TEAM 2 FINAL REPORT - School of Mathematics€¦ · 0 3.77 -0.22 ρ m-1 -0.55 Team 2 Final Report....
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TEAM 2 FINAL REPORT Chris Prouty Arun O Abraham, Fanda Yang, Feng Wan, Leigh E Lommen, Shu-Ting Chiu, Sundeep Bhimireddy, Yuchuan Li Constructing a no-arbitrage volatility surface in liquid and illiquid commodity markets https://sites.google.com/site/fmmodeling11/
Transcript of TEAM 2 FINAL REPORT - School of Mathematics€¦ · 0 3.77 -0.22 ρ m-1 -0.55 Team 2 Final Report....
TEAM 2 FINAL REPORT
Chris Prouty
Arun O Abraham, Fanda Yang, Feng Wan, Leigh E Lommen, Shu-Ting Chiu, Sundeep Bhimireddy, Yuchuan Li
Constructing a no-arbitrage volatility surface in liquid and illiquid commodity markets
https://sites.google.com/site/fmmodeling11/
CDigiSurfSDigi CDigiSurfSDigi
SVI Model
� Parameterization:
�
�
�
�
�
�
2 2var( ; , , , , ) { ( ) ( ) }k a b m a b k m k mσ ρ ρ σ= + − + − +
a gives the overall level of variance
b gives the angle between the left and right
asymptotes
σ determines how smooth the vertex is
ρ determines the orientation of the graph
m translates the graph
Team 2 Final Report
• Commodity Data
• Corn (liquid)
• Live Cattle (somewhere in between)
• Milk (illiquid)
In Commodity market
CDigiSurfSDigi CDigiSurfSDigi
Find Implied Volatility
� Bisection not Newton’s method
� Only out-of-money data
Corn Milk
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CDigiSurfSDigi CDigiSurfSDigi
Fitting the SVI Model
� Use the method of least square
�Minimize the objective function:
where
� No constraints imposed
( )2
SVI Mktσ σ−∑
{ }2 2var( ; , , , , ) ( ) ( )
SVIk a b m a b k m k mσ σ ρ ρ σ= = + − + − +
Team 2 Final Report
CDigiSurfSDigi CDigiSurfSDigi
Fitting the SVI Model
� Use the method of least square
�Minimize the objective function:
where
� No constraints imposed
( )2
SVI Mktσ σ−∑
{ }2 2var( ; , , , , ) ( ) ( )
SVIk a b m a b k m k mσ σ ρ ρ σ= = + − + − +
Team 2 Final Report
-0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.150.13
0.14
0.15
0.16
0.17
0.18
0.19
log(K/S)
Vol (%
)
LCJ1
Apr 11 Live Cattle
-0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.150.13
0.14
0.15
0.16
0.17
0.18
0.19
log(K/S)
Vol (%
)
LCJ1
CDigiSurfSDigi CDigiSurfSDigi
( )2
SVI Mktσ σ−∑
{ }2 2var( ; , , , , ) ( ) ( )
SVIk a b m a b k m k mσ σ ρ ρ σ= = + − + − +
� Use the method of least square
�Minimize the objective function:
where
� No constraints imposed
Fitting the SVI Model
Team 2 Final Report
Sep11 Corn
-1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4
0.35
0.4
0.45
0.5
0.55
0.6
0.65
log(K/S)
Vol (%
)
CU1
-1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4
0.35
0.4
0.45
0.5
0.55
0.6
0.65
log(K/S)
Vol (%
)
CU1
CDigiSurfSDigi CDigiSurfSDigi
Replicating digital options
� Bull call spread to replicate digital call
K1
K2
Buy Sell
Team 2 Final Report
CDigiSurfSDigi CDigiSurfSDigi
Replicating digital options
� Bull call spread to replicate digital call
� Bear put spread to reproduce digital put
K1
K2
Sell Buy
Team 2 Final Report
CDigiSurfSDigi CDigiSurfSDigi
Replicating digital options
� Equation for pricing digital call
�
21
21
KKVega
K
C
−
−⋅−
∂
∂−
σσ
Team 2 Final Report
CDigiSurfSDigi CDigiSurfSDigi
Replicating digital options
� Equation for pricing digital call
�
� Equation for pricing digital put
�
21
21
KKVega
K
C
−
−⋅−
∂
∂−
σσ
21
21
KKVega
K
P
−
−⋅+
∂
∂ σσ
Team 2 Final Report
CDigiSurfSDigi CDigiSurfSDigi
Replicating digital options
� Equation for pricing digital call
�
� Equation for pricing digital put
�
� Relation with Black-Scholes
� Call:
� Put:
21
21
KKVega
K
C
−
−⋅−
∂
∂−
σσ
21
21
KKVega
K
P
−
−⋅+
∂
∂ σσ
21
21
KKVegaDBS
−
−⋅−−
σσ
21
21
KKVegaDBS
−
−⋅+
σσ
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CDigiSurfSDigi
Formula difference
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Option Vega
Slope of volatility
skew
Method #1 Black Scholes
Method #2 Replication
CDigiSurfSDigi
Comparison
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CZ0 Volatility Skew ATM Corn Price Difference
10-3
10-2
10-1
100
101
102
0.2
0.25
0.3
0.35
0.4
TickSize
Dig
ital option p
rice
At the money (Corn)
Replication
Black Scholes
-2 -1.5 -1 -0.5 0 0.5 10
0.5
1
1.5
2
2.5
3
log(K/S)
Vol (%
)
CZ0
CDigiSurfSDigi
Comparison
Team 2 Final Report
CZ0 Volatility Skew OTM Corn Price Difference
-2 -1.5 -1 -0.5 0 0.5 10
0.5
1
1.5
2
2.5
3
log(K/S)
Vol (%
)
CZ0
10-3
10-2
10-1
100
101
102
0.01
0.015
0.02
0.025
0.03
0.035
TickSize
Dig
ital option p
rice
Moderately out of the money (Corn)
Replication
Black Scholes
CDigiSurfSDigi
Comparison
Team 2 Final Report
CZ0 Volatility Skew Deep OTM Corn Price Difference
-2 -1.5 -1 -0.5 0 0.5 10
0.5
1
1.5
2
2.5
3
log(K/S)
Vol (%
)
CZ0
10-3
10-2
10-1
100
101
102
1
1.5
2
2.5
3
3.5
4
x 10-3
TickSize
Dig
ital option p
rice
Deep out of the money (Corn)
Replication
Black Scholes
CDigiSurfSDigi
Comparison
Team 2 Final Report
LCM1 Volatility Skew ATM LCM1 Price Difference
-0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15
0.135
0.14
0.145
0.15
0.155
0.16
0.165
0.17
0.175
0.18
log(K/S)
Vo
l (%
)
LCM1
10-3
10-2
10-1
100
101
102
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
TickSize
Dig
ital option p
rice
At the money (Live Cattle)
Replication
Black Scholes
CDigiSurfSDigi
Comparison
Team 2 Final Report
DAJ1 Volatility Skew OTM LCM1 Price Difference
-0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15
0.135
0.14
0.145
0.15
0.155
0.16
0.165
0.17
0.175
0.18
log(K/S)
Vo
l (%
)
LCM1
10-3
10-2
10-1
100
101
102
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
TickSize
Dig
ital option p
rice
Moderately out of the money (Live Cattle)
Replication
Black Scholes
CDigiSurfSDigi
Comparison
Team 2 Final Report
DAJ1 Volatility Skew Deep OTM LCM1 Price Difference
-0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15
0.135
0.14
0.145
0.15
0.155
0.16
0.165
0.17
0.175
0.18
log(K/S)
Vo
l (%
)
LCM1
10-3
10-2
10-1
100
101
102
0
0.005
0.01
0.015
0.02
0.025
0.03
TickSize
Dig
ital option p
rice
Deep out of the money (Live Cattle)
Replication
Black Scholes
CDigiSurfSDigi
Comparison
Team 2 Final Report
DAJ1 Volatility Skew DAJ1 ATM Price Difference
-0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50.195
0.2
0.205
0.21
0.215
0.22
0.225
0.23
0.235
0.24
log(K/S)
Vol (%
)
DAJ1
10-3
10-2
10-1
100
101
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
TickSize
Dig
ital option p
rice
At the money (Milk)
Replication
Black Scholes
CDigiSurfSDigi
Comparison
Team 2 Final Report
DAJ1 Volatility Skew DAJ1 OTM Price Difference
-0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50.195
0.2
0.205
0.21
0.215
0.22
0.225
0.23
0.235
0.24
log(K/S)
Vol (%
)
DAJ1
10-3
10-2
10-1
100
101
0
0.05
0.1
TickSize
Dig
ital option p
rice
Deep out of the money (Milk)
Replication
Black Scholes
Where we have volatility surface
In equity market
CDigiSurfSDigi CDigiSurfSDigi
Fitting SVI curve
� Unconstrained Optimization
�MATLAB – fminsearch
a B σ
0.23 4.01 0.28
ρ m
-1.13 -0.68
Team 2 Final Report
CDigiSurfSDigi CDigiSurfSDigi
Fitting SVI curve
� Constrained Optimization
�MATLAB – fmincon
� a > 0, b > 0, |ρ| < 1
a b σ
0 3.77 -0.22
ρ m
-1 -0.55
Team 2 Final Report
CDigiSurfSDigi CDigiSurfSDigi
Fitting SVI curve
� Constrained Optimization
�MATLAB – fmincon
� |ρ| < 1
a b σ
-26.58 13.44 -2.05
ρ m
-0.26 -0.54
Team 2 Final Report
CDigiSurfSDigi
Implied Volatility Skews for Equities
Team 2 Final Report
CDigiSurfSDigi
Implied Volatility Skews for Equities
Team 2 Final Report
CDigiSurfSDigi
Implied Volatility Skews for Equities
Team 2 Final Report
CDigiSurfSDigi CDigiSurfSDigi
SVI Volatility Surface
Team 2 Final Report
� Forward Volatility
� Thanks Jason’s team
CDigiSurfSDigi CDigiSurfSDigi
Summary of what we have done
� Find implied volatility for commodities
� Parameterize SVI variance curves
� Price digital options with volatility skew
� Build SVI implied volatility skews for SPX
� Get the volatility surface from forward volatility
� Price SPX digital
� Compare between Black-Scholes pricing evaluation and replication pricing mechanism.
Team 2 Final Report
Constructing a no-arbitrage volatility surface in liquid and illiquid commodity markets
Team 2 Final Report
Constructing a no-arbitrage volatility surface in liquid and illiquid commodity markets
Team 2 Final Report
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