--- title: "3.4.6 Realized vs implied volatility, VIX and VSTOXX" description: "The VIX from the chain with Cboe's method, recomputed to the cent; realized vs implied volatility, the volatility risk premium and the VSTOXX." url: "https://pythonparatrading.com/en/python-trading-course/level-3/realized-vs-implied-volatility-vix/" lang: "en" image: "https://pythonparatrading.com/og/en/python-trading-course/level-3/realized-vs-implied-volatility-vix.png" --- # 3.4.6 · Realized vs implied volatility; volatility risk premium; VIX and VSTOXX > Implied volatility is a price and realized volatility, an outcome. Between the two lies a premium that option sellers collect almost always and give back all at once. This lesson builds a VIX-like index from the option chain with Cboe's formula, checks that it matches the published one, measures the premium on synthetic data with a known answer and compares the methodology with that of the VSTOXX. - Course: [Level 3: Derivatives: futures, spreads, forex and options](https://pythonparatrading.com/en/python-trading-course/level-3/) - Module: 3.4 · Option theory and pricing - Duration: 70 min - Prerequisites: [1.6.1 · Returns, volatility, rolling correlations and drawdown](https://pythonparatrading.com/en/python-trading-course/level-1/returns-volatility-correlation-and-drawdown/), [3.4.4 · Implied volatility: Newton-Raphson, Brent, py_vollib and QuantLib](https://pythonparatrading.com/en/python-trading-course/level-3/implied-volatility-newton-brent-quantlib/), [3.4.5 · Smile, skew and term structure; volatility surface with SVI](https://pythonparatrading.com/en/python-trading-course/level-3/volatility-smile-skew-svi-surface/) ## By the end of this lesson - Compute a 30-day volatility index from an option chain with the formula of Cboe's white paper: forward, K0, strike selection, variance per expiry and interpolation. - Explain why the VIX is the square root of a variance swap and not an ATM volatility, and measure the difference. - Measure realized volatility and the volatility risk premium (implied minus subsequent realized), with its mean, its tail and its worst episodes. - Distinguish the VIX and VSTOXX methodologies using their official documents. ## Two different volatilities The word “volatility” refers to two things that should not be confused: - **Realized volatility** is a statistical measure of the past: the annualized standard deviation of the returns of a period that has already happened. With daily log returns $r_t$ and $n$ sessions, $\sigma_{\text{real}} = \sqrt{(252/n)\sum r_t^2}$ (without subtracting the mean, as in variance contracts). There are better estimators using highs and lows (the site's “Volatility Estimators” research compares them), but the close-to-close one is the one derivatives settle on. - **Implied volatility** is a price: what the options market charges today for the volatility of a future period (lessons 3.4.4 and 3.4.5). It is not a neutral forecast, because it includes what buyers pay to protect themselves. The difference between today's implied and the realized that comes later is the **volatility risk premium**. If it is positive on average, selling volatility makes money on average; the question is in exchange for what risk. Measuring it requires a constant-tenor implied volatility, and the market standard is Cboe's **VIX**: the 30-day implied volatility of the S&P 500 computed with the whole option chain, not with a single option. ```mermaid flowchart TB O[OTM options] --> V[Variance at T1 and T2] V --> I[Interpolate to 30 d] I --> X[VIX = 100 sqrt] X --> P{Compare} R[30-day realized] --> P P --> VRP[Risk premium] ``` ## From the chain to the index: Cboe's formula The VIX was born in 1993 on ATM options on the S&P 100. In 2003 Cboe, with Goldman Sachs, rebuilt it on S&P 500 options with the current formula, which does not depend on any pricing model[^cboe]. The idea comes from the theory of variance swaps (Demeterfi, Derman, Kamal and Zou, 1999; Carr and Madan, 1998): a portfolio of out-of-the-money options with weights inversely proportional to the square of the strike replicates realized variance, whatever the level of the index. Its price, therefore, is the expected variance under the pricing measure. For an expiry $T$: $$ \sigma^2 = \frac{2}{T}\sum_i \frac{\Delta K_i}{K_i^2}\,e^{RT}\,Q(K_i) - \frac{1}{T}\left(\frac{F}{K_0} - 1\right)^2 . $$ Each term has its rule in Cboe's methodology: | Element | Rule | |---|---| | $T$ | Time to expiry in 365-day years (in the real VIX, in minutes) | | $R$ | Risk-free rate at tenor $T$ (in the real VIX, Treasury curve with a cubic spline) | | $F$ | Forward from parity at the strike where $\lvert C - P \rvert$ is smallest: $F = K + e^{RT}(C - P)$, with mids | | $K_0$ | First strike equal to or below $F$ | | Options | Puts below $K_0$ and calls above, from $K_0$ outward; those with a zero bid are excluded and selection stops after two consecutive zero bids; at $K_0$, the average of put and call | | $\Delta K_i$ | Average of the distance to the neighboring strikes; at the ends, the distance to the only neighbor | | $Q(K_i)$ | Mid of the bid and the ask | Then the two expiries are interpolated to 30 days in total variance and converted to volatility: $$ \text{VIX} = 100\sqrt{\left[T_1\sigma_1^2\,\frac{N_2 - N_{30}}{N_2 - N_1} + T_2\sigma_2^2\,\frac{N_{30} - N_1}{N_2 - N_1}\right] \frac{N_{365}}{N_{30}}} . $$ The real VIX today uses options with more than 23 and fewer than 37 days, monthly and weekly, and counts time in minutes. The 2009 white paper, with only monthly expiries, took the first two expiries with at least one week to go. That monthly version is the one used by the **VORX**, the VIX-like index of the course data on the fictitious ORX index (`vol3_indices.csv`): the first two expiries with at least 8 days, tenor in calendar days and the rate linearly interpolated on the curve in `opciones3_indice.csv`. Its known answer is that it can be recomputed exactly from the published files. ## One day's VORX, step by step ```python import math import numpy as np import pandas as pd cad = pd.read_parquet("opciones3_cadena.parquet") ind = pd.read_csv("opciones3_indice.csv", parse_dates=["fecha"]) \ .set_index("fecha") vx = pd.read_csv("vol3_indices.csv", parse_dates=["fecha"]) \ .set_index("fecha") ancha = cad.pivot_table(index=["fecha", "vencimiento", "strike"], columns="tipo", values=["bid", "ask"]) ancha.columns = [f"{a}_{t}" for a, t in ancha.columns] ancha = ancha.dropna().reset_index() # strikes with call and put PLAZOS = np.array([1 / 12, 0.25, 0.5, 1.0, 2.0]) CURVA = ["tipo_1m", "tipo_3m", "tipo_6m", "tipo_1a", "tipo_2a"] def tipo(fecha, T): """Continuous rate at tenor T: linear interpolation of the curve.""" return float(np.interp(T, PLAZOS, ind.loc[fecha, CURVA])) / 100 hoy = pd.Timestamp("2025-06-04") dia = ancha[ancha["fecha"] == hoy] print({f"{(v - hoy).days} days": len(g) for v, g in dia.groupby("vencimiento")}) ``` ```output {'16 days': 48, '44 days': 45, '72 days': 40} ``` `pivot_table` leaves one row per strike with the bid and the ask of the call and of the put. That day there are three expiries in the daily chain, at 16, 44 and 72 days, with 48, 45 and 40 strikes that have both legs. ```python def hasta_dos_ceros(bid): """From K0 outward: drop zero bids; stop on finding two in a row.""" nulo = bid <= 0 pares = np.flatnonzero(nulo[:-1] & nulo[1:]) fin = pares[0] if len(pares) else len(bid) usar = ~nulo usar[fin:] = False return usar def var_cboe(g, T, R, detalle=False): K = g["strike"].to_numpy() c = ((g["bid_C"] + g["ask_C"]) / 2).to_numpy() p = ((g["bid_P"] + g["ask_P"]) / 2).to_numpy() j = np.argmin(np.abs(c - p)) F = K[j] + math.exp(R * T) * (c[j] - p[j]) i0 = np.flatnonzero(K <= F)[-1] ip = np.arange(i0 - 1, -1, -1) # puts, downward ic = np.arange(i0 + 1, len(K)) # calls, upward ip = ip[hasta_dos_ceros(g["bid_P"].to_numpy()[ip])][::-1] ic = ic[hasta_dos_ceros(g["bid_C"].to_numpy()[ic])] idx = np.r_[ip, i0, ic] Q = np.r_[p[ip], (c[i0] + p[i0]) / 2, c[ic]] Ks = K[idx] dK = np.gradient(Ks) # centered; at the edges, the neighbor aporte = 2 / T * dK / Ks ** 2 * Q * math.exp(R * T) s2 = aporte.sum() - (F / K[i0] - 1) ** 2 / T if detalle: return s2, F, K[i0], pd.DataFrame( {"strike": Ks, "tipo": ["P"] * len(ip) + ["PC"] + ["C"] * len(ic), "Q": Q, "aporte": aporte}) return s2 for v, g in dia.groupby("vencimiento"): n = (v - hoy).days T = n / 365 s2, F, K0, tabla = var_cboe(g, T, tipo(hoy, T), detalle=True) puts = tabla.loc[tabla["tipo"] == "P", "aporte"].sum() print(f"{n} days: F = {F:.2f}, K0 = {K0:.0f}, " f"{len(tabla)} strikes, σ² = {s2:.6f}, " f"vol = {100 * math.sqrt(s2):.2f}, " f"puts {puts / tabla['aporte'].sum():.0%}") ``` ```output 16 days: F = 5143.89, K0 = 5100, 32 strikes, σ² = 0.065072, vol = 25.51, puts 56% 44 days: F = 5155.03, K0 = 5150, 40 strikes, σ² = 0.067723, vol = 26.02, puts 69% 72 days: F = 5166.23, K0 = 5150, 39 strikes, σ² = 0.069099, vol = 26.29, puts 69% ``` For each expiry, the forward comes from parity at the strike with the smallest $|C - P|$, $K_0$ is the strike just below and the selection leaves 32, 40 and 39 strikes: in the 16-day one the wings with zero bids are left out. Puts contribute 56%, 69% and 69% of the variance; with skew, the low part of the smile weighs more (and the $1/K^2$ weights favor it even more). The volatility of each expiry, 25.5%, 26.0% and 26.3%, is that of a variance swap at that tenor, not the ATM one: the 44-day ATM of that day was 22.2% (lesson 3.4.5). ```python def vorx(fecha, grupos): """30-day index with the first two expiries of 8 days or more (Cboe white paper, monthly version).""" vencs = sorted(v for v in grupos if (v - fecha).days >= 8)[:2] n1, n2 = [(v - fecha).days for v in vencs] s2 = [var_cboe(grupos[v], n / 365, tipo(fecha, n / 365)) for v, n in zip(vencs, (n1, n2))] w = (n1 / 365 * s2[0] * (n2 - 30) / (n2 - n1) + n2 / 365 * s2[1] * (30 - n1) / (n2 - n1)) return 100 * math.sqrt(max(w * 365 / 30, 0.0)) grupos_hoy = dict(tuple(dia.groupby("vencimiento"))) print(f"Computed VORX: {vorx(hoy, grupos_hoy):.2f}") print(f"Published VORX: {vx.loc[hoy, 'vorx']:.2f}") ``` ```output Computed VORX: 25.89 Published VORX: 25.89 ``` With $N_1 = 16$ and $N_2 = 44$, the 30 days fall halfway and the two expiries weigh the same in total variance. The result, 25.89, matches the published one. *Fig. — Contribution of each strike to the variance of the 44-day expiry of VORX, 4 June 2025, as % of the total: puts below K0 and calls above.* Bars by strike from 2,400 to 6,200 points. Puts, on the left, contribute 69% of the variance and extend down to 2,400; calls, on the right, 23%, over a much shorter range. The strike K0 = 5,150, highlighted, contributes the tallest bar. The figure breaks down the variance of the 44-day expiry by strike. Puts extend down to 2,400 points, 53% below the index, with small but not negligible contributions; calls end much earlier, because the far ones have a zero bid. The $K_0$ bar is the tallest because the put and the call (their average) add up there with the weight of a strike close to the money. ## Twelve years recomputed ```python grupos = {} for (f, v), g in ancha.groupby(["fecha", "vencimiento"]): grupos.setdefault(f, {})[v] = g calc = pd.Series({f: vorx(f, gv) for f, gv in grupos.items()}) dif = (calc.round(2) - vx["vorx"].reindex(calc.index)).abs() print(f"{len(calc)} days; maximum difference: {dif.max():.3f}") sin = vx.index.difference(calc.index) print("Index days without a chain:", [d.date() for d in sin]) ``` ```output 3017 days; maximum difference: 0.000 Index days without a chain: [datetime.date(2019, 7, 23)] ``` The 3,017 days with a chain give the published VORX to the cent: it is the known answer of the dataset, and the proof that the code applies each rule correctly. The only index day missing, 2019-07-23, is the `dia_sin_cadena` incident of lesson 3.6.4: that day the vendor did not deliver the chain and the index cannot be recomputed. ```python ver = pd.read_parquet("opciones3_verdad.parquet") def atm30(g): """True 30-day ATM vol: θ interpolated, θ(0) = 0.""" th = np.interp(30 / 365, np.r_[0, g["dias"] / 365], np.r_[0, g["theta"]]) return 100 * math.sqrt(th * 365 / 30) vx["atm30"] = ver.groupby("fecha").apply(atm30) d_mod = vx["vorx"] - vx["vorx_modelo"] d_atm = vx["vorx"] - vx["atm30"] print(f"VORX − model swap: mean {d_mod.mean():.2f}, " f"min {d_mod.min():.2f}, max {d_mod.max():.2f}") print(f"VORX − 30-day ATM: mean {d_atm.mean():.2f}, " f"median {d_atm.median():.2f}") vx[["vorx", "vorx_modelo", "atm30"]].describe().round(2) \ .loc[["mean", "50%", "min", "max"]] ``` ```output VORX − model swap: mean 0.17, min -2.36, max 1.39 VORX − 30-day ATM: mean 2.86, median 2.58 vorx vorx_modelo atm30 mean 19.80 19.63 16.94 50% 17.66 17.37 15.13 min 9.55 9.72 8.50 max 73.98 74.16 63.02 ``` The VORX is on average 0.17 points above the variance swap of the true surface (`vorx_modelo`), with deviations from −2.36 to 1.39: it is the error of discretizing the integral over strikes, truncating the wings and interpolating between two expiries. Against the 30-day ATM volatility, instead, it is 2.86 points above on average (median 2.58): the difference is not an error, it is the skew. A VIX-like index does not measure ATM volatility, but the square root of an average of the variances of the whole smile, and with skew that average exceeds the at-the-money one. ## Realized volatility and risk premium The fair comparison is today's implied against the realized **afterward**: the VORX of day $t$ looks at the following 30 calendar days, about 21 sessions. Using the realized of the previous 21 sessions would be comparing today's price with a figure that was already known. ```python r = np.log(ind["cierre"]).diff() real_post = 100 * np.sqrt(252 * (r ** 2).rolling(21).mean()) \ .shift(-21) # sessions t+1 … t+21 real_prev = 100 * np.sqrt(252 * (r ** 2).rolling(21).mean()) prima = (vx["vorx"] - real_post).dropna() print(f"VORX − realized premium: mean {prima.mean():.2f}, " f"median {prima.median():.2f}") print(f"Days with realized above: {(prima < 0).mean():.1%}") print(f"Worst day: {prima.idxmin().date()}, {prima.min():.1f} points") p_atm = (vx["atm30"] - real_post).dropna() print(f"With the 30-day ATM: mean {p_atm.mean():.2f}, " f"median {p_atm.median():.2f}") var_prima = (vx["vorx"] ** 2 - real_post ** 2).dropna() / 100 print(f"In variance (points² / 100): mean {var_prima.mean():.2f}") ``` ```output VORX − realized premium: mean 5.30, median 6.38 Days with realized above: 11.1% Worst day: 2020-03-06, -76.5 points With the 30-day ATM: mean 2.43, median 3.88 In variance (points² / 100): mean 1.53 ``` The mean premium is 5.30 volatility points and the median, 6.38. Realized exceeds the VORX on 11.1% of the days, and the worst case is March 6, 2020: the VORX stood at 23.2 and the realized of the following 21 sessions, which include the double-digit falls of March 12 and 16, was 99.7%, a difference of −76.5 points. With the 30-day ATM volatility, the premium is smaller (2.43 on average) because part of what the VORX charges is the skew. In variance, which is what is traded in a swap, the mean difference is 153 squared volatility points (0.0153 of annual variance). In the data the premium is planted: the implied variance is between 1.0 and 1.3 times the expected one (more in calm than in stress), according to the data README. The academic reference on the variance premium in real markets is Carr and Wu (2009), in the resources; no real figure is given here. ```python anual = prima.groupby(prima.index.year).agg( media="mean", mediana="median", minimo="min", negativa=lambda z: (z < 0).mean()) anual.round(2).T ``` ```output fecha 2014 2015 2016 2017 2018 2019 2020 2021 2022 2023 2024 2025 media 3.91 4.38 4.19 4.26 6.04 6.90 5.08 7.18 5.67 6.35 5.41 4.18 mediana 4.23 5.68 5.50 5.02 8.13 7.07 11.92 8.62 6.37 7.15 5.63 6.33 minimo -4.19 -15.67 -16.19 -8.09 -16.95 -1.40 -76.46 -7.03 -9.44 -1.15 -2.88 -24.94 negativa 0.11 0.22 0.15 0.17 0.16 0.02 0.10 0.09 0.08 0.04 0.06 0.14 ``` The median is positive every year, from 4.2 in 2014 to 11.9 in 2020; the fraction of negative days ranges from 2% (2019) to 22% (2015). The minimums tell another story: −76.5 in 2020, −24.9 in 2025, −17.0 in 2018 and −16.2 in 2016, the years with stress periods in the data script. The premium is stable in the center and very unstable in the tail. *Fig. — Daily distribution of the volatility risk premium in the data, VORX minus realized volatility over the next 21 sessions, in volatility points, 2014-2025 (number of days on a log scale).* Histogram with 2-point bars. The mass lies between 0 and 15 points, with the mode between 6 and 8; mean 5.3 and median 6.4. Left of zero (highlighted), 11.1% of days, with a tail reaching −76 points in March 2020. The figure shows the whole distribution, with the number of days on a log scale so that the tail is visible: the mass between 0 and 15 points and, to the left of zero (highlighted), a long tail with an isolated group between −76 and −62 that is the 16 sessions from February 13 to March 6, 2020. ```python import statsmodels.api as sm d = pd.concat({"real_post": real_post, "vorx": vx["vorx"], "real_prev": real_prev}, axis=1).dropna() d = d.iloc[::21] # no overlap: one every 21 sessions for x_cols in (["vorx"], ["real_prev"], ["vorx", "real_prev"]): m = sm.OLS(d["real_post"], sm.add_constant(d[x_cols])).fit() coef = ", ".join(f"{c} {m.params[c]:.2f} (t {m.tvalues[c]:.1f})" for c in x_cols) print(f"{coef} · R² {m.rsquared:.2f} · n {int(m.nobs)}") ``` ```output vorx 0.70 (t 7.9) · R² 0.31 · n 142 real_prev 0.41 (t 5.3) · R² 0.17 · n 142 vorx 0.91 (t 5.5), real_prev -0.19 (t -1.5) · R² 0.32 · n 142 ``` With one observation every 21 sessions (142, with no overlap), the VORX explains 31% of the variance of the subsequent realized, against 17% for the past realized; together, 32%, and the past one loses its significance (t = −1.5). The VORX contains the information of past volatility and something more. Its slope of 0.70 (alone) says that it is an upward-biased predictor: it carries the premium inside. ## The VSTOXX and the European index of the data The **VSTOXX** is the volatility index of the EURO STOXX 50, developed by Goldman Sachs and Deutsche Börse and calculated by STOXX with Eurex OESX options. Its variance per expiry uses the same formula as the VIX, but the STOXX guide[^stoxx] changes quite a few details: | Aspect | VIX (Cboe) | VSTOXX (STOXX) | |---|---|---| | Options | Monthly and weekly SPX, from 23 to 37 days | Eurex OESX; 8 sub-indices per expiry, from 1 to 24 months | | Main index | 30 days, interpolating two expiries | 30 days (and 11 other tenors up to 360), interpolating two sub-indices | | Price of each option | Mid of the bid and the ask | The most recent among trade, mid and settlement; the mid only if bid and ask are at least 0.1 and the spread does not exceed 8% of the bid (between 1.2 and 18 points in a normal market) | | Exclusions | Zero bids; stops after two in a row | Prices below 0.5 points; a sub-index needs at least 5 options | | Rates | US Treasury curve | €STR, EURIBOR and 2-year REX, interpolated | | Time | Minutes | Seconds; snapshots every 5 seconds from 9:15 to 17:30 CET | The practical consequence is that the two indices are not interchangeable even though they move together: with wide spreads (stress), the VSTOXX can use settlement or last-trade prices where the VIX uses mids, and the 0.5-point filter trims its wings earlier. In the course data, the fictitious European index **VGX** has its volatility index **VVGX**, computed as the square root of the 30-day variance swap of its own model (there is no published VGX chain, so it cannot be recomputed): ```python rg = np.log(vx["vgx"]).diff() real_post_g = 100 * np.sqrt(252 * (rg ** 2).rolling(21).mean()) \ .shift(-21) prima_g = (vx["vvgx"] - real_post_g).dropna() print(f"VVGX: mean {vx['vvgx'].mean():.1f}, " f"VORX: mean {vx['vorx'].mean():.1f}") print(f"VVGX premium: mean {prima_g.mean():.2f}, median " f"{prima_g.median():.2f}, negative {(prima_g < 0).mean():.1%}") print(f"ORX-VGX return correlation: {r.corr(rg):.2f}") print("VORX-VVGX daily change correlation: " f"{vx['vorx'].diff().corr(vx['vvgx'].diff()):.2f}") print("Correlation of ORX with VORX changes: " f"{r.corr(vx['vorx'].diff()):.2f}") ``` ```output VVGX: mean 21.3, VORX: mean 19.8 VVGX premium: mean 5.41, median 6.16, negative 11.0% ORX-VGX return correlation: 0.82 VORX-VVGX daily change correlation: 0.96 Correlation of ORX with VORX changes: -0.61 ``` The VVGX has a mean of 21.3 against 19.8 for the VORX, a similar premium (5.41 mean, 6.16 median, negative on 11.0% of the days) and daily changes almost identical to those of the VORX (correlation 0.96), because in the data model its variance is 1.2 times that of ORX with a small noise of its own. In real markets the relationship between the VIX and the VSTOXX is close, but not that close: each has its own options market, its trading hours and its rules. The negative correlation of −0.61 between the ORX return and the VORX change is the leverage effect: implied volatility rises when the index falls. ## Exercises 1. Compute the VORX for Wednesday, June 18, 2025, when the first expiry is two days away. Which expiries enter and why? Check the result against the published one. **Solution.** ```python f = pd.Timestamp("2025-06-18") print({(v - f).days: len(g) for v, g in grupos[f].items()}) print(f"Computed {vorx(f, grupos[f]):.2f}, " f"published {vx.loc[f, 'vorx']:.2f}") ``` ```output {2: 48, 30: 45, 58: 40} Computed 25.11, published 25.11 ``` The 2-day expiry has fewer than 8 days and is skipped: the 30- and 58-day ones enter. With $N_1 = 30$, all the weight falls on the first one, and the VORX is exactly the volatility of the 30-day swap: 25.11, the same as the published one. The 8-day rule avoids using options about to expire, whose price is dominated by noise and intrinsic value. 2. Repeat the premium with the realized of the **previous** 21 sessions instead of the following ones. How much do the mean and the fraction of negative days change? Why is that version useless for evaluating a strategy? **Solution.** ```python prima_prev = (vx["vorx"] - real_prev).dropna() print(f"Mean {prima_prev.mean():.2f} (subsequent {prima.mean():.2f})") print(f"Negative {(prima_prev < 0).mean():.1%} " f"(subsequent {(prima < 0).mean():.1%})") print(f"Minimum {prima_prev.min():.1f} on {prima_prev.idxmin().date()}") ``` ```output Mean 5.33 (subsequent 5.30) Negative 12.3% (subsequent 11.1%) Minimum -36.6 on 2020-04-06 ``` The mean barely changes (5.33 against 5.30), but the fraction of negative days rises to 12.3% and the worst day moves to April 6, 2020, with −36.6: it is the day on which the realized of the previous 21 sessions captures the whole March collapse while the VORX has already come down. That comparison is useless for evaluating a volatility sale, because the position opened today wins or loses with the volatility that comes afterward, not with what has already passed. 3. In the chain of June 4, 2025, the VORX uses 32 of the 48 strikes of the 16-day expiry. What would happen to its variance if all of them were used, with the mid of the zero-bid options? Reason it out with the formula and check it. **Solution.** ```python g = grupos_hoy[pd.Timestamp("2025-06-20")] T = 16 / 365 R = tipo(hoy, T) K = g["strike"].to_numpy() c = ((g["bid_C"] + g["ask_C"]) / 2).to_numpy() p = ((g["bid_P"] + g["ask_P"]) / 2).to_numpy() j = np.argmin(np.abs(c - p)) F = K[j] + math.exp(R * T) * (c[j] - p[j]) i0 = np.flatnonzero(K <= F)[-1] Q = np.r_[p[:i0], (c[i0] + p[i0]) / 2, c[i0 + 1:]] todos = (2 / T * np.gradient(K) / K ** 2 * Q * math.exp(R * T)).sum() - (F / K[i0] - 1) ** 2 / T regla = var_cboe(g, T, R) print(f"Cboe rule: σ² = {regla:.6f} " f"(vol {100 * math.sqrt(regla):.2f})") print(f"All strikes: σ² = {todos:.6f} " f"(vol {100 * math.sqrt(todos):.2f})") ``` ```output Cboe rule: σ² = 0.065072 (vol 25.51) All strikes: σ² = 0.065269 (vol 25.55) ``` Each term $\Delta K_i\,Q(K_i)/K_i^2$ is positive, so adding options can only raise the sum (the $\Delta K$ of the neighbors change little). In the data, the variance goes from 0.065072 to 0.065269 and the volatility, from 25.51 to 25.55: little, because those options are cheap and far away. But their “price” is not a price: with a zero bid nobody buys, and the mid of 0.025 is a convention. Cboe's rule excludes them and cuts the wing after two consecutive zero bids so that the index does not depend on quotes that cannot be executed. [^cboe]: History and current rules: Cboe, *Volatility Index Methodology: Cboe Volatility Index* (version 6.0) and *Cboe Volatility Index Mathematics Methodology*; monthly version with at least one week to go: 2009 white paper. Links in the lesson resources. [^stoxx]: STOXX, *Strategy Index Guide*, September 2024 edition, section 9 (EURO STOXX 50 Volatility). Link in the resources. ## Self-assessment 1. In the VIX formula, why does each option weigh ΔK/K²? - A. Because a portfolio of out-of-the-money options with 1/K² weights replicates a contract that pays realized variance (a variance swap), with a sensitivity to variance that does not depend on the index level - B. Because Cboe decided so to give more weight to cheap options - C. To correct put-call parity - D. Because high-strike options are more liquid *Answer A: Because a portfolio of out-of-the-money options with 1/K² weights replicates a contract that pays realized variance (a variance swap), with a sensitivity to variance that does not depend on the index level.* It is the result of Demeterfi, Derman, Kamal and Zou (1999) and of Carr and Madan: the expected variance under the risk-neutral measure is 2/T times the integral of the OTM options weighted by 1/K². The VIX is 100 times the square root of that 30-day variance. 2. The VORX in the data is on average 2.9 points above the model's 30-day ATM volatility. Why? - A. Because of quote noise - B. Because the VORX uses business days - C. Because the VORX includes the dividend - D. Because a variance swap integrates the whole smile, and with skew the out-of-the-money puts have more volatility than the ATM: the square root of the swap variance ends up above the ATM volatility *Answer D: Because a variance swap integrates the whole smile, and with skew the out-of-the-money puts have more volatility than the ATM: the square root of the swap variance ends up above the ATM volatility.* The swap variance is an average of the variances of the whole smile, with more weight on the low strikes (1/K²). With a skew like that of an index, that average exceeds the ATM variance. 3. The 30-day implied volatility has exceeded the realized volatility of the following 30 days on 89% of the days in the data. Is selling volatility a safe income? - A. Yes: it wins almost nine times out of ten - B. Yes, if it is done with futures instead of options - C. No: in the remaining 11% the realized can exceed the implied by tens of points (76 in March 2020 in the data), and an unhedged short loses in a few days what was earned over years - D. No, because the premium is negative on average *Answer C: No: in the remaining 11% the realized can exceed the implied by tens of points (76 in March 2020 in the data), and an unhedged short loses in a few days what was earned over years.* The volatility risk premium is the compensation for bearing that tail risk. Its mean is positive (5.3 points in the data) and its distribution has a very long left tail. Lesson 3.6.1 measures the result of selling it. 4. Which of these differences between the VSTOXX and the VIX is correct according to the STOXX and Cboe documents? - A. The VSTOXX uses only ATM options - B. The VSTOXX takes as the price of each option the most recent among the last trade, the mid (if the spread passes a filter) and the settlement; the VIX uses the mid of the bid and the ask - C. The VSTOXX does not interpolate between expiries - D. The VSTOXX is computed on the DAX *Answer B: The VSTOXX takes as the price of each option the most recent among the last trade, the mid (if the spread passes a filter) and the settlement; the VIX uses the mid of the bid and the ask.* The STOXX guide defines an inclusion price that can be the trade, the mid or the settlement, with spread filters; Cboe uses mids and excludes the options with a zero bid. The variance formula per expiry is the same. ## Key points - The VIX is 100 times the square root of the 30-day risk-neutral variance, estimated with a portfolio of out-of-the-money options weighted by ΔK/K² and interpolated between two expiries. - With Cboe's formula on the published chain, the VORX of the data is recomputed to the cent on the 3,017 days with a chain. - Puts contribute most of the index variance because of the skew: in the 44-day expiry of June 4, 2025, 69%. - Volatility risk premium in the data: VORX minus subsequent realized, mean of 5.3 points and median of 6.4; realized exceeds it on 11% of the days, with a maximum of 76 points in March 2020. - The VSTOXX uses the same variance formula per expiry, but different prices (trade, filtered mid or settlement), different rates (€STR, EURIBOR) and sub-indices per expiry. ## Resources ### Research - [Uses of Volatility and Current Volatility - VIX1D](https://pythonparatrading.com/en/uses-of-volatility/) · Research-0003 · 5 min ### Documentation - [Cboe: Volatility Index Methodology, Cboe Volatility Index (version 6.0, February 2026)](https://cdn.cboe.com/resources/indices/Volatility_Index_Methodology_Cboe_Volatility_Index.pdf) · cdn.cboe.com - [Cboe: Cboe Volatility Index Mathematics Methodology](https://cdn.cboe.com/resources/indices/Cboe_Volatility_Index_Mathematics_Methodology.pdf) · cdn.cboe.com - [Cboe (2009): The CBOE Volatility Index, VIX (white paper with monthly expiries; OIC copy)](https://www.optionseducation.org/getmedia/5d6a69e4-8a3a-449d-8fef-be10638919c7/vixwhite.pdf) · optionseducation.org - [STOXX: Strategy Index Guide, section 9, EURO STOXX 50 Volatility (VSTOXX) (September 2024)](https://www.stoxx.com/documents/stoxxnet/Documents/News/2024/September/Stoxx_strategy_guide.pdf) · stoxx.com ### Books - [Carr, P. and Wu, L. (2009): Variance Risk Premiums. Review of Financial Studies 22(3), 1311-1341](https://doi.org/10.1093/rfs/hhn038) · doi.org ### Data - [Level 3 data: daily VORX, VGX and VVGX (CSV)](https://pythonparatrading.com/curso/datos/nivel-3/vol3_indices.csv) · Download · vol3_indices.csv Previous lesson: [3.4.5 · Smile, skew and term structure; volatility surface with SVI](https://pythonparatrading.com/en/python-trading-course/level-3/volatility-smile-skew-svi-surface/) Next lesson: [3.4.7 · Advanced models: Heston, SABR and local volatility](https://pythonparatrading.com/en/python-trading-course/level-3/heston-sabr-local-volatility/)