Novel price models in the capital market

8/27/2019 26th Annual Technical Conference 2019
1
Novel Price Models for the Capital Market
Asoka Korale, Ph.D.

Motivations for Monitoring
Price Movements

Insights from modeling Price Movements
• A feature to construct alerts for market surveillance
• Vital in devising trading strategies
• A means to estimate Variance (Volatility) and Variation in the Variance
• A feature to characterize / classify a security
• Detect Anomalous / Outlier movements compared to benchmarks / historical behavior
• Estimate extreme value events consistent with historical behavior

Models on Prices
I. Stochastic Volatility model of Price
II. Prices simulated by estimating Random Process
III. Regression model on Consecutive Prices
IV. Sensitivity in estimated Variance to measurement Technique

Model I.
Stochastic Volatility Model of Price

Modeling prices as a Random Walk
In this type of model each new sample is modeled as a discrete jump from the previous value
Then the difference between the last and first elements of in a sequence of N events, is the sum of N IID RV
𝑃𝑛+1 = 𝑃𝑛 + 𝑋 𝑛+1
𝑃𝑛+2 = 𝑃𝑛+1 + 𝑋 𝑛+2= 𝑃𝑛 + 𝑋 𝑛+1+ 𝑋 𝑛+2
𝑃𝑛+𝑁 = 𝑃𝑛 + 𝑋 𝑛+1+ 𝑋 𝑛+2+…+ 𝑋 𝑛+𝑁
𝑃𝑛+𝑁 − 𝑃𝑛 =
𝑖=1
𝑁
𝑋 𝑛+𝑖
We can use this idea to estimate the mean and variance of the difference between the start and end prices in a sequence of N prices
In essence, a measure of the maximum mean change and the maximum variation about the mean

Estimating the deviation - N steps Away
We can estimate the parameters of the distribution underlying from data by
considering the difference equation relating prices and the IDD random variable 𝑋 𝑛
We consider a series (sequence) prices for which we wish to estimate certain quantities of interest –
The mean and variance of 𝑋 𝑛 should ideally be estimated over a “representative” interval –
so that estimations / predictions are based on statistics estimated on periods where the past is similar to the future
𝑋 𝑛~𝐹(𝜇 𝑥, 𝜎𝑥
𝑃𝑛 − 𝑃𝑛−1 = 𝑋 𝑛
𝐸 𝑃𝑛 − 𝑃𝑛−1 = 𝐸(𝑋 𝑛 = 𝜇 𝑥 𝑉𝑎𝑟 𝑃𝑛 − 𝑃𝑛−1 = 𝑉𝑎𝑟(𝑋 𝑛 = 𝜎𝑥
2
𝐸( 𝑃 𝑛+𝑁 − 𝑃𝑛 =
𝑖=1
𝑁
𝐸(𝑋 𝑛+𝑖 = 𝑁𝜇 𝑥 𝑉𝑎𝑟( 𝑃 𝑛+𝑁 − 𝑃𝑛 =
𝑖=1
𝑁
𝑉𝑎𝑟(𝑋 𝑛+𝑖 = 𝑁𝜎𝑥
2
The mean and variance of the difference is then the mean and variance of the sum of N IDD random variables
we have single realization of the random process from which we
estimate distribution or parameters of model
sample no n-3 n-2 n-1 n n+N
Pn-3 Pn-2 Pn-1 Pn Pn+N
…………..
𝑋 𝑛
𝑋 𝑛−1𝑋 𝑛−2
𝑖=1
𝑁
𝑋 𝑛+𝑖
Sequence of Prices

Results I:
Stochastic Volatility model of Price
Price movements on overlapping windows defined on fixed number of events

Estimating the Expected N Step change in Price
a segment of a time series of prices window length 10 samples –
captures the trend in the expected change

Estimating the Uncertainty in the Expected N Step change in Price
a segment of a price time series
greater uncertainty in estimating longer time horizons

Estimating the Expected N Step change in Price
A segment of a price time series
window length 3 samples –
estimated trend in expected change more sensitive to local conditions

Estimating the Uncertainty in the Expected N Step change in Price
a segment of a price time series
lower uncertainty in estimating shorter time horizons

Observations on the Trend and Variance
of an N Step Change in Price
has the interpretation of a trend
an average step size x N (N is the number of events in the window)
a measure of the maximum change
a measure of the difference between two prices (at two points) in a sequence
calculated as the variance of a sum of random variables, each with a variance measured
from the distribution of the first differences
also the uncertainty in the average step size x N (the number of events in a window)
a measure of the total uncertainty in the difference / deviation
𝑖=1
𝑁
𝐸(𝑋 𝑛+𝑖 = 𝑁𝜇 𝑥
𝑉𝑎𝑟( 𝑃 𝑛+𝑁 − 𝑃𝑛 =
𝑖=1
𝑁
2

Model II.
Simulating Prices by estimating Random Processes

Estimating the Stochastic Process underlying a Random Walk
First difference gives each step in the random walk – an IID random variable
Distribution ( F ) - of the first differences, estimated over a representative interval
𝑃𝑛 − 𝑃𝑛−1 = 𝑋 𝑛
• Random process underlying the movement in prices modeled as a random walk
• Estimated from the distribution of the first differences in the time series of prices.
𝑋 𝑛~𝐹(𝜇 𝑥, 𝜎 𝑥

Original and Simulated Prices
Time series of price segment used to estimate underlying random process
Simulated paths /
Realizations of the Random Process

Identical Distribution underlying all Paths
Statistic Original path 1 path 2 path 3 path 3 path 4
Min 179.54 180.31 180.65 179.85 183.51 180.33
Max 177.76 178.82 178.68 177.48 179.14 178.67
max - min 1.78 1.49 1.97 2.37 4.37 1.66
mean 178.55 179.47 179.71 178.82 181.10 179.53
variance 0.14 0.08 0.23 0.31 1.94 0.12
All simulated realizations have same underlying distribution of first differences as the original price segment
Each realization has widely different statistics from
the rest – but same distribution in first differences

Model III.
Modeling Price Change via Regression

A Linear Model on Consecutive Prices
“Typically price is modeled via a sequence of IID random variables drawn from a symmetric distribution (F) with zero mean and constant
variance 𝜎 𝑥
2
".
However this assumption is rarely valid and instead we estimate the random variable Xn underlying the prices
In this context we may estimate a linear relationship between and by considering a regression of the form
where y represents the price at time n, x the price at time n-1, “a” and “b” constants and an error term 𝜖 𝑛
𝑃𝑛 = 𝑃𝑛−1 + 𝑋 𝑛
𝑋 𝑛~𝐹(𝜇 𝑥, 𝜎𝑥
𝑦 𝑛 = 𝑎 + 𝑏𝑥 𝑛 + 𝜖 𝑛
𝑃𝑛𝑃𝑛−1
• A solution (fit) via regression is optimal in the least squares sense and finds the parameters “a” and “b” of a line of best fit
• The residual or error term can be used to gauge the goodness of fit between the estimated model (line) and data (price)
• we can estimate a “trend” in the prices - delimited by a window defined on consecutive events
• The modeling can provide an insight in to a “trend” between a minimum and maximum price
and we model it as a sequence driven by a series of shocks
where F is a distribution that can be estimated from data – as the first difference of the prices

Regression via Least Squares
Estimate parameters of line by minimizing error of prediction to obtain parameters optimal in least squares sense  2
)( eYY
Avye 











 11
....
1
Nn
n
P
P
A 






b
a
v













Nn
n
P
P
y ..
1
Thus minimizing the mean square error between the actual and estimated values corresponds to
2
min e
v
yy 
2
min Avy
v

          AvyyAvAvAvyyAvyAvyAvy TTTTT

2
AvyAvAvyy TTTT
2
where
  yAvAAAvy
v
TT
22
2



  022  yAvAA TT
  yAAAv TT 1

Differentiating w.r.t. v and setting to zero
equivalently
leading to the parameters
A solution most often can be found to this over determined system of equations when the columns of A are not dependent.

Modeling change in Consecutive Prices
𝑃𝑛
Monitor price over window defined over interval of time or window defined over a fixed number of events
Measure min and max prices values over the window and to model the change in price over the window
Fit linear regression to consecutive prices between min and max prices
Estimate Pn+1 at each Pn via regression line
Estimate “trend” between a “low price” and a “high price” and as a function of time or number of events n
Use regression error to detect outliers / with respect to trend
𝑃𝑛
𝑙𝑜𝑤
𝑃𝑛
ℎ𝑖𝑔ℎ
No of samples in window is N
𝑃𝑛
𝑙𝑜𝑤
𝑃𝑛
ℎ𝑖𝑔ℎ
𝑛𝑡𝑖𝑚𝑒 or
∆𝑡
𝑃𝑛
𝑙𝑜𝑤
𝑃𝑛
ℎ𝑖𝑔ℎ

Least Squares fit to Pn vs. Pn+1
𝑃𝑛+1 = a + 𝑏𝑃𝑛
𝜀 𝑛 = 𝑃𝑛+1 − 𝑃𝑛+1
𝑃𝑛+1
𝑃𝑛
𝑎
𝑏
x
x
x
x
x
x
𝜀 𝑛
𝑃𝑛
𝑃𝑛+1
𝑃𝑛+1
• Points close to the line (trend) can be considered “normal behavior”
• Points with large errors may be considered anomalies - with respect to the model (or line)
• Maximum change in price over a window can be measured in several ways
 Laterally and Vertically
• Insight in to conditional probability of observing next price Pn+1 given the current Pn
𝑃𝑛
𝑙𝑜𝑤
𝑃𝑛
ℎ𝑖𝑔ℎ
No of samples in window = N

Insight from distribution of the error
• The distribution in the error provides clues to the degree to which the price can vary – with respect to
• a deviation with respect to the “average behavior” – described by the model – the “line”
• deviation from the average behavior of a sample Pn to Pn+1
• different degrees of movement may be expected depending on the current price
• The distribution of the error is unlikely to be normal - but can be used to estimate the “likelihood” of the
magnitude of a particular “deviation” from the “norm”
𝜀 𝑛
𝜀 𝑛
𝑓𝜖(𝜀 𝑛

Results III.
Regression model on Consecutive Prices

Regression on Consecutive Prices
• Consecutive prices (of N events) – observe a pattern that can largely be described by a straight line.
• Trend indicates the degree to which consecutive prices differ as an (estimated) multiple of the current price
• detect outliers with respect to this trend
• Distribution of outliers with respect to estimated trend

Observations - Regression on Consecutive Prices
• A “trend” line between the minimum and maximum prices –
• that has been observed over an interval of N events
• A rate of change in the prices – between a min and max observed / measured over a period
• A function (trend line) that gives an estimate of the next price (Pn+1) given the current price (Pn)
• A measure of the error in such a prediction
• from the distribution of the sequence of errors estimated after the line is fit
• A method to identify outliers in price movement
• (points that deviate significantly from trend line)
• Detect anomalies in prices of on-book and off-book trading
• potential manipulations / deviations on agreed time –
• that could also manifest as outlier with respect to trend

Model IV.
Sensitivity of Variance to Measurement Technique
overlapping windows of time – with varying numbers of events
overlapping windows of a fixed number of events

Impact of overlapping window of time
• Overlapping measurement intervals of time
• Each window anchored on a consecutive trade
• Large variation in the number of events across windows
• due to the burst behavior in trades
• right shifting & flattening curves
• Large variation in the variance over different window lengths
• Estimated variance is less consistent
• made over a dissimilar number of samples
• the spread in the distribution is wider and the curves flatter

Impact of fixed number of events
• Overlapping intervals of measurement – move window by single sample
• each interval – a fixed number of consecutive trades
• less variation in the variance over different window lengths
• estimate is more consistent
• being made over a similar number of samples
• the spread in the distribution is narrower
• the curves less flat
• measured variance impacted by number of trades in the window of measurement
• random walk property of prices
• Each random variable in the walk contributes to variance

30
THANK YOU

Equivalence of Random Walk & ARIMA Process

Random Walk & ARIMA Process
In this type of model each new sample is modeled as a discrete jump from the previous value
• the difference between the last and first elements in a sequence of N prices, is the sum of N IID random variables
• this sum tends to a Normal Distribution for N large
𝑃𝑛+1 = 𝑃𝑛 + 𝑋 𝑛+1
𝑃𝑛+2 = 𝑃𝑛+1 + 𝑋 𝑛+2 = 𝑃𝑛 + 𝑋 𝑛+1 + 𝑋 𝑛+2
𝑃𝑛+𝑁 = 𝑃𝑛 + 𝑋 𝑛+1+ 𝑋 𝑛+2 +…+ 𝑋 𝑛+𝑁
𝑃𝑛+𝑁 − 𝑃𝑛 =
𝑖=1
𝑁
𝑋 𝑛+𝑖
Use this relationship to estimate the mean and variance of the difference between the start and end prices in a sequence of N prices
In essence, an estimate of the maximum variance and a measure of the maximum mean change
𝑃𝑛+𝑁 = 𝑃𝑛 + 𝑋 𝑛+1+ 𝑋 𝑛+2+ … + 𝑋 𝑛+𝑁
𝑋 𝑛~𝐹(𝜇 𝑥, 𝜎𝑥 F distribution estimated from historical Prices
…………
..
𝑋 𝑛
𝑖=1
𝑁
𝑋 𝑛+𝑖
Sequence of Prices

Random Walk & ARIMA Process
The first differences of the prices forms an Autoregressive Process
From equations on slide 7 we have estimate for price N steps away and a measure of the uncertainty in this estimate
𝑃𝑛−2 = 𝑃𝑛−3 + 𝑋 𝑛−2
𝑃𝑛−1 = 𝑃𝑛−2 + 𝑋 𝑛−1= 𝑃𝑛−3 + 𝑋 𝑛−2+ 𝑋 𝑛−1
𝑃𝑛 = 𝑃𝑛−1 + 𝑋 𝑛 = 𝑃𝑛−3 + 𝑋 𝑛+ 𝑋 𝑛−1+ 𝑋 𝑛−2
𝑖=1
𝑁
𝐸(𝑋 𝑛+𝑖 = 𝑁𝜇 𝑥 𝑉𝑎𝑟( 𝑃 𝑛+𝑁 − 𝑃𝑛 =
𝑖=1
𝑁
2
Instead of the expectation / average step size to estimate the future price as function of past differences
we may use a weighted average of the first differences in the immediate neighborhood of the current price Pn, using a sequence of weights w
the weights may also be interpreted as being representative of the PDF in the context of an expectation
We make estimate as N times an average step size and the uncertainty as N times the uncertainty in each step
𝑋 𝑛 = w1 𝑋 𝑛−1 + w2 𝑋 𝑛−2 + w3 𝑋 𝑛−3 ……
𝑋 𝑛 = w1 𝑋 𝑛−1 + w2 𝑋 𝑛−2 + w3 𝑋 𝑛−3 +… + 𝜀 𝑛 Xn is an AR process and Pn ARIMA
…………
..
𝑋 𝑛
𝑖=1
𝑁
𝑋 𝑛+𝑖
Sequence of Prices

THANKS

Novel price models in the capital market

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