plan providers emphasize predictability of wealth outcomes over adequacy of retirement wealth.