# An introduction to monte carlo method of option evaluation

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Since the underlying random process is the same, for enough price paths, the value of a european option here should be the same as under Black Scholes. More generally though, simulation is employed for path dependent exotic derivativessuch as Asian options.

In other cases, the source of uncertainty may be at a remove.

- Simulation can similarly be used to value options where the payoff depends on the value of multiple underlying assets [8] such as a Basket option or Rainbow option;
- The value is defined as the least squares regression against market price of the option value at that state and time -step;
- For each value of ST , one calculates the option payoff and then takes the average payoff as the fair value;
- Indeed, Monte Carlo simulation is really the only way to value derivatives with path dependent payoffs [ 1 ], since, contrary to other numerical methods, it steps forward through time rather than backwards.

For example, for bond options [3] the underlying is a bondbut the source of uncertainty is the annualized interest rate i. Here, for each randomly generated yield curve we observe a different resultant bond price on the option's exercise date; this bond price is then the input for the determination of the option's payoff.

The same approach is used in valuing swaptions[4] where the value of the underlying swap is also a function of the evolving interest rate. Whereas these options are more commonly valued using lattice based modelsas above, for path dependent interest rate derivatives — such as CMOs — simulation is the primary technique employed.

In all such models, correlation between the underlying sources of risk is also incorporated; see Cholesky decomposition Monte Carlo simulation.

Further complications, such as the impact of commodity prices or inflation on the underlying, can also be introduced. Since simulation can accommodate complex problems of this sort, it is often used in analysing real options [1] where management's decision at any point is a function of multiple underlying variables. Simulation can similarly be used to value options where the payoff depends on the value of multiple underlying assets [8] such as a Basket option or Rainbow option.

Here, correlation between asset returns is likewise incorporated.

## Monte Carlo methods for option pricing

As required, Monte Carlo simulation can be used with any type of probability distributionincluding changing distributions: Additionally, the stochastic process of the underlying s may be specified so as to exhibit jumps or mean reversion or both; this feature makes simulation the primary valuation method applicable to energy derivatives.

For example, in models incorporating stochastic volatilitythe volatility of the underlying changes with time; see Heston model.

- The first time such a simulation was used in a derivative valuation was in 1977 [ 2 ] and, since then, the techniques have become widespread;
- Secondly, when all states are valued for every timestep, the value of the option is calculated by moving through the timesteps and states by making an optimal decision on option exercise at every step on the hand of a price path and the value of the state that would result in;
- The generator takes a seed, from which it computes the starting value of the sequence, and this seed is in turn set internally to be different upon each calculation.

The technique works in a two step procedure. First, a backward induction process is performed in which a value is recursively assigned to every state at every timestep. The value is defined as the least squares regression against market price of the option value at that state and time -step. Option value for this regression is defined as the value of exercise possibilities dependent on market price plus the value of the timestep value which that exercise would result in defined in the previous step of the process.

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Secondly, when all states are valued for every timestep, the value of the option is calculated by moving through the timesteps and states by making an optimal decision on option exercise at every step on the hand of a price path and the value of the state that would result in. This second step can be done with multiple price paths to add a stochastic effect to the procedure. Application[ edit ] As can be seen, Monte Carlo Methods are particularly useful in the valuation of options with multiple sources of uncertainty or with complicated features, which would make them difficult to value through a straightforward Black—Scholes -style or lattice based computation.

- Simulation can similarly be used to value options where the payoff depends on the value of multiple underlying assets [8] such as a Basket option or Rainbow option.
- Indeed, for many derivatives, Monte Carlo simulation is the only feasible valuation technique. For each value of ST , one calculates the option payoff and then takes the average payoff as the fair value.
- Indeed, for many derivatives, Monte Carlo simulation is the only feasible valuation technique.

The technique is thus widely used in valuing path dependent structures like lookback- and Asian options [9] and in real options analysis. They are, in a sense, a method of last resort; [9] see further under Monte Carlo methods in finance.

- However, the more paths that are used, the closer the result will be;
- For such a path dependent structure, one simply discretizes the time axis, and computes intermediate values of ST at each relevant time point.

With faster computing capability this computational constraint is less of a concern.