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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.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%.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)
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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
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
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Last Week
Last week we talked about random number generators (RNG) and their
use in MC methods
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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
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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
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Model Risk I
Models are approximations to reality.
They are necessary for determining the price at which an instrument
They are also necessary for valuing and hedging a financial institution’s
position in an instrument once it has been traded.
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Model Risk I
Models are approximations to reality.
They are necessary for determining the price at which an instrument
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
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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
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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
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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?
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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.
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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.
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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.
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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 <- mean(R); s <- sd(R) x <- seq(min(R)-1,max(R)+1,length = 1000) y <- dnorm(x,m,s) lines(y~x, col = 2, lty = 2) legend("topleft",c("empirical","normal"), col = 1:2, lty = 1:2) skewness(R) 2 >
>
>
>
>
>
+
>
empirical
normal
0
library(quantmod)
library(lubridate)
library(moments)
v <- "SPY" P <- get(getSymbols(v,from = "2000-01-01")) P <- P[,6] P <- apply.monthly(P,function(x) x[nrow(x),] ) R <- na.omit(log(P/lag(P))) 12 SPY >
>
>
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>
>
>
>
-0.7482973
−0.20
−0.15
−0.10
−0.05
0.00
0.05
0.10
N = 224 Bandwidth = 0.01049
> kurtosis(R)
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)
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Financial Risk Management
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Let’s repeat the same for Google
GOOG
<<<<<- "GOOG" get(getSymbols(v,from = "2000-01-01")) P[,6] apply.monthly(P,function(x) x[nrow(x),] ) na.omit(log(P/lag(P))) >
>
>
>
>
>
+
>
plot(density(R), main = v)
m <- mean(R); s <- sd(R) x <- seq(min(R)-1,max(R)+1,length = 1000) y <- dnorm(x,m,s) lines(y~x, col = 2, lty = 2) legend("topleft",c("empirical","normal"), col = 1:2, lty = 1:2) skewness(R) empirical normal 5 v P P P R 3 2 1 Density 4 >
>
>
>
>
0
0.4964426
> kurtosis(R)
−0.2
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
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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 <- mean(R); s <- sd(R) x <- seq(min(R)-1,max(R)+1,length = 1000) y <- dnorm(x,m,s) lines(y~x, col = 2, lty = 2) legend("topleft",c("empirical","normal"), col = 1:2, lty = 1:2) skewness(R) 2 >
>
>
>
>
>
+
>
empirical
normal
Density
<<<<<- 1 v P P P R 4 AAPL >
>
>
>
>
0
-1.93906
> kurtosis(R)
−1.0
−0.8
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
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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)
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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
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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
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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
<<<<<- "SPY" get(getSymbols(v,from = "1960-01-01")) P[,6] na.omit(log(P/lag(P))) 25 vol_MA 1993−03−08 / 2018−09−14 0.05 0.05 0.04 0.04 0.03 0.03 0.02 0.02 0.01 0.01 The long way >
>
+
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>
var_MA <- numeric() for(d in D:nrow(R)) { d_seq <- (d-D+1):d R_window <- R[d_seq,] var_MA <- c(var_MA,mean(R_window^2)) } names(var_MA) <- date(R)[D:nrow(R)] var_MA <- as.xts(var_MA) vol_MA <- sqrt(var_MA) plot(vol_MA) Mar 08 1993 Jan 02 1996 Jan 04 1999 Jan 02 2002 Jan 03 2005 Jan 02 2008 Jan 03 2011 Jan 02 2014 Jan 03 2017 The short way > var_MA2 <- rollapply(R,D,function(x) mean(x^2) > vol_MA2 <- sqrt(var_MA2) ) 17/25 Spring 2019 (Stevens) Financial Risk Management FE535 17 / 25 Different Weighting The MA assigns equal weights to each return in the window However, one may wish to assign greater weight to the recent observation In a simple weighting scheme, one can define the following 2 σd+1 = D X 2 θi Rd−i+1 (10) i=1 where θi denotes the weight of the ith lagged return I I for instance, one can assign θi = θi for θ ∈ (0, 1) for θ = 0.5, the recent return has 50%, where as the second and third have 25% and 12.5%, etc. 18/25 Spring 2019 (Stevens) Financial Risk Management FE535 18 / 25 Different Weighting The MA assigns equal weights to each return in the window However, one may wish to assign greater weight to the recent observation In a simple weighting scheme, one can define the following 2 σd+1 = D X 2 θi Rd−i+1 (10) i=1 where θi denotes the weight of the ith lagged return I I for instance, one can assign θi = θi for θ ∈ (0, 1) for θ = 0.5, the recent return has 50%, where as the second and third have 25% and 12.5%, etc. Moreover, while volatility is time-variant, one may also consider an equilibrium point around which the volatility revolves To capture this, one can add an additional component to approximate the long-term stationary variance, i.e. 2 σd+1 = θ0 + D X 2 θi Rd−i+1 (11) i=1 I In fact, Equation (11) resembles what is known as the Autoregressive conditional heteroskedasticity (ARCH) model 18/25 Spring 2019 (Stevens) Financial Risk Management FE535 18 / 25 EWMA The idea behind (11) derives two important approaches to monitor the volatility The first one is the EWMA model in which 2 σd+1 = (1 − λ)Rd2 + λσd2 , ∀λ ∈ (0, 1) (12) 2 2 σd2 = (1 − λ)Rd−1 + λσd−1 (13) such that and so on Note that the volatility in EWMA is recursive, i.e. next day volatility depends on the previous one 19/25 Spring 2019 (Stevens) Financial Risk Management FE535 19 / 25 EWMA The idea behind (11) derives two important approaches to monitor the volatility The first one is the EWMA model in which 2 σd+1 = (1 − λ)Rd2 + λσd2 , ∀λ ∈ (0, 1) (12) 2 2 σd2 = (1 − λ)Rd−1 + λσd−1 (13) such that and so on Note that the volatility in EWMA is recursive, i.e. next day volatility depends on the previous one Nonetheless, if one repeats the above process multiple times, then it follows that 2 σd+1 = (1 − λ) D X 2 2 λi−1 Rd−i+1 + λD σn−D+1 (14) i=1 For a large D, note that λD → 0, since λ ∈ (0, 1) In addition, if we denote θi = (1 − λ)λi−1 , then (14) and (11) are consistent 19/25 Spring 2019 (Stevens) Financial Risk Management FE535 19 / 25 Note that the parameter λ denotes the decay factor I I Small λ implies that future volatility is more affected by recent returns (shocks) Large λ less attention is attributed to recent s ... Purchase answer to see full attachment

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