1.
Material
• All the slides: www.deep-analytics.org
• Working paper on arXiv: arxiv.org/abs/2005.02347
Differential Machine Learning
• GitHub: github.com/differential-machine-learning
• Demonstration notebooks: github.com/differential-machine-learning/notebooks
Designed for Google Colab
Discuss practical implementation details
Contain TensorFlow code
• Appendices: github.com/differential-machine-learning/appendices
Mathematical proofs
Implementation details
Extensions

2.
Automatic Differentiation (AAD)
• This is not strictly speaking a talk on AAD
• But we extensively apply AAD and backpropagation
• A good understanding of these techniques is a strong prerequisite
• We refer to:
The AAD textbook
amazon.com/Modern-
Computational-Finance-
Parallel-Simulations-dp-
1119539455/dp/1119539455
A 15min video tutotial
towardsdatascience.com/automatic-differentiation-15min-video-tutorial-with-
application-in-machine-learning-and-finance-
333e18c0ecbb?source=friends_link&sk=c11be895fffa9a8276d3e74a598dbbc3

3.
AAD tutorial on YouTubeyoutube.com/watch?v=IcQkwgPwfm4
slides and more here:
towardsdatascience.com/a
utomatic-differentiation-
15min-video-tutorial-with-
application-in-machine-
learning-and-finance-
333e18c0ecbb?source=frie
nds_link&sk=c11be895fffa
9a8276d3e74a598dbbc3

4.
AAD in finance
• AAD made differentials massively available
• Gave us realtime risk reports and instantaneous calibration
• Unlocked new research and development potential, with many powerful applications
• In this talk, we discuss a very special application:
we learn the pricing function of Derivatives from AAD pathwise differentials

5.
Pricing function approximation
• Pricing in closed-form a la Black & Scholes
is only available for simple instruments in simple models
• In all other cases, prices are computed with numerical methods, often Monte-Carlo
• Monte-Carlo is orders of magnitude slower than analytics
• We need fast analytics for:
 Risk management of option books
 Exotic risk reports in multiple scenarios
 Backtesting
 Value at Risk / Expected Loss
 Regulations like XVA, CCR, FRTB, SIMM-MVA
• Where trading books are repeatedly evaluated in different market scenarios
• Hence, the intensive research on pricing function approximation for 2-3 decades
 The objective is to derive approximate pricing functions of the market state
 With speed similar to analytics
 And accuracy similar to Monte-Carlo

6.
2000: SABR
• Market realizes the importance of stochastic volatility (SV) for risk management
• Challenge: no closed form formula, Monte-Carlo far too slow
• Pat Hagan works out a mathematical approximation in Managing Smile Risk (Wilmott, 2002)
• Resulting in the famous SABR formula
• SABR brings SV models to trading desks
and quickly becomes a market standard, which it remains to this day
dF F dW
d
dZ
dWdZ dt









  , ?T t tE F K F 

  
 

7.
2019: Deep Learning Volatility
• Age of AI: similar problem find different solution
• Example: ‘Rough volatility’ (RV) models
 New family of stochastic volatility models with excellent risk management properties
 (Read all about RV in Jim Gatheral’s papers and presentations)
 Unfortunately, no closed form or accurate approximation for European calls
• Deep Learning Volatility (Horvath and al. 2019)
 Proposed pricing of calls in RV models by neural network
 Trained on examples produced by Monte-Carlo
 Trained network is the approximation: fast pricing by feedforward induction, risk sensitivities by backpropagation
 Permits calibration and risk management in realistic time, making RV suitable for production
• Note:
 Pricing function is learned from data, not derived from underlying mathematics
 A very large training set is necessary to achieve accurate approximation
 Authors train the net with ground truth labels, where each training example is computed by Monte-Carlo
 Training set is obtained at massive computation cost
 Pricing function is learned once, offline, and reused forever

8.
2020: Deep Analytics
• Our ambition: generalize and automate the production of pricing approximations
 Learn the pricing function for arbitrary schedules of cashflows, not only calls:
options, exotics, hybrids, netting sets, trading books…
 In arbitrary simulation models:
multifactor, multiunderlying, multicurrency, stochastic volatility…
 Online, in realtime, as part of a risk computation
 From data alone, simulated in reasonable time
 With convergence guarantees
• Achieved by:
covered in the presentation
1. Training on payoff samples, a la Longstaff-Schwartz (2001)
2. Leveraging modern deep learning
3. Training on pathwise differentials obtained with AAD
not covered in the presentation, see appendix 4
github.com/differential-machine-learning/appendices/blob/master/App4-UnsupervisedTraining.pdf
4. Obtaining training convergence guarantees with special network architecture
5. Controlling asymptotics with specialized algorithms

9.
Context and notations
• We have a simulation model:
 Simulates a Markov state vector S under some martingale measure Q
 S represents the state of the market together with path dependencies (barriers, exercises, etc.)
• We represent transactions and trading books as collections of event driven cashflows:
 Cashflows CFp paid at time Tp are Tp-measurable variables
 We assume all cashflows are appropriately smoothed, we are going to differentiate them!
 For example, discontinuous digital cashflows are represented as tight call spread
 Smoothing is standard practice, necessary for the computation of Monte-Carlo risk reports
 We ignore discounting/numeraire to simplify notations
 We call payoff the sum of cashflow paid after some horizon date (maybe today)
• The price is the conditional expectation of the payoff
 It is therefore a deterministic (but unknown) function f of the current state S
 Could be evaluated by Monte-Carlo for every different input state
 Our objective is learn an accurate approximation of f that evaluates with analytic speed
  ,0p p t pCF g S t T  
 Q
t t tV E S f S    
p
p
T t
CF

 

10.
Context and notations (2)
• Assuming appropriate smoothing
• Differentiation and expectation commute:
• Risk sensitivities are conditional expectations of pathwise diferentials
 Pathwise differentials = differentials of final payoff wrt initial state along a path
 Random variables, measurable on maturity = payment date of the final cashflow
 Computed very efficiently with AAD, recall movie tutorial
• Monte-Carlo risk reports are computed by averaging pathwise differentials
• See detailed discussion in appendix 1
github.com/differential-machine-learning/appendices/blob/master/App1-LSM.pdf
  Q
t Q
t
t
E S xf x
E S x
x x S
          
   
pathwise
differentials
risk
sensitivities

11.
Example:
European call in Black & Scholes
• State vector: spot price
• Model: geometric Brownian motion
• State on horizon date: spot price at T1
• Payoff: European call, paid at T2
• Price at T1: expected payoff
• Given by Black & Scholes formula
• Pathwise differential:
 In Black & Scholes:
 Computed in closed form
 Effectively, together with payoff
 Random var with expectation = delta:
 Extend to general case:
 Always efficiently computed with AAD
 Always unbiased estimates of risk sensitivities
   1 1 2 1T T T Tf S E S E S K S

        
1TS
 2TS K

 
   
2
2 1 2 1
2 1
2
T TT T W W
T T
dS
dW S S e
S



   
 
 
   
   
2
2 1 2 1
1 1 1
2
1 2
T TT T W W
T T Tf S E S e K S N d KN d



   
  
         
S
   
 
   
 
2
2 1 2 12 2 2 2
2 2
1 1 2 1 1
2
1 1
T T
T T
T T W WT T T T
S K S K
T T T T T
S K S K S S
e
S S S S S


 
   
 
    
   
   
   
    12
1 1
2
1 1 1
1 2
11
T
TT
T TS K
T T T
S N d KN dS
E S E S N d
S S S


    
       
       

12.
Learning from samples

13.
Simulated dataset
• Simulate a training set a la Longstaff-Schwartz (2001)
• Each training example is simulated with one Monte-Carlo path:
 Training example (i) picked on simulated path (i)
 Training input i: the state vector on some horizon date (possibly today):
 Training label: the final payoff on the same path:
• The entire training set of m examples is simulated with m Monte-Carlo paths
in a time similar to one pricing by Monte-Carlo
• Hence, dataset is prepared in realistic time suitable for realtime training
   
ex
i i
TX S
   
p ex
ii
p
T T
Y CF

 

14.
Training example X (vec of dim n)
= state at Tex
Y (real number)
= payments at Tp > Tex
1
2
...
1. Sample state on horizon date
   1 1
exTX S
   1 1
exTX S

15.
2. Run Monte-Carlo path,
accumulate future payments
Training example X (vec of dim n)
= state at Tex
Y (real number)
= payments at Tp > Tex
1
2
...
   1 1
exTX S
   1 1
p ex
p
T T
Y CF

 
   1 1
exTX S
   1 1
p ex
p
T T
Y CF

 

16.
3. Repeat m times independently
Training example X (vec of dim n)
= state at Tex
Y (real number)
= payments at Tp > Tex
1
2
...
   1 1
exTX S
   1 1
p ex
p
T T
Y CF

 
   2 2
exTX S
   2 2
p ex
p
T T
Y CF

 
   2 2
exTX S
   2 2
p ex
p
T T
Y CF

 

17.
Training on samples
1. Practical
 Reuse the Monte-Carlo pricing engine, reconfigured as a training set simulator
2. Efficient
 The entire training set is simulated for the cost of one Monte-Carlo pricing
 If you are going to price your book, you get the training set for free
 Just store states and payoffs for every Monte-Carlo path
 Variance reduction methods (e.g. Sobol sequences) automatically reused
3. Consistent
 Dataset produced with the same model as front office pricing
4. General
 If you can price a transaction, then you can simulate a dataset
 Script your transaction
 Calibrate your model
 Simulate the training set

18.
Can we really
learn prices from samples?
• The machine is expected to learn a pricing function
• Without having seen a single price, only samples
• Example: European call in Black-Scholes
• Can the machine figure Black & Scholes formula from payoff samples?
learn this
     2 1 1 1 2T T TE S K S S N d KN d

   
  
from this
   
  1 2
,i i
T TS S K


               
 
2
2 1 2 1
1 2 1
2
and
i i
T TT T W Wi i i i i
T T TX S Y S K S e K



     
       
  

19.
Yes, we can
• Mathematical guarantee:
 A universal approximator with learnable weights w
 Capable of approximating any function when capacity (~number of weights) grows to infinity
 Includes classic basis function regression and neural networks
 Trained on sample payoffs by minimization of the mean squared error
 Converges to the true pricing function, a.k.a. conditional expectation
 Asymptotically in the size m of the training set and the capacity of the approximator
• Proof and discussion: appendix 1 github.com/differential-machine-learning/appendices/blob/master/App1-LSM.pdf
• Intuition:
 True price is the conditional expectation of the payoff
 Hence, payoff is an unbiased (noisy) estimate of the true price
 Independent noise diversified away in a growing dataset
 Approximators converge to the closest function to f they are able to represent
 Universal approximators converge exactly to f when capacity grows
     
 
2
1
1 ˆ ;
m
i i
i
MSE w Y f X w
m 
  
 
 ˆ ;f x w
 i
Y
 f X E Y X   
  where 0Y f X E X      
  Q
Tf x E Y X x E S x          
predictionssampled labels
independent noise

20.
Caveat 1
• The result is asymptotic in the size of the dataset size and the capacity
• Finite capacity approximators (e.g. finite size neural nets) do not converge to true prices
 When dataset grows, finite size neural nets converge
to the projection of the true pricing function onto the subspace of functions attainable by the net
 The difference is called bias
approximation error that remains when training with a dataset of infinite size
 The space of functions spanned by neural nets grow linearly in units and exponentially in layers
 Bias vanishes quickly with neural depth
 This is the main point of deep neural networks
 Excellent presentation
by Matthew Dixon @ Department of Mathematics, Illinois Institute of Technology, Nov 4th 2019
• With finite size datasets
 Noise is not fully diversified away
 Training may overfit noise in the training set and fail to approximate the true function
 Overfitting is related to variance: dependency of final result to noise of a particular training set

21.
Caveat 2
• The global minimum of the MSE converges to the true pricing function
• There is no guarantee that your optimization algorithm finds it in finite time
• With neural nets, MSE is a nonconvex function of weights
• There exists no algorithm guaranteed to find the global minimum
• Modern heuristics, see demonstration notebook
github.com/differential-machine-learning/notebooks/blob/master/DifferentialML.ipynb
 Xavier-Glorot weight initialization
 Normalization of inputs and labels
 ADAM optimizer
 Leslie Smith’s one-cycle learning rate schedule
• Help training quickly converge to “good” solutions in “most” situations
• But without hard guarantee, not good enough for online learning in a risk system
• In appendix 4, we introduce a neural architecture with worst case training guarantees
github.com/differential-machine-learning/appendices/blob/master/App4-UnsupervisedTraining.pdf

22.
Training with differentials
• In summary:
 We work with a sampled dataset
 Where is the initial state on path i, and is the final payoff on the same path
 The entire dataset is computed for the cost of one Monte-Carlo simulation
 With guaranteed convergence to true pricing function by training a universal approximator to minimize MSE
• We can do better than that:
 With AAD we can compute pathwise differentials for very little cost:
 Pathwise differentials are differential labels = differentials of labels wrt inputs
 We make them part of an extended training set to teach the machine the shape of the
function
   
  ,
i i
X Y
   
 
 
, ,
i
i i
i
Y
X Y
X
   
      
   
ex
i i
TX S    i i
Y 
 
 
 
 
ex
i i
i i
T
Y
S X
 

 
 i
Y  i
X

23.
Deep Learning

24.
• Results produced with simple self contained notebooks available on
github.com/asavine/CompFinLecture/tree/master/MLFinance
• Implement polynomial regression and simplistic neural nets on simulated samples
Simple example:
Black & Scholes
polynomial regression: instantaneous simplistic neural net: < one second

25.
High(er) dimensional example:
basket option
• Available on github.com/asavine/CompFinLecture/tree/master/MLFinance
• Toy basket option, correlated Gaussian model dimension 10
• Correct price known in closed form, (Bachelier) function of current basket
• We know its a 1d problem, but the machine sees it in dimension 10
polynomial regression: now ~one minute! simplistic neural net: still < one second

26.
Classic regression
in high dimension
• It looks like classic regression fails in high dimension
 Hence unsuitable for general trading books
 Where dimension is usually high
 Think Libor Market Models on all yield curves: Markov dimension grows in the 100s
• On the contrary, neural nets seem immune to dimensionality
• Let us try to understand this

27.
Curse of dimensionality
• Classic regression:
• The number d of basis functions g
increases exponentially with the dimension n of the input x
• Example: polynomial basis with degree up to p:
 Number d of monomials of the form
 Is
• So we have a large number d of regression parameters w and that makes regression:
 Expensive, hence computation time ~1min
 Vulnerable to overfitting, hence the poor performance:
Training tries to fit samples and ends up interpolating between them instead of learning the underlying pattern
 !
! !
n p
d
n p


   
1
ˆ ;
d
j j
j
f x w w g x

 
 
11
,ij
n n
k
j i ij
ii
g x x k p

 

28.
Classic LSM
• Least Square Method (LSM, Longstaff-Schwartz, Carriere, 1999-2001)
 Approximates prices by classic regression on samples
 Served the industry well for 20y for Bermudan options and other types of early exercisable transactions
 How is this consistent with our results?
• This is because implementations of LSM do not regress on the entire state
 They regress on a low number of regression variables
 Regression variables are fixed, hardcoded function of the state
 Derived from prior knowledge and analysis of the transaction and model
• Example: Bermudan options in the Libor Market Model (LMM)
 The reason why LSM was invented in the first place
 LMM is a high dimensional model
 The state vector contains all forward Libor rates up to the final maturity
 With 3m Libors and maturity 30y, this is dimension 120
 However, it is known that Bermudan options in LMM mainly depend on only two variables:
 Swap rate to maturity
 Discount rate to next call
 Typical implementations approximate continuation values by regression on these two variables

29.
LSM: manual feature extraction
3m
6m
9m
...
30y
df
1y
sw
20y
1x
2x
1x
2x
2
1x
2
2x
1 2x x
“input” layer
model state
all 3m fwd rates up to 30y
dimension 120
“hidden” layer
regression variables
swp to mat + df to call
dimension 2 “regression” layer
quadratic monomials
dimension 5
“output” layer
continuation value
by regression
fixed, hardcoded
transformation state to basis
learned linear regression

30.
Limitations and neural networks
• These particular regression variables only apply to standard Bermudas in LMM
• With different transactions or different models, we need different reg. variables
• E.g. with stochastic volatility, state of volatility is a 3rd variable, crucial in determination of value
• What about arbitrary schedules of cashflows in arbitrary models?
• Cannot use prior knowledge to turn a high dim state into a low dim regression vector
• Only simulated data
• We must learn regression features meaningful for this schedule of cashflows in this model
• Automatically, from data alone: called automatic feature extraction in Machine Learning
• Neural networks excel at it (hence their success in tasks like computer vision or NLP)

31.
Neural networks:
automatic feature extraction
...
“input” layer
model state
“hidden” layers
learn transformation
of state into regression vars
“regression” layer
learnt basis functions
“output” layer
still a linear regression
learned
transformation state -> regr. vector
learned linear regression

32.
• Neural networks learn useful regression features in their hidden layers
• And encode them in their connection weights
• This is how neural nets adapt regression features to data
• And break the curse of dimensionality
• Just like we break it manually in the case of e.g. Bermudan options
• But neural nets do it automatically, during training
• And adapt to arbitrary schedules of cashflows and simulation models
Neural networks:
breaking the curse

33.
Deep learning pricing functions
• Appendix 4 github.com/differential-machine-learning/appendices/blob/master/App4-UnsupervisedTraining.pdf
 Discusses the details of neural compared with classic regression
 Combines neural nets with classic regression to obtain training convergence guarantees
• Demonstration notebook github.com/differential-machine-learning/notebooks/blob/master/DifferentialML.ipynb
 Discusses practical implementation details: architecture, initialization, normalization, optimization
 TensorFlow implementation code

34.
Deep learning limitations
1. Requires unrealistically large datasets
2. Computes poor risk sensitivities
3. Prone to overfitting on smaller datasets

35.
Unrealistically large datasets
• Back to basket option example
 Dimension 30
 Results produced with demonstration notebook
github.com/differential-machine-learning/notebooks/blob/master/DifferentialML.ipynb

36.
Poor risk sensitivities
• Differentials of a good approximation are not always good approximations of differentials
 Risk sensitivities of the approximation converge very slowly
 Showing delta to the first stock in the basket

37.
Computational cost
• Most of the computation cost is in the simulation of the training set
 Not training, usually ~1sec on entry level GPU
 But the simulation of the model states
 And the evaluation of the many cashflows in a large trading book
• Hence, in realistic contexts
we train on small datasets
 In practice, 1,024 to 32,768 paths but no more
• Training with small datasets is vulnerable to overfitting

38.
Overfitting and variance
• High capacity ML models like neural nets have many parameters -- with small training sets
 Optimizers try to fit all the samples
 End up interpolating between samples instead of picking underlying patterns
 And fail to achieve meaningful approximation
• Simple example: fit degree 6 polynomial on 7 samples in Black & Scholes
 Perfect fit of the training set
 Completely missing the correct function
 With high variance: very different results depending on noise in training set

39.
Regularization
• Conventional regularization mitigates overfitting with constraints on learnable weights w
• Example: Tikhonov (also called ridge)
 Extend cost function with penalty on the size of weights:
 Expressing preference for small weights
 Hyperparameter lambda: regularization strength
• Other common regularizations include
 Lasso: L1 norm penalty
 Dropout: randomly drop connections when training neural nets
• Regularization constraints effectively stop optimizers from fitting samples
• Therefore mitigate overfitting and reduce variance
   
 
2
1
221 ˆ* arg min - ;
m
i i
w
i
X w ww Y f
m




 


40.
Tikhonov regularization

41.
Bias-variance tradeoff
• Regularization Bias
 Conventional regularization establishes arbitrary penalties
 Why should we prefer smaller weights?
 The only purpose is stopping optimizers from overfitting
 By reducing effective capacity
 Thereby introducing bias
• Regularization strength is key
 Too weak  overfitting – high variance
 Too strong  underfitting – high bias
 Finding the right amount of regularization is called bias-variance tradeoff
 Theory intensively analysed in classic ML, see e.g. Bishop
 Practically costly implementation, e.g. cross-validation

42.
• Reducing variance doesn’t have to increase bias
 Conventional regularization only introduces bias due to arbitrary preferences
 Variance can be reduced without bias
 E.g., trivially, by increasing the size of the training set
 Although this generally very costly
• Example: data augmentation in computer vision
 Training id recognition requires
a vast number of labeled images
 Labelling images is costly
and requires human supervision
 Data augmentation:
Make many labeled pictures out of one
zoom, crop, recolor, rotate…
 Increases dataset size at no cost
 And effectively reduces variance without bias
• Data augmentation is a more powerful form of regularization
• In finance, differential training is also a form of data augmentation
Overcoming tradeoff

43.
Differential Machine Learning

44.
Deep learning risk
• Risks are sensitivities of prices wrt market states
• Approximated by sensitivities of output (approx. price) to inputs (states)
• Efficiently computed by backpropagation
 Feed state into input layer
 Compute approx. price
By feedforward induction:
feedforward equation left to right
 Compute differentials of price wrt to state
By backpropagation:
adjoint equations right to left
   ˆ ˆ;t tt
t t t
f S f S wV Y
S S S X
  
  
   

45.
Unrolling backprop:
twin networks
• Combine feedforward induction and backpropagation in a single (twin) network of twice the depth:
• The twin net computes prices and risks for twice the computation expense

46.
Differential training
• Now we have a twin net, we approximate prices and risks in one computation
• But we have seen that the approximation of risks is poor and converges slowly
• The network struggles to learn the shape of the function from punctual examples
• Hence the idea: train the network on differentials
• Recall, from our AAD powered simulation engine
 We have high quality pathwise differentials for little cost
 These are unbiased, independent estimates of true risks
 So we could train our net to minimize errors on differentials
 With a asymptotic guarantee to represent true risks
   
 ˆ ;ˆ; ; ,
f X w
g X w f X w
X
 
  
  
 
 
 
 
 
i i
i
i i
t
Y
Z
X S
 
 
 
 
 
 
 
 
 
 
 
 
 
 
hence
p
i i
i
ti i iQ
ti i i
T t t t
f S f X
E Z E S Z
S S X



              

 
 
 
 
2
1
ˆ ;
w* arg min
i
m
i
w i
i
f X w
Z
X

 



47.
Differential regularization
• In practice, we train on a combination of value and differential MSEs:
• Note the similarity with regularization, e.g Tikhonov
• Differential training is a form of regularization
 Imposes penalty for wrong differentials
 Without changing capacity (~number of learnable weights)
 Effectively stopping optimizers from overfitting training values
 Hence, reducing variance
   
 
 
 
 
 
 
2
22
1 1
1 ˆw* arg min ;
ˆ ;m m
i i
i
iw i
i i
i
f X w
X
f
m m
Y
Y X
X
w

 
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
    
 


 
   
  2
1
2
21 ˆ* arg min ;
m
i i
w
i
f w ww Y X
m


 

 

differential labels
computed e.g. with AAD
fed to training set
differential predictions
computed with the twin net
hyperparameter expressing preference for correct differentialsusual MSE

48.
Differential regularization
is different
• We do not express preferences, whereby introducing a bias with arbitrary constraints
• We enforce correctness of differentials, thereby reducing variance without bias
• Proof:
 With infinite preference for differentials lambda, we train on differentials alone
 With asymptotic convergence guarantee towards a function with all the correct differentials
(by the same argument as before: pathwise differentials are unbiased estimates of true risks)
 That is, the correct function modulo additive constant!
• There is no bias-variance tradeoff: variance is reduced for free
• Hence, (virtually) no sensitivity to lambda, contrarily to other forms of regularization
• See:
 appendix 3 github.com/differential-machine-learning/appendices/blob/master/App3-Regression.pdf
 and second demonstration notebook github.com/differential-machine-
learning/notebooks/blob/master/DifferentialRegression.ipynb
 for a comparison between classical and differential regularization in the context of regression

49.
Toy Black-Scholes polynomial fit

50.
Basket option example
• Back to basket option
 Dimension 30
 Results produced with
demonstration notebook
github.com/differential-machine-
learning/notebooks/blob/master/D
ifferentialML.ipynb
prices deltas

51.
Differential training:
further benefits
• Empirically, differential training also stabilises sensitivity to other hyperparameters:
network architecture, weight initialization, optimization settings
• By construction, we solidly approximate not only prices but also risk sensitivities
• The neural net learns the shape of the pricing function from differential labels
• Thereby learning more effectively, and in a more stable manner
• Differential regularization is superior because it injects additional information
• Acting like additional datapoints, produced for very little cost with AAD
• In this regard, it is similar to data augmentation in computer vision

52.
• The examples enlighten the most critical benefit of differential training:
we can now train with a lot less samples
• This is because differential ML is a (particularly effective) form of data augmentation
Differential augmentation
Computer Vision
• Make many pictures out of one
• Increases dataset for negligible cost
• Teaches important invariances: this is all me
Finance
• Differentials inject additional information
• Act like additional data points (think finite difference)
• Also increase dataset for negligible cost (e.g. with AAD)
• And teach the machine the shape of the function
original point
2 new points
(think finite diff)

53.
Real life example:
A worst-of 4 autocallable
• 4 correlated local volatility models
• 8,192 paths
• Script (Superfly Analytics ”Jive” language):
choose(x,p,n,e)
=
0 /2-/2
n
p
x
explicit
smoothing

54.
A worst-of 4 autocallable
no derivatives
• Training: Starting
horizontal axis: targets with nested simulations
verticalaxis:networkprediction
test set: out of sample states

55.
A worst-of 4 autocallable
no derivatives
• Training: Running …
horizontal axis: targets with nested simulations
verticalaxis:networkprediction
test set: out of sample states

56.
A worst-of 4 autocallable
no derivatives
• Training: Running ……
horizontal axis: targets with nested simulations
verticalaxis:networkprediction
test set: out of sample states

57.
A worst-of 4 autocallable
no derivatives
• Training: Running ………
horizontal axis: targets with nested simulations
verticalaxis:networkprediction
test set: out of sample states

58.
A worst-of 4 autocallable
no derivatives
• Training: Complete
horizontal axis: targets with nested simulations
verticalaxis:networkprediction
test set: out of sample states

59.
A worst-of 4 autocallable
with derivatives
• Training: Starting
horizontal axis: targets with nested simulations
verticalaxis:networkprediction
test set: out of sample states

60.
A worst-of 4 autocallable
with derivatives
• Training: Running …
horizontal axis: targets with nested simulations
verticalaxis:networkprediction
test set: out of sample states

61.
A worst-of 4 autocallable
with derivatives
• Training: Running ……
horizontal axis: targets with nested simulations
verticalaxis:networkprediction
test set: out of sample states

62.
A worst-of 4 autocallable
with derivatives
• Training: Running ………
horizontal axis: targets with nested simulations
verticalaxis:networkprediction
test set: out of sample states

63.
A worst-of 4 autocallable
with derivatives
• Training: Complete
horizontal axis: targets with nested simulations
verticalaxis:networkprediction
test set: out of sample states

64.
A worst-of 4 autocallable
No derivatives With derivatives

65.
A real netting set
no derivatives
• Hybrid model with 20 state variables
• 8,192 paths
• Training: Starting

66.
A real netting set
no derivatives
• Training: Running …

67.
A real netting set
no derivatives
• Training: Running ……

68.
A real netting set
no derivatives
• Training: Running ………

69.
A real netting set
no derivatives
• Training: Complete

70.
A real netting set
no derivatives
• Performance stopped improving
• Midway during training
• When overfitting kicked in
• We finally managed to get (very) decent results
• But with at least 65,536 paths
• And some hyperparameter tweaking
• With differential regularization
• It just works with 8,192 paths
• And no tweaking at all

71.
A real netting set
with derivatives
• Training: Starting

72.
A real netting set
with derivatives
• Training: Running …

73.
A real netting set
with derivatives
• Training: Running ……

74.
A real netting set
with derivatives
• Training: Running ………

75.
A real netting set
with derivatives
• Training: Complete

76.
A real netting set
with derivatives
• Orders of magnitude improvement
• Almost perfect approximation
• Small error remains on right asymptotic
 Disappears with 16,384 samples
 More at the end of the talk

77.
A real netting set
No derivatives With derivatives

78.
Differential ML:
implementation in TensorFlow
• Demonstration code: github.com/differential-machine-learning/notebooks/blob/master/DifferentialML.ipynb
• Including practical details like initialization, normalization and optimization
• Further details on implementation code in this post:
https://towardsdatascience.com/differential-machine-learning-f207c158064d?source=friends_link&sk=f2325c6686c11e22286f1d1d6e4daf87

79.
• AI learns from your simulation system
 To reproduce what your system does
 Orders of magnitude faster
• Efficiently learns from samples
 Makes the problem tractable in practice
 Guaranteed solid mathematics
• Classic LSM regression is not good enough
 Suffers the curse of dimensionality
 Requires hardcoded features
 Unrealistic for general trading books
• Neural nets
 Overcome the shortcomings of classic regression
 Learn useful features from data, during training
 Just work on textbook examples
Conclusion:
pricing and risk with AI
• Neural nets are limited in real world situations
 Require unrealistically large datasets
 Prone to overfitting on small datasets
 Poorly predict risk sensitivities / shape
• The missing piece is differential ML
 Train twin networks
 On datasets augmented with differentials
 Very efficiently computed with AAD
 Sharply improves risk estimates
 Corrects the shape of the pricing function
 Regularizes and prevents overfitting without bias
 Improves speed and stability of training
 Makes training resilient to hyperparameters
 Therefore, trains effectively on small datasets

80.
System overview
production risk system
(parallel CPU)
tensorflow trainer
(GPU)
simulated training set
   
 
 
, ,
i
i i
i
Y
X Y
X
  
 
  
• states
• sample payments
• (path-wise differentials)
learned weightspricing/risk function ˆ ˆ,f f
XVA, CCR, FRTB, MVA…

81.
Conclusion:
unsupervised operation
• Automated training behind the scenes requires convergence guarantee
 Practical training of neural nets offers none: no algorithm is guaranteed to find minimum MSE
 Modern heuristics: Xavier-Glorot initialization, Dataset normalization, ADAM optimization, one-cycle scheduling
 Help converge to acceptable minima in most situations, but without strong guarantee
 Risk management is not build on faith or empirical evidence
 In appendix 4 https://github.com/differential-machine-learning/
appendices/blob/master/App4-UnsupervisedTraining.pdf
 We explore Google’s ‘wide and deep’ architecture
 And demonstrate hard worst case guarantees in this case
 Allowing for unsupervised training
• Some applications are dependent on asymptotics
 Value at risk, expected loss, FRTB…
 ML models struggle to learn correct asymptotics
for lack of substantial asymptotic data
 This is corrected with dedicated algorithms
 Also in appendix 4

82.
Conclusion:
extension to other ML models
• Differential ML is not limited to deep learning
• We obtained equally remarkable results with other kinds of machine learning models
• In the context of classic regression
 Differentials also provide effective regularization
 Without bias, contrarily to e.g. ridge/Tikhonov
 Solution remains analytic
 See appendix 3 github.com/differential-machine-learning/appendices/blob/master/App3-Regression.pdf
 And demonstration notebook github.com/differential-machine-learning/notebooks/blob/master/DifferentialRegression.ipynb
• In the context of PCA
 Differential PCA identifies the principal risk factors of a transaction or trading book
 Also provides an extremely effective data preparation and dimension reduction step
 See appendix 2 https://github.com/differential-machine-learning/appendices/blob/master/App2-Preprocessing.pdf

83.
Thank you for your attention
In-House system of the Year 2015
Winner: Superfly Analytics at Danske Bank
Excellence in Risk Management and Modelling, 2019
Winner: Superfly Analytics at Danske Bank