Page 1
Spread option pricing: implied volatility
implied from implied correlation
(and its implications!)
Michael Coulon
([email protected] )
University of Sussex
(work with Elisa Alòs, Universitat Pompeu Fabra, Barcelona)
Second International Conference on Computational FinanceICCF 2017, Lisbon, Portugal
Page 2
Energy or Commodity Spread Options
A general spread option payoff (at time T ) has the general form (butsometimes a = 1, b = 1 and/or K = 0):
(aXT − bYT −K)+
where XT and YT are different commodity prices (spot or forward): e.g.
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 2 / 34
Page 3
Energy or Commodity Spread Options
A general spread option payoff (at time T ) has the general form (butsometimes a = 1, b = 1 and/or K = 0):
(aXT − bYT −K)+
where XT and YT are different commodity prices (spot or forward): e.g.
Input / Output (e.g., ‘dark’ if XT is electricity, YT is coal)
Input / Output (e.g., ‘crack’ if XT is refined product, YT is crude)
Calendar (e.g., XT is Dec13 forward, YT is Jun13 forward)
Locational (e.g., XT is Henry Hub gas, YT is NorthEast gas)
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 2 / 34
Page 4
Energy or Commodity Spread Options
A general spread option payoff (at time T ) has the general form (butsometimes a = 1, b = 1 and/or K = 0):
(aXT − bYT −K)+
where XT and YT are different commodity prices (spot or forward): e.g.
Input / Output (e.g., ‘dark’ if XT is electricity, YT is coal)
Input / Output (e.g., ‘crack’ if XT is refined product, YT is crude)
Calendar (e.g., XT is Dec13 forward, YT is Jun13 forward)
Locational (e.g., XT is Henry Hub gas, YT is NorthEast gas)
Spread options are of utmost importance, due to their strong link withphysical assets (hence hedging and valuation tools). Examples above:
Coal power plant, Oil refinery, Gas storage facility, Pipeline, etc.
Optimal (unconstrained) operation mimics a string of spread options.
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 2 / 34
Page 5
Margrabe’s Approach
Margrabe (’78) introduced a formula for exchange options (K = 0):
Assume underlyings are both lognormal
Unfortunately difference (or sum) of lognormals not lognormal
Use change of numeraire approach to get a call on the price ratio:
Vt
F(2)t
= EQ̃t
[
VT
F(2)T
]
= . . . = EQ̃t
(
F(1)T
F(2)T
− 1
)+
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 3 / 34
Page 6
Margrabe’s Approach
Margrabe (’78) introduced a formula for exchange options (K = 0):
Assume underlyings are both lognormal
Unfortunately difference (or sum) of lognormals not lognormal
Use change of numeraire approach to get a call on the price ratio:
Vt
F(2)t
= EQ̃t
[
VT
F(2)T
]
= . . . = EQ̃t
(
F(1)T
F(2)T
− 1
)+
Then use Black’s formula with variance of log of ratio, e.g, for GBMs:
σ2ratio = Vart
{
log(
F(1)T /F
(2)T
)}
= (σ21 + σ2
2 − 2ρσ1σ2)(T − t)
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 3 / 34
Page 7
Margrabe’s Approach
Margrabe (’78) introduced a formula for exchange options (K = 0):
Assume underlyings are both lognormal
Unfortunately difference (or sum) of lognormals not lognormal
Use change of numeraire approach to get a call on the price ratio:
Vt
F(2)t
= EQ̃t
[
VT
F(2)T
]
= . . . = EQ̃t
(
F(1)T
F(2)T
− 1
)+
Then use Black’s formula with variance of log of ratio, e.g, for GBMs:
σ2ratio = Vart
{
log(
F(1)T /F
(2)T
)}
= (σ21 + σ2
2 − 2ρσ1σ2)(T − t)
Margrabe’s formula for options on forwards gives
Vt = e−r(T−t)[
F(1)T Φ(d+)− F
(2)T Φ(d−)
]
where Φ(d±) =log
(
F(1)T
/F(2)T
)
±12σ2ratio
σratioand σratio depends on our model.
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 3 / 34
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Spread Options - Extensions
What if K 6= 0? e.g. (F(1)T − F
(2)T −K)+
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 4 / 34
Page 9
Spread Options - Extensions
What if K 6= 0? e.g. (F(1)T − F
(2)T −K)+
No explicit formulas, but instead:
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 4 / 34
Page 10
Spread Options - Extensions
What if K 6= 0? e.g. (F(1)T − F
(2)T −K)+
No explicit formulas, but instead:
Various approximations have been proposed (e.g., ‘Kirk’s formula’)Numerical techniques: Monte Carlo, trees, PDE approaches, etc.
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 4 / 34
Page 11
Spread Options - Extensions
What if K 6= 0? e.g. (F(1)T − F
(2)T −K)+
No explicit formulas, but instead:
Various approximations have been proposed (e.g., ‘Kirk’s formula’)Numerical techniques: Monte Carlo, trees, PDE approaches, etc.
Kirk’s approximation is quite widely used, and can be understood asusing Margrabe with the second asset’s volatility adjusted:
σ̃2 = σ2
(
F(1)0
F(2)0 +K
)
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 4 / 34
Page 12
Spread Options - Extensions
What if K 6= 0? e.g. (F(1)T − F
(2)T −K)+
No explicit formulas, but instead:
Various approximations have been proposed (e.g., ‘Kirk’s formula’)Numerical techniques: Monte Carlo, trees, PDE approaches, etc.
Kirk’s approximation is quite widely used, and can be understood asusing Margrabe with the second asset’s volatility adjusted:
σ̃2 = σ2
(
F(1)0
F(2)0 +K
)
Similarly, the Levy approximation assumes F(2)T +K lognormal and
uses moment matching (identical to Kirk for small T ):
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 4 / 34
Page 13
Spread Options - Extensions
What if K 6= 0? e.g. (F(1)T − F
(2)T −K)+
No explicit formulas, but instead:
Various approximations have been proposed (e.g., ‘Kirk’s formula’)Numerical techniques: Monte Carlo, trees, PDE approaches, etc.
Kirk’s approximation is quite widely used, and can be understood asusing Margrabe with the second asset’s volatility adjusted:
σ̃2 = σ2
(
F(1)0
F(2)0 +K
)
Similarly, the Levy approximation assumes F(2)T +K lognormal and
uses moment matching (identical to Kirk for small T ):
See Carmona, Durrlemann (2003), Swindle (2014) for details more onthese and other approximations or pricing techniques, etc.
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 4 / 34
Page 14
Implied Correlations
Implied correlation ρimp is the number which makes the Margrabe formula(or potentially Kirk, etc.) match the true (market) price! Any problems?
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 5 / 34
Page 15
Implied Correlations
Implied correlation ρimp is the number which makes the Margrabe formula(or potentially Kirk, etc.) match the true (market) price! Any problems?
As we know, the prices are not lognormal!
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 5 / 34
Page 16
Implied Correlations
Implied correlation ρimp is the number which makes the Margrabe formula(or potentially Kirk, etc.) match the true (market) price! Any problems?
As we know, the prices are not lognormal!
Furthermore, prices may be outside Margrabe’s ρ ∈ [−1, 1] range
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 5 / 34
Page 17
Implied Correlations
Implied correlation ρimp is the number which makes the Margrabe formula(or potentially Kirk, etc.) match the true (market) price! Any problems?
As we know, the prices are not lognormal!
Furthermore, prices may be outside Margrabe’s ρ ∈ [−1, 1] range
=⇒ ρimp may be undefined!
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 5 / 34
Page 18
Implied Correlations
Implied correlation ρimp is the number which makes the Margrabe formula(or potentially Kirk, etc.) match the true (market) price! Any problems?
As we know, the prices are not lognormal!
Furthermore, prices may be outside Margrabe’s ρ ∈ [−1, 1] range
=⇒ ρimp may be undefined!
Finally, ρimp depends on choice of (implied) volatilities σ1, σ2
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 5 / 34
Page 19
Implied Correlations
Implied correlation ρimp is the number which makes the Margrabe formula(or potentially Kirk, etc.) match the true (market) price! Any problems?
As we know, the prices are not lognormal!
Furthermore, prices may be outside Margrabe’s ρ ∈ [−1, 1] range
=⇒ ρimp may be undefined!
Finally, ρimp depends on choice of (implied) volatilities σ1, σ2
=⇒ a ‘strike convention’ k1, k2 is needed
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 5 / 34
Page 20
Implied Correlations
Implied correlation ρimp is the number which makes the Margrabe formula(or potentially Kirk, etc.) match the true (market) price! Any problems?
As we know, the prices are not lognormal!
Furthermore, prices may be outside Margrabe’s ρ ∈ [−1, 1] range
=⇒ ρimp may be undefined!
Finally, ρimp depends on choice of (implied) volatilities σ1, σ2
=⇒ a ‘strike convention’ k1, k2 is needed
Building on Rebonato’s well known quote, here we have something like:
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 5 / 34
Page 21
Implied Correlations
Implied correlation ρimp is the number which makes the Margrabe formula(or potentially Kirk, etc.) match the true (market) price! Any problems?
As we know, the prices are not lognormal!
Furthermore, prices may be outside Margrabe’s ρ ∈ [−1, 1] range
=⇒ ρimp may be undefined!
Finally, ρimp depends on choice of (implied) volatilities σ1, σ2
=⇒ a ‘strike convention’ k1, k2 is needed
Building on Rebonato’s well known quote, here we have something like:
Implied correlation is the wrong number put in the wrong formula alongwith two other wrong numbers from wrong formulas to get the right price.
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 5 / 34
Page 22
Spread Options - Common Applications
Strips of spread options approximate physical assets, and may requiresumming over huge numbers of maturities (e.g. hourly over decades). e.g.,
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 6 / 34
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Spread Options - Common Applications
Strips of spread options approximate physical assets, and may requiresumming over huge numbers of maturities (e.g. hourly over decades). e.g.,
Power plant valuation / Tolling agreements (renting a power plant):
V0 ≈ EQ
[
N∑
i=1
e−rtiCmax(Sti −HGti −K)+
]
,
where Cmax is plant capacity, S is power price, G is gas price, H is‘heat rate’, K other operating costs (per MWh).
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 6 / 34
Page 24
Spread Options - Common Applications
Strips of spread options approximate physical assets, and may requiresumming over huge numbers of maturities (e.g. hourly over decades). e.g.,
Power plant valuation / Tolling agreements (renting a power plant):
V0 ≈ EQ
[
N∑
i=1
e−rtiCmax(Sti −HGti −K)+
]
,
where Cmax is plant capacity, S is power price, G is gas price, H is‘heat rate’, K other operating costs (per MWh).
Gas storage facility valuation (or rental):
V0 ≈ EQ
[
N∑
i=1
e−rti+1Cmax(
F (ti, ti+1)− er(ti+1−ti)F (ti, ti))+]
,
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 6 / 34
Page 25
Spread Options - Common Applications
Strips of spread options approximate physical assets, and may requiresumming over huge numbers of maturities (e.g. hourly over decades). e.g.,
Power plant valuation / Tolling agreements (renting a power plant):
V0 ≈ EQ
[
N∑
i=1
e−rtiCmax(Sti −HGti −K)+
]
,
where Cmax is plant capacity, S is power price, G is gas price, H is‘heat rate’, K other operating costs (per MWh).
Gas storage facility valuation (or rental):
V0 ≈ EQ
[
N∑
i=1
e−rti+1Cmax(
F (ti, ti+1)− er(ti+1−ti)F (ti, ti))+]
,
Approximation above sometimes called ‘virtual storage’ (assume canempty/fill instantly as often as we would like, limited only ∆t).
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 6 / 34
Page 26
Implied Correlation for Structural Power Models
Results from Carmona, Coulon & Schwarz (2013) for spark spread optionsusing a complex structural power price model:
7.5 8 8.5 9 9.5 10 10.5 11 11.5 12
0.65
0.7
0.75
0.8
0.85Implied Correlation (Dark Spread)
hc (covering range of allowed values)
implie
d c
oal pow
er
corr
ela
tion
µ
d=0.3
µd=0.5
µd=0.7
Implied correlation ‘frowns’ are observed in many cases (in simpler models,and in the market), BUT does this correlation structure make any sense?
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 7 / 34
Page 27
The Strike Convention
Goal: to price an exchange option via Magrabe, using the impliedvolatilities I1, I2 of vanilla options:
BS (T, x, y, γ) , where γ =√
I21 + I22 − 2ρI1I2
where BS denotes the classical Black-Scholes function in terms of thelog-prices x := logS1
0 , y := log S20 .
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 8 / 34
Page 28
The Strike Convention
Goal: to price an exchange option via Magrabe, using the impliedvolatilities I1, I2 of vanilla options:
BS (T, x, y, γ) , where γ =√
I21 + I22 − 2ρI1I2
where BS denotes the classical Black-Scholes function in terms of thelog-prices x := logS1
0 , y := log S20 .
Problem: I1 = I(x, k1) and I2 = I2(y, k2) depend on the strike!!!
Therefore, what is the optimal strike choice?
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 8 / 34
Page 29
The Strike Convention
Goal: to price an exchange option via Magrabe, using the impliedvolatilities I1, I2 of vanilla options:
BS (T, x, y, γ) , where γ =√
I21 + I22 − 2ρI1I2
where BS denotes the classical Black-Scholes function in terms of thelog-prices x := logS1
0 , y := log S20 .
Problem: I1 = I(x, k1) and I2 = I2(y, k2) depend on the strike!!!
Therefore, what is the optimal strike choice?
Solution: Defining γ̂ via the ‘true’ (model) price V0:
V0 = e−rTE(S1T − S2
T )+ = BS (T, x, y, γ̂(x, y)) ,
our problem reduces to find k1, k2 such that
γ(x, y) = γ̂(x, y).
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 8 / 34
Page 30
Implied Correlation and the Strike Convention
Note: As implied correlation ρ̂ is defined directly from ‘implied gamma’:
γ̂(x, y) =√
I21 + I22 − 2ρ̂I1I2
our problem is equivalent to finding k1, k2 such that
ρ(x, y) = ρ̂(x, y).
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 9 / 34
Page 31
Implied Correlation and the Strike Convention
Note: As implied correlation ρ̂ is defined directly from ‘implied gamma’:
γ̂(x, y) =√
I21 + I22 − 2ρ̂I1I2
our problem is equivalent to finding k1, k2 such that
ρ(x, y) = ρ̂(x, y).
Key idea: Choose the strike convention such that spread price isconsistent with known (historically estimated) correlation, if possible also:
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 9 / 34
Page 32
Implied Correlation and the Strike Convention
Note: As implied correlation ρ̂ is defined directly from ‘implied gamma’:
γ̂(x, y) =√
I21 + I22 − 2ρ̂I1I2
our problem is equivalent to finding k1, k2 such that
ρ(x, y) = ρ̂(x, y).
Key idea: Choose the strike convention such that spread price isconsistent with known (historically estimated) correlation, if possible also:
keeping methodology as model-independent as possible
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 9 / 34
Page 33
Implied Correlation and the Strike Convention
Note: As implied correlation ρ̂ is defined directly from ‘implied gamma’:
γ̂(x, y) =√
I21 + I22 − 2ρ̂I1I2
our problem is equivalent to finding k1, k2 such that
ρ(x, y) = ρ̂(x, y).
Key idea: Choose the strike convention such that spread price isconsistent with known (historically estimated) correlation, if possible also:
keeping methodology as model-independent as possible
pricing consistently across all moneyness... no more FROWNS! :)
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 9 / 34
Page 34
Implied Correlation and the Strike Convention
Note: As implied correlation ρ̂ is defined directly from ‘implied gamma’:
γ̂(x, y) =√
I21 + I22 − 2ρ̂I1I2
our problem is equivalent to finding k1, k2 such that
ρ(x, y) = ρ̂(x, y).
Key idea: Choose the strike convention such that spread price isconsistent with known (historically estimated) correlation, if possible also:
keeping methodology as model-independent as possible
pricing consistently across all moneyness... no more FROWNS! :)
Methodology: Expand γ and γ̂ (or ρ and ρ̂) to first order as functions of y,in the short-time limit. Then match terms to solve for k1, k2.
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 9 / 34
Page 35
The Underlying Model
Stochastic volatility model for the two assets (with r = 0 for simplicity):
dS1t
S1t
= λ1σtdW(1)t , x = log(S1
0)
dS2t
S2t
= λ2σtdW(2)t , y = log(S2
0)
with λ1, λ2 > 0 and σt is a non-negative square integrable process drivenby another Brownian motion Zt, with correlations
⟨
W 1t , Z
⟩
= ρ1,⟨
W 2t , Z
⟩
= ρ2,⟨
W 1t ,W
2t
⟩
= ρ.
(note: can use fractional volatility models as in Alòs, León, Vives (2007))
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 10 / 34
Page 36
The Underlying Model
Stochastic volatility model for the two assets (with r = 0 for simplicity):
dS1t
S1t
= λ1σtdW(1)t , x = log(S1
0)
dS2t
S2t
= λ2σtdW(2)t , y = log(S2
0)
with λ1, λ2 > 0 and σt is a non-negative square integrable process drivenby another Brownian motion Zt, with correlations
⟨
W 1t , Z
⟩
= ρ1,⟨
W 2t , Z
⟩
= ρ2,⟨
W 1t ,W
2t
⟩
= ρ.
(note: can use fractional volatility models as in Alòs, León, Vives (2007))
But for now a simple numerical example using the Heston Model:
dσ2t = κ
(
θ − σ2t
)
dt+ ν√
σ2t dZt
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 10 / 34
Page 37
Implied Correlation and the Strike Convention
Example: (F(1)T − F
(2)T )+, with T = 0.05,
Heston params κ = 1.5, θ = 0.15, ν = 0.5, σ0 = 0.15, λ1 = 1.5, λ2 = 1,
and correlations ρ = 0.5, ρ1 = −0.4, ρ2 = −0.5:
60 70 80 90 100 110 120 130 140
strike
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
implied vol skew - asset 1
implied vol skew - asset 2
1.1
15
1.2
1.3
1210
1.4
10
OTM spread option prices (varying strike convention)
moneyness F2
0/k
2
1.5
8
moneyness F1
0/k
1
6
1.6
54
20 0
Implied vol skews (left) and spread prices (right) for OTM: F(1)0 = 90,F
(2)0 = 100
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 11 / 34
Page 38
A Brief Intermission (sorry, no coffee)...
At this point, you may be wondering about one (or all) of the following:
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 12 / 34
Page 39
A Brief Intermission (sorry, no coffee)...
At this point, you may be wondering about one (or all) of the following:
1 What are we doing again?
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 12 / 34
Page 40
A Brief Intermission (sorry, no coffee)...
At this point, you may be wondering about one (or all) of the following:
1 What are we doing again?
2 What is the point of it anyway?
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 12 / 34
Page 41
A Brief Intermission (sorry, no coffee)...
At this point, you may be wondering about one (or all) of the following:
1 What are we doing again?
2 What is the point of it anyway?
3 What do people ‘normally’ do?
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 12 / 34
Page 42
1. What are we doing again?
An intuitive way of thinking about the aims / benefits:
‘Improving’ Margrabe to make it more consistent, both:
between different options (across strikes)with historical correlation estimates
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 13 / 34
Page 43
1. What are we doing again?
An intuitive way of thinking about the aims / benefits:
‘Improving’ Margrabe to make it more consistent, both:
between different options (across strikes)with historical correlation estimates
Without fitting a full model, we can translate information contained inthe two observed volatility smiles / skews into spread option pricing.
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 13 / 34
Page 44
1. What are we doing again?
An intuitive way of thinking about the aims / benefits:
‘Improving’ Margrabe to make it more consistent, both:
between different options (across strikes)with historical correlation estimates
Without fitting a full model, we can translate information contained inthe two observed volatility smiles / skews into spread option pricing.
Aside: Is an implied correlation frown logical? Swindle (2014) argues...
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 13 / 34
Page 45
1. What are we doing again?
An intuitive way of thinking about the aims / benefits:
‘Improving’ Margrabe to make it more consistent, both:
between different options (across strikes)with historical correlation estimates
Without fitting a full model, we can translate information contained inthe two observed volatility smiles / skews into spread option pricing.
Aside: Is an implied correlation frown logical? Swindle (2014) argues...
variations in implied correlation are “purely an artifact of theinteraction of skew with the Margrabe formulation."
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 13 / 34
Page 46
2. What is the point of it anyway?
These are European spread options... why try to use Margrabe anyway?
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 14 / 34
Page 47
2. What is the point of it anyway?
These are European spread options... why try to use Margrabe anyway?
Simpler approaches are still common, especially in energy industry.
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 14 / 34
Page 48
2. What is the point of it anyway?
These are European spread options... why try to use Margrabe anyway?
Simpler approaches are still common, especially in energy industry.
Computation times can still be a big problem:
if maturity frequency is high (e.g. hourly)or if a single valuation is only the first step (e.g., risk management)
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 14 / 34
Page 49
2. What is the point of it anyway?
These are European spread options... why try to use Margrabe anyway?
Simpler approaches are still common, especially in energy industry.
Computation times can still be a big problem:
if maturity frequency is high (e.g. hourly)or if a single valuation is only the first step (e.g., risk management)
Thompson (2016) uses a PDE approach with analytic radial basis function(RBF) expansions to study the valuation and associated credit risk (PFEand CVA) of natural gas storage facilities. He notes that
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 14 / 34
Page 50
2. What is the point of it anyway?
These are European spread options... why try to use Margrabe anyway?
Simpler approaches are still common, especially in energy industry.
Computation times can still be a big problem:
if maturity frequency is high (e.g. hourly)or if a single valuation is only the first step (e.g., risk management)
Thompson (2016) uses a PDE approach with analytic radial basis function(RBF) expansions to study the valuation and associated credit risk (PFEand CVA) of natural gas storage facilities. He notes that
for a moderately sized firm, “the number of individual deal level valuationsthat need to be done each day to price and manage counter-party credit
risk can easily number into the many trillions."
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 14 / 34
Page 51
3. What do people ‘normally’ do?
Recalling that x = log(S10), y = log(S1
0), what ‘strike conventions’ mightwe try? (for strikes ek1 , ek2)
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 15 / 34
Page 52
3. What do people ‘normally’ do?
Recalling that x = log(S10), y = log(S1
0), what ‘strike conventions’ mightwe try? (for strikes ek1 , ek2)
1 k1 = x, k2 = y (use ATM vol for both assets)
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 15 / 34
Page 53
3. What do people ‘normally’ do?
Recalling that x = log(S10), y = log(S1
0), what ‘strike conventions’ mightwe try? (for strikes ek1 , ek2)
1 k1 = x, k2 = y (use ATM vol for both assets)
2 k1 = y, k2 = x (fix opposite leg as strike to look up the imp vol)
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 15 / 34
Page 54
3. What do people ‘normally’ do?
Recalling that x = log(S10), y = log(S1
0), what ‘strike conventions’ mightwe try? (for strikes ek1 , ek2)
1 k1 = x, k2 = y (use ATM vol for both assets)
2 k1 = y, k2 = x (fix opposite leg as strike to look up the imp vol)
3 k1 = k2 =x+y2 (a compromise?)
Which seems more logical?
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 15 / 34
Page 55
3. What do people ‘normally’ do?
Recalling that x = log(S10), y = log(S1
0), what ‘strike conventions’ mightwe try? (for strikes ek1 , ek2)
1 k1 = x, k2 = y (use ATM vol for both assets)
2 k1 = y, k2 = x (fix opposite leg as strike to look up the imp vol)
3 k1 = k2 =x+y2 (a compromise?)
Which seems more logical? Hard to say! What do people do?
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 15 / 34
Page 56
3. What do people ‘normally’ do?
Recalling that x = log(S10), y = log(S1
0), what ‘strike conventions’ mightwe try? (for strikes ek1 , ek2)
1 k1 = x, k2 = y (use ATM vol for both assets)
2 k1 = y, k2 = x (fix opposite leg as strike to look up the imp vol)
3 k1 = k2 =x+y2 (a compromise?)
Which seems more logical? Hard to say! What do people do?Very little academic literature on this issue:
Alexander and Venkatramanan (2011): mention using first approachabove, and testing a number of alternatives, but in a different context.
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 15 / 34
Page 57
3. What do people ‘normally’ do?
Recalling that x = log(S10), y = log(S1
0), what ‘strike conventions’ mightwe try? (for strikes ek1 , ek2)
1 k1 = x, k2 = y (use ATM vol for both assets)
2 k1 = y, k2 = x (fix opposite leg as strike to look up the imp vol)
3 k1 = k2 =x+y2 (a compromise?)
Which seems more logical? Hard to say! What do people do?Very little academic literature on this issue:
Alexander and Venkatramanan (2011): mention using first approachabove, and testing a number of alternatives, but in a different context.
Swindle (2014): discusses issue at length, suggests ‘vol look upheutristic’ (2nd above) and investigates size of impact. He notes:
“skew risk can manifest itself as spurious correlation risk simply due tothe look-up heuristic."
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 15 / 34
Page 58
3. What do people ‘normally’ do?
Recalling that x = log(S10), y = log(S1
0), what ‘strike conventions’ mightwe try? (for strikes ek1 , ek2)
1 k1 = x, k2 = y (use ATM vol for both assets)
2 k1 = y, k2 = x (fix opposite leg as strike to look up the imp vol)
3 k1 = k2 =x+y2 (a compromise?)
Which seems more logical? Hard to say! What do people do?Very little academic literature on this issue:
Alexander and Venkatramanan (2011): mention using first approachabove, and testing a number of alternatives, but in a different context.
Swindle (2014): discusses issue at length, suggests ‘vol look upheutristic’ (2nd above) and investigates size of impact. He notes:
“skew risk can manifest itself as spurious correlation risk simply due tothe look-up heuristic."
Can we investigate this in the Heston model example?Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 15 / 34
Page 59
Implied Correlation and the Strike Convention
Comparing the three strike conventions for different F 20 (moneyness):
90 92 94 96 98 100 102 104 106 108 110
F2
0
0.49
0.495
0.5
0.505
0.51
0.515implied correlation (varying strike convention)
1. ATM
2. vol look-up
3. midpoint
So perhaps the midpoint idea is the best?Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 16 / 34
Page 60
Implied Correlation and the Strike Convention
Comparing the three strike conventions for different F 20 (moneyness):
90 92 94 96 98 100 102 104 106 108 110
F2
0
0.49
0.495
0.5
0.505
0.51
0.515implied correlation (varying strike convention)
1. ATM
2. vol look-up
3. midpoint
So perhaps the midpoint idea is the best? NOT ALWAYS!Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 16 / 34
Page 61
What about other parameter sets?
(i) Left: ρ1 = −0.5, ρ2 = 0.4; (ii) Right: ρ1 = ρ2 = 0.1, ν = 1.5
0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25
moneyness F/k
0.35
0.4
0.45
0.5
0.55
0.6
0.65
implied vol skew - asset 1
implied vol skew - asset 2
0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25
moneyness F/k
0.35
0.4
0.45
0.5
0.55
0.6
0.65
implied vol skew - asset 1
implied vol skew - asset 2
4.2
1.4
4.3
4.4
4.5
1.2 1.3
4.6
1.2
4.7
ATM spread option prices (varying strike convention)
moneyness F2
0/k
2
1 1.1
4.8
moneyness F1
0/k
1
1
4.9
0.8 0.90.8
0.6 0.7
4.4
1.3
4.6
1.2
1.31.1
4.8
1.2
moneyness F2
0/k
2
11.1
ATM spread option prices (varying strike convention)
moneyness F1
0/k
1
5
10.9
0.90.8
5.2
0.80.7 0.7
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 17 / 34
Page 62
Implied Correlation and the Strike Convention
Key Observations:
The best choice of convention can vary significantly across cases
In some cases, best choice is NOT between the two simplest choices(ATM vs vol look up)
High vol of vol case (steep smiles) leads to implied correlation frowns.
Pricing differences can end up large, especially for OTM case.
90 92 94 96 98 100 102 104 106 108 110
F2
0
0.42
0.44
0.46
0.48
0.5
0.52
0.54
0.56implied correlation (varying strike convention)
1. ATM
2. vol look-up
3. midpoint
90 92 94 96 98 100 102 104 106 108 110
F2
0
0.46
0.465
0.47
0.475
0.48
0.485
0.49
0.495
0.5
0.505implied correlation (varying strike convention)
1. ATM
2. vol look-up
3. midpoint
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 18 / 34
Page 63
Towards an Optimal Strike Convention
Recall idea: seek k1(x, y), k2(x, y) such that ρ̂(k1, k2) = ρ.
A first order Taylor expansion (around the ATM point x = y) gives us:
ρ̂ (T, x, y, k1, k2) ≈ ρ̂ (T, x, x, k1, k2) +∂ρ̂
∂y(T, x, y, k1, k2)|x=y (y − x)
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 19 / 34
Page 64
Towards an Optimal Strike Convention
Recall idea: seek k1(x, y), k2(x, y) such that ρ̂(k1, k2) = ρ.
A first order Taylor expansion (around the ATM point x = y) gives us:
ρ̂ (T, x, y, k1, k2) ≈ ρ̂ (T, x, x, k1, k2) +∂ρ̂
∂y(T, x, y, k1, k2)|x=y (y − x)
Results for short T ATM vol imply limT→0 ρ̂ (T, x, x, k1, k2) = ρ.
Hence seek to minimize the quantity limT→0∂ρ̂∂y (T, x, y, k1, k2)|x=y .
Or equivalently rewrite the problem in terms of γ instead of ρ...
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 19 / 34
Page 65
Short-time expansion for γ (and γ̂)
Using the fact that
limT→0
I1(x, x) = λ1σ0, limT→0
I2(x, x) = λ2σ0
and a first-order Taylor expansion we get (with C =√
λ21 + λ2
2 − 2ρλ1λ2):
γ(x, y) ≈ Cσ0+1
C
[
(λ1 − ρλ2)∂I1∂z
∂k1∂y
+ (λ2 − ρλ1)
(
∂I2∂z
∂k2∂y
+∂I2∂y
)]
(x−y),
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 20 / 34
Page 66
Short-time expansion for γ (and γ̂)
Using the fact that
limT→0
I1(x, x) = λ1σ0, limT→0
I2(x, x) = λ2σ0
and a first-order Taylor expansion we get (with C =√
λ21 + λ2
2 − 2ρλ1λ2):
γ(x, y) ≈ Cσ0+1
C
[
(λ1 − ρλ2)∂I1∂z
∂k1∂y
+ (λ2 − ρλ1)
(
∂I2∂z
∂k2∂y
+∂I2∂y
)]
(x−y),
Malliavin Calculus techniques (see Alòs, León and Vives (2007)) give
limT→0
Tα∂I1∂y
(x, x) = −ρ1
2λ21σ
20
limT→0
1
T 2−αE
(∫ T
0
∫ T
sDZ
s σ2ududs
)
,
where α := H − 1/2 (for diffusion models H = 1/2 and α = 0) and Ddenotes the Malliavin derivative operator (e.g. D = ν
2 for Heston).
Then via similar expressions for limT→0 Tα ∂I2
∂y (x, x), limT→0 Tα ∂γ̂∂y (x, x)...
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 20 / 34
Page 67
Optimal Strike Convention - Key Result
Main theoretical result: By matching terms in the expansions for γ andγ̂, we have (under suitable integrability conditions, and for λ1 6= 0)
∂k1∂y
∂I1∂z
(
1−ρλ2
λ1
)
+∂k2∂y
∂I2∂z
(
λ2
λ1− ρ
)
=∂I1∂z
− ρ∂I2∂z
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 21 / 34
Page 68
Optimal Strike Convention - Key Result
Main theoretical result: By matching terms in the expansions for γ andγ̂, we have (under suitable integrability conditions, and for λ1 6= 0)
∂k1∂y
∂I1∂z
(
1−ρλ2
λ1
)
+∂k2∂y
∂I2∂z
(
λ2
λ1− ρ
)
=∂I1∂z
− ρ∂I2∂z
Notice that:
This equation relating ∂k1∂y and ∂k2
∂y does not depend on the specificmodel for σt (can even include fractional volatility).
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 21 / 34
Page 69
Optimal Strike Convention - Key Result
Main theoretical result: By matching terms in the expansions for γ andγ̂, we have (under suitable integrability conditions, and for λ1 6= 0)
∂k1∂y
∂I1∂z
(
1−ρλ2
λ1
)
+∂k2∂y
∂I2∂z
(
λ2
λ1− ρ
)
=∂I1∂z
− ρ∂I2∂z
Notice that:
This equation relating ∂k1∂y and ∂k2
∂y does not depend on the specificmodel for σt (can even include fractional volatility).
Can substitute for key market observables from vanilla options,∂I1∂z ,
∂I2∂z from ATM skews, and λ2
λ1from ratio of ATM vol levels.
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 21 / 34
Page 70
Optimal Strike Convention - Key Result
Main theoretical result: By matching terms in the expansions for γ andγ̂, we have (under suitable integrability conditions, and for λ1 6= 0)
∂k1∂y
∂I1∂z
(
1−ρλ2
λ1
)
+∂k2∂y
∂I2∂z
(
λ2
λ1− ρ
)
=∂I1∂z
− ρ∂I2∂z
Notice that:
This equation relating ∂k1∂y and ∂k2
∂y does not depend on the specificmodel for σt (can even include fractional volatility).
Can substitute for key market observables from vanilla options,∂I1∂z ,
∂I2∂z from ATM skews, and λ2
λ1from ratio of ATM vol levels.
Next step: Choose an appropriate form for k1(x, y), k2(x, y) and simplifythe expression above.
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 21 / 34
Page 71
Optimal Log-Linear Strike Conventions
We assume a symmetric log-linear strike convention of the form:
k1(x, y) = (1− a)x+ ay
k2(x, y) = ax+ (1− a) y,
typically for a ∈ [0, 1], (but can also let a ∈ R).
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 22 / 34
Page 72
Optimal Log-Linear Strike Conventions
We assume a symmetric log-linear strike convention of the form:
k1(x, y) = (1− a)x+ ay
k2(x, y) = ax+ (1− a) y,
typically for a ∈ [0, 1], (but can also let a ∈ R).
Earlier result simplifies to give an optimal choice of a (if finite) given by:
a =(λ1ρ1 − λ2ρ2)
ρ1 (λ1 − ρλ2)− ρ2 (λ2 − ρλ1)
(recall that this choice ensures limT→0∂ρ̂∂x (T, x, y, k1, k2)|x=y = 0).
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 22 / 34
Page 73
Optimal Linear Strike Conventions
What does a look like in a simple case?
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 23 / 34
Page 74
Optimal Linear Strike Conventions
What does a look like in a simple case? Notice that ρ = 0 gives (why?):
a =(λ1ρ1 − λ2ρ2)
ρ1 (λ1 − ρλ2)− ρ2 (λ2 − ρλ1)=
(λ1ρ1 − λ2ρ2)
(λ1ρ1 − λ2ρ2)= 1
If instead ρ1 = 0, then a = λ1λ1−ρλ2
. Intuition when ρ > 0? Or ρ < 0?
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
correlation ρ
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
optim
al a
optimal a (versus ρ)
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 23 / 34
Page 75
Numerical Tests of Optimal Strike Convention
Adding the optimal a line (in purple) to our plots from earlier:
Example 1 (left): two downward skews =⇒ a = 0.364
Example 2 (right): one downward, one upward =⇒ a = 1.917
As hoped, ρimp is close to flat across moneyness and matching ‘true’ ρ.
90 92 94 96 98 100 102 104 106 108 110
F2
0
0.49
0.495
0.5
0.505
0.51
0.515implied correlation (varying strike convention)
a=0
a=1/2
a=1
optimal a
90 92 94 96 98 100 102 104 106 108 110
F2
0
0.42
0.44
0.46
0.48
0.5
0.52
0.54
0.56implied correlation (varying strike convention)
a=0
a=1/2
a=1
optimal a
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 24 / 34
Page 76
More Numerical Tests - Price Ratios (% pricing errors)
OTM % errors dominate (here ρ1 = −0.7, ρ2 = −0.8, varying ρ):
80 90 100 110 1200.96
0.98
1
1.02
1.04
price
ra
tio
ρ=0.3
80 90 100 110 1200.9
0.95
1
1.05
price
ra
tio
ρ=0.5
80 90 100 110 1200.8
0.85
0.9
0.95
1
1.05
1.1
price
ra
tio
ρ=0.7
80 90 100 110 1200.2
0.4
0.6
0.8
1
1.2
1.4
price
ra
tio
ρ=0.9
a=0
a=1/2
a=1
optimal a
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 25 / 34
Page 77
More Numerical Tests - Price Differences
ITM and OTM absolute errors similar (same parameters as above):
80 90 100 110 120-0.04
-0.02
0
0.02
0.04
price
diffe
ren
ce
ρ=0.3
80 90 100 110 120-0.04
-0.02
0
0.02
0.04
0.06
price
diffe
ren
ce
ρ=0.5
80 90 100 110 120-0.05
0
0.05
0.1
price
diffe
ren
ce
ρ=0.7
80 90 100 110 120-0.1
-0.05
0
0.05
0.1
price
diffe
ren
ce
ρ=0.9
a=0
a=1/2
a=1
optimal a
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 26 / 34
Page 78
More Numerical Tests - Varying ρ2, T , and ν
Higher ν (vol of vol) doesn’t change the optimal a, but does increase error:
80 90 100 110 120-0.15
-0.1
-0.05
0
0.05
0.1
0.15
price
diffe
ren
ce
ρ2=-0.3, T=0.05, ν=0.5
80 90 100 110 120-0.1
-0.05
0
0.05
0.1
price
diffe
ren
ce
ρ2=-0.5, T=0.05, ν=0.5
80 90 100 110 120-0.2
-0.1
0
0.1
0.2
0.3
price
diffe
ren
ce
ρ2=-0.3, T=0.1, ν=0.5
80 90 100 110 120-0.3
-0.2
-0.1
0
0.1
0.2
0.3
price
diffe
ren
ce
ρ2=-0.3, T=0.05, ν=1
a=0
a=1/2
a=1
optimal a
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 27 / 34
Page 79
More Numerical Tests - Varying T
Extending to longer maturities also produces encouraging results so far...
80 90 100 110 1200.95
1
1.05
1.1
1.15
price
ra
tio
T=0.2
80 90 100 110 1200.95
1
1.05
1.1
1.15
price
ra
tio
T=0.4
80 90 100 110 120-0.4
-0.2
0
0.2
0.4
0.6
price
diffe
ren
ce
T=0.2
80 90 100 110 120-1
-0.5
0
0.5
1
price
diffe
ren
ce
T=0.4
a=0
a=1/2
a=1
optimal a
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 28 / 34
Page 80
More Extensive Numerical Tests - Parameters
Instead of picking sample cases, we now attempt to test a wide range ofparameters:
S10 = 100, S2
0 ∈ [80, 84, . . . , 100, . . . , 116, 120]
λ1 = 1, λ2 = [0.72, 1.02, 1.32]
Heston parameters (as before): κ = 1.5, θ = 0.15, ν = 0.5, σ0 = 0.15
ρ ∈ [−0.5,−0.3,−0.1, 0.1, 0.3, 0.5, 0.7]
ρ1 ∈ [−0.72,−0.42,−0.12, 0.18, 0.48]
ρ2 ∈ [−0.61,−0.31,−0.01, 0.29, 0.59]
T = 0.1
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 29 / 34
Page 81
More Extensive Numerical Tests - Parameters
Instead of picking sample cases, we now attempt to test a wide range ofparameters:
S10 = 100, S2
0 ∈ [80, 84, . . . , 100, . . . , 116, 120]
λ1 = 1, λ2 = [0.72, 1.02, 1.32]
Heston parameters (as before): κ = 1.5, θ = 0.15, ν = 0.5, σ0 = 0.15
ρ ∈ [−0.5,−0.3,−0.1, 0.1, 0.3, 0.5, 0.7]
ρ1 ∈ [−0.72,−0.42,−0.12, 0.18, 0.48]
ρ2 ∈ [−0.61,−0.31,−0.01, 0.29, 0.59]
T = 0.1
Note: challenging to pick a ‘fair’ set, because of how our optimal a can changequickly, especially if its denominator gets close to zero. Also need positive definitecorrelation matrices (some cases thus omitted).
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 29 / 34
Page 82
More Extensive Numerical Tests - Average Errors
Mean Absolute Errors (MAE) for different cases (best convention in red):
ρ a = 0 a = 1/2 a = 1 Optimal a ATM error
λ2=
0.72
-0.5 0.2279 0.0738 0.1275 0.048 0.0449-0.3 0.2255 0.0967 0.0907 0.0558 0.0542-0.1 0.2333 0.1177 0.0644 0.0552 0.05410.1 0.2496 0.1437 0.0584 0.0709 0.04430.3 0.2499 0.1642 0.088 0.0868 0.04120.5 0.2224 0.1662 0.1195 0.1268 0.04410.7 0.2136 0.1758 0.1538 0.0977 0.0383
λ2=
1.32
-0.5 0.2625 0.117 0.158 0.0949 0.0956-0.3 0.262 0.14 0.1326 0.1076 0.1061-0.1 0.2677 0.1579 0.111 0.1176 0.10850.1 0.2806 0.1778 0.1042 0.1028 0.09930.3 0.2886 0.1986 0.1227 0.1303 0.09360.5 0.2764 0.2053 0.1504 0.1411 0.08410.7 0.2809 0.2218 0.194 0.1175 0.0707
Note: right most column (ATM error) is the ‘best case’ we can hope for.Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 30 / 34
Page 83
More Extensive Numerical Tests - Impact of Exteme a
Recall that
a =(λ1ρ1 − λ2ρ2)
ρ1 (λ1 − ρλ2)− ρ2 (λ2 − ρλ1)
How much are we hurt by cases which produce high |a|? Left plot averagesall results while right plot excludes cases with a < −1 or a > 2:
-0.5 0 0.5 1
ρ
0.05
0.1
0.15
0.2
0.25
0.3avg MAE (from Table 1)
a=0
a=1/2
a=1
optimal a
ATM error
-0.5 0 0.5 1
ρ
0.05
0.1
0.15
0.2
0.25
0.3avg MAE (after exclusions)
a=0
a=1/2
a=1
optimal a
ATM error
-0.5 0 0.5 1-4
-2
0
2
4
Note: more could also be done to improve extrapolation of OTM implied volsMichael Coulon (University of Sussex) Spread Options Sept 7th, 2017 31 / 34
Page 84
Average Errors for Different Error Measures
Results for Mean Absolute Errors (MAE), Mean Absolute Percentage Error(MAPE), Root Mean Squared Error (RMSE), Maximum Absolute Error (MaxAE), and mean standard deviation of errors across moneyness grid (MStd):
Measure a = 0 a = 1/2 a = 1 Optimal a ATM error
no
excl
usion MAE 0.2512 0.153 0.118 0.1 0.0672
MAPE 0.0275 0.0183 0.0141 0.013 0.0051RMSE 0.3264 0.1927 0.1476 0.1352 0.081Max AE 1.0097 0.5714 0.4221 0.5319 0.1574MStd 0.2727 0.1408 0.0912 0.0316 n/a
−1≤
a≤
2 MAE 0.2457 0.1424 0.1073 0.0771 0.0663MAPE 0.0258 0.0156 0.0114 0.008 0.0051RMSE 0.3194 0.1798 0.1342 0.093 0.0806Max AE 0.9897 0.5421 0.3864 0.2464 0.1571MStd 0.2727 0.1408 0.0912 0.0316 n/a
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 32 / 34
Page 85
Average Errors versus Moneyness
Here a clear consistency benefit when using the optimal a. (purple lines)
80 90 100 110 120
S2
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4Mean Absolute Error (MAE)
80 90 100 110 120
S2
0
0
0.02
0.04
0.06
0.08
0.1
0.12Mean Absolute Percentage Error (MAPE)
a=0
a=1/2
a=1
optimal a
80 90 100 110 120
S2
0
0
0.1
0.2
0.3
0.4
0.5Root Mean Squard Error (RMSE)
80 90 100 110 120
S2
0
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8Max Absolute Error (Max AE)
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 33 / 34
Page 86
Conclusions and Further Work
Key contributions:
A rigorously justified optimal choice of strike convention, which adaptsto the covariance structure of the two assets and volatility process.
Model-independent inputs via market observables (and historical ρ)
A tool to correct for the mispecification caused by Margrabe and skew.
Numerical experiments to test and confirm results.
Investigations into when the issue matters most.
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 34 / 34
Page 87
Conclusions and Further Work
Key contributions:
A rigorously justified optimal choice of strike convention, which adaptsto the covariance structure of the two assets and volatility process.
Model-independent inputs via market observables (and historical ρ)
A tool to correct for the mispecification caused by Margrabe and skew.
Numerical experiments to test and confirm results.
Investigations into when the issue matters most.
Many further questions to explore and more work possible:
Error bounds on the price; relevance of second order terms
Investigation of possible extensions of results to other cases:
Kirk instead of Margrabebehaviour for larger T
Empirical analysis, if spread data magically becomes available!
Michael Coulon (University of Sussex) Spread Options Sept 7th, 2017 34 / 34