Pricing Convertible Bonds with Interest Rate, Equity, Credit and FX Risk

Convertible bonds are hybrid securities whose pricing relies on a set of complex inter-dependencies due to the sensitivity to interest rate risk, underlying (equity) risk, FX risk, and credit risk, and due to the convertible bond’s early exercise American feature. We present a two factor model of interest rate and equity risk that is implemented using the Crank-Nicholson technique on the discretized pricing equation with projective successive over-relaxation. This paper extends a methodology proposed in the literature (TF[98]) to deal with credit risk in a self- consistent way, and proposes a new methodology to deal with FX sensitive cross-currency convertibles. A technique for extracting the price of vanilla options struck on a synthetic asset, the foreign equity in domestic currency, is employed to obtain the implied volatility for these options. These implied volatilities are then used to obtain the local volatility for use in the numerical routine. The model is designed to deal with most of the usual contractual features such as coupons, dividends, continuous and/or Bermudan call and put clauses. We suggest that credit spread adjustments in the boundary conditions can be made, to account for the negative correlation between spreads and equity. Detailed description of the numerical methods and the discretization schemes, together with their accuracy, are provided.cross-currency convertibles, credit spread, interest rate risk, American feature, local volatility, Crank-Nicholson.

Pricing Convertible Bonds with Interest Rate, Equity, Credit and FX Risk

Convertible bonds are hybrid securities whose pricing relies on a set of complex inter-dependencies due to the sensitivity to interest rate risk, underlying (equity) risk, FX risk, and credit risk, and due to the convertible bond’s early exercise American feature. We present a two factor model of interest rate and equity risk that is implemented using the Crank-Nicholson technique on the discretized pricing equation with projective successive over-relaxation. This paper extends a methodology proposed in the literature (TF[98]) to deal with credit risk in a self-consistent way, and proposes a new methodology to deal with FX sensitive cross-currency convertibles. A technique for extracting the price of vanilla options struck on a synthetic asset, the foreign equity in domestic currency, is employed to obtain the implied volatility for these options. These implied volatilities are then used to obtain the local volatility for use in the numerical routine. The model is designed to deal with most of the usual contractual features such as coupons, dividends, continuous and/or Bermudan call and put clauses. We suggest that credit spread adjustments in the boundary conditions can be made, to account for the negative correlation between spreads and equity. Detailed description of the numerical methods and the discretization schemes, together with their accuracy, are provided. cross-currency convertibles, credit spread, interest rate risk. American feature, local volatility, Crank-Nicholson

Valuing credit spread options under stochastic volatility/interest rates.

This thesis studies the pricing of credit spread options in a continuous time setting. Our main examples are credit spreads between US government bonds and highly risky emerging market bonds, such as Argentina, Brazil, Mexico, etc. Based on empirical findings we model the credit spread options as a geometric Brownian Motion with stochastic volatility. We implement and compare several one-factor stochastic volatility models, namely the Vasicek, Cox-Ingersoll-Ross and Ahn/Gao. As a stochastic model for the credit risk free interest rate, we use the Vasicek model. As a further new ingredient we introduce dependence between the spread rate and interest rate in our pricing model (stochastic volatility is assumed to be independent of the other factors). The mean reverting property of the short rate models enables us to view the mean reverting stochastic volatility models as moment generating function of a time integral of positive diffusion. The moment generating function of the average variance of the credit spread price process is evaluated. The Numerical Laplace inversion method is used to invert the moment generating function to obtain the density of the average variance. This average variance density is then used in the analytic pricing formulae. We compare the credit spread option prices under the closed form and the numerical formula in the cases of no correlation and some correlation between the credit spreads and the short rate under the Vasicek, Cox/Ross and Ahn/Gao(Alternative) mean reverting stochastic volatility model. We also look at the delta hedge parameters for the credit spread options under the various stochastic volatility models. Further analysis is carried out on the effects of correlation between the credit spread, the short rate and various mean reversion parameters on the pricing and hedging of the credit spread options. We finally compare our credit spread option price/hedging stochastic volatility model with the Longstaff and Schwartz model on mean reverting credit spreads under constant volatility

Arbitrage-Free Pricing of XVA for Options in Discrete Time

The goal of this project is to develop XVA pricing methods for options with discrete time settings. Particularly, this project focuses on risk valuation adjustments pertaining to funding spread and counterparty credit risk, and applies them to the binomial tree model. The final model incorporates both risk valuation adjustments, and numerical examples are provided

Integrating credit and market risk: An empirical study for the swap market

This dissertation proposes an integrated measure of credit and market risk for interest rate swap portfolios. Our research is based upon the use of EUR interest rate and credit spread data for the period 2001 - 2004. There are three self-contained but seemingly related projects in this dissertation. The objectives of this research are: 1) to price interest rate sensitive and credit spread options under the Longstaff & Schwartz 1992 framework; 2) to devise an integrated measurement approach of credit and market risk; 3) to extent the proposed integrated approach in measuring economic loss and compare it with the current standard approach. The mean reverting and GARCH characteristics of EUR credit spread indices were investigated between 2001 and 2004. We find evidence of significant GARCH effects in the EUR credit spread indices and mean reversion which is dependent on the frequency of the time series. These properties of the EUR credit spread indices suggest a stochastic term structure volatility model would be suitable to model their evolution. The model of choice was used to estimate discount curves based upon the observed interest rate term structure. The range of yield curve shapes fitted accurately was extensive suggesting the model would be suitable in fitting the more complex credit spread curves. The estimation of yield curves over a period of time suggested that the volatility of the model parameters is reduced substantially with the use of weekly data. The model was able to match the market implied volatility in pricing interest rate options with greater accuracy in the pricing of short-term options. The LS model was quite successful in the fitting of various credit spread curves Although the pricing of credit spread options using the LS model is internally inconsistent evidence suggested that it prices short term spread options with good accuracy. The direct link between credit spreads and default probabilities was fully exploited by estimating implied default probabilities. Evidence suggested that the implied default probabilities did not violate the no-arbitrage conditions of credit risk pricing. A time series examination showed only in one occasion that a lower rating had a lower probability of default than its immediate higher rating. Also the historical transition matrix of S&P proved to be quite far from the expectations of the credit markets. A dynamic approach to manage the risks associated with 10 different rated hypothetical interest rate swap portfolios was proposed based upon a hedging methodology. The proposed dynamic hedging of the swaps default risk is done by taking offsetting exposure related positions in respective credit spread index options. The efficiency of the hedging methodology shows strong linkages between the swap exposures and the credit spread index options. The integrated measure is proved to be higher at all times than the market VaR of swaps. Evidence suggest that the credit risk part of the integrated measure is not correlated with its respective market risk. The approach illustrates in a single overall market VaR measure both the market and the implicit credit risk run by a portfolio of swaps over a specified time horizon and confidence level. The proposal of the integrated measure was put further to the test by performing a comparison between the existing methodology of integrated credit risk measurement and the proposed analytic “integrated” methodology. The comparison was performed on an actual swap portfolio taken from a medium sized European Bank. The comparison yielded similar results with the integrated approach measuring higher economic loss. The link of the expected credit exposure to the credit spread index option was evident. The use of historical simulation (HS) over a multi-step Monte Carlo (MC) simulation to measure expected credit exposures over a 3 month period is proved to be less accurate but not substantially different across all cases suggesting that over small time intervals the HS method is a fast and efficient way of measuring expected credit exposures

Three Essays on Default Risk in Capital Markets

This dissertation comprises three essays on default risk in capital markets exploring (a) failure risk of hedge funds, (b) pricing in equity option markets, and (c) relationship between option and credit default swap markets, respectively, with a particular focus on the recent financial crisis. The first essay “The role of Excess Leverage in Hedge Funds Failure” investigates the role of financial leverage, including the use of margins and derivatives, in the hedge funds failure during the 2008 financial crisis. Motivated by failure of the two Bear Sterns hedge funds, this paper examines why some hedge funds failed during and after the recent financial crisis, and why some also survived. Using a 15-year panel dataset of 17,202 hedge funds from the Lipper TASS Hedge Fund database, the empirical analysis shows that during the crisis period, financial leverage is more significant in increasing the probability of failure, whereas it becomes insignificant during non-crisis periods. Moreover, hedge funds following specific styles such as “Emerging Markets”, “Equity Market”, “Long/Short Equity Hedge”, and “Multi-strategy” are more likely to fail during the financial crisis. The second essay “Is Default Risk Priced in Equity Options?” explores the impact of default risk on equity option pricing. The impact is studied in detail by empirically examining to what extent the firm-specific default risk matters in pricing individual equity options. Since credit default swaps (CDS) are similar to put options in that both offer a low cost and effective protection against downside risk, we use CDS spread as credit risk proxy to investigate the effects of default risk on put pricing. By examining an exhaustive sample of US-listed firms with both CDS and put options data available over the period from 2002 to 2010, and studying the primary determinants of option implied volatility (IV) cross-sectionally and over time, the findings show that default risk is a significant factor in the prices of equity options. The results remain significant after controlling for firm-specific and macroeconomic factors, and endogeneity. The third essay “The Impact of Using CDS in Forecasting Option-Implied Volatility” addresses the issue of forecasting IV which is of interest to option market participants, who routinely formulate volatility and option price forecasts for trading and hedging purposes. Credit risk matters for option pricing since options are valued on firms with significant trading liquidity, yet subject to default risk, similar to liquidity risk. This essay particularly explores whether better out-of-sample forecasts for IV can be developed using lagged credit risk measures. Various time-series IV forecasts show that inclusion of default risk as measured by CDS can significantly improve out-of-sample performance, measured through decreased mean squared error (MSE) as well as smaller root mean squared error (RMSE)

Two-factor capital structure models for equity and credit

We extend the now classic structural credit modeling approach of Black and Cox to a class of "two-factor" models that unify equity securities such as options written on the stock price, and credit products like bonds and credit default swaps. In our approach, the two sides of the stylized balance sheet of a firm, namely the asset value and debt value, are assumed to follow a two dimensional Markov process. Amongst models of this type we find examples that lead to derivative pricing formulas that are capable of reproducing the main features of well known equity models such as the variance gamma model, and at the same time reproducing the stylized facts about default stemming from structural models of credit risk. Moreover, in contrast to one-factor structural models, these models allow for much more flexible dependence between equity and credit markets. Two main technical obstacles to efficient implementation of these pricing formulas are overcome in our paper. The first obstacle stems from the barrier condition implied by the non-default of the firm, and is overcome by the idea of time-changing Brownian motion in a way that preserves the reflection principle for Brownian motion. The second obstacle is the difficulty of computing spread options: this is overcome by using results in recent papers that make efficient use of the two dimensional Fast Fourier Transform.Comment: 26 pages, 9 figures, 2 table