Reading 6
MODULE 6.1: LOGNORMAL DISTRIBUTIONS AND SIMULATION TECHNIQUES
Explain the relationship between normal and lognormal distributions and why the lognormal distribution is used to model asset prices when using continuously compounded asset returns.
The lognormal distribution is generated by the function \(e^x\), where \(x\) is normally distributed. Because the natural logarithm, \(\ln\), of \(e^x\) is \(x\), the logarithms of lognormally distributed random variables are normally distributed, thus the name.
Figure 6.1 illustrates the differences between a normal distribution and a lognormal distribution.
A normal distribution is symmetric and bell-shaped, centered on its mean \(\mu\), and may take values from \(-\infty\) to \(+\infty\). A lognormal distribution is bounded below by zero and is skewed to the right.
Recall from our reading about rates and returns that we use the natural logarithm to calculate continuously compounded returns. The lognormal distribution is useful for modeling asset prices if we think of an asset's future price as the result of a continuously compounded return on its current price. That is:
\[P_T = P_0 \, e^{r_{0,T}}\]
where:
- \(P_T\) = asset price at time \(T\)
- \(P_0\) = asset price at time 0 (today)
- \(r_{0,T}\) = continuously compounded return on the asset from time 0 to time \(T\)
Because continuously compounded returns are additive, we can divide the time from 0 to \(T\) into shorter periods and state that \(r_{0,T}\) is the sum of the continuously compounded returns over each of these shorter periods. Then, if we assume each of these returns is normally distributed, we can state that \(r_{0,T}\) is normally distributed. Even if they are not normally distributed, the central limit theorem implies that their sum (\(r_{0,T}\)) is approximately normally distributed. This allows us to say \(P_T\) is lognormally distributed, because it is proportional to the logarithm of a normally distributed variable.
In many of the pricing models that we will see in the CFA curriculum, we assume returns are independently and identically distributed. If returns are independently distributed, past returns are not useful for predicting future returns. If returns are identically distributed, their mean and variance do not change over time (a property known as stationarity that is important in time series modeling, a Level II topic).
對數常態分配（lognormal distribution）是由函數 \(e^x\) 所生成的，其中 \(x\) 服從常態分配。由於 \(e^x\) 取自然對數 \(\ln\) 後等於 \(x\)，因此「對數常態分配隨機變數」取對數後是常態分配——這也是它名字的由來。
Figure 6.1 展示了常態分配與對數常態分配的差異。
Figure 6.1：常態分配 vs. 對數常態分配
常態分配是對稱的鐘形曲線，以平均數 \(\mu\) 為中心，可取值範圍從 \(-\infty\) 到 \(+\infty\)。對數常態分配則受限於下界為零，且向右偏（right-skewed）。
回想我們在「利率與報酬率」單元提過：自然對數可用來計算連續複利報酬。如果把資產的未來價格視為「以當前價格為基礎、經過連續複利報酬後的結果」，那麼對數常態分配就很適合用來建模資產價格。也就是：
\(P_T = P_0 \, e^{r_{0,T}}\)
其中：
- \(P_T\)＝在 \(T\) 時點的資產價格
- \(P_0\)＝在 0 時點（今天）的資產價格
- \(r_{0,T}\)＝資產從時點 0 到時點 \(T\) 的連續複利報酬率
因為連續複利報酬具有可加性（additive），我們可以把時點 0 到 \(T\) 切成若干個較短的子期間，並把 \(r_{0,T}\) 表示為「這些短期間連續複利報酬的總和」。如果再假設每個短期間的報酬都服從常態分配，那麼 \(r_{0,T}\) 也是常態分配。即使它們不是常態分配，依中央極限定理（central limit theorem），其總和 \(r_{0,T}\) 仍會近似常態分配。也因此 \(P_T\) 為對數常態分配——因為它與「常態分配變數的對數」成比例。
在 CFA 課綱許多定價模型中，我們會假設報酬是獨立且同分配（independently and identically distributed, i.i.d.）。若報酬獨立分配（independently distributed），代表「過去報酬無法用來預測未來報酬」。若報酬同分配（identically distributed），則代表「報酬的平均數與變異數不隨時間改變」——這個性質稱為定態（stationarity），是 Level II 時間序列分析的重要概念。
Describe Monte Carlo simulation and explain how it can be used in investment applications.
Monte Carlo simulation is a technique based on the repeated generation of one or more risk factors that affect security values to generate a distribution of security values. For each of the risk factors, the analyst must specify the parameters of the probability distribution that the risk factor is assumed to follow. A computer is then used to generate random values for each risk factor based on its assumed probability distributions. Each set of randomly generated risk factors is used with a pricing model to value the security. This procedure is repeated many times (100s, 1,000s, or 10,000s), and the distribution of simulated asset values is used to draw inferences about the expected (mean) value of the security—and possibly the variance of security values about the mean as well.
As an example, consider the valuation of stock options that can only be exercised on a particular date. The main risk factor is the value of the stock itself, but interest rates could affect the valuation as well. The simulation procedure would be to do the following:
- Specify the probability distributions of stock prices and of the relevant interest rate, as well as the parameters (e.g., mean, variance, skewness) of the distributions.
- Randomly generate values for both stock prices and interest rates.
- Value the options for each pair of risk factor values.
- After many iterations, calculate the mean option value and use that as your estimate of the option's value.
Monte Carlo simulation is used to do the following:
- Value complex securities.
- Simulate the profits/losses from a trading strategy.
- Calculate estimates of value at risk (VaR) to determine the riskiness of a portfolio of assets and liabilities.
- Simulate pension fund assets and liabilities over time to examine the variability of the difference between the two.
- Value portfolios of assets that have nonnormal return distributions.
An advantage of Monte Carlo simulation is that its inputs are not limited to the range of historical data. This allows an analyst to test scenarios that have not occurred in the past. The limitations of Monte Carlo simulation are that it is fairly complex and will provide answers that are no better than the assumptions about the distributions of the risk factors and the pricing/valuation model that is used. Also, simulation is not an analytic method, but a statistical one, and cannot offer the insights provided by an analytic method.
蒙地卡羅模擬（Monte Carlo simulation）是一種透過反覆生成一個或多個「會影響證券價值的風險因子（risk factors）」，藉此產生證券價值分配的方法。對每一個風險因子，分析師必須先指定其所假設的機率分配參數。接著由電腦依照這些假設分配，為每個風險因子隨機生成數值。每一組隨機生成的風險因子都會搭配定價模型計算證券價值。這個程序會重複多次（數百次、數千次、甚至上萬次），最後利用模擬出的資產價值分配，來推論證券的期望（平均）價值——也可能推論價值相對於平均的變異程度。
舉例來說，考慮一檔只能在特定日期行使的股票選擇權的評價。主要的風險因子是股票本身的價值，但利率也可能影響評價。模擬程序如下：
- 指定股價與相關利率的機率分配，並設定其參數（例如平均數、變異數、偏態）。
- 隨機生成股價與利率的數值。
- 對每一組（股價、利率）配對計算選擇權的價值。
- 多次迭代後計算選擇權價值的平均，作為對該選擇權價值的估計。
蒙地卡羅模擬可用於下列情境：
- 對複雜證券進行評價。
- 模擬某交易策略的損益。
- 計算風險值（value at risk, VaR）以衡量資產與負債組合的風險。
- 長期模擬退休基金的資產與負債，檢視兩者差距的變動性。
- 對「報酬呈非常態分配」的資產組合進行評價。
蒙地卡羅模擬的優點是「輸入不受歷史資料範圍的限制」，這讓分析師可以測試過去未曾出現過的情境。它的限制包括：方法相當複雜，且結果好壞完全取決於「風險因子分配的假設」與「所用的定價／評價模型」之品質。此外，模擬屬於統計方法而非解析方法（analytic method），因此無法提供解析方法可帶來的洞見。
Describe the use of bootstrap resampling in conducting a simulation based on observed data in investment applications.
Resampling is another method for generating data inputs to use in a simulation. Often, we do not (or cannot) have data for a population, and can only approximate the population by sampling from it. (For example, we may think of the observed historical returns on an investment as a sample from the population of possible return outcomes.) To conduct resampling, we start with the observed sample and repeatedly draw subsamples from it, each with the same number of observations. From these samples, we can infer parameters for the population, such as its mean and variance.
In our reading on Estimation and Inference, we will describe some of the available resampling techniques. One of these is known as bootstrap resampling. In bootstrap resampling, we draw repeated samples of size \(n\) from the full dataset, replacing the sampled observations each time so that they might be redrawn in another sample. We can then directly calculate the standard deviation of these sample means as our estimate of the standard error of the sample mean.
Simulation using data from bootstrap resampling follows the same procedure as Monte Carlo simulation. The difference is the source and scope of the data. For example, if a simulation uses bootstrap resampling of historical returns data, its inputs are limited by the distribution of actual outcomes.
重抽樣（resampling）是另一種為模擬產生輸入資料的方法。實務上我們通常沒有（或無法取得）整體母體的資料，只能透過從母體中抽樣來近似它。（例如，我們可以把某項投資觀察到的歷史報酬，視為「可能報酬結果母體」的一個樣本。）進行重抽樣時，我們從已觀察到的樣本出發，反覆從其中抽取觀察值數量相同的子樣本。再藉由這些子樣本，推論母體的參數，如平均數與變異數。
在「估計與推論（Estimation and Inference）」單元中，我們會介紹幾種可用的重抽樣技術。其中之一稱為自助重抽樣（bootstrap resampling）。在自助重抽樣中，我們從完整資料集裡反覆抽取大小為 \(n\) 的樣本，每次抽取後都把觀察值放回（with replacement），這樣同一筆觀察值有可能在後續樣本中被重複抽到。接著我們可直接計算這些「樣本平均數」的標準差，作為樣本平均數標準誤（standard error of the sample mean）的估計。
使用 bootstrap 重抽樣資料的模擬，其程序與蒙地卡羅模擬相同。差別在於資料的來源與範圍。例如，若模擬使用「歷史報酬資料的 bootstrap 重抽樣」作為輸入，那麼這些輸入就會受限於「實際發生過的結果」之分配。
- A. mean equals the median.
- B. probability of a negative outcome is zero.
- C. probability of a positive outcome is 50%.
- A. Monte Carlo simulation is a statistical rather than an analytic method.
- B. Results of the analysis are no better than the assumptions used to generate it.
- C. Monte Carlo simulation is unable to provide answers to "what if" questions.
- A. Offers a representation of the statistical features of a population.
- B. Provides only statistical estimates, not exact results.
- C. Inputs may be limited by the distribution of actual outcomes.
If \(x\) is normally distributed, \(e^x\) follows a lognormal distribution. The lognormal distribution is useful for modeling an asset's future price as the result of a continuously compounded return on its current price.
If investment returns are independently distributed, past returns are not useful for predicting future returns.
If investment returns are identically distributed, their mean and variance do not change over time.
Monte Carlo simulation uses randomly generated values for risk factors, based on their assumed distributions, to produce a distribution of possible security values. Its limitations are that it is fairly complex and will provide answers that are no better than the assumptions used.
Bootstrap resampling involves drawing repeated samples from a sample that represents the population, replacing the sampled observations each time so that they might be redrawn in another sample. The standard deviation of these sample means is an estimate of the standard error of the sample mean. As with all simulation techniques, its answers are no better than the assumptions used.
LOS 6.a
若 \(x\) 服從常態分配，則 \(e^x\) 服從對數常態分配（lognormal）。對數常態分配適合用來建模資產未來價格——把它視為「以當前價格為基礎、經連續複利後的結果」。
若投資報酬「獨立分配」，則「過去報酬無法用來預測未來報酬」。
若投資報酬「同分配」，則平均數與變異數不隨時間改變。
LOS 6.b
蒙地卡羅模擬（Monte Carlo simulation）以隨機生成的風險因子數值（依其假設分配）為輸入，產生「可能證券價值」的分配。其限制是方法本身相當複雜，且結果優劣取決於所用的假設。
LOS 6.c
自助重抽樣（bootstrap resampling）從「代表母體的樣本」中反覆抽取子樣本，每次抽完都把觀察值放回，因此同一筆觀察值在後續樣本中有機會被重複抽到。這些樣本平均數的標準差，就是樣本平均數標準誤（standard error of the sample mean）的估計。與所有模擬技術一樣，其結果不會比所用的假設更好。