```{r} #| label: setup #| include: false set.seed(2026) library(ggplot2) theme_set(theme_minimal(base_size = 18)) sp <- read.csv("data/sp500.csv") sp$SP500 <- suppressWarnings(as.numeric(sp$SP500)) # FRED marks holidays "." sp <- na.omit(sp) r_all <- 100 * diff(log(sp$SP500)) # daily log returns, % d_all <- as.Date(sp$observation_date[-1]) ojs_define(ret = round(r_all, 4)) ``` ## π¬ The Idea in 3 Minutes ::: {style="text-align:center"} ```{=html} ``` [β¬
Previous lecture: 16 Β· Structural Breaks and Parameter Stability](../16-structural-breaks/structural_breaks_lecture16.html){style="font-size:28px"} [A short film with the intuition. Watch it before the slides, or after. Captions: CC button.]{style="font-size:28px"} ::: --- ## π― Learning Objectives ::: {style="font-size: 34px"} By the end of this lecture, you will be able to: - **Distinguish** observable, model-free volatility measures from model-based ones, and **show** that RiskMetrics is an IGARCH - **Explain** why realized volatility converges to integrated volatility, and what microstructure noise does to it - **Write** a multiplicative error model and the ACD model for durations - **Model** the direction of change (autologit) and conditional quantiles (VaR, CAViaR) - **Backtest** a VaR forecast with Kupiec's test on real data ::: --- ## π Warm-up from last time {.quiz-question} You search over the middle 70% of the sample for one break in a mean and find $\sup W = 7.9$. At 5%, what do you conclude? - [No evidence of a break: 7.9 is below Andrews' critical value 8.68]{.correct data-explanation="β
The date was searched for, so sup W must be compared with the critical value of sup Q(r) over [0.15, 0.85], which is 8.68 for one coefficient, not with the ΟΒ²β value 3.84."} - A break: 7.9 exceeds the $\chi^2_1$ critical value 3.84 - A break, since the CUSUM test is never needed after sup-W - Nothing can be said until the number of breaks is fixed --- ## πΊοΈ Where We Are ::: {style="font-size: 32px"} Lecture 15 modelled the conditional variance $\sigma_t^2$ with GARCH. Lecture 16 asked whether such parameters are stable. $\sigma_t^2$ is never observed: it exists only inside a model, and is only as good as the model's mean and variance equations. ::: {.fragment} This last lecture asks three questions. - Can volatility be **measured** rather than modelled? - How do we model other positive quantities, such as the time between trades? - How do we forecast what risk managers actually report: a **quantile** of tomorrow's return? ::: ::: --- ## β The Question ::: {.callout-important} ## The number the regulator sees Every day a bank reports its 1-day 99% Value-at-Risk: a loss it expects to exceed on only 1 day in 100. Every quarter the regulator counts the days on which the loss was larger. ::: ::: {style="font-size: 30px"} Too many such days and the bank holds more capital. To get the count right the bank needs a good volatility measure, a good model of the tails, and a way to **test** its own quantile forecasts. ::: --- ## π Measures of Daily Volatility ::: {style="font-size: 30px"} **Observable, model-free:** - $|r_t|$ and $r_t^2$: the earliest measures. $E_{t-1}r_t^2 = \sigma_t^2$ when $\mu_t = 0$, but a single squared return is very noisy - JPMorgan's **RiskMetrics**, an exponentially weighted average of past squared returns: $$RM_t = (1 - 0.94)\sum_{i=0}^{\infty} 0.94^{\,i}\, r_{t-i}^2$$ **Not observable, model-based:** - $\hat\sigma_t^2$ from an ARCH-type model. It is only as good as the model's conditional mean and variance equations ::: --- ## π RiskMetrics Is an IGARCH ::: {style="font-size: 30px"} Split off the first term of the sum. With $\lambda = 0.94$, $$\sigma_{t+1 \mid t}^2 = (1-\lambda)\sum_{i \ge 0}\lambda^i r_{t-i}^2 = (1-\lambda)\, r_t^2 + \lambda\,\sigma_{t \mid t-1}^2 .$$ This is GARCH(1,1) without a constant: $$\omega = 0, \qquad \alpha = 1 - \lambda = 0.06, \qquad \beta = \lambda = 0.94, \qquad \alpha + \beta = 1 .$$ - Integrated GARCH: shocks to the variance never die out, and there is no long-run level to revert to - Nothing is estimated: $\lambda = 0.94$ is fixed for all daily series - Weights halve every $\ln 0.5 / \ln 0.94 = 11.2$ days ::: --- ## π Realized Volatility ::: {style="font-size: 29px"} The daily return $r_t = \log p_{\text{close}} - \log p_{\text{open}}$ sees only two prices. Very different intraday paths give the same $r_t$. Split day $t$ into $m$ intervals with intraday returns $r_{t,i}$, $i = 1, \dots, m$, and define $$RV_t = \sum_{i=1}^m r_{t,i}^2 \qquad \text{(Andersen, Bollerslev, Diebold and co-authors, 2000s).}$$ Like $|r_t|$ and $r_t^2$, it is **observable** and **model-free**. Unlike them, it is far less noisy, as the next two slides show. ::: --- ## π Same Return, Different Days ```{r} #| label: two-paths #| echo: false #| fig-width: 10.5 #| fig-height: 3.8 set.seed(2026) m <- 390; s <- (0:m) / m bridge <- function(sd) { b <- c(0, cumsum(rnorm(m, 0, sd / sqrt(m)))); b - s * b[m + 1] } calm <- 0.5 * s + bridge(0.4) # log price in %, close - open = 0.5 storm <- 0.5 * s + bridge(2.0) rv <- function(x) sum(diff(x)^2) df <- rbind(data.frame(s = s, p = calm, day = sprintf("calm day: RV = %.2f", rv(calm))), data.frame(s = s, p = storm, day = sprintf("stormy day: RV = %.2f", rv(storm)))) ggplot(df, aes(s, p, colour = day)) + geom_line(linewidth = 0.9) + scale_colour_manual(values = c("#8ba3c7", "#8b2635"), name = NULL) + labs(x = "time of day (open = 0, close = 1)", y = "log price β log open (%)") + theme_minimal(base_size = 20) + theme(legend.position = "top") ``` ::: {style="font-size: 28px"} Both days open at 0 and close at $+0.5\%$, so $r_t = 0.5$ and $r_t^2 = 0.25$ on each. The one-minute returns tell them apart: realized variance differs by a factor of about 25. ::: --- ## π Why Realized Volatility Works ::: {style="font-size: 30px"} Suppose the log price within the day follows $$d\log p_{t+s} = \mu_{t+s}\,ds + \sigma_{t+s}\,dW_{t+s}, \qquad s \in [0, 1],$$ with $\sigma^2_{t+s}$ the **instantaneous volatility**. Then as $m \to \infty$ $$RV_t \xrightarrow{p} IV_t = \int_0^1 \sigma_{t+s}^2\,ds \qquad \text{(integrated volatility)},$$ with a mixed-normal CLT at rate $\sqrt m$. - The drift $\mu_{t+s}$ contributes only $O(m^{-1})$ and can be neglected - A GARCH, by contrast, must get the conditional mean right before its $\hat\sigma_t^2$ means anything ::: --- ## π Covariance, Noise, and the Choice of $m$ ::: {style="font-size: 30px"} **Multivariate analogue.** For a vector of intraday returns, the realized covariance matrix $$RC_t = \sum_{i=1}^m r_{t,i}\, r_{t,i}' .$$ **But $m$ is finite**, and one would like it large. At very high frequencies prices bounce between bid and ask, and ticks are discrete. This **market microstructure noise** adds a term to every $r_{t,i}^2$, and the bias grows with $m$. In practice the sampling interval is about **5 to 15 minutes**: fine enough for $RV_t$ to be precise, coarse enough for the noise to be small. ::: --- ## π» Noise and the Sampling Frequency ```{r} #| label: signature #| code-summary: "π¦ 200 simulated days, second by second" set.seed(2026) n_sec <- 23400; days <- 200; s <- (1:n_sec) / n_sec sig <- 0.01 * sqrt(0.6 + 3.2 * (s - 0.5)^2) # U-shaped intraday vol IV <- mean(sig^2) # integrated variance per day grid <- c(13, 26, 78, 390, 1170, 4680, 23400) # 30 min ... 1 second res <- replicate(days, { lp <- cumsum(sig * rnorm(n_sec) / sqrt(n_sec)) # efficient log price obs <- lp + rnorm(n_sec, 0, 0.0001) # + noise, 1 basis point sapply(grid, function(m) { k <- seq(n_sec / m, n_sec, by = n_sec / m) c(clean = sum(diff(c(0, lp[k]))^2), noisy = sum(diff(c(0, obs[k]))^2)) }) }) round(cbind(m = grid, clean = rowMeans(res[1, , ]) / IV, noisy = rowMeans(res[2, , ]) / IV), 2) ``` ::: {style="font-size: 28px"} Average $RV_t / IV_t$ over 200 simulated days. Without noise it is 1 at every $m$. With a one-basis-point noise, it is fine at 5 minutes ($m = 78$) and more than six times too large at one second. ::: --- ## π Multiplicative Error Models ::: {style="font-size: 30px"} Many quantities in finance are **positive**: $RV_t$, the daily range, $|r_t|$, and the **durations** $d_t = t_i - t_{i-1}$ between transactions. Durations cluster: busy periods have short gaps. ::: {.callout-note} ## MEM (Engle, 2002) $$x_t = \psi_t\,\varepsilon_t, \qquad \psi_t = E_{t-1}x_t, \qquad \varepsilon_t \sim \text{iid } \mathcal D(1),\ \ \varepsilon_t > 0,\ E\varepsilon_t = 1 .$$ ::: It is the positive analogue of $r_t = \sigma_t\eta_t$, $\eta_t \sim (0, 1)$: there the shock scales a **standard deviation**, here it scales a **mean**. ::: --- ## π The ACD Model ::: {style="font-size: 30px"} For durations, Engle & Russell (1998): **A**utoregressive **C**onditional **D**uration, $$d_t = \psi_t\varepsilon_t, \qquad \text{ACD}(1,1):\ \ \psi_t = \omega + \alpha\, d_{t-1} + \beta\,\psi_{t-1}, \qquad \omega > 0,\ \alpha, \beta \ge 0,$$ with $\alpha + \beta < 1$ for a finite unconditional mean duration $\omega/(1-\alpha-\beta)$. | Choice of $\mathcal D$ | Method | Analogue in GARCH | |---|---|---| | Standard exponential | QMLE, $\ell_t = -\log\psi_t - d_t/\psi_t$ | Gaussian QMLE | | Weibull$(\lambda)$, Burr$(\lambda, \mu)$ | MLE | Student-$t$ MLE | The exponential QMLE is consistent whenever $\psi_t$ is correctly specified, whatever the true $\mathcal D$. ::: --- ## π Beyond Mean and Variance ::: {style="font-size: 30px"} So far dynamic models have targeted - β the conditional **mean** (ARMA, VAR, TAR) - β the conditional **variance** (GARCH, SV, RV) - β the conditional **density**: conditional skewness and kurtosis (ARCD) ::: {.fragment} Two more objects are often what a user actually needs: - the **direction** of tomorrow's move, $\mathbb 1\{r_t > 0\}$ - a **quantile** of tomorrow's return: Value-at-Risk ::: ::: --- ## π Direction of Change: Autologit ::: {style="font-size: 30px"} The sign indicator has conditional mean equal to a conditional probability: $$E_{t-1}\,\mathbb 1\{r_t > 0\} = \Pr_{t-1}\{r_t > 0\}.$$ **Autologit(1,1):** $$\Pr_{t-1}\{r_t > 0\} = \frac{1}{1 + \exp\{-w_t\}}, \qquad w_t = \varphi + \alpha\,\mathbb 1\{r_{t-1} > 0\} + \gamma\, r_{t-1} + \beta\, w_{t-1} .$$ - $w_t$ plays the role of $\sigma_t^2$ in GARCH: a recursion driven by yesterday's data and its own lag - Estimated by maximum likelihood with the Bernoulli likelihood $\sum_t \big[\mathbb 1\{r_t>0\}\log P_t + \mathbb 1\{r_t \le 0\}\log(1 - P_t)\big]$ ::: --- ## π Conditional Quantiles: Value-at-Risk ::: {style="font-size: 30px"} ::: {.callout-note} ## Value-at-Risk The $(1-p)\cdot 100\%$ VaR is the conditional $p$-quantile of the return: $$\Pr_{t-1}\big\{r_t \le \mathrm{VaR}_t\big\} = p .$$ ::: **Indirect modelling.** If $(r_t - \mu_t)/\sigma_t = \eta_t \sim \text{iid } \mathcal D(0, 1, \vartheta)$, estimate $\hat\mu_t$, $\hat\sigma_t$ and the density $\hat f_{\mathcal D}$, then $$\widehat{\mathrm{VaR}}_t = \hat\mu_t + \hat\sigma_t\, q_p\big(\hat f_{\mathcal D}\big),$$ with $q_p$ the $p$-quantile of the fitted standardised distribution. Normal: $q_{0.01} = -2.326$, $q_{0.05} = -1.645$. ::: --- ## π Direct Modelling: CAViaR ::: {style="font-size: 30px"} Model the quantile itself, with no distribution (Engle & Manganelli, 2004): $$r_t = q_t^p + e_t, \qquad \text{Quantile}_p\big(e_t \mid I_{t-1}\big) = 0,$$ $$q_t^p = \omega + \beta\, q_{t-1}^p + \alpha\, g(r_{t-1}), \qquad \text{e.g. } g(r) = |r| .$$ Estimate by **quantile regression**: minimise $$\sum_t \rho_p\big(r_t - q_t^p\big), \qquad \rho_p(u) = u\,\big(p - \mathbb 1\{u < 0\}\big).$$ The check function $\rho_p$ penalises a return below the quantile with weight $1-p$ and one above it with weight $p$; its minimiser is the $p$-quantile. ::: --- ## π§ͺ Backtesting a VaR ::: {style="font-size: 29px"} Define the **hit** $I_t = \mathbb 1\{r_t < \widehat{\mathrm{VaR}}_t\}$. If the VaR is correct, the hits are iid Bernoulli($p$). With $x$ hits in $n$ days and $\hat\pi = x/n$, Kupiec's unconditional-coverage test is $$LR_{uc} = -2\log\big[(1-p)^{n-x}p^{x}\big] + 2\log\big[(1-\hat\pi)^{n-x}\hat\pi^{x}\big] \xrightarrow[H_0]{d} \chi^2_1 .$$ - Reject at 5% if $LR_{uc} > 3.84$: too many hits **or** too few - Christoffersen (1998) adds a test that the hits are **independent**: clustered hits mean the VaR reacts too slowly [Not in the 2016 notes: added from Kupiec (1995) and Christoffersen (1998).]{.gapnote} ::: --- ## βοΈ Worked Example: RiskMetrics VaR ::: {style="font-size: 28px"} Yesterday $\sigma_{t \mid t-1}^2 = 1.20$ ($\%^2$) and the index fell $r_t = -2.5\%$. A fund holds 10 million USD. ::: {.fragment} **1. Update.** $\sigma_{t+1 \mid t}^2 = 0.06 \times 6.25 + 0.94 \times 1.20 = 0.375 + 1.128 = 1.503$, so $\sigma_{t+1 \mid t} = 1.226\%$. ::: ::: {.fragment} **2. VaR at 99%.** $\mathrm{VaR}_{t+1} = -2.326 \times 1.226 = -2.85\%$: a loss above 285,000 USD on 1 day in 100. ::: ::: --- ## βοΈ Worked Example: Backtesting It ::: {style="font-size: 28px"} Over $n = 1000$ days the same VaR is exceeded $x = 18$ times, against $10$ expected. So $\hat\pi = 0.018$ and ::: {.fragment} $$\begin{aligned} LR_{uc} &= -2\,[982 \ln 0.99 + 18 \ln 0.01] + 2\,[982 \ln 0.982 + 18 \ln 0.018] \\ &= 185.53 - 180.30 = 5.23 > 3.84 . \end{aligned}$$ Reject: the VaR is exceeded too often. With 6 hits instead, $LR_{uc} = 1.89$: fewer hits than expected, but not significantly so. ::: ::: --- ## π§ Think-Pair-Share ```{r} #| label: tps-timer #| echo: false countdown::countdown(minutes = 3, seconds = 0, top = 0, right = 0, font_size = "2em", warn_when = 30) ``` ::: {style="font-size: 30px"} 1. A colleague writes RiskMetrics as $\sigma_t^2 = \omega + \alpha r_{t-1}^2 + \beta\sigma_{t-1}^2$ with $\alpha = 0.94$. What should $\omega$, $\alpha$, $\beta$ be? Is the model covariance stationary? 2. A bank's 99% VaR was exceeded on 4 of the last 250 days. Does Kupiec's test reject? 3. Why is it hopeless to backtest a 99% VaR on one month of data? ::: --- ## β
Think-Pair-Share: Solution ::: {style="font-size: 29px"} 1. $\omega = 0$, $\alpha = 1 - \lambda = 0.06$, $\beta = \lambda = 0.94$. Since $\alpha + \beta = 1$ it is an IGARCH: **not** covariance stationary, and there is no unconditional variance. 2. $\hat\pi = 0.016$ and $$\begin{aligned} LR_{uc} &= -2[246\ln 0.99 + 4\ln 0.01] + 2[246\ln 0.984 + 4\ln 0.016] \\ &= 41.79 - 41.02 = 0.77 < 3.84 . \end{aligned}$$ No rejection: 4 hits against 2.5 expected is within sampling error. 3. In 21 days the expected number of hits is $0.21$. Zero hits has probability 0.81 if the VaR is right, and remains likely if it is badly wrong, so the test has almost no power. ::: --- ## ποΈ Explore: The RiskMetrics Decay $\lambda$ ```{ojs} //| echo: false viewof lam = { const input = Inputs.range([0.80, 0.995], {value: 0.94, step: 0.005, label: "Ξ» (decay)"}); ['pointerdown','touchstart','mousedown','click','wheel','pointermove','touchmove'] .forEach(e => input.addEventListener(e, ev => ev.stopPropagation())); return input; } ``` ```{ojs} //| echo: false ewma = { const r = ret, n = r.length, out = []; let s2 = r.slice(0, 60).reduce((a, b) => a + b * b, 0) / 60, hits = 0, cnt = 0; for (let t = 1; t < n; t++) { s2 = lam * s2 + (1 - lam) * r[t - 1] ** 2; const v = -2.326 * Math.sqrt(s2); const hit = t >= 250 && r[t] < v; if (t >= 250) { cnt++; if (hit) hits++; } out.push({t, r: r[t], v, hit}); } const p = 0.01, ph = hits / cnt; const LR = -2 * ((cnt - hits) * Math.log(1 - p) + hits * Math.log(p)) + 2 * ((cnt - hits) * Math.log(1 - ph) + hits * Math.log(ph)); return {out, hits, cnt, LR}; } ``` ```{ojs} //| echo: false Plot.plot({ width: 1100, height: 300, marginLeft: 70, marginBottom: 50, marginTop: 30, style: {fontSize: "18px"}, x: {label: "trading day since Oct 2016"}, y: {label: "return (%)", domain: [-13, 10], grid: true}, marks: [ Plot.lineY(ewma.out, {x: "t", y: "r", stroke: "#c9d4e6", strokeWidth: 1}), Plot.lineY(ewma.out, {x: "t", y: "v", stroke: "#2f4b7c", strokeWidth: 2}), Plot.dot(ewma.out.filter(d => d.hit), {x: "t", y: "r", r: 4, fill: "#8b2635"}) ] }) ``` ```{ojs} //| echo: false html`Blue: normal 99% VaR. Red: hits. ${ewma.hits} hits in ${ewma.cnt} days (${(100 * ewma.hits / ewma.cnt).toFixed(2)}%, expected 1%) Β· Kupiec LR = ${ewma.LR.toFixed(1)} ${ewma.LR > 3.84 ? "Β· reject" : "Β· do not reject"}
` ``` --- ## π° Case Study: Backtesting S&P 500 VaR :::: columns ::: {.column width="50%"} ::: {.callout-note} ## π The question Three ways to forecast tomorrow's 1% and 5% quantile of the S&P 500 return. 1. Does RiskMetrics with normal quantiles pass Kupiec's test? 2. Does model-free historical simulation do better? 3. Does replacing the normal quantile by the empirical one (indirect modelling with $\hat f_{\mathcal D}$) fix the tails? ::: ::: ::: {.column width="50%"} ::: {.callout-tip} ## π The data **Series**: S&P 500 close, FRED `SP500` **Period**: Oct 2016 β Oct 2026; VaR evaluated on the last 2,261 days **Transformation**: $r_t = 100\,\Delta\log P_t$; $\mu_t = 0$; first 250 days used only to start the forecasts ::: ::: :::: ```{.r .display-only} sp <- read.csv("https://fred.stlouisfed.org/graph/fredgraph.csv?id=SP500") ``` --- ## π° Case Study: Hit Rates and Kupiec ```{r} #| label: case-var #| code-summary: "π¦ RiskMetrics, historical simulation, filtered HS; Kupiec LR" r <- r_all; n <- length(r); ev <- 251:n s2 <- numeric(n); s2[1] <- mean(r[1:60]^2) for (t in 2:n) s2[t] <- 0.94 * s2[t - 1] + 0.06 * r[t - 1]^2 # sigma^2_{t|t-1} z <- r / sqrt(s2) # standardised returns kupiec <- function(hit, p) { x <- sum(hit); m <- length(hit); ph <- x / m LR <- -2 * ((m - x) * log(1 - p) + x * log(p)) + 2 * ((m - x) * log(1 - ph) + x * log(ph)) c(hit_rate = 100 * ph, LR = LR, p_value = pchisq(LR, 1, lower.tail = FALSE)) } var_fc <- function(p) list( RiskMetrics_normal = qnorm(p) * sqrt(s2[ev]), historical_250d = sapply(ev, function(t) quantile(r[(t - 250):(t - 1)], p)), filtered_HS = sapply(ev, function(t) sqrt(s2[t]) * quantile(z[1:(t - 1)], p))) V1 <- var_fc(0.01); V5 <- var_fc(0.05) round(rbind(t(sapply(V1, function(v) kupiec(r[ev] < v, 0.01))), t(sapply(V5, function(v) kupiec(r[ev] < v, 0.05)))), 3) |> `rownames<-`(paste(rep(c("1%:", "5%:"), each = 3), names(V1))) ``` ::: {style="font-size: 28px"} Hit rates in %, against 1 and 5. Read the 1% rows first. ::: --- ## π° Case Study: The Normal Tail Fails ```{r} #| label: case-fig #| echo: false #| fig-width: 10 #| fig-height: 3.9 dd <- d_all[ev]; v <- V1$RiskMetrics_normal; vf <- V1$filtered_HS lab_n <- "99% VaR, RiskMetrics normal"; lab_f <- "99% VaR, filtered HS" lines <- rbind(data.frame(d = dd, v = v, what = lab_n), data.frame(d = dd, v = vf, what = lab_f)) hits <- data.frame(d = dd, r = r[ev])[r[ev] < v, ] ggplot() + geom_line(data = data.frame(d = dd, r = r[ev]), aes(d, r), colour = "#d5dde9", linewidth = 0.3) + geom_line(data = lines, aes(d, v, colour = what), linewidth = 0.7) + geom_point(data = hits, aes(d, r), colour = "#8b2635", size = 2) + scale_colour_manual(values = setNames(c("#2f4b7c", "#e08a2c"), c(lab_n, lab_f)), name = NULL) + coord_cartesian(ylim = c(-8, 6)) + labs(x = NULL, y = "Daily return (%)") + theme_minimal(base_size = 20) + theme(legend.position = "top") ``` ::: {style="font-size: 28px"} **What the data say:** at 5% all three forecasts pass. At 1% the normal RiskMetrics VaR is exceeded on 2.3% of days (red dots), and Kupiec rejects decisively; historical simulation, which reacts slowly, is also rejected. Keeping RiskMetrics for $\sigma_t$ but taking the quantile from the standardised returns passes. The failure is the normal tail, not the volatility model. ::: --- ## π VIX: Volatility Looking Forward ```{r} #| label: vix #| echo: false #| fig-width: 10 #| fig-height: 3.5 vix <- read.csv("data/vix.csv"); vix$VIXCLS <- suppressWarnings(as.numeric(vix$VIXCLS)) mm <- merge(data.frame(date = d_all, r = r_all, rm = sqrt(252 * s2)), data.frame(date = as.Date(vix$observation_date), vix = vix$VIXCLS)) mm <- na.omit(mm); N <- nrow(mm) mm$rv_fwd <- c(sapply(1:(N - 21), function(i) sqrt(252 / 21 * sum(mm$r[(i + 1):(i + 21)]^2))), rep(NA, 21)) # next 21 days, annualised ok <- !is.na(mm$rv_fwd) st <- c(mean(mm$vix[ok]), mean(mm$rv_fwd[ok]), cor(mm$vix[ok], mm$rv_fwd[ok]), cor(mm$rm[ok], mm$rv_fwd[ok]), mean(mm$vix[ok] > mm$rv_fwd[ok])) long <- rbind(data.frame(date = mm$date, v = mm$vix, what = "VIX (implied, next 30 days)"), data.frame(date = mm$date, v = mm$rv_fwd, what = "realized vol, next 21 trading days")) ggplot(long, aes(date, v, colour = what)) + geom_line(linewidth = 0.6) + scale_colour_manual(values = c("#8ba3c7", "#8b2635"), name = NULL) + coord_cartesian(ylim = c(0, 90)) + labs(x = NULL, y = "Annualised vol (%)") + theme_minimal(base_size = 20) + theme(legend.position = "top") ``` ::: {style="font-size: 28px"} The VIX is the volatility priced into one-month S&P 500 options, a risk-neutral expectation: forward-looking, but not a forecast under the real-world measure. It averages `r sprintf("%.1f", st[1])`% against `r sprintf("%.1f", st[2])`% realized, and exceeds realized vol on `r round(100 * st[5])`% of days: investors pay a **variance risk premium**. Its correlation with future realized vol is `r sprintf("%.2f", st[3])`, against `r sprintf("%.2f", st[4])` for the backward-looking RiskMetrics. ::: --- ## π Quiz #1: RiskMetrics {.quiz-question} RiskMetrics with $\lambda = 0.94$ is a GARCH(1,1) with - [$\omega = 0$, $\alpha = 0.06$, $\beta = 0.94$]{.correct data-explanation="β
ΟΒ²(t+1|t) = (1 β Ξ») rΒ²(t) + Ξ» ΟΒ²(t|tβ1): no constant, weight 1 β Ξ» = 0.06 on news and Ξ» = 0.94 on the previous variance. Ξ± + Ξ² = 1, an IGARCH."} - $\omega = 0$, $\alpha = 0.94$, $\beta = 0.06$ - $\omega = 0.06$, $\alpha = 0.94$, $\beta = 0$ - $\omega > 0$ estimated from the data, $\alpha + \beta = 0.94$ --- ## π Quiz #2: Realized Volatility {.quiz-question} Why do practitioners compute $RV_t$ from 5-minute returns rather than from every tick? - [Microstructure noise inflates RV at very high frequencies]{.correct data-explanation="β
RV converges to integrated volatility as m grows only for the efficient price. Bidβask bounce and discrete ticks add noise to every intraday return, and its contribution to RV grows with m."} - Because $RV_t$ diverges as $m \to \infty$ even without noise - Because the drift term dominates at high frequency - Because tick data are not observable --- ## π Quiz #3: Backtesting {.quiz-question} A 95% VaR is exceeded on 0 of 500 days. Kupiec's test: - [Rejects: 0 hits is far fewer than the 25 expected, so the VaR is too conservative]{.correct data-explanation="β
With x = 0, LR = β2Β·500Β·ln 0.95 = 51.3 > 3.84. Kupiec's test is two-sided: too few hits means the bank holds a VaR far beyond the 5% quantile."} - Does not reject: no hits is the best possible outcome - Cannot be computed, because $\log \hat\pi$ is undefined - Rejects only if the hits are clustered --- ## π Key Formulas ::: {style="font-size: 28px"} | | | |---|---| | RiskMetrics | $\sigma^2_{t+1\mid t} = 0.06\,r_t^2 + 0.94\,\sigma^2_{t\mid t-1}$, an IGARCH | | Realized volatility | $RV_t = \sum_{i=1}^m r_{t,i}^2 \to IV_t = \int_0^1\sigma^2_{t+s}\,ds$ | | MEM / ACD(1,1) | $x_t = \psi_t\varepsilon_t$, $E\varepsilon_t = 1$; $\ \psi_t = \omega + \alpha d_{t-1} + \beta\psi_{t-1}$ | | Autologit(1,1) | $P_t = 1/(1+e^{-w_t})$, $\ w_t = \varphi + \alpha\mathbb 1\{r_{t-1}>0\} + \gamma r_{t-1} + \beta w_{t-1}$ | | VaR, indirect | $\widehat{\mathrm{VaR}}_t = \hat\mu_t + \hat\sigma_t q_p(\hat f_{\mathcal D})$ | | CAViaR | $q_t^p = \omega + \beta q_{t-1}^p + \alpha|r_{t-1}|$, $\ \min\sum_t\rho_p(r_t - q_t^p)$ | | Kupiec | $LR_{uc} = -2\log[(1-p)^{n-x}p^x] + 2\log[(1-\hat\pi)^{n-x}\hat\pi^x] \to \chi^2_1$ | : {tbl-colwidths="[24,76]"} ::: --- ## π Summary ::: {style="font-size: 30px"} - $|r_t|$, $r_t^2$ and RiskMetrics are observable and model-free; GARCH $\hat\sigma_t^2$ is model-based - RiskMetrics is an IGARCH with $\omega = 0$, $\alpha = 0.06$, $\beta = 0.94$ - Realized volatility converges to integrated volatility; noise sets the sampling interval at 5β15 minutes - MEM and ACD model positive series as a conditional mean times a unit-mean shock - Direction (autologit) and quantiles (VaR, CAViaR) are dynamic models in their own right - On the S&P 500 the 99% normal VaR fails Kupiec's test; the tail, not the volatility, is the problem ::: --- ## π Further Reading ::: {style="font-size: 28px"} - J.P. Morgan/Reuters (1996), *RiskMetrics β Technical Document* - Andersen, Bollerslev, Diebold & Labys (2003), *Econometrica*; Barndorff-Nielsen & Shephard (2002), *JRSS B* - Engle & Russell (1998), *Econometrica*: ACD; Engle (2002), *J. Applied Econometrics*: MEM - Engle & Manganelli (2004), *JBES*: CAViaR; Kupiec (1995), *J. Derivatives*; Christoffersen (1998), *IER* - Tsay (2010), ch. 3, 5 and 7 This is the last lecture of the course. ::: 