Week 1 Quantitative Analysis of Financial Markets Distributions B

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1 Week 1 Quantitative Analysis of Financial Markets Distributions B Christopher Ting Christopher Ting : : : LKCSB 5036 October 16, 2017 Christopher Ting QF 603 October 16, /14

2 Table of Contents 1 Introduction 2 Central Limit Theorem 3 Mixture Distributions 4 Takeaways Christopher Ting QF 603 October 16, /14

3 Introduction It is important to distinguish between discrete distribution versus continuous distribution. Probability distributions can be divided into two broad categories: parametric distributions, which is described by a mathematical function nonparametric distributions, which are not described by a mathematical formula A major advantage of dealing with nonparametric distributions is that assumptions required are minimum. Data speak for themselves. Christopher Ting QF 603 October 16, /14

4 Learning Outcomes of QA03 Chapter 4. Michael Miller, Mathematics and Statistics for Financial Risk Management, 2nd Edition (Hoboken, NJ: John Wiley & Sons, 2013). Apply the Central Limit Theorem in simulations. Describe the properties of independent and identically distributed (i.i.d.) random variables. Describe a mixture distribution and explain the creation and characteristics of mixture distributions. Christopher Ting QF 603 October 16, /14

5 IID E Statistically identical random variables (same probability distributions) E statistically independent random variables E Independent and identically distributed (i.i.d.) is an ideal condition. E If you add two i.i.d. normal distributions together you get a normal distribution. But this is a special case. Most other distribution do not obtain the same distribution. Christopher Ting QF 603 October 16, /14

6 Central Limit Theorem E Consider n i.i.d. random variables, X 1, X 2,..., X n, each with mean µ and standard deviation σ, and we define S n as the sum of those n variables, then, lim n S n N ( nµ, nσ 2). E When n, ( ) Sn n n µ d N(0, σ 2 ). E The central limit theorem is often utilized to justify the approximation that financial random variables follow a normal distribution. Christopher Ting QF 603 October 16, /14

7 What is Monte Carlo Simulation? E A Monte Carlo simulation consists of a number of trials. For each trial we feed random inputs into a system of equations. By collecting the outputs from the system of equations for a large number of trials, we can estimate the statistical properties of the output variables. E Example: Pricing an Asian option on an underlying stock with price S t and the strike price is X. ( ) 1 T V = max S t X, 0. T t=1 Christopher Ting QF 603 October 16, /14

8 Algorithm of Monte Carlo Simulation E Suppose the log returns are i.i.d. random variables with (annualized) mean 10% and standard deviation 20%. E The input to the Monte Carlo simulation would be normal variables with the pre-supposed mean and standard deviation. Compute the daily mean by dividing the mean by 365. Compute the daily standard deviation by dividing it by 365 E For each trial, we generate 200 random daily log returns, use the returns to calculate a series of random prices, calculate the average of the price series, and use the average to calculate the value of the option. E Repeat this process again and again, using a different realization of the random returns each time, and each time calculating a new value for the option. Christopher Ting QF 603 October 16, /14

9 IID E How do we create uncorrelated normally distributed random numbers to start with? E Answer: By adding together a large number of i.i.d. uniform distributions and then multiplying and adding the correct constants, a good approximation to any normal variable can be formed. E A classic approach is to simply add 12 standard uniform variables U i together, and subtract 6: X = 12 i=1 U i 6. Christopher Ting QF 603 October 16, /14

10 Mixture Distribution r Imagine a stock whose log returns follow a normal distribution with low volatility 90% of the time, and a normal distribution with high volatility 10% of the time. r The combined density function is written as f(x) = w L f L (x) w H f H (x), where w L = 0.90 is the probability of the return coming from the low-volatility distribution, f L (x), and w H = 0.10 is the probability of the return coming from the high-volatility distribution f H (x). r The distribution that results from a weighted average distribution of density functions is known as a mixture distribution. Christopher Ting QF 603 October 16, /14

11 Mixture Distribution (cont d) r The mixture distribution approach is semi-parametric: the component distributions are parametric, but the weights are based on empirical data, which is nonparametric. r Mixture distributions can be extremely useful in risk management. Securities whose return distributions are skewed or have excess kurtosis are often considered riskier than those with normal distributions, since extreme events can occur more frequently. r Mixture distributions provide a ready method for modeling these attributes. Christopher Ting QF 603 October 16, /14

12 Skewed Mixture Distribution Christopher Ting QF 603 October 16, /14

13 Bimodal Mixture Distribution Christopher Ting QF 603 October 16, /14

14 Summary All the parametric probability distributions, whether discrete or continuous, are useful models for risk management and quantitative analysis of investment or trading. Central Limit Theorem: The arithmetic mean of a sufficiently large number of iterates of independent random variables, will be approximately normally distributed, regardless of their underlying distribution. An application of central limit theorem is in Monte Carlo simulation. Mixing two mixtures is to take p% of the time from one distribution and (1 p)% of the time from another distribution. Christopher Ting QF 603 October 16, /14

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