Prediction of Rare Volatility Surface Patches using

Artificial Neural Network

Regression based model vs. Neural networks based model

• Regression based modeldoesn’t consider infrequently available patches as less reliable

• Neural networks based model– can consider the frequency of data by varying the

learning rate between observations– easily implementable in today’s fast computers– can better model the underlying relationship between

input and output data sets than regression based models

Outline of the Algorithm

1. Generation of data with missing values

2. Fill the missing values with a regression algorithm to construct the data set.

3. Train the neural network with the above data set with a modified learning rate using incremental (online) approach.

4. Simulate the above neural network with some testing data set and find results.

Generation of data

Data Format:

V=(v1, v2, v3,…, v27)

1. v1 .. v20 represent frequently available surface patches

2. v21 .. v22 represent macro-economic factors

3. v23 .. v27 represent infrequently available surface patches

Generation of data (contd.)Facts:

• No real life data

• patches and macro economic factors correspond to a volatility surface

Data generation using simulated surfaces :

• Generate surfaces (with random parameters) to simulate volatility surfaces, and

• Obtain data (set of Vs) from these generated surfaces

Randomized data generation (2D view)

Randomized data generation (3D view)

Generated data

Generating Data with Missing Values

• Randomly miss values from v23… v27 with the following rates

Variable name Missing percentages

v23 No miss

v24 75%

v25 65%

v26 45%

v27 25%

Data after missing

Fill the missing values with some regression algorithm

Regularized EM Algorithm

• Steps 1. Estimate mean values and covariance

matrices of incomplete datasets. 2. Impute missing values in incomplete

datasets.• Based on iterated analyses of linear regressions

of variables with missing values on variables with available values

• Regression coefficients are estimated by Ridge Regression

Data after missing values filled by Regularized EM Algorithm

Train neural network with modified learning rate

Structure of Neural Network

1

2

3

22

.

.

.

Input

v

v

v

v

… …

23

24

25

26

27

output

v

v

v

v

v

Weight, Gradient & Learning Rate (LR)

( ) ( 1) ( ), where

weight of a single link

momentum

learning rate

( ) gradient function

w t w t g t

w

g t

ERROR

Weight of a single link

( )slope g t

Effect of small and large LR

Effect of change in Learning rate

Observation LR CHANGE

Effect

Has missing value

Decrease 1. Change of Ws will be low

2. Relatively less effect on training

No missing value

Increase 1. Change of Ws will be high

2. Relatively more effect on training

• Learning rate of an observation = lr*normalized multiplier of an observation, where lr is some constant.

• Calculating multiplier of a observation

1. Frequency norm

2. Log frequency norm

Learning rate of an observation

Frequency norm

• Multiplier of an observation, mfn =

• Frequency of an output node is the number of observations, where a value for that output node is available (i.e., not missing)

• mfn is between [0,1]

Learning rate of an observation (contd.)

Log frequency norm

• Map mfn within [lr/2, 2*lr] using logarithmic scaling

Learning rate of an observation (contd.)

Log frequency norm,

exp log 2 log log2 2

lfn

fn

m

lr lrm lr

0 1

0 log 2 log2

lrlr

log2

lr log 2lr

2

lr2lr

Incorporating mfn and mlfn in training of

NN using Incremental Mode

Constant LR

Unmodified Learning Rate

Modified LR with Frequency NORM

Modified Learning Rate

with Frequency Norm

Modified LR with Log Frequency NORM

Modified Learning Rate

with Log Frequency Norm

Simulation Result

• 3 different methods have been tested

1.Modified learning rate with frequency norm

2.Modified learning rate with log frequency norm

3.ANN with constant learning rate• Each method is run for 10 times and 100 epochs• Result shows that the method with modified

learning rate using frequency norm outperforms the other two methods consistently.

Simulation result (Graph)