Modeling Risk Factors (20 Points) In this task, you will need to download daily data for the S&P 500 index (not the ETF). The data should date between to Jan 1960 and Feb 2019. Given this time series, compute the daily returns using the adjusted prices and address the following: 1. Your first task to compute the realized volatility. To do so, for each week in the time series, compute the realized variance as the average squared return and scale it by 5. Hence, the realized volatility is given by the squared root of the realized variance. For instance, if you have Dw days in week w, then the realized volatility of that week is given by v u Dw u 1 X σw = t5 × Rd2 Dw (1) d=1 with Rd denoting the daily return. As a summary provide a plot of the realized volatility over time (5 Points). 2. Your second task is to construct a forward estimate (forecast) for the realized volatility. You will need convert the returns from daily to weekly. Given the weekly returns, you are required to use two models: a 50-week moving average (MA) and an EWMA model with λ = 90%. In the former case, the w + 1 week volatility forecast is given by v u u 1 σ̂w+1 = t W w X Rt2 (2) t=w−W +1 with W = 50 weeks and Rt is the return in week w. For the EWMA, the forecast is given by σ̃w+1 = q 2 + λσ̃ 2 (1 − λ)Rw w (3) with λ = 0.9 and σ̃w denotes the former estimate. Note that in the EWMA case, you need start with an initial volatility. I suggest using the sample estimate using the first 50 weeks as an initial estimate. Hence, the EWMA model will require an initial sample of 50 weeks to derive the first forecast. After deriving a time series forecast using each model merge the two altogether and provide a figure summarizing both on the same plot over the whole sample (5 Points). In addition, provide a summary statistics in a well-organized table (5 Points). Note that the sample period in this case should start from late Dec 1960, and the frequency is weekly. 3. Finally, merge the realized volatility along with the forecast of each model. As a summary of performance, plot the realized against the forecast (1 Points). Moreover, regress the realized volatility on the each of the forecasts and report the following measures: coefficient of determination R2 , mean-squared error (MSE), the intercept, and the slope. How do you compare between the two models? Which one provides a better approach to model risk? (4 Points) 6 Agenda A quick recap from last week Model Risk I I Model Error Case Study Violation of Normality I I iid Assumption Fat Tails Intro to time-varying volatility I I I Moving Average Exponentially Weighted Moving Average (EWMA) Intro GARCH Model 2/25 Spring 2019 (Stevens) Financial Risk Management FE535 2 / 25 Last Week Last week we talked about random number generators (RNG) and their use in MC methods 3/25 Spring 2019 (Stevens) Financial Risk Management FE535 3 / 25 Last Week Last week we talked about random number generators (RNG) and their use in MC methods We spoke about Brownian Motion (geometric) and its relevance to simulating stock prices 3/25 Spring 2019 (Stevens) Financial Risk Management FE535 3 / 25 Last Week Last week we talked about random number generators (RNG) and their use in MC methods We spoke about Brownian Motion (geometric) and its relevance to simulating stock prices Finally, we discussed the application of MC to portfolio risk management I I Value at Risk (VaR) Sensitivity to model specification, i.e. volatility 3/25 Spring 2019 (Stevens) Financial Risk Management FE535 3 / 25 Model Risk I Models are approximations to reality. They are necessary for determining the price at which an instrument should be traded They are also necessary for valuing and hedging a financial institution’s position in an instrument once it has been traded. 4/25 Spring 2019 (Stevens) Financial Risk Management FE535 4 / 25 Model Risk I Models are approximations to reality. They are necessary for determining the price at which an instrument should be traded They are also necessary for valuing and hedging a financial institution’s position in an instrument once it has been traded. There are two main types of model risk: I I Model returning wrong price at the time a product is bought/sold Wrong hedging model, e.g. wrong Greeks 4/25 Spring 2019 (Stevens) Financial Risk Management FE535 4 / 25 Model Risk II Physics Vs. Finance The differential equation such as the GBM is the heat-exchange equation that has been used by physicists for many years. The main difference is that the models of physics describe physical processes and are highly accurate. The models of finance describe the behavior of market variables I which are the result of human interactions and behavior At best the finance models can give us an approximation of reality I For this reason, Model Risk is inevitable in Finance 5/25 Spring 2019 (Stevens) Financial Risk Management FE535 5 / 25 Model Risk II Physics Vs. Finance The differential equation such as the GBM is the heat-exchange equation that has been used by physicists for many years. The main difference is that the models of physics describe physical processes and are highly accurate. The models of finance describe the behavior of market variables I which are the result of human interactions and behavior At best the finance models can give us an approximation of reality I For this reason, Model Risk is inevitable in Finance One major concern in Finance is model parameters The parameters of models in Physics generally do not change Whereas in parameters in Finance change daily 5/25 Spring 2019 (Stevens) Financial Risk Management FE535 5 / 25 Model Risk III Model risk can refer to I I Model error Implementing a model incorrectly Model Error For instance, the pricing of derivatives relies heavily on mathematical and simulation models I I error model calibration error in deriving solution A more common and dangerous risk relates to assumption errors on the underlying stock process I I does GBM process fully reflect stock prices? in other words, do returns follow iid normal? 6/25 Spring 2019 (Stevens) Financial Risk Management FE535 6 / 25 Niederhoffer’s Case Study1 A well-established hedge fund ran by Victor Niederhoffer I I a star on Wall Street his fund was wiped out in November 1997 What happened? 1 See Box 14.1 from this chapter. Spring 2019 (Stevens) Financial Risk Management 7/25 FE535 7 / 25 Niederhoffer’s Case Study1 A well-established hedge fund ran by Victor Niederhoffer I I a star on Wall Street his fund was wiped out in November 1997 What happened? Victor wrote (sold) “naked” options on the S&P 500 index His strategy was the following I I 1 collect many put option premiums for a small price the chances of losses were small See Box 14.1 from this chapter. Spring 2019 (Stevens) Financial Risk Management 7/25 FE535 7 / 25 Niederhoffer’s Case Study1 A well-established hedge fund ran by Victor Niederhoffer I I a star on Wall Street his fund was wiped out in November 1997 What happened? Victor wrote (sold) “naked” options on the S&P 500 index His strategy was the following I I collect many put option premiums for a small price the chances of losses were small Nonetheless, his main assumption was that the market won’t drop more than 5% percent in a day During the Asian market crisis, the S&P 500 dropped more than 7% in a single day To meet margin calls, Victor had to liquidate his position in a fire-sale 1 See Box 14.1 from this chapter. Spring 2019 (Stevens) Financial Risk Management 7/25 FE535 7 / 25 Let’s take a closer look at what happened If Rd denotes the return on the S&P 500 in a single day d and Rd ∼ N(µd , σd2 ) then P (Rd < −0.05) = P  −0.05 − µd σd  (1) The probability should be small as long as σd (µd ) is small (large) enough 8/25 Spring 2019 (Stevens) Financial Risk Management FE535 8 / 25 Let’s take a closer look at what happened If Rd denotes the return on the S&P 500 in a single day d and Rd ∼ N(µd , σd2 ) then P (Rd < −0.05) = P  −0.05 − µd σd  (1) The probability should be small as long as σd (µd ) is small (large) enough However, in October 1997, the market exhibited a sudden increase in volatility due to worries about possible spillovers from Asian Financial Crisis For instance, compared to Sep, 1997, the S&P 500 volatility more than doubled in Oct, 1997 Probability of 5% drop I which, as a result, significantly increased the probability 1997−01−02 / 1997−12−31 0.010 0.010 0.008 0.008 0.006 0.006 0.004 0.004 0.002 0.002 Jan 02 1997 Mar 03 1997 May 01 1997 Jul 01 1997 Sep 02 1997 Nov 03 1997 Dec 31 1997 8/25 Spring 2019 (Stevens) Financial Risk Management FE535 8 / 25 Why Normal? If daily returns follow an iid normal distribution, i.e. Rd ∼ N(µ, σ) ∀d = 1, ..., T , then we know that V[ D X Rd ] = D × σ 2 (2) d=1 This is the result of an iid assumption for time aggregation (see Section 5.1.2 of the Jorion) 9/25 Spring 2019 (Stevens) Financial Risk Management FE535 9 / 25 Why Normal? If daily returns follow an iid normal distribution, i.e. Rd ∼ N(µ, σ) ∀d = 1, ..., T , then we know that V[ D X Rd ] = D × σ 2 (2) d=1 This is the result of an iid assumption for time aggregation (see Section 5.1.2 of the Jorion) In terms of VaR, under normal distribution, it follows that VaR(Rd , c) = E[Rd ] − Q(Rd , c) = µ − Q(Rd , c) (3) Under normal distribution the c quantile of Rd Q(Rd , c) = µ + σZc (4) VaR(Rd , c) = µ − [µ + σZc ] = −σZc = σZ1−c (5) such that where Z1−c is the 1 − c percentile of the standard normal distribution, i.e. P(Z < Z1−c ) = 1 − c with Z ∼ N(0, 1) (6) 9/25 Spring 2019 (Stevens) Financial Risk Management FE535 9 / 25 Equation (5) indicates that the daily VaR at the 1−c level of confidence is mainly determined by the assessment of the daily volatility Obviously, a daily monitoring of the VaR requires a daily monitoring of the volatility If returns were iid, then the D multiple periods VaR is given by VaR D X √ ! Rd , c = D × σ × Z1−c (7) d=1 The above results are relevant if returns were normal and iid. See examples 5.1 through 5.5 from Jorion’s for further discussion on this 10/25 Spring 2019 (Stevens) Financial Risk Management FE535 10 / 25 Normal Assumption So far, we have assumed that stock returns exhibit normal distribution Normality assumption greatly simplifies computations and pricing, especially for derivatives However, there is also trade-off between simplicity and accuracy 11/25 Spring 2019 (Stevens) Financial Risk Management FE535 11 / 25 Normal Assumption So far, we have assumed that stock returns exhibit normal distribution Normality assumption greatly simplifies computations and pricing, especially for derivatives However, there is also trade-off between simplicity and accuracy In practice, the behavior of stock returns tends to contradict normal distribution In particular, the empirical distribution tends to I I have heavier tails than the normal distribution be more peaked than the normal distribution 11/25 Spring 2019 (Stevens) Financial Risk Management FE535 11 / 25 Normal Assumption So far, we have assumed that stock returns exhibit normal distribution Normality assumption greatly simplifies computations and pricing, especially for derivatives However, there is also trade-off between simplicity and accuracy In practice, the behavior of stock returns tends to contradict normal distribution In particular, the empirical distribution tends to I I have heavier tails than the normal distribution be more peaked than the normal distribution This means that small changes and large changes are more likely than the normal distribution would suggest Many market variables have this property, known as excess kurtosis 11/25 Spring 2019 (Stevens) Financial Risk Management FE535 11 / 25 Let’s consider the monthly returns of the SPY since 2000 10 8 Density 6 4 plot(density(R), main = v) m > > + > empirical normal 0 library(quantmod) library(lubridate) library(moments) v > > > SPY.Adjusted -0.7482973 −0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 N = 224 Bandwidth = 0.01049 > kurtosis(R) SPY.Adjusted 4.664545 > shapiro.test(as.numeric(R)) Shapiro-Wilk normality test data: as.numeric(R) W = 0.96368, p-value = 1.744e-05 Spring 2019 (Stevens) 12/25 Financial Risk Management FE535 12 / 25 Let’s repeat the same for Google GOOG > + > plot(density(R), main = v) m > 0 GOOG.Adjusted 0.4964426 > kurtosis(R) −0.2 GOOG.Adjusted 4.79023 0.0 0.2 0.4 N = 169 Bandwidth = 0.02291 > shapiro.test(as.numeric(R)) Shapiro-Wilk normality test data: as.numeric(R) W = 0.97167, p-value = 0.001562 13/25 Spring 2019 (Stevens) Financial Risk Management FE535 13 / 25 Apple "AAPL" get(getSymbols(v,from = "2000-01-01")) P[,6] apply.monthly(P,function(x) x[nrow(x),] ) na.omit(log(P/lag(P))) 3 plot(density(R), main = v) m > > + > empirical normal Density 0 AAPL.Adjusted -1.93906 > kurtosis(R) −1.0 −0.8 GOOG.Adjusted 4.79023 −0.6 −0.4 −0.2 0.0 0.2 0.4 N = 224 Bandwidth = 0.02949 > shapiro.test(as.numeric(R)) Shapiro-Wilk normality test data: as.numeric(R) W = 0.97167, p-value = 0.001562 14/25 Spring 2019 (Stevens) Financial Risk Management FE535 14 / 25 Stationarity The most common error assumes that the return process is stationary I I such as the case for σ constant over time whereas, derivatives traders know very well that volatility is not constant In fact, fat tails can occur when returns are drawn from a distribution with a time varying volatility A more realistic case, therefore, is to assume that the volatility is stochastic and, hence, develop a relevant return distribution (price path) 15/25 Spring 2019 (Stevens) Financial Risk Management FE535 15 / 25 Time Variation in Risk 16/25 Moving Average (MA) To model time-varying volatility in a very simple way is to consider the moving average (MA) of squared returns over D periods For instance, let Ri denote the return on a stock in day i, for i = d, d −1.., d −D +1, then the next day variance can be modeled as 2 σd+1 = D 1 X 2 Rd−i+1 D (8) i=1 Note that above is a similar expression to the variance of returns, however, assuming an average daily return of zero 16/25 Spring 2019 (Stevens) Financial Risk Management FE535 16 / 25 Moving Average (MA) To model time-varying volatility in a very simple way is to consider the moving average (MA) of squared returns over D periods For instance, let Ri denote the return on a stock in day i, for i = d, d −1.., d −D +1, then the next day variance can be modeled as 2 σd+1 = D 1 X 2 Rd−i+1 D (8) i=1 Note that above is a similar expression to the variance of returns, however, assuming an average daily return of zero After realizing the returns for day d + 1, the window is rolled over, and the variance for d + 2 is given by 2 σd+2 = D 1 X 2 Rd−i+2 D (9) i=1 For this reason, it is referred to as MA 16/25 Spring 2019 (Stevens) Financial Risk Management FE535 16 / 25 In R, this can be achieved in the long way or the short way In either case, we to start with a window of D days and roll it on a daily basis > > > > > v P P R D > var_MA + 2007−01−03 / 2009−12−31 RM
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