A unified approach to pricing and risk management of equity and credit risk

We propose a unified framework for equity and credit risk modeling, where the default time is a doubly stochastic random time with intensity driven by an underlying affine factor process. This approach allows for flexible interactions between the defaultable stock price, its stochastic volatility and the default intensity, while maintaining full analytical tractability. We characterize all risk-neutral measures which preserve the affine structure of the model and show that risk management as well as pricing problems can be dealt with efficiently by shifting to suitable survival measures. As an example, we consider a jump- to-default extension of the Heston stochastic volatility model

Relationship of Ankle Plantar Flexor Strength and Step/Stride Length in the Elderly Population

Purpose: To investigate if a relationship exists between the number of attempted heel rises and the step/stride lengths in the elderly population, using the standing heel rise test as assessment of ankle plantar flexor strength. Also, to investigate what influence other factors such as age, gender, weight, and height have on this relationship. Subjects: 25 participants (13 females, 12 males) over 65 years of age, with no previously diagnosed impairments that affected their gait. Methods: Standing heel rise test was the plantar flexor strength assessment. Step/stride lengths were measured using a quantitative gait analysis method, which measured number of steps and stride lengths for a 53ft distance. Pearson\u27s bivariate correlation coefficients calculated for dependent and independent variables. Results: A strong significant correlation and a positive relationship were found between the amount of heel rises that could be achieved, and the step and stride length. As participants heel rise numbers increased, so did the step and stride lengths. Moderately significant correlations and negative relationships were found between age and heel rise numbers, as well as age and step/stride lengths. As participants\u27 ages increased, their heel rise numbers and step/stride lengths decreased. No significant correlations were found between gender, height, or weight, to heel rise numbers, nor step/stride lengths. Conclusion: Positive correlations and possible ranges between heel rise numbers and step/stride lengths could help assess risks of falling and lead to additional intervention strategies for the prevention of falls

FOREIGN EXCHANGE OPTIONS UNDER STOCHASTIC VOLATILITY AND STOCHASTIC INTEREST RATES

In this paper, we present a stochastic volatility model with stochastic interest rates in a Foreign Exchange (FX) setting. The instantaneous volatility follows a mean-reverting Ornstein–Uhlenbeck process and is correlated with the exchange rate. The domestic and foreign interest rates are modeled by mean-reverting Ornstein–Uhlenbeck processes. The main result is an analytic formula for the price of a European call on the exchange rate. It is derived using martingale methods in arbitrage pricing of contingent claims and Fourier inversion techniques.Foreign exchange options, Ornstein–Uhlenbeck process, stochastic volatility

Pricing foreign exchange options with stochastic volatility

This article is a survey and contains new results in the area of foreign exchange quantitative research. The foreign exchange OTC (over the counter) derivative market is one of the largest and most liquid markets in the world. There are only a few currencies contributing to this enormous turnover, namely the US dollar, Japanese yen, Euro and Sterling, followed by the Swiss franc, Australian dollar and Canadian dollar

Foreign exchange options under stochastic volatility and stochastic interest rates

In this paper, we present a stochastic volatility model with stochastic interest rates in a Foreign Exchange (FX) setting. The instantaneous volatility follows a mean-reverting Ornstein-Uhlenbeck process and is correlated with the exchange rate. The domestic and foreign interest rates are modeled by mean-reverting Ornstein-Uhlenbeck processes. The main result is an analytic formula for the price of a European call on the exchange rate. It is derived using martingale methods in arbitrage pricing of contingent claims and Fourier inversion techniques

Stochastic volatility and stochastic interest rates with mean-reverting Ornstein-Uhlenbeck and square root processing

In this paper, we extend the stochastic volatility model of Stein and Stein [25], where the volatility is given by a mean-reverting Ornstein-Uhlenbeck process to include stochastic interest rate given by mean-reverting square root process independent of stock returns. A closed-form pricing solution for European options is derived

Pricing FX Options in the Heston/CIR Jump-Diffusion Model with Log-Normal and Log-Uniform Jump Amplitudes

We examine foreign exchange options in the jump-diffusion version of the Heston stochastic volatility model for the exchange rate with log-normal jump amplitudes and the volatility model with log-uniformly distributed jump amplitudes. We assume that the domestic and foreign stochastic interest rates are governed by the CIR dynamics. The instantaneous volatility is correlated with the dynamics of the exchange rate return, whereas the domestic and foreign short-term rates are assumed to be independent of the dynamics of the exchange rate and its volatility. The main result furnishes a semianalytical formula for the price of the foreign exchange European call option

Forward start options under stochastic volatility and stochastic interest rates

Forward start options are examined in Heston's (Review of Financial Studies 6 (1993) 327-343) stochastic volatility model with the CIR (Econometrica 53 (1985) 385-408) stochastic interest rates. The instantaneous volatility and the instantaneous short rate are assumed to be correlated with the dynamics of stock return. The main result is an analytic formula for the price of a forward start European call option. It is derived using the probabilistic approach combined with the Fourier inversion technique, as developed in Carr and Madan (Journal of Computational Finance 2 (1999) 61-73)