 Errata

Tail index in dependent data

Mar 28, 2019

Antonella Altamura and Marco Bee spotted that the language of the discussion on tail index for ARCH type data was not correct. It said that \begin{equation*} \Gamma(\iota/2+1/2)=\sqrt{\pi}(2\alpha)^{-\iota/2} \end{equation*} was the unconditional distribution of $Y$ which of course does not make sense.

Instead it should say that the value of $\iota$ can be found by solving for $\iota$ in the equation.

Arch kurtosis

Jan 28, 2018

Stefano Soccorsi pointed out that the one of the Kurtosis equations on page 37 is wrong. The rest of of it is correct, as is the final Kurtosis value. The typo was in the second equation below:

\begin{align*} E(Y^4) &= 3 E\left(\left(\omega+\alpha Y_{t-1}^2\right)^2\right) \\ &= 3 \omega^2+ 6\alpha \omega E(Y^2)+3 \alpha^2 E(Y^4)\\ &= 3 \omega^2+ 6\alpha \omega \frac{\omega}{1-\alpha}+3 \alpha^2 E(Y^4)\\ \end{align*}

Sequential moments

Jan 22, 2018

Stefano Soccorsi pointed out that the sequential moments equation on page 19 is wrong. It should be:

\begin{equation*} \frac{1}{t}\sum\limits_{i=1}^{t}x_i^m. \end{equation*}

Testing the independence of violations, Section 8.3.2 p. 155-6

Dec 27, 2017

My FM320 students Hongshen Chen, Yida Li and Yanfei Zhou pointed out that the discussion in Section 8.3.2 could be more clear, so I repeat the relevant parts of the section here with more clarifications.

We need to calculate the probabilities of two consecutive violations, $p_{11}$, as well the probability of a violation, if there was no violation on the previous day, i.e. $p_{01}$. More generally, where $i$ and $j$ are either 0 or 1: \begin{equation*} p_{ij}=\Pr \left( \eta_{t}=j|\eta_{t-1}=i\right). \end{equation*} The violation process can be represented as a Markov chain with two states, so the first order transition probability matrix is defined as: \begin{equation*} \Pi_1=\left( \begin{array}{cc} 1-p_{01} & p_{01} \\ 1-p_{11} & p_{11} \end{array} \right) . \end{equation*} The likelihood function is: \begin{equation} L_1(\Pi_1) =\left( 1-p_{01}\right) ^{v_{00}}p_{01}^{v_{01}}\left( 1-p_{11}\right) ^{v_{10}}p_{11}^{v_{11}} \tag{8.5}\label{eq:risk2:lik:bt:int} \end{equation} where $v_{ij}$ is the number of observations where $j$ follows $i$.

The maximum likelihood (ML) estimates are obtained by maximizing the likelihood function which is simple since the parameters are the ratios of the counts of the outcomes: \begin{gather*} \hat{\Pi}_{1}= \begin{pmatrix} \frac{v_{00}}{v_{00}+v_{01}} & \frac{v_{01}}{v_{00}+v_{01}} \\\-2mm] \frac{v_{10}}{v_{10}+v_{11}} & \frac{v_{11}}{v_{10}+v_{11}} \\ \end{pmatrix} . \end{gather*} Under the null hypothesis of no clustering, the probability of a violation tomorrow does not depend on today being a violation, then $p_{01}=p_{11}=p$ and the transition matrix is simply: \begin{align*} \Pi_{2} & =\left( \begin{array}{cc} 1-p & p \\ 1-p & p \end{array} \right) \end{align*} and the ML estimate is: \[ \hat{p} =\frac{v_{01}+v_{11}}{v_{00}+v_{10}+v_{01}+v_{11}}. so \begin{align*} \hat{\Pi}_{2} & =\left( \begin{array}{cc} 1-\hat{p} & \hat{p} \\ 1-\hat{p} & \hat{p} \end{array} \right) \end{align*}

The likelihood function then is \begin{equation} L_2(\Pi_2) =\left( 1-p\right) ^{v_{00}+v_{10}} p^{v_{01}+v_{11}} .\tag{8.6} \label{eq:risk2:lik:bt:int2} \end{equation}

Note in \eqref{eq:risk2:lik:bt:int2} we impose independence but do not in \eqref{eq:risk2:lik:bt:int}. Replace the $\Pi$ by the estimated numbers, $\hat{\Pi}$. The LR test is then: \begin{equation*} LR=2\left( \log L_1\left( \hat{\Pi}_{1}\right) -\log L_2\left( \hat{\Pi}_{2}\right) \right) \overset{\rm asymptotic}{\sim}\chi _{\left( 1\right) }^{2}. \end{equation*}

Example 4.4

May 16, 2017

Example 4.4 could be more clear, it is not strictly wrong, but could be better, i.e. have weights in the right side of the inequality, i.e. $$VaR^{5\%}(0.5 X+ 0.5 Y) \approx 50 > 0.5 VaR^{5\%}(X) + 0.5 VaR^{5\%}(Y) = 0+0.$$

Example 4.5

Oct 30, 2016

My FM320 student Emily Wong spotted a typo in line 3, Example 4.5. The VaRs in the equation are missing a minus, and should be $$0 > - VaR_1 > -VaR_0$$

E and ES and Q

Jun 17, 2015

My FM320 student and summer intern Yiying Zhong spotted a typo in Chapter 4 page 86. The ES equation at the bottom of the page says $$ES = - [Q|Q \le -VaR(p)]$$ but is missing the expectation $$ES = - E[Q|Q \le -VaR(p)]$$

Listings 3.3, 3.4

Mar 19, 2015

It had been pointed out that Listings 3.3, 3.4 might be better if they had  y[i-1]  inside the loop. It is not wrong as it is, but this is better

few issues

Feb 5, 2015

My FM447 student Gevorg Saakyan pointed out a few typos

The tail index on page 1 uses the letter $t$ where it  should use $\iota$.

Chapter 1, page 8. The date in the code comment does not correspond to the date in the code.

Chapter 8.2 Backtesting the S&P 500, pages 147-148. The dates in the text do not correspond with the dates in the code, the code should use February 2, this means that the  number of observations is not 4000.

Table 2.2

May 19, 2014

My Fm320 student, Chetan Varsani, noted that it might be better to reverse the first line of the table, i.e. ARCH(1) and ARCH(4)

Listings 8.9 to 8.12

May 12, 2013

the code in Listings 8.9 to 8.12 is correct, but it can have numerical problems when sample size is large. It is better to do the code in logs. Here is Listing 8.9 with that alternative, and it would be straightforward to do same adjustment to the other 3.

Section 8.3.2

May 6, 2013

Gian Giacomo pointed our that the likelihoods in the independence discussion are mislabeled, so restricted is unrestricted, and vice versa.

page 189, endogenous price section

Apr 20, 2013

My FM320 student Richard Dunn spotted that below the equation the $\Delta P$ was incorrectly defined. It should be: $$\Delta P_2=P_2-P_1$$

Table 8.3 and 8.4

Apr 20, 2013

My FM320 students Jocelyn Tete and Chi Li independently spotted incorrect references where the reference to the four VaR models reported in Table 8.2/8.3 should have been 8.1 and 8.2 instead.

Figure 8.1

Apr 20, 2013

My FM320 student Chi Li spotted yet another problem in the cursed Figure 8.1, where MW should be MA.

page 48.

Apr 20, 2013

My FM320 student Chi Li spotted that section 2.6.4 second paragraph, last sentence: This is consistent with the residual analysis in Table 2.4. This should instead be Table 2.5.

More on ES for the normal

Apr 20, 2013

My FM320 student Richard Dunn spotted that second equation under section 5.3.4 is missing a minus in front.

page 90

Nov 8, 2012

my FM320 student Richard Dunn spotted a typo on page 90. The 10th word on the 5th line should be 'variance' instead of 'variable

page 44, 149

Jun 14, 2012

My FM320 student Alexander Stampfer found a number of small typos

The last paragraph ion page 44 “ The restricted log-likelihood minus the unrestricted log-likelihood” which should be reversed.

Page 149. There is a bracket missing in the Matlab code in the very last line of the page for EWMA

4 and 20 on page 96

Jun 9, 2012

My FM320 student Akash Jhunjhunwala points out that on page 96, the last line on the first paragraph, the 1% and 5% levels corresponds to the 4th and 20th values respectively, not the 5th and 25th as suggested in the brackets.

$p_{ij}$ and $p_{ji}$ bottom of page 155

Jun 5, 2012

My FM320 student, Bide Liu, spotted that I should have $p_{ji}$ and not $p_{ij}$ in the penultimate equation on page 155. It should read like $$p_{ji}= \Pr (\eta_t=i | \eta_{t-1} = j)$$

LR ratio in (8.4) page 154

Jun 5, 2012

My FM320 student, Bide Liu, spotted that the LR test in (8.4) has the wrong order in the first line. Should be unrestricted - the restricted, i.e. something like $$LR=2(\log L_U (\hat{p})- \log L_R (p))$$

standard deviation not variance on page 44

Jun 5, 2012

My FM320 student Amith Bhattacharyya spotted that on the penultimate line on page 42 standard deviation should replace variance.

Var(WE+1) and not Var(WT+1) on page 144

Jun 5, 2012

My class teacher Marcela Valenzuela and Ehsan Ramezanifar, University of Tehran, both spotted a typo in the middle of page 144, where the subscript T should be E.

example 4.3

Nov 16, 2011

My FM320 student Daniel Payne pointed out that in example 4.3 in third line from the bottom the subscript on the weight is wrong, its right in the preceding line, so in both cases it should be:$$(w_X \sigma_X + w_Y \sigma_Y)^2$$

vol and mean numbers page 104

Nov 16, 2011

My MF320 student Ken Starling pointed out that I refer to Table 1.2 on page 104, but cite different numbers, so instead of 0.021% and 1.1% for mean and vol, use 0.019% and 1.16%.

page 85 ( and not [

Nov 16, 2011

My MF320 student Ken Starling pointed out that in the penultimate line on page 85 the left side of the interval should be ( since infinity is not included, so $$(- \infty ,-VaR(p)]$$

Wrong word order on page 44

Oct 26, 2011

My FM320 student Daniel Payne pointed out that the words restricted and unrestricted are reversed on the bottom of page 44. It should read: "The unrestricted log-likelihood minus the restricted log-likelihood"

Multiperiod volatility

Oct 22, 2011

My FM320 student Dominic Clark pointed out that I could have been more clear on page 39 in the last text paragraph. Its not wrong, but a better way is: where $E_t(Y^2_{t+1})$ indicates volatility...

3rd equation from the bottom on page 38

Oct 22, 2011

My FM320 student Ken Starling pointed out a missing + in the 3rd equation from the bottom on page 38, it should be $$\sigma^2= E(\omega+\alpha Y_{t-1}^2 +\beta \sigma_{t-1}^2) =\omega+\alpha \sigma^2 +\beta \sigma^2.$$

Equation on top of page 37

Oct 22, 2011

My FM320 student Yong Bin Ng pointed out a typo in the equation on top of page 37. It should be:$$E(Y^4)=3E\left[(\omega+\alpha Y_{t-1}^2)^2\right]=3(\omega^2+2\alpha \omega \sigma^2 + \alpha^2 E(Y^4))$$

Also, in case you were wondering how to derive the equation we use previous results on page 36, independence of Y's and Z and properties of the normal distribution, and it's done as follows. \begin{aligned} E(Y^4)&=E(Y_t^4)\\ &=E(\sigma_t^4 Z_t^4)\\ &=E(\sigma_t^4)E(Z_t^4)\\ &=E(\sigma_t^4)3(E(Z_t^2))^2\\ &=3E((\sigma_t^2)^2)\\ &=3E\left[(\omega+\alpha Y_{t-1}^2)^2\right]\\ &=3(\omega^2+2\alpha \omega \sigma^2 + \alpha^2 E(Y_{t-1}^4))\\ &=3\omega^2+6\alpha \omega \sigma^2 + 3\alpha^2 E(Y^4)\\ &=3\omega^2+6\alpha \omega \frac{\omega}{1-\alpha} + 3\alpha^2 E(Y^4) \end{aligned} then, $$E(Y^4)(1-3\alpha^2)(1-\alpha) =3\omega^2(1-\alpha)+6\alpha \omega^2$$ and $$E(Y^4)=\frac{3\omega^2(1+\alpha)}{(1-3\alpha^2)(1-\alpha)}$$

Table 1.5

Oct 22, 2011

My FM320 student Han Wang pointed out that Table 1.5 is not right. It is supposed to have a two tailed probability of outcomes, but the %1 number is 1-the one tailed prob, and the rest are one tailed. So, here are the correct numbers. Set the volatility to 1.16 as per Table 1.2, and get

 1% 0.38865 2% 0.084683 3% 0.00970387 5% 1.63e-05 15% 3.00745e-38 23% 1.72103e-87
with the R code to do the calculation 2*pnorm(-23,sd=1.16)

Monte Carlo VaR with one basic asset

May 24, 2011

item 2 on page 133 has an incorrect second index for the y. It should be $y_{t+1,i}$ not $y_{t+1,t}$

Figure 8.1 backtesting

May 24, 2011

Philippe Mueller spotted a bug in the legend of Figure 8.1. It is an unwieldy figure, with almost too much going on, and hard to see in black and white. The color plot below is much clearer, and hopefully correct. Also, I called returns, volatility. Guess the pic was cursed. Finally, one could specify the probability, 1%. In any case, here is the correct. Typo in equation for ES for the normal

Feb 7, 2011

I do thank Oliver Linton for spotting a typo in the equation of ES for the normal at the bottom of page 103 and top of page 104. The setup and derivation is correct, but somehow the $\sigma$ became $\sigma^2$. The correct equation (bottom page 103) $$\text{ES}=-\frac{\sigma \phi(-\text{VaR}(p))}{p}$$ and the corresponding equation at the top of 104 $$\text{ES}=-\varphi \frac{\sigma \phi(-\text{VaR}(p))}{p}$$