### Monte Carlo Methods Basics - Used to value and analyse complex financial instruments - Simulates the various sources of uncertainty affecting their value and then determining their average value over a large range of resultant outcomes or paths - We build a computer simulations of the moving parts and see with enough runs of the model what the fair value of the financial instrument is - Monte Carlo Options Pricing Methods was Developed by Phelim Boyle in 1977 - Particularly useful in the valuation of options with multiple sources of uncertainty or with complicated features that would make them difficult or impossible to value through a Black Scholes style partial differential equation or through lattice based approaches like the binomial tree - Widely used in valuing path dependent structures like look back and Asian options and in real options analysis - Simulating the underling process followed by the various risk factors affecting the price of the derivative we are trying to price. ### Simple Method 1. Generate a price path for the underlying based on the random movements of the various risk factors 2. Calculate the payoffs from the derivative based based on that pat Repeat above steps to generate numerous sample values of the payoff from the derivatives in the future. 1. Calculate the average of the sample payoffs giving you the estimate fo the derivatives expected payoff in a risk neutral world 2. Discount the payoff at the risk free rate --> fair value of the options today. Number of iterations depends on the operator and depends on the required accuracy. Usual to calculate the standard deviation of the discounted payoffs generated by the simulation. The uncertainty of the value of the derivative is inversely proportional to the square root of the no. of iterations we run. ##### Monte Carlo method has great flexibility: 1. Complex stochastic processes including jumps, mean reversion or both can be accommodated. 2. Different distributions including changing distributions can be assumed. Used when there are 3 or more stochastic variables that make using a PDE or lattice based approach, extremely difficult or impossible Time taken to run a Monte Carlo increases linearly with no. of variables.. A brute force approach for pricing options. ### Assumptions Accounted: 1. Distribution of the driving asset 2. Structure for its volatility 3. Absence or existence of jumps --- Black Scholes and Lattice based approaches give you a fair value for the option and specify a trading strategy like delta hedging Options that require the monte Carlo method are **very difficult to hedge accurately or impossible to hedge** The option is usually sold when the buyer will pay well **above fair value** as the seller usually has to keep the option on their books for the lifespan and can only roughly hedge it. Monte Carlo can be used to price options only when: - payoff depends on the **final price** of the underlying at the expiration date - payoff depends on the **price path followed** by the underlying - payoff depends on the **value of multiple underlying assets** such as basket options or rainbow options Allows for compounding of uncertainties when a joint probability distribution is used.. - 2 random variables - bivariate distribution - generalizes to any number of random variables