Pricing Stock Options with Stochastic Interest Rate

Abstract

This paper constructs a closed-form generalization of the Black-Scholes model for the case where the short-term interest rate follows a stochastic Gaussian process. Capturing this additional source of uncertainty appears to have a considerable effect on option prices. We show that the value of the stock option increases with the volatility of the interest rate and with time to maturity. Our empirical tests support the theoretical model and demonstrate a significant pricing improvement relative to the Black-Scholes model.The magnitude of the improvement is a positive function of the option's time to maturity, the largest improvement being obtained for around-the-money options. JEL Classification: G12, G13

A Semi-Analytical Pricing Formula for European Options Under the Rough heston-Cir model

“…Although the rough Heston model remarkably admits a semi-analytical solution that can be written in a similar form as the Heston pricing formula, the assumption of constant interest rate is sometimes not appropriate, as a lot of empirical evidence has demonstrated that stochastic interest rate can lead to significantly improved model performance [2,28]. Therefore, we further introduce the CIR stochastic interest rate into the rough Heston model to formulate the rough Heston-CIR model, so that our model follows the dynamics of…”

“…Based on this, a rough Heston model was proposed [13,14], which admits a semi-analytical pricing formula for European options. Another popular approach in modifying stochastic volatility models is to make the interest rate another random variable, as improved model performance has been shown after incorporating stochastic interest rate into option pricing models [2,28]. Examples in this category include the Stein-Stein-Hull-White hybrid model [19] and the Heston-CIR hybrid model [18,22].…”

A closed-form pricing formula for European options under the Heston model with stochastic interest rate

“…In this section, we will mainly discuss the specific model we adopt for European option pricing. Although the Black-Scholes model is very popular among market traders, some of the unrealistic assumptions made to achieve analytical tractability are inappropriate, such as the constant volatility assumption [8] and the constant interest rate assumption [1]. As a result, a number of modifications to the Black-Scholes model have been proposed to incorporate the effect of stochastic volatility and stochastic interest rate [6,24].…”

“…and many attempts are made to improve its pricing performance in real markets, such as the introduction of the time-dependent Heston models [10] and the regime-switching Heston models [14]. One of the most popular approaches is to incorporate the stochastic interest rate into stochastic volatility models to form a hybrid model since there are a lot of empirical evidence suggesting that introducing stochastic interest rate into option pricing models can lead to better model performance [1,20], and a number of authors have worked on this area. For instance, a combination of the correlated Stein-Stein model [22] and the Hull-White interest rate model [18] is adopted in [12] with European options evaluated under the Fourier cosine expansion framework.…”

The Effects of Negative Nominal Rates on the Pricing of American Calls: Some Theoretical and Numerical Insights

“…From the financial standpoint, it is therefore necessary to check to what extent the existing pricing models can be adapted to incorporate negative nominal rates. This aspect has been already investigated in some research papers: [3] and [4] discuss the issue for options written on interest rates, both from the practical and the theoretical viewpoint; [5], focusing on foreign exchange and index options investigate whether the use of models allowing for negative interest rates can improve option pricing and implied volatility forecasting; [6], discusses a new closed form for option pricing that leads to sensitively lower the error in European options pricing. Besides, [7] adapts the Nelson-Siegel model [8] to include the negative interest.…”