The lecture Bisection Method, Newton Raphson Method, Option Pricing by Edu Pristine is from the course ARCHIV Numerical Methods. It contains the following chapters:
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... Other Methods (Not in Learning Objectives for PRM): Regula falsi method, Finite difference method, Constrained numerical optimization, Unconstrained numerical optimization. Various methods in MS Excel can be used for numerical methods like Goal Seek, Solver, etc., which can be found under the Tools menu and use an iterative approach which is a modification of bisection ...
... Used when upper and lower bounds of solution exists, range in which solution lies. The method converges very slowly Algorithm. Step 1: Find f(X 0) and f(X 1), if [f(X 0) × f(X 1)] < 0, then solution lies in that range. Step 2: Find X 2= (X 0+ X 1) / 2 and divide the interval in two ...
... The method is quite fast as compared to bisection method. Order of convergence: Quadratic. Requires only lower bound/ starting point. Uses to find the value of x after every iteration Algorithm. Step 1: convert the function in form of f(x) = 0 Step 2: Use the formulae to find the value after every ...
... Step 5: If a matrix is negative definite, we have local minima & if a matrix is positive definite, we have local maxima. For functions involving two variables ...
... Optimization Subject to Constraints. E.g. minimum risk achievable, subject to return. Can be solved using Lagrange´s multiplier method. Let f(x) be the function to be optimize using constraint g(x), then. Step 1: define new function L(x, y) = f(x, y). Step 2: find partial derivative with respect to each variable. Step 3: solve the equation for x, y and find the functional value. ...
... Maximization/minimization without any constraints. Can be calculated using finding stationary point/ hessian matrix. Step 1: Find partial derivative with respect to each variable. Step 2: Equate each partial derivative with zero, to get the stationary points. Step 3: Differentiate again to obtain, partial 2nd order derivatives. Step 4: Construct the Hessian matrix. Step 5: If a matrix is negative definite, we have local minima & if a matrix is positive definite, we have local maxima. For functions involving two variables ...
... Step 2: find partial derivative with respect to each variable. Step 3: solve the equation for x, y and find the functional value. Example: Maximize 2x + 3x 2 + 3y subject to constraints x + y = 10 Solving these three equations we get (1/6, 59/6, 3) ...
... the valuation date and the option's expiration date. Assumption. Discrete-time jump for pricing of option based on the underlying financial instrument. No arbitrage. Efficient market hypothesis ...
... Widely used, because of simplicity and flexibility. Particularly effective for options with long time to maturity and dividend paying options. Disadvantages. Slower than Black-Scholes Model. Ineffective for Real Options, Options with complicated structures, etc. ...