I. Stochastic Models and Processes

I. Stochastic Models and Processes · 17. juin 2024

Si vous considérez un processus de Wiener Wₜ, et le multipliez par son intégrale ∫ de 0 à t Wₛ ds, vous obtenez un produit de deux processus stochastiques : Wₜ ⋅ ∫ de 0 à t Wₛ ds. Le produit Wₜ ⋅ ∫ de 0 à t Wₛ ds est une fonction non linéaire du processus de Wiener. En calcul stochastique, traiter des fonctions non linéaires de processus stochastiques nécessite généralement des outils comme le lemme d'Itô, qui permet la différenciation et l'intégration de...

I. Stochastic Models and Processes · 10. juin 2024

La formule de Black-Scholes pour une option d'achat est donnée par : C = S * N(d1) - K * e^(-rt) * N(d2). Dans cette formule, C représente le prix de l'option d'achat, S est le prix actuel de l'action, et K est le prix d'exercice de l'option. Les termes N(d1) et N(d2) proviennent de la fonction de distribution cumulative (CDF) de la distribution normale standard, où la CDF indique la probabilité qu'une variable soit inférieure ou égale à une valeur particulière, résumant l'accumulation...

I. Stochastic Models and Processes · 19. novembre 2023

Multiplying a Wiener process \( W_t \) by its integral creates a complex stochastic process, combining an instantaneous, "memoryless" state with its cumulative history. This nonlinear product, needing tools like Itô's lemma for analysis, reveals interactions between the current state and past values, crucial in financial mathematics for pricing path-dependent options. #StochasticProcesses #ItôsLemma #StochasticCalculus #QuadraticCovariation #BrownianMotion

I. Stochastic Models and Processes · 14. novembre 2023

A caplet is a financial derivative, akin to a call option, used for hedging against interest rate increases. It pays out if the interest rate exceeds a predetermined rate (K) at the end of a period. The payout, calculated as α * max(LT - K, 0), depends on the period's interest rate (LT) and the day count fraction (α), reflecting the time span of the caplet. It effectively caps the borrower's interest rate costs, ensuring they don't exceed the strike rate K

I. Stochastic Models and Processes · 13. novembre 2023

Bond convexity describes the curve-like relationship between bond prices and interest rates, causing prices to rise more when rates drop than they fall when rates rise. This curvature means bond price changes are not linear and convexity corrects pricing models, especially for large rate moves. #BondConvexity

I. Stochastic Models and Processes · 12. novembre 2023

The Merton model, essential in credit risk analysis, views a company's equity as a call option on its assets, crucial for default probability assessment. Using the Black-Scholes formula, it combines equity with zero-coupon debt for valuation. Despite its innovativeness, the model's reliance on market data and idealistic market assumptions limit its applicability. This has spurred alternative approaches like reduced form models, addressing these shortcomings in credit risk evaluation.

I. Stochastic Models and Processes · 03. novembre 2023

The Cheyette Model is a complex financial tool for predicting interest rate movements, accounting for time-varying mean reversion and volatility. It's more intricate than simpler models like Vasicek due to its detailed parameters, which makes it robust but computationally intensive and less commonly used in practice.

I. Stochastic Models and Processes · 01. novembre 2023

The Bjerksund-Stensland model modifies Black-Scholes-Merton to value American options with dividends. It calculates when to exercise early, using an "early exercise boundary." If the stock's below this, exercising might be wise; if above, holding on could be better. It factors in discrete dividends, unlike the continuous assumption in Black-Scholes.

I. Stochastic Models and Processes · 01. novembre 2023

The Vasicek model predicts interest rates using mean reversion, volatility, and the speed of reversion. Its equation, `dr(t) = κ(θ - r(t)) dt + σ dW(t)`, models rates' return to a mean (θ) with volatility (σ) and randomness (dW(t)). It's vital for financial strategies and simulations.

I. Stochastic Models and Processes · 22. octobre 2023

The Cox-Ingersoll-Ross (CIR) model is essential for modeling interest rate evolution with mean reversion, variable volatility, and a square root process that precludes negative rates. Used for valuing financial instruments sensitive to rate changes, its parameters guide simulations of rate behavior. #CIRModel #InterestRates #Finance