Upload
dorie
View
45
Download
0
Embed Size (px)
DESCRIPTION
Prediction of Rare Volatility Surface Patches using Artificial Neural Network. Regression based model vs. Neural networks based model. Regression based model doesn’t consider infrequently available patches as less reliable Neural networks based model - PowerPoint PPT Presentation
Citation preview
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)