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Introduction of Monte Carlo Simulation 1. 單變數模擬 2. 多變數模擬 3. Contract pricing with the Monte Carlo simulation 4. 信賴區間 5. 變異數削減技巧 6. 範例 7. Homework 4 1 1. 單變數模擬 模擬抽樣方法: 均勻分配抽樣 + 欲模擬分配之累積機率函數 的反函數 均勻分配 (uniform distribution) ※ RAND() provided by Excel or RND() provided by VBA can draw random samples from the uniform distribution 2 欲模擬分配之累積機率函數 𝑁(𝑥) ※ 𝑁 −1 (𝑦)為欲模擬分配之累積機率函數的反函數,即輸入𝑦為一個介於0與1之間 的數,則可以得到一個𝑥之值 ※ 當𝑦符合均勻分配,依上述方法所產生的𝑥會服從欲模擬之分配 ※ NORMSINV() in Excel is the inverse function of the standard normal distribution ※ To generate random samples from the stand normal distribution, one can use NORMSINV(RAND()) in Excel and Application.WorksheetFunction.NormSInv (Rnd()) in VBA ※ Please refer to “Count.xls” 3 對於沒提供欲模擬分配之累積機率函數的反函數之程式語言, 例如C或C++,一般使用Box-Muller Method來產生標準常態 分配的抽樣 Box-Muller Method: Step 1: Draw random samples 𝑥1 and 𝑥2 from the uniform distribution Step 2: 𝑦1 = −2 ln 𝑥1 cos(2𝜋𝑥2 ) and 𝑦2 = −2 ln 𝑥1 sin(2𝜋𝑥2 ) 𝑦1 and 𝑦2 are independent random samples drawn from the standard normal distribution 4 2. 多變數模擬 對於bivariate normal distribution 𝑟1 𝑟2 ~ 𝜎12 0 , 0 𝜌𝜎1 𝜎2 𝜌𝜎1 𝜎2 𝜎22 如何抽樣𝑟1 與𝑟2 以滿足上述分配? Step 1: Draw random samples 𝑧1 and 𝑧2 from the bivariate standard normal distribution, i.e., 𝑧1 𝑧2 ~ 0 1 0 , 0 0 1 (Since 𝑧1 and 𝑧2 are independently normal distribution, in practice, we draw 𝑧1 and 𝑧2 from the standard normal distribution separately) Step 2: 𝑟1 = 𝜎1 𝑧1 𝑟2 = 𝜎2 𝑧1 𝜌 + 𝑧2 1 − 𝜌2 5 對於multivariate normal distribution,使用Cholesky decomposition method來完成抽樣 Step 1: Decompose the covariance matrix 𝐶 = 𝐴𝑇 𝐴, where 𝐴 is an upper triangular matrix Step 2: 𝑟1 𝑟2 ⋯ 𝑟𝑛 = 𝑧1 𝑧2 ⋯ 𝑧𝑛 𝐴, where 𝑧1 𝑧2 ⋯ 𝑧𝑛 are random samples from multivariate standard normal distribution Take the bivariate normal distribution for example: 𝛼 𝐴= 0 𝜎12 𝛽 ⇒𝐶= 𝜙 𝜌𝜎1 𝜎2 𝜌𝜎1 𝜎2 𝛼2 = 2 𝛼𝛽 𝜎2 𝛼𝛽 𝛽2 + 𝜙 2 One can solve 𝛼 = 𝜎1 , 𝛽 = 𝜎2 𝜌, and 𝜙 = 𝜎2 1 − 𝜌2 and obtain random samples through 𝑟1 𝑟2 = 𝑧1 𝑧2 𝐴 = 𝑧1 𝑧2 𝜎1 𝜎2 𝜌 0 𝜎2 1 − 𝜌2 ※ The results of the Cholesky decomposition method are exactly the same as the method in the previous slide 6 The reason behind the Cholesky decomposition method var 𝑟1 𝑟 = 𝐸 1 𝑟1 𝑟2 𝑟2 𝑟2 = 𝐴𝑇 𝐸 = 𝐸 𝐴𝑇 𝑧1 𝑧 𝑧 𝐴 𝑧2 1 2 𝑧1 𝑧1 𝑧2 𝐴 = 𝐴𝑇 𝐼𝐴 = 𝐴𝑇 𝐴 =C 𝑧2 ※ In fact, if we can decompose the covariance matrix in any way to be 𝐶 = 𝑀𝑇 𝑀, then we can generate the random samples from multivariate normal distribution through 𝑟1 𝑟2 ⋯ 𝑟𝑛 = 𝑧1 𝑧2 ⋯ 𝑧𝑛 𝑀 7 3. Contract pricing with the Monte Carlo Simulation (MCS) How to price a bonus contract with the following payoff $10,000,000 × max 𝑟 − 𝐾, 0 , where 𝑟 is the growth rate of your firm and follows a normal distribution with the mean to be 5% and the standard deviation to be 20%, and 𝐾 = 4% represents the threshold to earn the bonus? Step 1: Draw 𝑁 random samples for 𝑟 following the designated normal distribution, i.e., normally distributed 𝑥1 , 𝑥2 , …, 𝑥𝑁 with the mean to be 5% and the standard deviation to be 20% Step 2: For each 𝑥𝑖 (which represents one possible scenario), calculate the corresponding payoff to be 𝑝𝑖 = $10,000,000 × max 𝑥𝑖 − 𝐾, 0 Step 3: The expected value of this contract is the average of all payoffs, i.e., 𝑁 𝑖=1 𝑝𝑖 /𝑁 ※ Please refer to “Contract pricing with MCS.xlsx” 8 4. 信賴區間 Standard Deviation and Standard Error Standard deviation is a simple measure of the variability or dispersion of a population, a data set, or a probability distribution. For 𝑁 simulated random samples, the standard deviation of this set can be calculated through 𝜎 = 𝑁 𝑖=1 𝑥𝑖 − 𝑥 2 /𝑁 Standard error 𝑠𝐸 : The standard error of an estimation method is the standard deviation of the sample distribution associated with the estimation method A sampling distribution is the probability distribution, under repeated sampling of the population, of a given estimation – Suppose we repeatedly draw sample sets with a given size from a population and calculate the sample mean for each sample sets – Different sample sets leads to different sample means – The distribution of these means is the “sampling distribution of the sample mean” ※ For the estimates of the sample variance, skewness, kurtosis, etc., it is also possible to derive the “sampling distributions” of these estimates 9 Standard errors for the MCS, which in essence a method for estimating the mean Method 1: The standard error of the mean is usually estimated by the sample standard deviation divided by the square root of the sample size, 𝜎 i.e., 𝑠𝐸,1 = 𝑁 Method 2: Generate the sampling distribution of the results of MCS by repeating the MCS for many times, e.g., 20-30 times, and next calculate the standard deviation of the sampling distribution to be 𝑠𝐸,2 ※ Please refer to “Standard Error.xlsx” Confidence interval (CI) for the MCS Method 1: 𝑁 = 10000 MCS for calculating 𝑠𝐸,1 , and CI = [mean of 𝑁 = 10000 MCS – 2× 𝑠𝐸,1 , mean of 𝑁 = 10000 MCS + 2× 𝑠𝐸,1 ] Method 2: Repeat 𝑁 = 10000 MCS for 𝑀 times for calculating 𝑠𝐸,2 , and CI = [mean of 𝑀 repetitions – 2× 𝑠𝐸,2 , mean of 𝑀 repetitions + 2× 𝑠𝐸,2 ] ※ According to the central limit theorem, the sampling distribution of the mean is asymptotically normal, and 𝑍2.5% = −1.96 and 𝑍97.5% = +1.96. So, the use of ± 2 × 𝑠𝐸 is approximately correct and common in practice 10 Method 1 vs. 2 for standard errors and CIs Method 2, which follows the definition to derive the standard error, is more intuitive Method 1 does not work sometimes after conducting variance reduction techniques. The possible reason may be that the random samples are no more independent after using the antithetic variate approach ※ Please refer to “Effect of Antithetic.xlsx” 11 5. 變異數削減技巧 (Variance-Reduction Techniques) To improve the accuracy (or said to decrease the standard errors) of the MCS, one should consider a larger value of 𝑁 and thus the MCS become more time consuming Variance-reduction techniques are approaches which can narrow the CIs of the MCS without increasing the number of simulation 𝑁 Antithetic variate approach (反向變異法): satisfying the zero mean and the symmetric feature of the standard normal distribution 𝑧1 , 𝑧2 , …, 𝑧𝑁 , 𝑧𝑁+1 , 𝑧𝑁+2 , …, 𝑧𝑁 2 2 2 || || || −𝑧1 −𝑧2 −𝑧𝑁 2 12 Moment matching: matching the first two moments of the standard normal distribution Step 1: Draw random samples 𝑧1 , 𝑧2 ,…, 𝑧𝑁 Step 2: Suppose the mean and standard deviation of 𝑧1 , 𝑧2 ,…, 𝑧𝑁 are 𝑚 and 𝑠, respectively Step 3: Generate 𝑦𝑖 = 𝑧𝑖 −𝑚 𝑠 (The mean of standard deviation of 𝑦1 , 𝑦2 ,…, 𝑦𝑁 will be 0 and 1, respectively) Antithetic variate approach + Moment matching: – – Use the antithetic variate approach to generate 𝑧1 , 𝑧2 ,…, 𝑧𝑁 in Step 1 in the moment matching method Since the mean of 𝑧1 , 𝑧2 ,…, 𝑧𝑁 based on the antithetic variate approach 𝑧 is 0, Step 3 can be reduced to 𝑦𝑖 = 𝑖 𝑠 – This method is very common in practice 13 6. 範例 Calculate N(c) by the Monte Carlo Simulation Monte Carlo Simulation: generate some random scenarios and conduct further analysis based on these scenarios By definition, N(c) = prob (x c) for x ~ N(0,1) Draw, for example, 1000 random samples from a standard normal distribution x ~ N(0,1) Then N(c) can be estimated with #(x c) / 1000 ※ Please refer to “MCS applications.xls” 14 Analyze a project with expanding and aborting options by the Monte Carlo Simulation Time 0: initial investment V0 Time t1: the value of the project is V1 ~ N (V0 (1 k )t , 1 ), and the firm can double the investment if V1 u (i.e., V1 2V1), abort the investment if V1 d (i.e., V1 0), or otherwise maintain the same investment amount such that V1 V1 , where k is the expected growth rate of the investment Time t2: the value of the project is V2 ~ N (V1(1 k )(t t ) , 2 ) 1 2 1 (Through the Monte Carlo Simulation, you can generate thousands of scenarios for the final payoff of the project, and further analyze the associated mean and s.d. or even the whole distribution of the payoff of the project) ※ Please refer to “MCS applications.xls” 15 7. Homework 4 Estimate the expected value of the following bonus contract $10,000,000 × max 𝑟 − 𝑟𝐼 , 0 , where 𝑟 is the growth rate of your firm and follows a normal distribution with the mean to be 𝜇 and the standard deviation to be 𝜎, and 𝑟𝐼 is the growth rate of the industry where your firm belongs and follows a normal distribution with the mean to be 𝜇𝐼 and the standard deviation to be 𝜎𝐼 . The correlation between 𝑟 and 𝑟𝐼 is 𝜌. Given the number of simulations to be 𝑁 and the number of repetitions to be 𝑀, calculate the expected value and the 95% confidence interval of this contract Inputs: 𝜇, 𝜎, 𝜇𝐼 , 𝜎𝐼 , 𝜌, 𝑁, and 𝑀; Output (with one CommandButton): expected contract value and the 95% confidence interval ※ Total points for this homework is 10 16