Collection of Mathematical financial models with performance ratio
Few Key takeaways
- The Heath-Jarrow-Morton Model (HJM Model) use a differential equation that incorporates randomness to model forward interest rates.
- These rates are then modeled against an existing interest rate term structure to determine appropriate prices for interest-rate sensitive securities such as bonds or swaps.
- It is now primarily used by arbitrageurs looking for arbitrage opportunities, as well as analysts pricing derivatives.
- Formula for the HJM Model
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$df(t,T) = \alpha(t,T)dt + \sigma(t,T)dW(T)$ -
$df(t,T)$ : The stochastic differential equation shown above is assumed to satisfy the instantaneous forward interest rate of a zero-coupon bond with maturity T. -
$\alpha, \sigma$ : Adapted -
$W$ : Under the risk-neutral assumption, a Brownian motion (random-walk).
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- More details: https://en.wikipedia.org/wiki/Heath%E2%80%93Jarrow%E2%80%93Morton_framework
Few Key Takeaways
- The Crank-Nicolson method is a finite difference method used to solve the heat equation and other partial differential equations numerically. It's a second-order method.
- The Crank-Nicolson method has also been used in those areas. The differential equation of the Black-Scholes option pricing model, in particular, can be transformed into the heat equation, and thus numerical solutions for option pricing can be obtained using the Crank-Nicolson method.
- You can find a seprate github repo for Black-Scholes pricing model (https://github.com/white07S/Black-Scholes)
- By using Crank-Nicolson method we compare the American and European option pricing with spreadhttps://github.com/white07S/Black-Scholes
- More details: https://en.wikipedia.org/wiki/Crank%E2%80%93Nicolson_method