Pricing S&P 500 Variance Futures based on the instantaneous variance of the S&P 500 index with double jump processes

Pricing S&P 500 Variance Futures Based on the Instantaneous Variance of S&P 500 Index with Double

Jump Processes

Shu Su¹, Jiling Cao², Xinfeng Ruan³, Wenjun Zhang²

1 School of Applied Business

Unitec Institute of Technology - Part of Te P¯ukenga 2 School of Engineering, Computer and Mathematical Sciences

Auckland University of Technology 3 Otago Business School

University of Otago

December 7, 2022

1 Introduction Background Literature review

2 Prcing S&P 500 Varaince Futures Data

Variance Futures Valuation Variance Risk Premium

Model Calibration and Estimation Process, and Results

3 Conclusion and Future Work

4 Reference

Overview

Motivation:

To better understand the changes in the U.S. stock market.

An alternative way to observe and forecast the volatility of the U.S.

stock market to reduce investment risk, leverage volatility, and diversify a portfolio.

Pricing the variance futures contract with new specifications has not been studied in the literature.

What did we do?

Explore the feature of the Standard & Poor’s 500 index (SPX) price evolution.

Discover the relationship among the S&P 500 index, CBOE volatility index (VIX), and the price of S&P 500 variance future (VA).

Deduce an accurate VA pricing formula based on the returns of SPX.

Background - SPX and VIX

Standard & Poor’s 500 Index (SPX)

Market-capitalization-weighted index of 500 leading publicly traded companies in the U.S.

Launched in 1957 by the credit rating agency Standard and Poor’s One of the best gauges of large U.S. stocks, even the entire equities market

CBOE volatility index (VIX)

The first benchmark index to measure the market’s expectation of future (30-day) volatility (Market’s ”fear gauge”)

Based on real-time prices of options on the S&P 500 Index Not tradable but can be speculated through the VIX derivatives.

SPX VS VIX

Source from: www.cboe.com/us/indices/dashboard/vix/

Background - VA

Summary product specifications for S&P 500 variance future (VA) Exchange-traded futures contracts based on the realized variance of SPX.

Launched in 2012 and listed Chicago Board Options Exchange (CBOE).

The final settlement value: the realized variance of the S&P 500 measured from the time of initial listing until the expiration of the contract.

Contract size: the contract multiplier for the S&P 500 Variance futures contract is$1 per variance unit

Literature Review

[Zhang and Huang(2010)]

Derived a formula for pricing S&P 500 3-month variance futures Found a linear relationship between the variance futures price and the value of VIX squared

The volatility risk premium is negative

[Chang et al.(2013)Chang, Jimenez-Martin, McAleer, and Amaral]

Three conditional volatility models (GARCH, GJR, and EGARCH) with three different probability densities (Gaussian, Student-t, and

Generalized Normal distribution) to value the 3- and 12-month variance futures

The value of VIX is highly correlated to the prices of 3- and 12-month variance futures

S&P 500 Variance Futures Data

42 S&P 500 variance future contracts

Contract expiration date: between 17/01/2014 and 18/12/2020 Trading period: from 24/07/2013 to 14/02/2020

Excluding the number of days between issue data and trade date is less than 90 days.

Average daily settle price: between $113.77 and$441.92

The contract with a longer trading period has a higher daily average settle price, and vice versa

Table: Descriptive statistics of S&P 500 variance futures contracts

Code No. of Obs. No. of Days Settle Price

Mean Std.

F14 54 144 188.9080 44.3120

H14 97 187 235.8399 50.3069

M14 250 369 289.3102 82.4285

U14 160 250 231.2168 67.7068

X14 87 339 228.5410 85.4947

Z14 377 496 319.3339 84.3104

F15 14 105 297.5783 146.1982

H15 159 250 313.5196 78.8818

M15 120 211 293.6429 102.3710

U15 159 250 348.1140 181.1643

Source from: www.cboe.com

Variance Futures Valuation

Issue date t0

Trade date t

Settlement date(Maturity date) T

Realized variance Implied variance strike

Let F_t^T be the price of variance future at timet, then we have, F_t^T

10,000 = 1 T −t₀

h

(t−t₀) RV +E^Qt

QV_t,Ti

. (1)

where

RV = annualized realized variance = 252×PN−1 i=1

R_i² N−1

R = log return of the S&P 500 index = ln(S /S)

Let {lnSt} be the log return of SPX log return, under physical measure P, dlnS_t =

a−Λ^P_tk¯^P+η_Sv_t−1 2v_t

dt+√

v_tdW_S,t^P +dJ_S,t^P , dvt =κv(mt−vt)dt+σv

√vtdW_v,t^P +dJ_v,t^P , dm_t= [κ_m(θ_m−m_t) +η_mm_t]dt+σ_m√

m_tdW_m,t^P ,

(2)

Under Qmeasure, we have dlnS_t =

r−q−Λ^Q_tk¯^Q− 1 2v_t

dt+√

v_tdW_S,t^Q +dJ_S,t^Q , dv_t = [κ_v(m_t−v_t) +η_vv_t]dt+σ_v√

v_tdW_v,t^Q +dJ_v,t^Q , dm_t =κ_m(θ_m−m_t)dt +σ_m√

m_tdW_m,t^Q ,

(3)

Risk premiums for the index return (η^S_J) and the variance (η^v_J) are η^S_J = Λ^P_t

µ^P_J +ρ_Jµ^P_v

−Λ^Q_t

µ^Q_J +ρ_Jµ^Q_v

, (4)

η^v_J = Λ^P_tµ^P_v −Λ^Q_tµ^Q_v. (5)

Variance Risk Premium

The variance risk premium (VRP) measures the average difference between the realized variance and the variance swap strike.

VRPt,T = 1 T −t

h

α_QV^P −α^Q_QV

mt+

β_QV^P −β_QV^Q

vt+

γ_QV^P −γ_QV^Q i

. where

α^Q_QV=

σ²J+ρ²J(µ^Qv)²+

µ^Q_J +ρJµ^Qv

2

λ^Q₁α^Qv +λ^Q₂α^Qm

+α^Qv, β^Q_QV=βv^Q

1 +λ^Q₁

σJ²+ρ²J(µ^Qv)²+

µ^Q_J +ρJµ^Qv

2 , γ_QV^Q =

σ²_J+ρ²_J(µ^Q_v)²+

µ^Q_J +ρJµ^Q_v2 h

λ^Q₀(T −t) +λ^Q₁γ^Q_v +λ^Q₂γ_m^Qi +γ_v^Q, α^P_QV=

σ²_J+ρ²_J(µ^P_v)²+

µ^P_J+ρJµ^P_v2

λ^P₁α^P_v +λ^P₂α^P_m +α^P_v, β^PQV=βv^P

1 +λ^P1

σJ²+ρ²J(µ^Pv)²+

µ^PJ+ρJµ^Pv

2 ,

Model Calibration and Estimation Process, and Results

Model Calibration and Estimation Process

Using the joint data set, including the S&P 500 index, the VIX, and the variance futures data.

Applying Euler approximation on the latent processes, {m}_t≥0 and {v}t≥0 in Eq. (2)

Applying joint estimation method which combines the Maximum Log Likelihood (MLE) and Unscented Kalman filter (UKF)

Estimation Results

Table:Parameters in Eq. (2)

Parameter ρ κm κv θm µ^Qv σm σv σJ

Value -0.64 0.08 7.01 0.00 4.05 0.01 0.35 0.43 ηv ηm λ^Q₀ λ^Q₁ λ^Q₂ µ^Q_J ρJ σe

-5.61 0.28 0.00 1.17 20.95 4.05 -0.94 0.01

Market Data VS. Estimation Data

Jul 2013 Jan 2014 Jul 2014 Jan 2015 Jul 2015 Jan 2016

Trade Date 0

200 400 600 800 1000

Settle price

Z15 Market data

SVSCJ SVSMJ SVSMCJ-R SVSMCJ

Jan 2016 Jul 2016 Jan 2017 Jul 2017 Jan 2018 Jul 2018 Jan 2019

Trade Date 200

300 400 500 600 700

Settle price

Z18

Market data SVSCJ SVSMJ SVSMCJ-R SVSMCJ

400 600 800

Settle price

Z19 Market data

SVSCJ SVSMJ SVSMCJ-R SVSMCJ

Conclusion and Future Work

Sudden and quick changes occur in the evolution processes of both the SPX returns and its variance.

The long-term mean of the SPX variance changes randomly during the selected period.

The rates of occurring sudden and quick changes depend on all latent processes.

Our pricing model yields more accurate estimation results for the variance futures contracts with a time-to-maturity that is more than or equal to 180 days.

The model cannot capture the size of the sudden decrease in the prices of VF contracts well, but can capture the decreasing tendency Apply the particle filter to improve the model estimation performance.

Reference

Chang, C., J. Jimenez-Martin, M. McAleer, and T. Amaral, 2013, The rise and fall of S&P 500 variance futures, North American Journal of Economics and Finance 25, 151–167.

Zhang, J., and Y. Huang, 2010, The CBOE S&P 500 three-month variance futures, Journal of Futures Markets 30, 48–70.