Simulation Methods

Why asset prices are modelled as lognormal when continuously compounded returns are normal, how returns and volatility scale with time, and how Monte Carlo simulation and bootstrap resampling generate thousands of scenarios to measure risk and value complex securities.

1. 1. The lognormal distributionA variable is lognormal when its natural log is normal; that makes it never negative and skewed to the right, which is exactly how asset prices behave.Free13 min
2. 2. From continuous returns to lognormal pricesContinuously compounded returns add up across periods, so the multi-period return is (at least approximately) normal and the price it produces is lognormal.Video · 7 min14 min
3. 3. Estimating and annualising volatilityVolatility is the annualised standard deviation of continuously compounded returns, usually estimated from daily prices and scaled up by $\sqrt{250}$.12 min
4. 4. Monte Carlo simulation: what it is and what it is forMonte Carlo simulation draws a very large number of random scenarios from distributions you specify, then reads the answer off the resulting distribution of outcomes.12 min
5. 5. Running a Monte Carlo simulation, step by stepEvery Monte Carlo valuation follows six steps: three to specify the simulation and three to run it, ending with the average of the discounted payoffs.14 min
6. 6. Bootstrap resamplingBootstrapping treats the observed sample as if it were the population and resamples it, with replacement, to build distributions and to run simulations from real data.13 min