Monte Carlo is a very flexible numerical method which can model and price complex instruments when other methods can not. But it has the strong disadvantage of being very time consuming. The pricing of complex exotic products can take a few seconds when we have to simulate a high number of paths with numerous time […]
Merton Jump Diffusion Model The Merton Jump Diffusion model proposed by Merton in 1976 is an extension of the Black-Scholes model (link). It contains: μ: annualised expected return on the asset price, σ: annualised volatility of the asset price, Wt: Brownian motion. Qt: compound Poisson process Yi: price ratio associated with the i-th jump happening […]
The Poisson process N(t) is a counting process used to describe the occurrence of events in a time interval of length t. It satisfies the following conditions: The waiting time between two consecutive events occurring in a Poisson process is exponentially distributed with parameter λ. The average waiting time between two consecutive events is 1 […]
The Cox-Ingersoll-Ross (CIR) model is a stochastic interest rate model used in finance to describe the evolution of interest rates. The model was introduced in 1985 as an alternative to the Vasicek model (The Vasicek Model). It assumes that the short-term interest rate follows a mean-reverting stochastic process, it does not allow negative interest rates while […]
We present here two methods for calibrating the Vasicek model (link) to historical data: The Python code is available below. Presentation Save 10% on All Quant Next Courses with the Coupon Code: QuantNextBlog10 For students and graduates: We offer a 50% discount on all courses, please contact us if you are interested: contact@quant-next.com Python Code […]
The Vasicek model is a mathematical model used in finance to describe the movement of interest rates over time. It was developed by Oldrich Vasicek in 1977. It describes the evolution of interest rates by assuming that the short-term interest rate follows a mean-reverting stochastic process. It is an example of Ornstein-Uhlenbeck process, the instantaneous interest-rate […]
The Brownian motion is one of the most famous and important stochastic process. Let’s start with a bit of history. Robert Brown discovered the Brownian motion unintentionally while observing movements of grains of pollen through a microscope in 1827. He noticed that pollen seeds suspended in water moved in an irregular zigzag and random manner. […]
Foundations of Stochastic Calculus Stochastic Calculus is a branch of mathematics that deals with random processes. Beyond probabilities it also has links with differential equations, and is widely used in finance particularly for option modelling. Kiyosi Itô (1915-2008) pioneered the field inventing the concept of stochastic integral and stochastic differential equations. Itô Stochastic Integral In […]
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