MLX 2018 - Thomas F. Coleman, University of Waterloo

Thomas Coleman , University of Waterloo Machine Learning Fintech Conference 2.0. May 2, 2018
Delta Hedging via Machine Learning
(w/o parametric models).*
Thomas F. Coleman
Director, WatRISQ
University of Waterloo
* Learning Minimum Variance Discrete Hedging Directly from the Market
Journal of Quantitative Finance, 2017. Ke Nian, Yuying Li, Thomas F. Coleman

Seminar last week on Wall St : machine learning, text, arbitrage…
Speaker: Arun Verma, Bloomberg
“Machine Learning & Sentiment Analysis in Finance for Statistical
Arbitrage”

Seminar last week on Wall St : machine learning, text, arbitrage…

The Big Picture Message
Machine learning technology, w/o any finance knowledge or built-in
finance models, can (sometimes) outperform highly developed
specialized finance models in important (predictive) finance tasks.

We illustrate this on one of the most important and well-studied problems
in finance : option hedging.

We illustrate this on one of the most important and well-studied problems
in finance : option hedging.
The success we report is definitive.

Hedging options is a major task in computational finance to remove
market risk .
Typical process (delta hedge):
1. Choose an option model
2. Calibrate the model (using real price(s)) to estimate model
parameters
3. Differentiate the model (with parameters now determined) at
current point to determine delta hedge

Hedging options is a major task in computational finance to remove
market risk .
Typical process (delta hedge):
1. Choose an option model
2. Calibrate the model (using real price(s)) to estimate model
parameters
3. Differentiate the model (with parameters now determined) at
current point to determine delta hedge
BUT: - fails to capture parameter dependence on underlying
- does not minimize variance of hedge risk

Practitioner's BS delta hedging for an option:
1. Assume that the underlying follows a BS underlying price model:
( )t t t tdS r q S dt S dW  

1. Assume that the underlying follows a BS underlying price model:
2. Calibrate BS implied volatility
from the market option price
MKTV
( ; )BS MKTV S V 

1. Assume that the underlying follows a BS underlying model:
2. Calibrate BS implied volatility
from the market option price
3. Compute the BS delta using implied vol

MKTV
( ; )BS MKTV S V 
BSV
S



0In Coleman et al (2001) we illustrated the dependence of on S

Minimum Variance Hedging
Goal: minimize option risk minimize variance (MV) in hedging error.
Let
: instantaneous change in underlying
: instantaneous change in option value
: position in underlying for a MV strategy
Hedging error:
MV delta hedging:
dS
dV
MV
MVdV dS
min variance ( ) (1)MV MVdV dS 

For example, Hull and White [2017] directly attack (1) by assuming
Using market data to determine parameters a,b,c
Hull and White approximate
E( )BS BS
MV
V V
S S



  
  
  
2
E( ) ( )BS BS
a b c
S S T
    



For example, Hull and White [] directly attack (1) by assuming
Using market data to determine parameters a,b,c
Hull and White approximate
Improved results are obtained using this correction to BS-delta:
E( )BS BS
MV
V V
S S



  
  
  
2
E( ) ( )BS BS
a b c
S S T
    


2
( )
( )
BS BS
BS BS
MV
a b c
vega
S T
 
 
 
 

Our new data-driven approach:
Solve
directly from the market data : no option models: use (standard) ML
techniques directly

Our new approach:
solve
directly from the market data : no option models: use (standard) ML
techniques directly
learn

A Machine Learning Primer
Given m training points:
Where
x: feature vector
y: target (or outcome)
A regularized kernel method learns a nonlinear function
to capture the dependence between target y and feature vector x.
is then used to ‘predict’ y for feature selection x
1 1{( , ),...,( , )}m mx y x y
( )f x
( )f x

Where
x: feature vector
Choose a positive definite kernel: e.g.,
1 1{( , ),...,( , )}m mx y x y
( )f x
( )f x
2
2
2
'
2
( , ')
x x
k x x e 




Where
x: feature vector
Choose a positive definite kernel: e.g.,
A regularized regression problem:
1 1{( , ),...,( , )}m mx y x y
( )f x
( )f x
2
2
2
'
2
( , ')
x x
k x x e 




A regularized (nonlinear) regression problem (to determine (learn)
parameters that define ) :
2
1
1
min ( , ( )) ,
e.g., ( , ( )) ( )
m
f i i
i
p
i i i i
L y f x f
m
L y f x y f x
 

 
 
 
 

Defined by choice of K
f

A regularized (nonlinear) regression problem:
2
1
1
min ( , ( )) ,
e.g., ( , ( )) ( )
m
f i i
i
p
i i i i
L y f x f
m
L y f x y f x
 

 
 
 
 


where m-vector minimizes the pos def quadratic:
2
1
1
min ( , ( )) ,
e.g., ( , ( )) ( )
m
f i i
i
p
i i i i
L y f x f
m
L y f x y f x
 

 
 
 
 

1
( ) ( , )
m
i i
i
f x k x x

 

2
1 1 1 1
min { ( ( , )) ( , )}m
m m m m
i j i j i j i j
i j i j
y k x x k x x
   
   
   

or
2
1
1
min ( , ( )) ,
e.g., ( , ( )) ( )
m
f i i
i
p
i i i i
L y f x f
m
L y f x y f x
 

 
 
 
 

1
( ) ( , )
m
i i
i
f x k x x

 

2
1 1 1 1
min { ( ( , )) ( , )}m
m m m m
i j i j i j i j
i j i j
y k x x k x x
   
   
   
2
2
min { }, where ( , )T
ij i jy K K K k x x     

or
or
2
1
1
min ( , ( )) ,
e.g., ( , ( )) ( )
m
f i i
i
p
i i i i
L y f x f
m
L y f x y f x
 

 
 
 
 

1
( ) ( , )
m
i i
i
f x k x x

 

2
1 1 1 1
min { ( ( , )) ( , )}m
m m m m
i j i j i j i j
i j i j
y k x x k x x
   
   
   
2
2
min { }, where ( , )T
ij i jy K K K k x x     
1
( )K I y  
 

Recipe for determining ML model
1. Choose feature vector x , kernel k
2. Get data
3. Form m-by-m matrix
4. Determine (cross validation)
5. Solve
( , ), 1:i ix y i m
( ( , ))i jK k x x
( )K I y  


Recipe for determining ML model
1. Choose feature vector x , kernel k
2. Get data
3. Form m-by-m matrix
4. Determine (cross validation)
5. Solve
Prediction:
( , ), 1:i ix y i m
( ( , ))i jK k x x
( )K I y  

1
( ) ( , )
m
pred i i
i
y x k x x

 

To apply to our problem:
1. What is feature vector x?
2. Kernel function k?
3. Data?
4. Choose
And most importantly


To apply to our problem:
1. What is feature vector x?
2. Kernel function k?
3. Data?
4. Choose
And most importantly, what is objective function:

2
1
1
min ( , ( ))
m
f i i
i
L y f x f
m
 

 
 
 

min variance ( )MV MVdV dS 
Key!

Learn hedging function directly from the market using ML
approach: let be a positive definite kernel
where denotes change in (observed) market price
Solution is
( )f x
22
1
1
min ( ( ))
m
f i i i
i
V S f x f
m
 

 
    
 

t
i i iV V V
  
1
( ) ( , )
m
i i
i
f x k x x

 
( , ')k x x

Learn hedging function directly from the market using ML
approach: let be a positive definite kernel.
where denotes change in (observed) market price
Solution is
Where solves the pos def. quadratic minimization:
Features x = moneyness , time-to-expiry;
( )f x
22
1
1
min ( ( ))
m
f i i i
i
V S f x f
m
 

 
    
 

t
i i iV V V
  
1
( ) ( , )
m
i i
i
f x k x x

 
( , ')k x x

2
1 1 1 1
min { ( ( ( , )) ( , )}m
m m m m
i i i i j i j i j
i i i j
V S k x x k x x
   
   
     
BS
( )
S
strike

Empirical Results
Data: S&P 500 index option data from the OptionMetric Database
data: January 2, 2004 to August 31, 2015

Empirical Results
Methods compared:
MV: Hull-White's minimum variance hedging based on formulation
LVF: MV hedging using correction based on LVF analysis
BS: implied volatility BS delta hedging
SABR : MV hedging from SABR stochastic volatility.
DKLSPL : direct learning a spline kernel hedging function

Empirical Results
Methods compared:
MV: Hull-White's minimum variance hedging based on formulation
LVF: MV hedging using correction based on LVF analysis
BS: implied volatility BS delta hedging
SABR : MV hedging from SABR stochastic volatility.
DKLSPL : direct learning a spline kernel hedging function
Performance Measure:
2
1
2
1
( )
( )
m
M
i i
i
m
BS
i i
i
V S
GAIN
V S




  

  



Empirical Testing
Sliding window:
3 years data 1 mth-out-of sample
learning model testing
9 buckets, 9 models

9 buckets, 9 models

Concluding remarks
We have proposed to learn the hedge function directly from market data
(using ML techniques).

Concluding remarks
We have proposed to learn the hedge function directly from market
data (using ML techniques).
Empirical results show
- proposed approach definitively outperforms existing MV hedge
methods in daily hedging

Concluding remarks
- proposed approach definitively outperforms implied vol BS hedge
in daily, weekly and monthly hedging

Concluding remarks
- proposed approach definitively outperforms implied vol BS hedge
in daily, weekly and monthly hedging
Question: What other important (predictive) problems in finance,
traditionally solved with highly developed specialized finance models,
are amenable to general data driven machine learning techniques?

REFERENCES
T. F. Coleman, Y. Kim, Y. Li, A. Verma, Dynamic hedging with a deterministic
local volatility function model, The Journal of Risk 5 (2001) 63:89.
J. Hull, A. White, Optimal delta hedging for options Journal of Banking &
Finance, 82:180-190, 2017
G. Bakshi, C. Cao, Z. Chen, Empirical performance of alternative option pricing
models, The Journal of Finance 52 (1997) 2003.
S. Crepey, Delta-hedging vega risk, Quantitative Finance 4 (2004) 559{579.
G. Bakshi, C. Cao, Z. Chen, The Journal of Finance 52 (1997) 2003{2049.
K. P. R., R. Schenk-Hoppe, C.-O. Ewald, Risk minimization in stochastic
volatility models: model risk and empirical performance, Quantitative Finance 9
(2009) 693{704.
G. Bakshi, C. Cao, Z. Chen, The Journal of Finance 52 (1997) 2003.
J. Hutchinson, A. Lo, T. Poggio. A nonparametric approach to pricing
and hedging derivative securities via learning networks. The Journal
of Finance. 49(3):851-889, 1994

MLX 2018 - Thomas F. Coleman, University of Waterloo

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