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.     

HOW CHANGES IN BOND CALL FEATURES AFFECT COUPON RATES

In April 1998, Level 3, a telecommunications company, sold $2 billion of 9.19%, ten-year bonds to finance the building of a fiber-optic network. Like most below-investment-grade issues, as well as many investment-grade issues, the Level 3 issue contained an embedded call option that gave the company the right to repurchase the bonds after five years at par value plus the (semi-annual) coupon rate, with the call price declining to par two years before maturity.
Because issuers must pay for the call provision in the form of a higher coupon rate, the choice of whether or not to include a call option can be a difficult one. And, once management decides to include a call option, it must then decide how to structure the call. The most important call structure decisions are how long to make the call protection period and how to set the call price—both of which can have a significant impact upon the coupon yields required to attract investors. Using a well-known option pricing model, the authors of this article summarize their recent research on how variations in bond call features can be expected to affect par coupon yields of new issues under different market circumstances—circumstances that include market conditions relevant to option valuation such as the shape of the term structure and the volatility of interest rates.     

Leverage, options liabilities, and corporate bond pricing

The two major problems with typical structural models are the failure to attain a positive credit spread in the very short term, and overestimation of the overall level of the credit spread. We recognize the presence of option liabilities in a firm’s capital structure and the effect they have on the firm’s credit spread. Including option liabilities and employing a regime switching interest rate process to capture the business cycle resolves the above-mentioned drawbacks in explaining credit spreads. We find that the credit spread overestimation problem in one of the structural models, Collin-Dufresne and Goldstein (J Finan 56:1929–1957, 2001), can be resolved by combining option liabilities and the regime-switching interest rate process when dealing with an investment grade bond, whereas with junk bonds, only the regime-switching interest rate process is needed. We also examine vulnerable option values, debt values, and zero-coupon bond values with different model settings and leverage ratios.      

Two Closed-Form Formulas for the Futures Price in the Presence of a Quality Option

The paper derives closed-form formulas for the futures price in the presence of a multi-asset quality option. This is done for two cases: In the first one the underlying assets are zero coupon bonds with different maturities in the single-factor Vasicek model. In the second one these are commodities in a multi-factor setting, again with Vasicek interest rate uncertainty.     

On the range of options prices

In this paper we consider the valuation of an option with time to expiration and pay-off function which is a convex function (as is a European call option), and constant interest rate , in the case where the underlying model for stock prices is a purely discontinuous process (hence typically the model is incomplete). The main result is that, for “most” such models, the range of the values of the option, using all possible equivalent martingale measures for the valuation, is the interval , this interval being the biggest interval in which the values must lie, whatever model is used.     

Improved lower and upper bound algorithms for pricing American options by simulation

This paper introduces new variance reduction techniques and computational improvements to Monte Carlo methods for pricing American-style options. For simulation algorithms that compute lower bounds of American option values, we apply martingale control variates and introduce the local policy enhancement, which adopts a local simulation to improve the exercise policy. For duality-based upper bound methods, specifically the primal–dual simulation algorithm, we have developed two improvements. One is sub-optimality checking, which saves unnecessary computation when it is sub-optimal to exercise the option along the sample path; the second is boundary distance grouping, which reduces computational time by skipping computation on selected sample paths based on the distance to the exercise boundary. Numerical results are given for single asset Bermudan options, moving window Asian options and Bermudan max options. In some examples the computational time is reduced by a factor of several hundred, while the confidence interval of the true option value is considerably tighter than before the improvements.     

Bias in an Empirical Approach to Determining Bond and Mortgage Risk Premiums

Empirical studies of bond and commercial mortgage performance often quantify a required risk premium by examining the difference between the promised yield and the realized yield as adjusted for default occurrence. These studies omit the effects of various other sources of risk, however, including collateral asset market risk, interest rate risk, and possibly call risk. These omissions downwardly bias the empirical risk premium estimate on the debt. In this paper, we disentangle and quantify the sources of this bias by modeling secured coupon debt (the commercial mortgage) as used in the calculation of a realized investment return. We consider deterministic and stochastic interest rate economies with mortgage contracts that are either noncallable or subject to a temporary prepayment lockout period. Given realistic parameter values associated with the term structure, underlying asset dynamics, and debt contracting, we show that the magnitude of the bias can be significant.     

Callable Bonds: Better Value Than Advertised?

Callable bonds allow issuers to manage interest rate risk in the sense that if rates decline, the bonds can be redeemed and replaced with lower‐cost debt. Investors demand a coupon premium for giving issuers this option; and when deciding whether to issue callable or noncall‐able bonds, the issuing companies must determine whether it's worth paying the coupon premium. This article addresses two main questions about the structuring and refunding of callable bonds. The first concerns the value of the call option: At the time of issuance, does it make sense to accept the coupon premium for the option being acquired? The second concerns the optimal timing of a refunding: At refunding, do the cash flow savings provide adequate compensation for the option that is being exercised and hence given up? In perfect markets with no taxes or transactions costs, the average corporate issuer should be indifferent between issuing callable bonds or their noncallable equivalent. But corporate taxes, together with risk management considerations, can lead some issuers to prefer callable bonds, possibly with coupons that otherwise would be unacceptably high. Refunding decisions should be made using the concept of “call efficiency,” which compares the savings (net of transactions costs) from calling to the loss of option value. The latter should also account for any option that is built into the replacement issue. Transaction costs that occur when refunding diminish the value of the call option, and their effect should be factored in at the time of issuance. One way of avoiding such costs is to issue “ratchet” bonds—essentially one‐way floaters that automatically reset lower when rates decline, thus delivering the benefits of callable bonds while eliminating transaction costs.     

A Two-Factor Model for Low Interest Rate Regimes

This paper derives a two-factor model for the term structure of interest rates that segments the yield curve in a natural way. The first factor involves modelling a non-negative short rate process that primarily determines the early part of the yield curve and is obtained as a truncated Gaussian short rate. The second factor mainly influences the later part of the yield curve via the market index. The market index proxies the growth optimal portfolio (GOP) and is modelled as a squared Bessel process of dimension four. Although this setup can be applied to any interest rate environment, this study focuses on the difficult but important case where the short rate stays close to zero for a prolonged period of time. For the proposed model, an equivalent risk neutral martingale measure is neither possible nor required. Hence we use the benchmark approach where the GOP is chosen as numeraire. Fair derivative prices are then calculated via conditional expectations under the real world probability measure. Using this methodology we derive pricing functions for zero coupon bonds and options on zero coupon bonds. The proposed model naturally generates yield curve shapes commonly observed in the market. More importantly, the model replicates the key features of the interest rate cap market for economies with low interest rate regimes. In particular, the implied volatility term structure displays a consistent downward slope from extremely high levels of volatility together with a distinct negative skew. 1991 Mathematics Subject Classification: primary 90A12; secondary 60G30; 62P20 JEL Classification: G10, G13     

Machine learning for pricing American options in high-dimensional Markovian and non-Markovian models

In this paper we propose two efficient techniques which allow one to compute the price of American basket options. In particular, we consider a basket of assets that follow a multi-dimensional Black–Scholes dynamics. The proposed techniques, called GPR Tree (GRP-Tree) and GPR Exact Integration (GPR-EI), are both based on Machine Learning, exploited together with binomial trees or with a closed form formula for integration. Moreover, these two methods solve the backward dynamic programing problem considering a Bermudan approximation of the American option. On the exercise dates, the value of the option is first computed as the maximum between the exercise value and the continuation value and then approximated by means of Gaussian Process Regression. The two methods mainly differ in the approach used to compute the continuation value: a single step of the binomial tree or integration according to the probability density of the process. Numerical results show that these two methods are accurate and reliable in handling American options on very large baskets of assets. Moreover we also consider the rough Bergomi model, which provides stochastic volatility with memory. Despite that this model is only bidimensional, the whole history of the process impacts on the price, and how to handle all this information is not obvious at all. To this aim, we present how to adapt the GPR-Tree and GPR-EI methods and we focus on pricing American options in this non-Markovian framework.     

Optimal Long-Term Financial Contracting

We develop an agency model of financial contracting. We derivelong-term debt, a line of credit, and equity as optimal securities,capturing the debt coupon and maturity; the interest rate andlimits on the credit line; inside versus outside equity; dividendpolicy; and capital structure dynamics. The optimal debt-equityratio is history dependent, but debt and credit line terms areindependent of the amount financed and, in some cases, the severityof the agency problem. In our model, the agent can divert cashflows; we also consider settings in which the agent undertakeshidden effort, or can control cash flow risk.     

Paying for Minimum Interest Rate Guarantees: Who Should Compensate Who?

Defined contribution pension schemes often have a mandatory minimum interest rate guarantee as an integrated part of the contract. The guarantee is an embedded put option issued by the institution to the individual who is forced to invest in the option. As argued in this paper, the individual may in this way face a constraint on the feasible set of portfolio choices. We quantify the effect of the minimum interest rate guarantee constraint and demonstrate that guarantees may induce a significant utility loss. We also consider the effects of the interest rate guarantee in the case of heterogenous investors sharing a common portfolio on a pro rata basis.     

Generic market models

Currently, there are two market models for valuation and risk management of interest rate derivatives: the LIBOR and swap market models. We introduce arbitrage-free constant maturity swap (CMS) market models and generic market models featuring forward rates that span periods other than the classical LIBOR and swap periods. We develop generic expressions for the drift terms occurring in the stochastic differential equation driving the forward rates under a single pricing measure. The generic market model is particularly apt for pricing of, e.g., Bermudan CMS swaptions and fixed-maturity Bermudan swaptions.     

Stochastic durations,the convexity effect,and the impact of interest rate changes

This article shows that the equilibrium models of bond pricing do not preclude arbitrage opportunities caused by convexity. Consequently, stochastic durations derived from these models are limited in their ability to act as interest rate risk measures. The research of the present article makes use of an intertemporal utility maximization framework to determine the conditions under which duration is an adequate interest rate risk measure. Additionally, we show that zero coupon bonds satisfy those equilibrium conditions, whereas coupon bonds or bond portfolios do not as a result of the convexity effect. The results are supported by empirical evidence, which confirms the influence of convexity on the deviation of coupon bond returns from equilibrium.     

Exact solutions for bond and option prices with systematic jump risk

A variety of realistic economic considerations make jump-diffusion models of interest rate dynamics an appealing modeling choice to price interest-rate contingent claims. However, exact closed-form solutions for bond prices when interest rates follow a mixed jump-diffusion process have proved very hard to derive. This paper puts forward two new models of interest-rate dynamics that combine infrequent, discrete changes in the interest-rate level, modeled as a jump process, with short-lived, mean reverting shocks, modeled as a diffusion process. The two models differ in the way jumps affect the central tendency of interest rates; in one case shocks are temporary, in the other shocks are permanent. We derive exact closed-form solutions for the price of a discount bond and computationally tractable schemes to price bond options.     

基于分数维随机利率的分数跳-扩散外汇期权定价

假设利率为分数维随机利率,外汇汇率服从分数跳一扩散过程,并且波动率为常数,期望收益率为时间的非随机函数,本文利用保险精算方法,得出了看涨、看跌外汇欧武期权的一般定价公式,并建立了平价公式。     

Effects of taxation for option writers: an Australian perspective

Writing an option is a taxable event for Australian investors. This method of taxation penalizes investors who hold open short option positions over the tax year end by accelerating their tax liability relative to the timing of the economic gain from writing options. This paper examines the levels of open interest in the Australian Stock Exchange over the change in financial year to determine whether investors time their transactions to avoid this tax acceleration. The results show that level of open interest is lower in the last month of the financial year after controlling for non‐tax determinants of option demand.     

Pricing interest rate options in a two-factor Cox-Ingersoll-Ross model of the term structure

Solutions are presented for prices on interest rate optionsin a two-factor version of the Cox-Ingersoll-Ross model of theterm structure. Specific solutions are developed for caps onfloating interest rates and for European options on discountbonds, coupon bonds, coupon bond futures, and Euro-dollar futures.The solutions for the options are expressed as multivariateintegrals, and we show how to reduce the calculations to univariatenumerical integrations, which can be calculated very quickly.The two-factor model provides more flexibility in fitting observedterm structures, and the fixed parameters of the model can beset to capture tie variability of the term structure over time.     

A contingent claims analysis of the interest rate risk characteristics of corporate liabilities

This paper provides a contingent claims analysis of the interest rate risk characteristics of corporate liabilities by identifying Merton's (1973) option pricing model with Vasicek's (1977) mean reverting term structure model. Only a non-zero positive range of duration values for the firms' assets is shown to be consistent with the previous empirical evidence on the interest rate sensitivity of corporate stocks and bonds. Chance's (1990) duration measure is shown to be biased downward under empirically realistic conditions. Theoretical conditions are derived under which the duration of a default-prone zero coupon bond can be either higher or lower than the duration of the corresponding default-free bond. The duration of the default-prone bond of a firm with high (low) interest rate sensitive assets is shown to be an increasing (decreasing) function of the bond's default-risk.     

Presentation of the English translation of Ettore Majorana's paper: The value of statistical laws in physics and social sciences

We consider a consistent pricing model of government bonds, interest-rate swaps and basis swaps in one currency within the no-arbitrage framework. To this end, we propose a three yield-curve model, one for discounting cash flows, one for calculating LIBOR deposit rates and one for calculating coupon rates of government bonds. The derivation of the yield curves from observed data is presented, and the option prices on a swap or a government bond are studied. A one-factor quadratic Gaussian model is proposed as a specific model, and is shown to provide a very good fit to the current Japanese low-interest-rate environment.