Content uploaded by Tim Xiao
Author content
All content in this area was uploaded by Tim Xiao on Apr 08, 2023
Content may be subject to copyright.
Futures Fair Value Model
The proposal for the new methodology to calculate the fair value of equity-index futures was
reviewed. The proposed method attempts to improve traditional method by taking into consideration
the dynamic rebulanching of the position (tail hedge) over the life of futures contract. We find the
model adequate for the purposes of the fair value adjustment. We present our conclusions below.
We briefly outline the mathematics of the model including the modeling process, details of the
testing method, and results.
Notation:
t
t = 0 to N where N is the futures expiry.
t
I
The value of the equity index on day t.
r
The prevailing money market interest rate to the term of the future.
t
r
One-day funding rate on day t
t
T
The number of days from t to expiry of the futures contract.
t
H
The hedge ratio on day t.
t
FV
The fair value of a futures contract on day t.
The following conditions should be satisfied for a fully hedged short futures/long index position
1111111 360/ −−−−−−− −=+− ttttttttt FVIHrHIFVIH
(1),
with the boundary condition at the contract expiry
NN IF =
(2),
and at t=N-1 (assuming the full-carry)
( )
360/1 111 −−− += NNN rIFV
(3).
Using above equations and the definition that the value at time t of the fully-hedged portfolio
established at time t-1 is indifferent with the value of index at time t, one obtains
( )
−
=
+=
1
0
00 360/1
N
i
i
rIFV
(4).
Assuming a constant funding rate (r), the above equation simplifies to the equation given in the
proposal.
( )
N
rIFV 360/1
00 +=
(5).
The two values of central concern are the expected present value of the fees to be paid to the
seller of the credit derivative and the expected present value of the protection offered to the
buyer.
It is possible to identify several sources of error between the two models to account for these small
discrepancies. These include date roll conventions, yield curve interpolation, and numerical noise.
In the construction of the test model’ accurate date roll conventions, taking into account holidays
and weekends were not implemented, rather the different maturities were simply represented as
fractions of years.
It is known that the small differences in daycount fractions due to date rolls can easily account for
the very small discrepancies between the test model and the GCD model. Thus we are completely
confident that the GCD model is performing correctly.
The identical contract priced from both curves reveals substantial differences in protection value.
Since the model relies on fee information from the market in order to compute the mark to
market of a derivative contract, it is reliable only to the extent that the input fee information is
reliable. Thus an important criterion is the efficiency of the relevant market.
Contracts that are illiquid and for which the bid-ask spreads are large will be very difficult to
price accurately with this model. In relation to the above paragraph, data of the correct type
should be used, but the priority should be to use high quality data. Thus if data for both contract
types exists, but that for one type is much inferior to that of the other, the better quality data
should be used.