machine learning for quantitative finance€¦ · machine learning for quantitative finance: fast...

Fast derivative pricing

Machine Learningfor Quantitative Finance

Sofie Reyners

Joint work with Jan De Spiegeleer, Dilip Madan andWim Schoutens

Derivative pricing is time-consuming...

I Vanilla option pricing• European-type→ Fast Fourier transform

• American-type→ Tree methods

I Exotic option pricing→ Monte Carlo simulations

1 Machine Learning for Quantitative Finance

... but time is money!

time-consuming algorithms

continuously moving markets

→ prices are outdated when available, overnight calculationscannot be performed in one night, ...


Let a machine learn the pricing function

product, market andmodel parameters model price

time-consuming method

machine learning

Expensive pricing function is summarized with machine learning.


Let a machine learn the pricing function



machine learning

When training is completed, prediction is extremely fast!


Gaussian process regression (GPR)

Consider a training set

(X,y) = {(xi, yi) | i = 1, . . . , n}.

Find a relation between inputs and outputs:

yi = f(xi) + εi

where f(x) is a Gaussian process and εi ∼ N (0, σ2n) are i.i.d. random

variables representing the noise in the data.


Gaussian process

A Gaussian process f(x) is a, possibly infinite, collection of randomvariables, any finite subset of it having a joint Gaussian distribution.

I Mean function: m(x) = E[f(x)

]I Kernel function: k(x,x′) = Cov(f(x), f(x′))

=⇒ f(x) ∼ GP (m(x), k(x,x′))


Gaussian process

If f(x) ∼ GP (0, k(x,x′)), then

f ∼ N (0,K(X,X))

where (X,f) = {(xi, fi) | i = 1, . . . , n} is a sample from f(x) and

K(X,X) =

k(x1,x1) . . . k(x1,xn)... . . . ...

k(xn,x1) . . . k(xn,xn)


GPR: a Bayesian method

I Don’t model the relation as one function, but as a distributionover functions.

I Procedure:

1 Start from a prior GP→ prior knowledge: smooth function, periodic function, ...→ prior distribution over functions

2 Include observed data points

3 Compute a posterior GP


Posterior distribution

Only consider functions that agree with the data.

I Take new inputs X∗, with corresponding (unknown) functionvalues f∗

I Joint distribution of training outputs and function values:[yf∗

]∼ N

(0 ,

[K(X,X) + σ2

nI K(X,X∗)K(X∗, X) K(X∗, X∗)

])


Posterior distribution

I Condition on the observations:

f∗|X∗, X,y ∼ N(µ,Σ

)with

µ = K(X∗, X)[K(X,X) + σ2

nI]−1y

Σ = K(X∗, X∗)−K(X∗, X)[K(X,X) + σ2

nI]−1

K(X,X∗)


Kernel function

Squared exponential kernel function

k(x,x′) = σ2f exp

(−|x− x′|2

2`2

)

with hyperparameters σf and `:I σ2

f = signal varianceI ` = length-scale parameter

→ Hyperparameters (including σn) are estimated from the trainingdata, usually with MLE.


Mean function

Often set to zero, but can be modelled using basis functions h(x).

g(x) = f(x) + h(x)Tβ ∼ GP (h(x)Tβ, k(x,x′))

whereI f(x) ∼ GP (0, k(x,x′))I β should be estimated from the training data

Common choice:h(x) = (1, x, x2)


Application set-up

I Construct a training set:


sample n randomcombinations xi

compute ncorresponding prices yi


I Fit a Gaussian process regression (GPR) model.

I Fast prediction of new model prices.


Pricing European call options

Training set:

Product/market VG Heston

K ∈ [40%, 160%] σ ∈ [0.05, 0.45] κ ∈ [1.4, 2.6]T ∈ [11M, 1Y ] ν ∈ [0.55, 0.95] ρ ∈ [−0.85,−0.55]r ∈ [1.5%, 2.5%] θ ∈ [−0.35,−0.05] θ ∈ [0.45, 0.75]q ∈ [0%, 5%] η ∈ [0.01, 0.1]

v0 ∈ [0.01, 0.1]

→ sample n values of each parameter→ calculate n FFT-based model prices


Pricing European call options

I Fit the GPR model

K, T , r − q,model parameters

FFTmodel price

GPR

I Construct a test set:• Similarly as training set• Slightly smaller parameter intervals


Out-of-sample prediction

(a) Variance Gamma (b) Heston

→ model trained on 10 000 points, tested on 100 000 points.


Performance summary

VG Heston

Size of training set 5000 10 000 20 000 5000 10 000 20 000

In-sample predictionMAE 0.0016 0.0017 0.0013 0.0036 0.0027 0.0033AAE 2.5763e-04 1.9627e-04 1.4747e-04 5.2260e-04 4.0347e-04 3.4524e-04

Out-of-sample predictionMAE 0.0028 0.0022 0.0016 0.0060 0.0054 0.0048AAE 2.2508e-04 1.6942e-04 1.2828e-04 5.8991e-04 4.4112e-04 3.6623e-04

Speed-up × 30 × 15 × 7 × 40 × 20 × 10

withMAE = max {|ECF F T (i)− ECGP R(i)|, i = 1, . . . , n}

AAE = 1n

n∑i=1

|ECF F T (i)− ECGP R(i)|


Pricing American options

I Put options with strike K and maturity T

I Use binomial tree model (daily steps) with volatility

σ ∈ [0.05, 0.55]

K, T , r, q, σ binomial treemodel price

GPR


Out-of-sample performance

MAE 0.0086

AAE 9.1684e-04

Speed-up ×70

→ model trained on 10 000 points, tested on 100 000 points.


Pricing barrier options

I Down-and-out barrier put options with barrier level H, strike Kand maturity T with

H ∈ [55%, 99%]

I Use Monte Carlo simulation, according to Heston’s model

H, K, T , r, q,κ, ρ, θ, η, v0

MC Hestonmodel price

GPR


Out-of-sample performance

MAE 0.0086

AAE 6.7386e-04

Speed-up ×5850

→ model trained and tested on 10 000 points.


Conclusion

I Time-consuming pricing methods

I Gaussian process regression• Matrix inversion• Hyperparameter optimization

I Apply GPR on existing methods• Speed-up of several orders of magnitude• Some trade-off with accuracy


Thank you!

More information:

De Spiegeleer, J., Madan, D. B., Reyners, S. and Schoutens, W. (2018),Machine Learning for Quantitative Finance: Fast Derivative Pricing, Hedgingand Fitting, Quantitative Finance, forthcoming.


machine learning for quantitative finance€¦ · machine learning for quantitative finance: fast...

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