Valuing spread options with counterparty risk and jump risk

In this paper, we investigate spread options with counterparty risk in a jump-diffusion model. Due to the fact that there is no closed-form formula of spread options with counterparty risk, we obtain analytical expressions of lower and upper bounds by employing the measure-change technique. Finally, we numerically check the accuracy of the bounds and analyze the impacts of counterparty risk and jump risk on spread option prices.     

Pricing of vulnerable exchange options with early counterparty credit risk

The exchange option is one of the most popular options in the over-the-counter (OTC) market, which enables the holder of two underlying assets to exchange one with another. In OTC markets, with the increasing apprehension of credit default risk in the case of option pricing since the global financial crisis, it has become necessary to consider the counterparty credit risk while evaluating the option price. In this study, we combine the vulnerable exchange option and early counterparty default risk to obtain the closed-form formula for the vulnerable exchange option with early counterparty credit risk by using the method of dimension reduction, Mellin transform, and the method of images. Moreover, we examine the pricing accuracy of the option value by comparing our closed-form solution with the formula derived by the Monte-Carlo simulation.     

Exchange options for catastrophe risk management

In this paper, we investigate the pricing issue and catastrophe risk management of exchange options. Exchange options allow the holder to exchange its stocks for another at maturity and can be seen as an extended version of catastrophe equity put options with another traded asset price as strike prices. Since option holders have to issue new shares to exercise the option, we illustrate the differences between option prices calculated using pre-exercise and post-exercise share prices. The effects of default risk on option prices and risk management are also considered. Finally, risk management analysis shows that exchange options can effectively hedge catastrophe risk.     

Pricing catastrophe equity puts with counterparty risks under Markov-modulated,default-intensity processes

According to the observation of the catastrophic events with regime-switching phenomena and default rate varying with economic condition, we extend the results of Chang et al. (2011) and also take the default rate varying with economic condition into consideration by using the Markov-modulated reduced-form model. In order to value options under stochastic interest rates and a default intensity environment, we employ Girsanov’s theorem to obtain two different forward measures and to derive a pricing formula. We also conduct numerical analyses using Monte Carlo simulations to illustrate the influence of the recovery rate, the time to maturity, the frequency of catastrophic events, and the effect of counterparties’ default intensity on the catastrophe equity put price.     

Valuation of options on the maximum of two prices with default risk under GARCH models

In this paper, we work under GARCH models to value options on the maximum or the minimum of two prices. In addition, we consider not only two underlying asset prices but also geometric average ones. Further, default risk is also incorporated in a reduced-form model. In the proposed framework, closed-form pricing formulae of options on the maximum with or without default risk are derived and then used to perform numerical examples.     

Valuation of piecewise linear barrier options

This paper discusses the valuation of piecewise linear barrier options that generalize classical barrier options. We establish formulas for joint probabilities of the logarithmic returns of the underlying asset and its partial running maxima when the process has a piecewise constant drift. In particular, we show that our results embrace the famous reflection principle as a special case, and that our established proposition delivers useful scalability for computing desired probabilities related to various types of barriers. We derive the closed-form prices of piecewise linear barrier options under the Black–Scholes framework, which are obtainable with little effort by relying on the derived probabilities. In addition, we provide numerical examples and discuss how option prices respond to several types of piecewise linear barriers.     

A simulation analysis of systemic counterparty risk in over-the-counter derivatives markets

Journal of Economic Interaction and Coordination - In this paper, we propose a simulation framework to assess systemic risk in over-the-counter derivatives markets. We incorporate credit valuation...     

Valuation of stock loans with jump risk

A stock loan is a special loan with stocks as collateral, which offers the borrowers the right to redeem the stocks on or before the maturity (Xia and Zhou, 2007, Dai and Xu, 2011). We investigate pricing problems of both infinite- and finite-maturity stock loans under a hyper-exponential jump diffusion model. In the infinite-maturity case, we derive closed-form formulas for stock loan prices and deltas by solving the related optimal stopping problem explicitly. Moreover, we obtain a sufficient and necessary condition under which the optimal stopping time is finite with probability one. In the finite-maturity case, we provide analytical approximations to both stock loan prices and deltas by solving an ordinary integro-differential equation as well as a complicated non-linear system. Numerical experiments demonstrate that the approximation methods for both prices and deltas are accurate, fast, and easy to implement.     

The effect of systematic risk factors on counterparty default and credit risk of interest rate swaps

This research investigates the effect of specific systematic risk factors on credit risk pricing and capital allocation of interest rate swaps. Because of the stochastic nature of uncertain future cash flows and interest rates, practitioners typically employ the Black-Scholes option pricing model in combination with a simulation analysis to establish capital requirements and estimate the shadow price of an interest rate swap. However, this practice of pricing swap risk excludes systematic risk factors that affect the risk shadow price, thereby underestimating the capital allocation required for financial institutions. This research demonstrates the effect of risk mispricing when simulation models ignore systematic risk factors such as model risk, convexity risk, and parameter risk on the pricing of interest rate swaps.     

Pricing vulnerable options with stochastic liquidity risk

In this paper, we consider vulnerable options with stochastic liquidity risk. We employ liquidity-adjusted pricing models to describe the underlying stock price and option issuer’s assets. In addition, the correlation between these assets is stochastic, depending on the market liquidity measures. In the proposed framework, we derive closed forms of vulnerable European options with stochastic liquidity risk and then use them to illustrate the effects of stochastic liquidity risk on vulnerable option prices. Numerical results show that the effects of liquidity risk on the prices of out-of-the-money options or the options with a short maturity are not negligible.     

Insuring against the shortfall risk associated with real options

We would like to insure against the risk that a geometric Brownian motion, correlated with the price process of a certain traded asset, is in a set E at time T. In this paper it is shown that the best action one can take to insure against this risk is to buy a binary option on the traded asset. We give explicit formulas in the case that E is an infinite interval. The setting of all our investigations is the Black-Scholes model. Mathematics Subject Classification (2000): 91B28, 60J65, 62P05, 91B30, 62F03 Journal of Economic Literature Classification: G31     

A discrete-time algorithm for pricing double barrier options

Decisions in Economics and Finance -     

Reinsurance or securitization: The case of natural catastrophe risk

We investigate the suitability of securitization as an alternative to reinsurance for the purpose of transferring natural catastrophe risk. We characterize the conditions under which one or the other form of risk transfer dominates using a setting in which reinsurers and traders in financial markets produce costly information about catastrophes. Such information is useful to insurers: along with the information produced by insurers themselves, it reduces insurers’ costly capital requirements. However, traders who seek to benefit from trading in financial markets may produce ‘too much’ information, thereby making risk transfer through securitization prohibitively costly.     

Optimal corporate hedging using options with basis and production risk

We investigate the optimal hedging strategy for a firm using options, where the role of production and basis risk are considered. Contrary to the existing literature, we find that the exercise price which minimizes the shortfall of the hedged portfolio is primarily affected by the amount of cash spent on the hedging. Also, we decompose the effect of production and basis risk showing that the former affects hedging effectiveness while the latter drives the choice of the optimal contract. Fitting the model parameters to match a financial turmoil scenario confirms that suboptimal option moneyness leads to a non-negligible economic loss.     

Probability of multiple crossings and pricing of double barrier options

This paper derives pricing formulas of standard double barrier option, generalized window double barrier option and chained option. Our method is based on probabilitic approach. We derive the probability of multiple crossings of curved barriers for Brownian motion with drift, by repeatedly applying the Girsanov theorem and the reflection principle. The price of a standard double barrier option is presented as an infinite sum that converges very rapidly. Although the price formula of standard double barrier option is the same with Kunitomo and Ikeda (1992), our method gives an intuitive interpretation for each term in the infinite series. From the intuitive interpretation we present the way how to approximate the infinite sum in the pricing formula and an error bound for the given approximation. Guillaume (2003) and Jun and Ku (2013) assumed that barriers are constant to price barrier options. We extend constant barriers of window double barrier option and chained option to curved barriers. By employing multiple crossing probabilities and previous skills we derive closed formula for prices of 16 types of the generalized chained option. Based on our analytic formulas we compute Greeks of chained options directly.     

The Cox,Ross and Rubinstein tree model which includes counterparty credit risk and funding costs

The binomial asset pricing model of Cox, Ross and Rubinstein (CRR) is extensively used for the valuation of options. The CRR model is a discrete analog of the Black–Scholes–Merton (BSM) model. The 2008 credit crisis exposed the shortcomings of the oversimplified assumptions of the BSM model. Burgard and Kjaer extended the BSM model to include adjustments such as a credit value adjustment (CVA), a debit value adjustment (DVA) and a funding value adjustment (FVA). The aim of this paper is to extend the CRR model to include CVA, DVA and FVA and to prove that this extended CRR model coincides with the model that results from discretising the Burgard and Kjaer model. Our results are numerically implemented and we also show that as the number of time-steps increase in the derived tree structure model, the model converges to the model developed by Burgard and Kjaer.     

Retrieving the implicit risk neutral density of WTI options with a semi-nonparametric approach

This paper contributes to the literature on the estimation of the Risk Neutral Density (RND) function by proposing a log-semi-nonparametric (log-SNP) distribution as the implicit RND when the Gram-Charlier model is used for option pricing. The performance of the model is compared to the lognormal (Black Scholes) benchmark for a sample of option prices for West Texas Intermediate (WTI) crude oil that were traded in the period between January 2016 and December 2017. Results show that the lognormal specification tends to systematically undervalue option prices and that the proposed log-SNP distribution, which explicitly adjusts for negative skewness and excess kurtosis, results in markedly improved accuracy, especially in periods of market instability. As a result, the implied skewness and excess kurtosis are relevant sources of information on market expectations that should be used for hedging and risk management purposes.     

Abnormal returns, risk, and options in large data sets

Large data sets in finance with millions of observations have become widely available. Such data sets enable the construction of reliable semi-parametric estimates of the risk associated with extreme price movements. Our approach is based on semi-parametric statistical extreme value analysis, and compares favorably with the conventional finance normal distribution based approach. It is shown that the efficiency of the estimator of the extreme returns may benefit from high frequency data. Empirical tail shapes are calculated for the German Mark—US Dollar foreign exchange rate, and we use the semi-parametric tail estimates in combination with the empirical distribution function to evaluate the returns on exotic options.     

Valuing lookback options with barrier

In this paper, we introduce a new class of exotic options, termed lookback-barrier options, which literally combine lookback and barrier options by incorporating an activating barrier condition into the European lookback payoff. A prototype of lookback-barrier option was first proposed by Bermin (1998), where he intended to reduce the expensive cost of lookback option by considering lookback options with barrier. However, despite his novel trial, it has not attracted much attention yet. Thus, in this paper, we revisit the idea and extend the horizon of lookback-barrier option in order to enhance the marketability and applicability to equity-linked investments. Devising a variety of payoffs, this paper develops a complete valuation framework which allows for closed-form pricing formulas under the Black–Scholes model. Our closed-form pricing formulas provide a substantial advantage over the method of Monte Carlo simulation, because the extrema appearing in both of the lookback payoff and barrier condition would require a large number of simulations for exact calculation. Complexities involved in the derivation process would be resolved by the Esscher transform and the reflection principle of the Brownian motion. We illustrate our results with numerical examples.