TYVIX Futures Fair Value Spreadsheet Documentation
CBOE Futures Exchange
February 1, 2016
1
Scope of the document
This document describes the methodology underlying the real-time Bloomberg data-based spreadsheet for calculating non-parametric estimates for the fair value of near- and next-month futures
contracts (“TYVIX futures”) on the CBOE/CBOT 10-Year U.S. Treasury Note Volatility Index
(“TYVIX Index”).
2
Background
The methodology is based on the works of Carr and Wu (2006) and Dupire (2006), which rely on
the key principle that variance is additive in time. Consider, for example, the realized variance of
T-Note future returns over the next two months:
Realized variance (0, 2) = Realized variance (0, 1) + Realized variance (1, 2) .
This is simply the sum of two variances: the variance occurring in the first month plus that
occurring in the second month. By re-arranging terms and annualizing,
Realized variance (1, 2)
Realized variance (0, 2) Realized variance (0, 1)
=2×
−
.
1/12
2/12
1/12
A one-month TYVIX future delivers the (square root of the) left-hand side of the previous
equation that will be realized one month from now. Then, an upper bound estimate for the fair
value of a one-month TYVIX futures conrtact value is
q
F1 = 2 × TYVIX22/12 − 1 × TYVIX21/12 ,
where TYVIX1/12 and TYVIX2/12 are implied volatilities calculated using the same methodology
as the constant 30-day TYVIX, except based only on the one-month and two-month option series,
respectively, which we refer to herein as “single-series TYVIX” indexes. The theoretical derivation
and approximations underlying this result can be found in the Appendix.
3
“Single-series TYVIX” index
Because traded options do not commonly have round months to expiry, we generalize the concept
above and let the single-series TYVIX level X based on options expiring at TO be defined as
v
(
u
2 )
u
TO −t X ∆K
1
F
i
X = 100 × t
2er( 365 )
Q(Ki ) −
−1
((TO − t)/365)
K0
Ki2
i∈I
1
where
• t is the current time
O −t
• r is the ( T365
)-year zero-coupon Treasury yield, which we define for the sake of expediency
as:
– if: |TO − t − 28| < |TO − t − 91| (time in days), then: r = 4-week T-bill yield
– else: r = 3-month T-bill yield
• F is the underlying T-Note futures price
• K0 is the at-the-money (ATM) strike, defined as the first available strike below or equal to F
• I is the set of options to be included. An option with strike K ∗ belongs to I if and only if:
– K ∗ = K0
– OR K ∗ < K0 if:
∗ The option is a put with a bid price of at least $1/64
∗ No two consecutive puts with bid < $1/64 (N/A included) between K ∗ and K0
– OR K ∗ > K0 if:
∗ The option is a call with a bid price of at least $1/64
∗ No two consecutive calls bid < $1/64 (N/A included) between K ∗ and K0
• Q(K ∗ ) is the option price with strike K ∗ , defined as:
– if: K ∗ = K0 , then: Q(K ∗ ) = (Put bid + Put ask + Call bid + Call ask) / 4
– else if: K ∗ < K0 , then: Q(K ∗ ) = (Put bid + Put ask) / 2
– else if: K ∗ > K0 , then: Q(K ∗ ) = (Call bid + Call ask) / 2
• ∆K ∗ is the change in strike at level K ∗ , defined as:
– if: K ∗ = maxi∈I Ki , then: ∆K ∗ = K ∗ − max{Ki |i ∈ I, Ki 6= K ∗ }
– else if: K ∗ = mini∈I Ki , then: ∆K ∗ = min{Ki |i ∈ I, Ki 6= K ∗ } − K ∗
– else: ∆K ∗ = (min{Ki |i ∈ I, Ki > K ∗ } − max{Ki |i ∈ I, Ki < K ∗ }) /2
Calculations for the near-, next-, and third-month single-series TYVIX indexes can be found in
the worksheets titled “ssTYVIX1,” “ssTYVIX2,” and “ssTYVIX3,” respectively. NB: the underlying T-Note futures contract, and therefore price, is not always the same across all three option
series. Users should not edit any of these worksheets.
4
Upper bound on TYVIX futures prices
We now extend the generalization to fair value calculations for the near- and next-term TYVIX
futures prices, which are implemented in the worksheet titled “getUppeBound.” Users should not
edit this worksheet.
2
4.1
Notation for dealing with day count
To deal with day count and the uneven number of days between serial TY option maturities, let
us define the following notation, to be interpreted in terms of calendar days:1
• t is the current time
• F1 is the price of the near-term TYVIX futures with maturity date TF1 , upper bound t UF1 ,
and current fair value t VF1 .
• F1 is the price of the next-term TYVIX futures with maturity date TF2 , upper bound t UF2 ,
and current fair value t VF2 .
• O1 is the near-term T-Note future option series with maturity TO1 and corresponding singleseries TYVIX value XO1
• O2 is the next-term T-Note future option series with maturity TO2 and corresponding singleseries TYVIX value XO2
• O3 is the third-term T-Note future option series with maturity TO3 and corresponding singleseries TYVIX value XO3
4.2
Constraints imposed by contract design
• TF1 = TO2 − 30
• TF2 = TO3 − 30
• F1 is the forecast of the Government bond volatility from TF1 to TF1 + 30 = TO2 , and should
converge to TYVIX index at TF1
4.3
Upper bound on F2
With notation out of the way, we start with the calculation of the upper bound on the next-term
TYVIX futures price, which is the simpler case as it does not involve any super short-dated options:
s
2 × (T
2
XO
O3 − t) − XO2 × (TO2 − t)
3
U
=
t F2
TO3 − TO2
4.4
Upper bound on F1
Calculating the upper bound for F1 involves a slight added complexity when the near-term options
become super short-dated. To be consistent with the TYVIX calculation methodology, when the
near-term option series comes within a week to expiry, we extrapolate forward using the next- and
third-term single-series TYVIX values. NB: there is no theoretical justification for this practice
and is meant as an expedient to be used with much caution.
• If TO1 − t ≥ 8
s
t UF1 =
1
2 × (T
2
XO
O2 − t) − XO1 × (TO1 − t)
2
TO2 − TO1
In the actual spreadsheet implementation, time is measured in minutes instead of days.
3
(1)
eF such that
• If TO1 − t < 8, define X
1
e 2 = max TF1 − TO2 ((TO − t) × X 2 ) + TO3 − TF1 ((TO − t) × X 2 ),
(TF1 − t) × X
F1
O3
O2
3
2
TO3 − TO2
TO3 − TO2
0 ,
which ensures convergence at TF1 .
Then the upper bond is
s
t UF1 =
2 × (T
e2
XO
O2 − t) − XF1 × (TF1 − t)
2
TO2 − TF1
Note that our approach here deviates from that in Varma (2004), which documents Bloomberg’s
implementation of fair value calculations for VIX futures in its FVD calculator. We prefer our
approach for its simplicity and seamless ability to handle differences between the various maturity
cycles involved in VIX and TYVIX futures fair value calculations. For all intents and purposes,
however, the numerical difference between the two approaches should be negligible.
5
Fair value of TYVIX futures prices
As explained in Appendix A, we must subtract the upper bound by a small adjustment term to
arrive at the fair value. To this end, we follow the approach taken by Verma (2004) for calculating
the fair value of VIX futures.
Based on the assumption that T Y V IXTF is log-normally distributed,
2
TYVIXTF = UF e−σl (TF −t)+σl
√
TF −tN (0,1)
(2)
we get the adjustment equation of TYVIX future fair value as t VF = Et [TYVIXTF ], where
t VF
= t UF e −
σl2 (TF −t)
2
(3)
TF is the futures maturity and σl is the log normal volatility of TYVIX given by
σl2 =
−log(1 −
Vart (TYVIXTF )
)
2
t UF
TF − t
(4)
TYVIXt is the spot TYVIX level, which is assumed to have dynamics
dTYVIXt = κs (θs − TYVIXt )dt + σs dWt
Then we know that the variance of the TYVIX process at TF evaluated at t is given by
Vart (TYVIXTF ) = σs2
1 − e−2κs (TF −t)
2κs
We use an autoregressive model to estimate parameters using data of the past year
TYVIXi = aTYVIXi−1 + b + i ,
4
i = 1, 2, . . . , 252
(5)
The values of a, b are given by an ordinary least square regression
P251
−1 P251
P251
a
TYVIX2i
TYVIXi
TYVIXi × TYVIXi+1
i=0
i=0
i=0
P
P
=
251
252
b
252
TYVIXi
i=0 TYVIXi
i=1
According to the solution of the OU process, the corresponding parameters are given by
∆t = 1/252
log(a)
∆t
b
θs =
1−a
2κs Var()
σs2 =
1 − e−2κs ∆t
κs = −
This is implemented in the worksheets titled “getAdjustment” and “varTYVIX” and the Bloomberg
data in the latter worksheet should be refreshed manually at the beginning of each day by clicking
the “Refresh Worksheet” button in the Bloomberg plug-in tab. Otherwise, users should not edit
these worksheets.
References
Carr, Peter and Liuren Wu, 2006. “A Tale of Two Indices.” The Journal of Derivatives 16, 13-29.
Dupire, Bruno, 2006. “Model Free Results on Volatility Derivatives.” Bloomberg LP, Unpublished
manuscript.
Mele, Antonio and Yoshiki Obayashi, 2014a. “Interest Rate Variance Swaps and the Pricing of
Fixed Income Volatility.” GARP Risk Professional: Quant Perspectives, March 1-8.
Mele, Antonio and Yoshiki Obayashi, 2014b. The Price of Fixed Income Market Volatility. Book
manuscript, Swiss Finance Institute.
Varma, Arun, 2004. VIX Futures pricing : Fair Value and Upper bound TREQ 115331. Bloomberg
LP Internal Technical Document.
Appendices
A
Derivation and Approximations
We provide a comprehensive derivation of TYVIX fair value calculations and the approximations
involved. We rely on a calendar spreads approach proposed by Carr and Wu (2006) and Dupire
(2006) in the context of equity volatility.
1
Consider a generic ∆-forward looking volatility gauge (e.g., ∆ = 12
) for a generic risk,
Vx2t,∆
≡
E∗t
1
∆
5
Z
t
t+∆
στ2 dτ
,
where στ2 denotes the instantaneous variance of this risk, and the conditional expectation E∗t is
taken under some numéraire probability (Mele and Obayashi, 2014a,b).
Consider the price of a future on Vx expiring at t + δ,
Ft,t+δ ≡ Et (Vxt+δ,∆ ) ,
where now Et denotes the expectation under the risk-neutral probability. Let us, first, determine
the expected squared Vx at t + δ,
Z t+δ+∆
1
∗
2
2
Et Vxt+δ,∆ = Et Et+δ
στ dτ
∆ t+δ
Z t+δ+∆
1
2
≈ Et Et+δ
στ dτ
∆ t+δ
Z t+δ+∆
1
2
= Et
στ dτ
∆ t+δ
Z t+δ+∆
Z t+δ
1
2
2
στ dτ −
στ dτ
= Et
∆
t
t
δ+∆ 2
δ
=
(A.1)
Vxt,δ+∆ − Vx2t,δ .
∆
∆
The second line is only an approximation because the risk-neutral probability and the numéraire
probability are generally not the same, e.g., the numéraire probability is the forward probability in
our context.
1
Next, fix ∆ = 12
, and set the futures month maturity equal to δ = n∆ for integer n, such that
the approximation in (A.1) is:
Et Vx2t+n∆,∆ ≈ (n + 1) Vx2t,(n+1)∆ − nVx2t,n∆ .
(A.2)
Note that Ft+n∆,t+n∆ = Vxt+n∆,∆ , such that:
Et (Ft+n∆,t+n∆ )2 = Et Vx2t+n∆,∆ − vart (Ft+n∆,t+n∆ )
≈ (n + 1) Vx2t,(n+1)∆ − nVx2t,n∆ − vart (Ft+n∆,t+n∆ )
where the second line follows by the approximation in (A.2). Therefore, we have the following
approximations:
Ft,t+n∆ = Et (Ft+n∆,t+n∆ )
q
≈ (n + 1) Vx2t,(n+1)∆ − nVx2t,n∆ − vart (Ft+n∆,t+n∆ )
q
≈ (n + 1) Vx2t,(n+1)∆ − nVx2t,n∆ ,
where the second approximation follows by that underlying (A.1) and the third from disregarding
the variance term, vart (Ft+n∆,t+n∆ ). In the implementation of the fair value, we estimate this
variance term assuming that TYVIX is generated by an autoregressive model and by estimating
its conditional one month-ahead variance.
6
Figure 1: Screenshot of MAIN worksheet: the warning sign shows that the maturity date of TYVIX
Fut 1 is not 30 days before that of TY Opt 2.
B
Description of the worksheets
B.1
“MAIN” worksheet
This is the only worksheet users need to be familiar with.2 It displays:
1. TYVIX Index level
2. Near-, next-, and third-month single-series TYVIX index levels
3. Bid, ask, and last prices for near- and next-term TYVIX futures
4. Upper bound and fair value of near- and next-term TYVIX futures
5. Fair value spread to bid, ask, and last prices for near- and next-term TYVIX futures
The only task required of users is to maintain correct tickers for the near- and next-month
TYVIX futures (cells F5 and F6) and the near-, next-, and third-month TY options (cells F7, F8,
and F9).
The difference between the TYVIX Fut 1 and TY Opt 2 maturities should generally be 30
days, and a warning sign will appear in cell I5 when this condition does not hold to flag a potential
problem to the user.3 The same goes for TYVIX Fut 1 and TY Opt 3 maturities.
2
The one exception being that the user needs to refresh the Bloomberg data feed in the “getAdjustment” worksheet
at the beginning of each trading day. Otherwise all other worksheets are locked and not meant to be edited by the
user.
3
If the TYVIX future maturity falls on an exchange holiday, then the difference may be less than 30 days.
7
B.2
“time” worksheet
This worksheet calculates minutes to maturity for the various relevant contracts and also fetches
the four-week and three-month T-bill yields from Bloomberg to be used as proxies for zero coupon
yields corresponding to the three TY option maturity dates. All cells should be locked for editing.
B.3
“optPx” worksheet
This worksheet fetches bid and ask prices for the three TY call and put option series for a fixed
strike range of 90 to 150. If strikes outside of this range are being quoted in the market, the fair
value calculations will no longer be accurate. All cells should be locked for editing.
B.4
“ssTYVIX1” worksheet
This worksheet fetches the price of futures underlying the near-month TY option and calculates
the near-month single-series TYVIX index. All cells should be locked for editing.
B.5
“ssTYVIX2” worksheet
This worksheet fetches the price of futures underlying the next-month TY option and calculates
the next-month single-series TYVIX index. All cells should be locked for editing.
B.6
“ssTYVIX3” worksheet
This worksheet fetches the price of futures underlying the third-month TY option and calculates
the third-month single-series TYVIX index. All cells should be locked for editing.
B.7
“getUpperBound” worksheet
This worksheet calculates the upper bound for the price of the near- and next-month TYVIX
futures. All cells should be locked for editing.
B.8
“getAdjustment” worksheet
This worksheet calculates variance adjustments to convert upper bounds into fair values of nearand next-month TYVIX futures. All cells should be locked for editing.
B.9
“varTYVIX” worksheet
This worksheet estimates the AR parameters and calculates the model-implied volatility of TYVIX
to be used to calculate the variance adjustment in “getAdjustment.” At the beginning of each
trading day, users should refresh the Bloomberg data in columns A and B on this worksheet by
pressing the “Refresh Worksheet” button in the Bloomberg Plug-In menu. All cells except for those
in columns A and B should be locked for editing.
LEGAL DISCLAIMER: The TYVIX futures fair value calculator is a reference tool that provides data to the user
to facilitate the user’s own investment decisions. The calculator is not intended to provide, and should not be relied on for
investment advice or recommendations. Use of this calculator is subject to the Terms and Conditions for Use of
CBOE Websites at http://www.cboe.com/Common/TermsConditions.aspx
8