This is the assembly point: random numbers (random number generation) drive paths (simulating SDEs), and here the paths become prices. The note covers the basic algorithm with its statistical guarantee (LLN + CLT ⟹ a confidence interval, not just a number), the careful accounting of the two error sources — statistical error and bias, combined in the mean square error — and the variance-reduction techniques (antithetic and control variates) that buy accuracy far more cheaply than raw path count. The arithmetic of cost is brutal and worth internalising: one extra decimal of accuracy by brute force costs a factor of 100 in computation.
Learning objectives. By the end of this note you should be able to:
- justify Monte Carlo integration from the LLN and attach CLT confidence intervals;
- write the European MC pricing algorithm for both one-draw and path-simulation cases;
- distinguish statistical error from bias, give examples of each, and state the MSE decomposition;
- derive the variance formulas for antithetic variates and the optimal control-variate coefficient;
- choose sensible control variates for an Asian option.
In this lesson
- 4.1 Monte Carlo integration
- 4.2 Basic Monte Carlo for European options
- 4.3 Accuracy: statistical error, bias, MSE
- 4.4 Variance reduction
- 4.5 MC Greeks
The full lesson, with worked examples and exercise solutions, is for members.
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