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Quantitative Trading as a Mathematical Science
QuantCon Singapore 2016
Haksun Li
haksun.li@numericalmethod.com
www.numericalmethod.com
Abstract
2
 Quantitative trading is distinguishable from other trading methodologies like
technical analysis and analysts’ opinions because it uniquely provides justifications
to trading strategies using mathematical reasoning. Put differently, quantitative
trading is a science that trading strategies are proven statistically profitable or even
optimal under certain assumptions. There are properties about strategies that we
can deduce before betting the first $1, such as P&L distribution and risks. There are
exact explanations to the success and failure of strategies, such as choice of
parameters. There are ways to iteratively improve strategies based on experiences of
live trading, such as making more realistic assumptions. These are all made possible
only in quantitative trading because we have assumptions, models and rigorous
mathematical analysis.
 Quantitative trading has proved itself to be a significant driver of mathematical
innovations, especially in the areas of stochastic analysis and PDE-theory. For
instances, we can compute the optimal timings to follow the market by solving a
pair of coupled Hamilton–Jacobi–Bellman equations; we can construct sparse mean
reverting baskets by solving semi-definite optimization problems with cardinality
constraints and can optimally trade these baskets by solving stochastic control
problems; we can identify statistical arbitrage opportunities by analyzing the
volatility process of a stochastic asset at different frequencies; we can compute the
optimal placements of market and limit orders by solving combined singular and
impulse control problems which leads to novel and difficult to solve quasi-
variational inequalities.
Speaker Profile
 Dr. Haksun Li
 CEO, NM LTD.
 (Ex-)Adjunct Professors, Industry Fellow, Advisor,
Consultant with the National University of Singapore,
Nanyang Technological University, Fudan University,
the Hong Kong University of Science and Technology.
 Quantitative Trader/Analyst, BNPP, UBS
 Ph.D., Computer Science, University of Michigan Ann
Arbor
 M.S., Financial Mathematics, University of Chicago
 B.S., Mathematics, University of Chicago
3
What is Quantitative Trading?
4
Quantitative Trading?
5
 Quantitative trading is the buying and selling of assets
following the instructions computed from a set of
proven mathematical models.
 The differentiation from other trading methodologies
or the emphasis is on how a strategy is proven and
not on what strategy is created.
 It applies (rigorous) mathematical reasoning in all
steps during trading strategy construction from the
start to the end.
Moving Average Crossover as a TA
6
 A popular TA signal: Moving Average Crossover.
 A crossover occurs when a faster moving average (i.e. a
shorter period moving average) crosses above/below a
slower moving average (i.e. a longer period moving
average); then you buy/sell.
 In most TA book, it is never proven only illustrated
with an example of applying the strategy to a stock for
a period of time to show profits.
Technical Analysis is Not Quantitative Trading
7
 TA books merely describes the mechanics of strategies
but never prove them.
 Appealing to common sense is not a mathematical
proof.
 Conditional probabilities of outcomes are seldom
computed. (Lo, Mamaysky, & Wang, 2000)
 Application of TA is more of an art (is it?) than a
science.
 How do you choose the parameters?
 For any TA rule, you almost surely can find an asset
and a period that the rule “works”, given the large
number of assets and many periods to choose from.
Fake Quantitative Models
 Data snooping
 Misuse of mathematics
 Assumptions cannot be quantified
 No model validation against the current regime
 Ad-hoc take profit and stop-loss
 why 2?
 How do you know when a model is invalidated?
 Cannot explain winning and losing trades
 Cannot be analyzed (systematically)
8
The Quantitative Trading Research
Process
9
NM Quantitative Trading Research Process
10
1. Translate a vague trading intuition (hypothesis) into
a concrete mathematical model.
2. Translate the mathematical symbols and equations
into a computer program.
3. Strategy evaluation.
4. Live execution for making money.
Step 1 - Modeling
11
 Where does a trading idea come from?
 Ex-colleagues
 Hearsays
 Newspapers, books
 TV, e.g., Moving Average Crossover (MA)
 A quantitative trading strategy is a math function, f,
that at any given time, t, takes as inputs any
information that the strategy cares and that is
available, Ft, and gives as output the position to take,
f(t,Ft).
Step 2 - Coding
12
 The computer code enables analysis of the strategy.
 Most study of a strategy cannot be done analytically.
 We must resort to simulation.
 The same piece of code used for research and
investigation should go straight into the production
for live trading.
 Eliminate the possibility of research-to-IT translation
errors.
Step 3 – Evaluation/Justification
13
 Compute the properties of a trading strategy.
 the P&L distribution
 the holding time distribution
 the stop-loss
 the maximal drawdown
 http://redmine.numericalmethod.com/projects/publi
c/repository/svn-
algoquant/show/core/src/main/java/com/numericalm
ethod/algoquant/execution/performance
Step 4 - Trading
14
 Put in capitals incrementally.
 Install safety measures.
 Monitor the performance.
 Regime change detection.
Mathematical Analysis of Moving
Average Crossover
15
Moving Average Crossover as a Quantitative
Trading Strategy
16
 There are many mathematical justifications to Moving
Average Crossover.
 weighted Sum of lags of a time series
 Kuo, 2002
 Whether a strategy is quantitative or not depends not
on the strategy itself but
 entirely on the process to construct it;
 or, whether there is a scientific justification to prove it.
Step 1 - Modeling
 Two moving averages: slower (𝑛) and faster (𝑚).
 Monitor the crossovers.
 𝐵𝑡 =
1
𝑚
∑ 𝑃𝑡−𝑗
𝑚−1
𝑗=0 −
1
𝑛
∑ 𝑃𝑡−𝑗
𝑛−1
𝑗=0 , 𝑛 > 𝑚
 Long when 𝐵𝑡 ≥ 0.
 Short when 𝐵𝑡 < 0.
How to Choose 𝑛 and 𝑚?
 It is an art, not a science (so far).
 They should be related to the length of market cycles.
 Different assets have different 𝑛 and 𝑚.
 Popular choices:
 (250, 5)
 (250 , 20)
 (20 , 5)
 (20 , 1)
 (250 , 1)
Two Simplifications
19
 Reduce the two dimensional problem to a one
dimensional problem.
 Choose 𝑚 = 1. We know that m should be small.
 Replace arithmetic averages with geometric averages.
 This is so that we can work with log returns rather than
prices.
GMA(n , 1)
 𝐵𝑡 ≥ 0 iff 𝑃𝑡 ≥ ∏ 𝑃𝑡−𝑗
𝑛−1
𝑗=0
1
𝑛
 𝑅𝑡 ≥ − ∑
𝑛− 𝑗+1
𝑛−1
𝑅𝑡−𝑗
𝑛−2
𝑗=1 (by taking log)
 𝐵𝑡 < 0 iff 𝑃𝑡 < ∏ 𝑃𝑡−𝑗
𝑛−1
𝑗=0
1
𝑛
 𝑅𝑡 < − ∑
𝑛− 𝑗+1
𝑛−1
𝑅𝑡−𝑗
𝑛−2
𝑗=1 (by taking log)
What is 𝑛?
 𝑛 = 2
 𝑛 = ∞
Acar Framework
 Acar (1993): to investigate the probability distribution
of realized returns from a trading rule, we need
 the explicit specification of the trading rule
 the underlying stochastic process for asset returns
 the particular return concept involved
Knight-Satchell-Tran Intuition
 Stock returns staying going up (down) depends on
 the realizations of positive (negative) shocks
 the persistence of these shocks
 Shocks are modeled by gamma processes.
 Asymmetry
 Fat tails
 Persistence is modeled by a Markov switching process.
Knight-Satchell-Tran 𝑍𝑡
Zt = 0
DOWN
TREND
𝑓𝛿 𝑥
=
𝜆2
𝛼2
𝑥 𝛼2−1
Γ 𝛼2
𝑒−𝜆2 𝑥
Zt = 1
UP TREND
𝑓𝜀 𝑥
=
𝜆1
𝛼1
𝑥 𝛼1−1
Γ 𝛼1
𝑒−𝜆1 𝑥
q p
1-q
1-p
Knight-Satchell-Tran Process
 𝑅𝑡 = 𝜇𝑙 + 𝑍𝑡 𝜀𝑡 − 1 − 𝑍𝑡 𝛿𝑡
 𝜇𝑙: long term mean of returns, e.g., 0
 𝜀𝑡, 𝛿𝑡: positive and negative shocks, non-negative, i.i.d
 𝑓𝜀 𝑥 =
𝜆1
𝛼1 𝑥 𝛼1−1
Γ 𝛼1
𝑒−𝜆1 𝑥
 𝑓𝛿 𝑥 =
𝜆2
𝛼2 𝑥 𝛼2−1
Γ 𝛼2
𝑒−𝜆2 𝑥
Step 3 – Evaluation/Justification
 Assume the long term mean is 0, 𝜇𝑙 = 0.
 When 𝑛 = 2,
 𝐵𝑡 ≥ 0 ≡ 𝑅𝑡 ≥ 0 ≡ 𝑍𝑡 = 1
 𝐵𝑡 < 0 ≡ 𝑅𝑡 < 0 ≡ 𝑍𝑡 = 0
GMA(2, 1) – Naïve MA Trading Rule
 Buy when the asset return in the present period is
positive.
 Sell when the asset return in the present period is
negative.
Naïve MA Conditions
 The expected value of the positive shocks to asset
return >> the expected value of negative shocks.
 The positive shocks persistency >> that of negative
shocks.
𝑇 Period Returns
 𝑅𝑅 𝑇 = ∑ 𝑅𝑡 × 𝐼 𝐵𝑡−1≥0
𝑇
𝑡=1
Sell at this time point
𝑇
𝐵 𝑇 < 0
0 1
hold
Holding Time Distribution
 𝑃 𝑁 = 𝑇
 = 𝑃 𝐵 𝑇 < 0, 𝐵 𝑇−1 ≥ 0, … , 𝐵1 ≥ 0, 𝐵0 ≥ 0
 = 𝑃 𝑍 𝑇 = 0, 𝑍 𝑇−1 = 1, … , 𝑍1 = 1, 𝑍0 = 1
 = 𝑃 𝑍 𝑇 = 0, 𝑍 𝑇−1 = 1, … , 𝑍1 = 1|𝑍0 = 1 𝑃 𝑍0 = 1
 = �
Π𝑝 𝑇−1
1 − 𝑝 , T ≥1
1 − Π, T=0
 Stationary state probability:
 Π =
1−𝑞
2−𝑝−𝑞
Conditional Returns Distribution (1)
 Φ 𝑅𝑅 𝑇|𝑁=𝑇 𝑠 = E 𝑒
𝑖 ∑ 𝑅 𝑡×𝐼 𝐵 𝑡−1≥0
𝑇
𝑡=1 𝑠
|𝑁 = 𝑇
 = E 𝑒
𝑖 ∑ 𝑅 𝑡×𝐼 𝐵 𝑡−1≥0
𝑇
𝑡=1 𝑠
|𝐵 𝑇 < 0, 𝐵 𝑇−1 ≥ 0, … , 𝐵0 ≥ 0
 = E 𝑒 𝑖 ∑ 𝑅 𝑡
𝑇
𝑡=1 𝑠
|𝑍 𝑇 = 0, 𝑍 𝑇−1 = 1, … , 𝑍1 = 1
 = E 𝑒 𝑖 𝜀1+⋯+𝜀 𝑇−1−𝛿 𝑇 𝑠
 = �
Φ 𝜀
𝑇−1
𝑠 Φ 𝛿 −𝑠 , T ≥1
Φ 𝛿 −𝑠 , T =0
Unconditional Returns Distribution (2)
 Φ 𝑅𝑅 𝑇
𝑠 =
∑ E 𝑒
𝑖 ∑ 𝑅 𝑡×𝐼 𝐵 𝑡−1≥0
𝑇
𝑡=1 𝑠
|𝑁 = 𝑇 𝑃 𝑁 = 𝑇∞
𝑇=0
 =
∑ Π𝑝 𝑇−1
1 − 𝑝 Φ 𝜀
𝑇−1
𝑠 Φ 𝛿 −𝑠∞
𝑇=1 + 1 − Π Φ 𝛿 −𝑠
 = 1 − Π Φ 𝛿 −𝑠 + Π 1 − 𝑝
Φ 𝛿 −𝑠
1−𝑝Φ 𝜀 𝑠
Expected Returns
 E 𝑅𝑅 𝑇 = −𝑖Φ 𝑅𝑅 𝑇
′
0
 =
1
1−𝑝
Π𝑝𝜇 𝜀 − 1 − 𝑝 𝜇 𝛿
 When is the expected return positive?
 𝜇 𝜀 ≥
1−𝑝
Π𝑝
𝜇 𝛿, shock impact
 𝜇 𝜀 ≫ 𝜇 𝛿, shock impact
 Π𝑝 ≥ 1 − 𝑝, if 𝜇 𝜀 ≈ 𝜇 𝛿, persistence
GMA(∞,1) Rule
 𝑃𝑡 ≥ ∏ 𝑃𝑡−𝑗
𝑛−1
𝑗=0
1
𝑛
 ln 𝑃𝑡 ≥
1
𝑛
∑ ln 𝑃𝑡−𝑗
𝑛−1
𝑗=0
 ln 𝑃𝑡 ≥ 𝜇1
GMA(∞,1) Expected Returns
 Φ 𝑅𝑅 𝑇
𝑠 =
1 − Π 𝑞 Φ 𝛿 𝑠 + Φ 𝛿 −𝑠 +
1 − 𝑝 1 − Π Φ 𝜀 𝑠 + Φ 𝜀 −𝑠
 E 𝑅𝑅 𝑇 = − 1 − 𝑝 1 − Π 𝜇 𝜀 + 𝜇 𝛿
MA Using the Whole History
 An investor will always expect to lose money using
GMA(∞,1)!
 An investor loses the least amount of money when the
return process is a random walk.
Optimal MA Parameters
 So, what are the optimal 𝑛 and 𝑚?
Step 2: AR(1)
Step 2 : ARMA(1, 1)
no systematic
winner
optimal
order
Step 2 : ARIMA(0, d, 0)
Live Results of Quantitative Trading
Strategies
41
Unique Guiding Principle
 What Others Do:
 Start with a trading strategy.
 Find the data that the
strategy works.
 Result:
 Paper P&L looks good.
 Live P&L depends on luck.
 Trading strategies are
results of a non-scientific,
a pure data snooping
process.
 What We Do:
 Start with simple assumptions
about the market.
 Compute the optimal trading
strategy given the
assumptions.
 Result:
 Can mathematically prove
that no other strategy will
work better in the same
market conditions.
 Trading strategies are results
of a scientific process.
42
Optimal Trend Following (TREND)
 We make assumptions that the market is a two (or
three) state model. The market state is either up,
down, (or sideway).
 In each state, we assume a random walk with positive,
negative, or zero drift.
 We use math to compute what the best thing to do is
in each of the states.
 We estimate the conditional probability, 𝑝, of that the
market is going up given all the available information.
 When 𝑝 is big enough, i.e., most certainly that the
market is going up, we buy.
0.00%
20.00%
40.00%
60.00%
80.00%
100.00%
120.00%
140.00%
160.00%
trading period 2015/1/2 - 2016/5/2
assets traded
Hang Seng china enterprises
index futures
annualized
return 107.00%
max
drawdown 6.61%
Sharpe ratio 4.79
Result:
Optimal Trend Following (Math)
 Two state Markov model for a stock’s prices: BULL and
BEAR.
 𝑑𝑆𝑟 = 𝑆𝑟 𝜇 𝛼 𝑟
𝑑𝑑 + 𝜎𝜎𝐵𝑟 , 𝑡 ≤ 𝑟 ≤ 𝑇 < ∞
 The trading period is between time 𝑡, 𝑇 .
 𝛼 𝑟 = 1,2 are the two Markov states that indicates
the BULL and BEAR markets.
 𝜇1 > 0
 𝜇2 < 0
 𝑄 =
−𝜆1 𝜆1
𝜆2 −𝜆2
, the generator matrix for the Markov
chain.
 When 𝑖 = 0, expected return is
 E0,𝑡 𝑅𝑡 =
E 𝑡 𝑒 𝜌 𝜏1−𝑡 ∏
𝑆 𝜈 𝑛
𝑆 𝜏 𝑛
1−𝐾𝑠
1+𝐾 𝑏
𝐼 𝜏 𝑛<𝑇
𝑒 𝜌 𝜏 𝑛+1−𝜐 𝑛∞
𝑛=1
 We are long between 𝜏 𝑛 and 𝜈 𝑛 and the return is
determined by the price change discounted by the
commissions.
 We are flat between 𝜈 𝑛 and 𝜏 𝑛+1 and the money
grows at the risk free rate.
 Value function:
 J0 𝑆, 𝛼, 𝑡, Λ0 =
E 𝑡
𝜌 𝜏1 − 𝑡 +
∑ log
𝑆 𝜈 𝑛
𝑆 𝜏 𝑛
+ 𝐼 𝜏 𝑛<𝑇 log
1−𝐾𝑠
1+𝐾 𝑏
+ 𝜌 𝜏 𝑛+1 − 𝜐 𝑛
∞
𝑛=1
 J1 𝑆, 𝛼, 𝑡, Λ1 =
E 𝑡
log
𝑆 𝜈1
𝑆
+ 𝜌 𝜏2 − 𝜈1 + log 1 − 𝐾𝑠 +
∑ log
𝑆 𝜈 𝑛
𝑆 𝜏 𝑛
+ 𝐼 𝜏 𝑛<𝑇 log
1−𝐾𝑠
1+𝐾 𝑏
+ 𝜌 𝜏 𝑛+1 − 𝜐 𝑛
∞
𝑛=2
 Find an optimal trading sequence (the stopping times) so that
the value functions are maximized.
 𝑉𝑖 𝑝, 𝑡 = sup
Λ 𝑖
𝐽𝑖 𝑆, 𝑝, 𝑡, Λ 𝑖
 𝑉𝑖: the maximum amount of expected returns
 �
𝑉0 𝑝, 𝑡 = sup
𝜏1
𝐸𝑡 𝜌 𝜏1 − 𝑡 − log 1 + 𝐾𝑏 + 𝑉1 𝑝𝜏1
, 𝜏1
𝑉1 𝑝, 𝑡 = sup
𝜈1
𝐸𝑡 log
𝑆 𝜈1
𝑆𝑡
+ log 1 − 𝐾𝑠 + 𝑉0 𝑝 𝜈1
, 𝜈1
 Hamilton-Jacobi-Bellman Equations
 �
min −ℒ𝑉0 − 𝜌, 𝑉0 − 𝑉1 + log 1 + 𝐾𝑏 = 0
min −ℒ𝑉1 − 𝑓 𝜌 , 𝑉1 − 𝑉0 − log 1 − 𝐾𝑠 = 0
 with terminal conditions: �
𝑉0 𝑝, 𝑇 = 0
𝑉1 𝑝, 𝑇 = log 1 − 𝐾𝑠
 ℒ = 𝜕𝑡 +
1
2
𝜇1−𝜇2 𝑝 1−𝑝
𝜎
2
𝜕 𝑝𝑝 + − 𝜆1 + 𝜆2 𝑝 + 𝜆2 𝜕 𝑝
 Based on: M Dai, Q Zhang, QJ Zhu, "Trend following trading under a regime switching
model," SIAM Journal on Financial Mathematics, 2010.
Optimal Mean Reversion (MR)
45
 Basket construction problem:
 Select the right financial instruments.
 Obtain the optimal weights for the
selected financial instruments.
 Basket trading problem:
 Given the portfolio can be modelled
as a mean reverting OU process,
dynamic spread trading is a stochastic
optimal control problem.
 Given a fixed amount of capital,
dynamically allocate capital over a
risky mean reverting portfolio and a
risk-free asset over a finite time
horizon to maximize a general
constant relative risk aversion (CRRA)
utility function of the terminal
wealth .
 Allocate capital amongst several
mean reverting portfolios.
 Based on: Mudchanatongsuk, S., Primbs, J.A., Wong, " Optimal Pairs Trading: A
Stochastic Control Approach," Dept. of Manage. Sci. & Eng., Stanford Univ., CA.
Optimal Mean Reversion (Math)
46
 Assume a risk free asset 𝑀𝑡, which satisfies
 𝑑𝑀𝑡 = 𝑟𝑀𝑡 𝑑𝑑
 Assume two assets,𝐴 𝑡 and 𝐵𝑡.
 Assume 𝐵𝑡follows a geometric Brownian
motion.
 𝑑𝐵𝑡 = 𝜇𝐵𝑡 𝑑𝑑 + 𝜎𝐵𝑡 𝑑𝑧𝑡
 𝑥 𝑡is the spread between the two assets.
 𝑥 𝑡 = log 𝐴 𝑡 − log 𝐵𝑡

𝑑𝑉𝑡
𝑉𝑡
= ℎ 𝑡
𝑑𝐴 𝑡
𝐴 𝑡
+ ℎ 𝑡
� 𝑑𝐵𝑡
𝐵𝑡
+
𝑑𝑀𝑡
𝑀𝑡
 = ℎ 𝑡 𝑘 𝜃 − 𝑥 𝑡 +
1
2
𝜂2 + 𝜌𝜌𝜌 + 𝑟 𝑑𝑑 +
ℎ 𝑡 𝜂𝜂𝜔 𝑡
 max
ℎ 𝑡
𝐸 𝑉𝑇
𝛾
, s.t.,
 𝑉 0 = 𝑣0, x 0 = 𝑥0
 𝑑𝑥𝑡 = 𝑘 𝜃 − 𝑥 𝑡 𝑑𝑑 + 𝜂𝜂𝜔 𝑡
 𝑑𝑉𝑡 = ℎ 𝑡 𝑑𝑥 𝑡 = ℎ 𝑡 𝑘 𝜃 − 𝑥 𝑡 𝑑𝑑 + ℎ 𝑡 𝜂𝜂𝜔 𝑡
 ℎ 𝑡 ∗
=
𝑉𝑡
1−𝛾
−
𝑘
𝜂2 𝑥 𝑡 − 𝜃 + 2𝛼 𝑡 𝑥 𝑡 + 𝛽 𝑡
covariance selection
Intraday Volatility Trading (VOL)
 In mid or high frequency trading, or
within a medium or short time
interval, prices tend to oscillate.
 If there are enough oscillations
before prices move in a direction,
arbitrage exists.
47
profit region
loss region
-20.00%
0.00%
20.00%
40.00%
60.00%
80.00%
100.00%
120.00%
140.00%
trading period 2014/2/27 - 2015/2/27
assets traded
Hang Seng china enterprises
index futures
annualized
return 122.32%
max
drawdown 10.24%
Sharpe ratio 18.45
Live Result:
Intraday Volatility Trading (Math)
 For a continuous price process 𝑋𝑡, we
define H-variation
 𝑉𝑇 𝐻, 𝑋 = sup
𝑇
∑ 𝑋 𝑡𝑙 − 𝑋 𝑡𝑙−1
𝐿
𝑙=1
 It can be shown that for any H, there
exists a sequence 𝜏 𝑛
∗
, 𝜏 𝑛 𝑛=0,1,…,𝑁 such
that 𝜏 𝑛 𝑛=0,1,…,𝑁 are Markovian and 𝜏 𝑛
∗
are defined by 𝑋𝑡 on intervals 𝜏 𝑛−1, 𝜏 𝑛 .
And they satisfy the equality:
 𝑉𝑇 𝐻, 𝑋 = ∑ 𝑋 𝜏 𝑛
∗ − 𝑋 𝜏 𝑛−1
∗𝑁
𝑙=1
 𝑁 𝑇 𝐻, 𝑋 is the number of KAGI-
inversion in the T-interval.
 H-volatility:
 𝜂 𝑇 𝐻, 𝑋 =
𝑉 𝑇 𝐻,𝑋
𝑁 𝑇 𝐻,𝑋
 For an no-arbitrage Wiener process, we
have
 lim
𝑇→∞
𝜂 𝑇 𝐻, 𝜎𝜎 = 𝐾𝐾 = 2𝐻
 Define a trading strategy such that the
position of X is:
 𝛾�𝑡
𝐾
𝐻, 𝑋 =
∑ sign 𝜒 𝜏 𝑛−1 − 𝜒 𝜏 𝑛−1
∗
𝜒 𝜏 𝑛−1,𝜏 𝑛
𝑡
𝑁 𝑇 𝐻,𝑋
𝑛=1
 The trend following P&L is:
 𝑌�𝑡
𝐾
𝐻, 𝑋 = ∫ 𝛾� 𝑢
𝐾 𝐻, 𝑋 𝑑𝑑 𝑢
𝑡
0
 = 𝜂 𝑇 𝐻, 𝑋 − 2𝐻 𝑁 𝑇 𝐻, 𝑋 + 𝜀
 The expected income per trade is:
 𝑌𝑡
𝐾
𝐻, 𝑋 = ∫ 𝛾𝑢
𝐾 𝐻, 𝑋 𝑑𝑑 𝑢
𝑡
0
 𝑦𝑡
𝐾
𝐻, 𝑋 =
𝑌𝑡
𝐾 𝐻,𝑋
𝑁𝑡
𝐾 𝐻,𝑋
 lim
𝑇→∞
E 𝑦𝑡
𝐾
𝐻, 𝑋 = 𝐾 − 2 𝐻
48
 Based on: SV Pastukhov, "On some probabilistic-statistical methods in technical
analysis," Theory of Probability & Its Applications, SIAM, 2005.
Optimal Market Making (MM)
 We optimally place limit and market
orders depending on the current
inventory and spread.
49
the best market making strategy:
trading period 2015/7/16 - 2016/3/1
assets traded
rebar + iron ore commodity
futures
annualized
return 65%
max
drawdown 0.90%
Sharpe ratio 16.71
Live Result:
0.00%
5.00%
10.00%
15.00%
20.00%
25.00%
30.00%
35.00%
40.00%
Optimal Market Making (Math)
 State variable:
 𝑋, 𝑌, 𝑃, 𝑆
 cash, inventory, mid price, spread
 Objective:
 max
𝛼
E 𝑈 𝑋 𝑇 − 𝛾 ∫ 𝑔 𝑌𝑡 𝑑𝑑
𝑇
0
 𝑌𝑇 = 0, e.g., don’t hold position overnight
 𝑈: utility function
 𝑋 𝑇: terminal wealth
 𝛾: penalty for holding inventory
 Liquidation function (how much we get by selling
everything):
 𝐿 𝑥, 𝑦, 𝑝, 𝑠 = 𝑥 − 𝑐 −𝑦, 𝑝, 𝑠 = 𝑥 + 𝑦𝑦 − 𝑦
𝑠
2
− 𝜀
 Equivalent problem (get rid of 𝑌𝑇 = 0):
 max
𝛼
E 𝑈 𝐿 𝑋 𝑇, 𝑌𝑇, 𝑃 𝑇, 𝑆 𝑇 − 𝛾 ∫ 𝑔 𝑌𝑡 𝑑𝑑
𝑇
0
 Value function:
 𝑣 𝑡, 𝑧, 𝑠 = sup 𝛼 E
𝑡,𝑧,𝑠
𝑈 𝐿 𝑍 𝑇, 𝑆 𝑇 − 𝛾 ∫ 𝑔 𝑌𝑢 𝑑𝑑
𝑇
𝑡
 𝑧 = 𝑥, 𝑦, 𝑝
 This is a mixed regular/impulse control problem in a
regime switching jump-diffusion model.
 Quasi-Variational Inequality
 min −
𝜕𝜕
𝜕𝜕
− sup ℒ 𝑞,𝑙
𝑣 + 𝛾𝛾, 𝑣 − ℳ𝑣 = 0
 Terminal condition:
 𝑣 𝑇, 𝑥, 𝑦, 𝑝, 𝑠 = 𝑈 𝐿 𝑥, 𝑦, 𝑝, 𝑠
 For each state 𝑖, we have
 min
−
𝜕𝑣 𝑖
𝜕𝜕
− 𝒫𝑣𝑖 − ∑ 𝑟𝑖𝑖 𝑡 𝑣𝑗 𝑡, 𝑥, 𝑦, 𝑝 − 𝑣𝑖 𝑡, 𝑥, 𝑦, 𝑝𝑚
𝑗=1
− sup 𝜆𝑖
𝑏
𝑞 𝑏
𝑣𝑖 𝑡, 𝑥 − 𝜋𝑖
𝑏
𝑞 𝑏
, 𝑝 𝑙 𝑏
, 𝑦 + 𝑙 𝑏
, 𝑝 − 𝑣𝑖 𝑡, 𝑥, 𝑦, 𝑝
− sup 𝜆𝑖
𝑎
𝑞 𝑎
𝑣𝑖 𝑡, 𝑥 + 𝜋𝑖
𝑎
𝑞 𝑎
, 𝑝 𝑙 𝑎
, 𝑦 − 𝑙 𝑎
, 𝑝 − 𝑣𝑖 𝑡, 𝑥, 𝑦, 𝑝
+𝛾𝛾,
𝑣𝑖 𝑡, 𝑥, 𝑦, 𝑝 − sup 𝑣𝑖 𝑡, 𝑥 − 𝑐𝑖 𝑒, 𝑝 , 𝑦 + 𝑒, 𝑝
= 0
 𝑣𝑖 𝑇, 𝑥, 𝑦, 𝑝 = 𝑈 𝐿𝑖 𝑥, 𝑦, 𝑝
 Assumptions:
 𝑈 𝑥 = 𝑥; we care about only how much money made.
 𝑃𝑡 𝑡 is a martingale; we know nothing about where the market will move.
 Solution:
 𝑣𝑖 𝑡, 𝑥, 𝑦, 𝑝 = 𝑥 + 𝑦𝑦 + 𝜙𝑖 𝑡, 𝑦
 𝜙𝑖 𝑡, 𝑦 is the solution to the system of integro-differential equations
(IDE):
 min
−
𝜕𝜙 𝑖
𝜕𝜕
− ∑ 𝑟𝑖𝑖 𝑡 𝜙𝑗 𝑡, 𝑦 − 𝜙𝑖 𝑡, 𝑦𝑚
𝑗=1
− sup 𝜆𝑖
𝑏
𝑞 𝑏
𝜙𝑖 𝑡, 𝑦 + 𝑙 𝑏
− 𝜙𝑖 𝑡, 𝑦 +
𝑖𝑖
2
− 𝛿1 𝑞 𝑏=𝑃𝑡
𝑏+ 𝑙 𝑏
− sup 𝜆𝑖
𝑎
𝑞 𝑎
𝜙𝑖 𝑡, 𝑦 − 𝑙 𝑎
− 𝜙𝑖 𝑡, 𝑦 +
𝑖𝑖
2
− 𝛿1 𝑞 𝑎=𝑃𝑡
𝑎− 𝑙 𝑎
+𝛾𝛾 𝑦 ,
𝜙𝑖 𝑡, 𝑦 − sup 𝜙𝑖 𝑡, 𝑦 + 𝑒 −
𝑖𝑖
2
𝑒 − 𝜀
= 0
 𝜙𝑖 𝑇, 𝑦 = − 𝑦
𝑖𝑖
2
− 𝜀
50
 Based on: F Guilbaud, H Pham, "Optimal high-frequency trading with limit and market
orders," Quantitative Finance, 2013.
Conclusions
51
FinMath Infrastructure Support
 All these mathematics and simulations are possible only with a
finmath technology that serves as the modeling infrastructure.
52
• Linear algebra • Calculus
� 𝑓 𝑥
𝑏
𝑎
= 𝐹 𝑏 − 𝐹 𝑎
• Unconstrained
optimization
• Statistics • Differential
Equations
local minimumglobal minimum
Financial
Mathematics
Advanced
Mathematics
Foundation
Mathematics
• Parallelization
• Cointegration
• Optimization (LP, QP,
SQP, SDP, SOCP, IP,
GA)
• Digital Signal Processing• Time Series Analysis
Applications
• Optimal Trading Strategies • Portfolio Optimization • Extreme Value Theory
• Trading Signals • Asset Allocation • Risk Management
The Essential Skills
53
 Financial intuitions, market understanding, creativity.
 Mathematics.
 Computer programming.
An Emerging Field
54
 It is a financial industry where mathematics and
computer science meet.
 It is an arms race to build
 more reliable and faster execution platforms (computer
science);
 more comprehensive and accurate prediction models
(mathematics).

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"Quantitative Trading as a Mathematical Science" by Dr. Haksun Li, Founder and CEO, Numerical Method Inc.

  • 1. Quantitative Trading as a Mathematical Science QuantCon Singapore 2016 Haksun Li haksun.li@numericalmethod.com www.numericalmethod.com
  • 2. Abstract 2  Quantitative trading is distinguishable from other trading methodologies like technical analysis and analysts’ opinions because it uniquely provides justifications to trading strategies using mathematical reasoning. Put differently, quantitative trading is a science that trading strategies are proven statistically profitable or even optimal under certain assumptions. There are properties about strategies that we can deduce before betting the first $1, such as P&L distribution and risks. There are exact explanations to the success and failure of strategies, such as choice of parameters. There are ways to iteratively improve strategies based on experiences of live trading, such as making more realistic assumptions. These are all made possible only in quantitative trading because we have assumptions, models and rigorous mathematical analysis.  Quantitative trading has proved itself to be a significant driver of mathematical innovations, especially in the areas of stochastic analysis and PDE-theory. For instances, we can compute the optimal timings to follow the market by solving a pair of coupled Hamilton–Jacobi–Bellman equations; we can construct sparse mean reverting baskets by solving semi-definite optimization problems with cardinality constraints and can optimally trade these baskets by solving stochastic control problems; we can identify statistical arbitrage opportunities by analyzing the volatility process of a stochastic asset at different frequencies; we can compute the optimal placements of market and limit orders by solving combined singular and impulse control problems which leads to novel and difficult to solve quasi- variational inequalities.
  • 3. Speaker Profile  Dr. Haksun Li  CEO, NM LTD.  (Ex-)Adjunct Professors, Industry Fellow, Advisor, Consultant with the National University of Singapore, Nanyang Technological University, Fudan University, the Hong Kong University of Science and Technology.  Quantitative Trader/Analyst, BNPP, UBS  Ph.D., Computer Science, University of Michigan Ann Arbor  M.S., Financial Mathematics, University of Chicago  B.S., Mathematics, University of Chicago 3
  • 4. What is Quantitative Trading? 4
  • 5. Quantitative Trading? 5  Quantitative trading is the buying and selling of assets following the instructions computed from a set of proven mathematical models.  The differentiation from other trading methodologies or the emphasis is on how a strategy is proven and not on what strategy is created.  It applies (rigorous) mathematical reasoning in all steps during trading strategy construction from the start to the end.
  • 6. Moving Average Crossover as a TA 6  A popular TA signal: Moving Average Crossover.  A crossover occurs when a faster moving average (i.e. a shorter period moving average) crosses above/below a slower moving average (i.e. a longer period moving average); then you buy/sell.  In most TA book, it is never proven only illustrated with an example of applying the strategy to a stock for a period of time to show profits.
  • 7. Technical Analysis is Not Quantitative Trading 7  TA books merely describes the mechanics of strategies but never prove them.  Appealing to common sense is not a mathematical proof.  Conditional probabilities of outcomes are seldom computed. (Lo, Mamaysky, & Wang, 2000)  Application of TA is more of an art (is it?) than a science.  How do you choose the parameters?  For any TA rule, you almost surely can find an asset and a period that the rule “works”, given the large number of assets and many periods to choose from.
  • 8. Fake Quantitative Models  Data snooping  Misuse of mathematics  Assumptions cannot be quantified  No model validation against the current regime  Ad-hoc take profit and stop-loss  why 2?  How do you know when a model is invalidated?  Cannot explain winning and losing trades  Cannot be analyzed (systematically) 8
  • 9. The Quantitative Trading Research Process 9
  • 10. NM Quantitative Trading Research Process 10 1. Translate a vague trading intuition (hypothesis) into a concrete mathematical model. 2. Translate the mathematical symbols and equations into a computer program. 3. Strategy evaluation. 4. Live execution for making money.
  • 11. Step 1 - Modeling 11  Where does a trading idea come from?  Ex-colleagues  Hearsays  Newspapers, books  TV, e.g., Moving Average Crossover (MA)  A quantitative trading strategy is a math function, f, that at any given time, t, takes as inputs any information that the strategy cares and that is available, Ft, and gives as output the position to take, f(t,Ft).
  • 12. Step 2 - Coding 12  The computer code enables analysis of the strategy.  Most study of a strategy cannot be done analytically.  We must resort to simulation.  The same piece of code used for research and investigation should go straight into the production for live trading.  Eliminate the possibility of research-to-IT translation errors.
  • 13. Step 3 – Evaluation/Justification 13  Compute the properties of a trading strategy.  the P&L distribution  the holding time distribution  the stop-loss  the maximal drawdown  http://redmine.numericalmethod.com/projects/publi c/repository/svn- algoquant/show/core/src/main/java/com/numericalm ethod/algoquant/execution/performance
  • 14. Step 4 - Trading 14  Put in capitals incrementally.  Install safety measures.  Monitor the performance.  Regime change detection.
  • 15. Mathematical Analysis of Moving Average Crossover 15
  • 16. Moving Average Crossover as a Quantitative Trading Strategy 16  There are many mathematical justifications to Moving Average Crossover.  weighted Sum of lags of a time series  Kuo, 2002  Whether a strategy is quantitative or not depends not on the strategy itself but  entirely on the process to construct it;  or, whether there is a scientific justification to prove it.
  • 17. Step 1 - Modeling  Two moving averages: slower (𝑛) and faster (𝑚).  Monitor the crossovers.  𝐵𝑡 = 1 𝑚 ∑ 𝑃𝑡−𝑗 𝑚−1 𝑗=0 − 1 𝑛 ∑ 𝑃𝑡−𝑗 𝑛−1 𝑗=0 , 𝑛 > 𝑚  Long when 𝐵𝑡 ≥ 0.  Short when 𝐵𝑡 < 0.
  • 18. How to Choose 𝑛 and 𝑚?  It is an art, not a science (so far).  They should be related to the length of market cycles.  Different assets have different 𝑛 and 𝑚.  Popular choices:  (250, 5)  (250 , 20)  (20 , 5)  (20 , 1)  (250 , 1)
  • 19. Two Simplifications 19  Reduce the two dimensional problem to a one dimensional problem.  Choose 𝑚 = 1. We know that m should be small.  Replace arithmetic averages with geometric averages.  This is so that we can work with log returns rather than prices.
  • 20. GMA(n , 1)  𝐵𝑡 ≥ 0 iff 𝑃𝑡 ≥ ∏ 𝑃𝑡−𝑗 𝑛−1 𝑗=0 1 𝑛  𝑅𝑡 ≥ − ∑ 𝑛− 𝑗+1 𝑛−1 𝑅𝑡−𝑗 𝑛−2 𝑗=1 (by taking log)  𝐵𝑡 < 0 iff 𝑃𝑡 < ∏ 𝑃𝑡−𝑗 𝑛−1 𝑗=0 1 𝑛  𝑅𝑡 < − ∑ 𝑛− 𝑗+1 𝑛−1 𝑅𝑡−𝑗 𝑛−2 𝑗=1 (by taking log)
  • 21. What is 𝑛?  𝑛 = 2  𝑛 = ∞
  • 22. Acar Framework  Acar (1993): to investigate the probability distribution of realized returns from a trading rule, we need  the explicit specification of the trading rule  the underlying stochastic process for asset returns  the particular return concept involved
  • 23. Knight-Satchell-Tran Intuition  Stock returns staying going up (down) depends on  the realizations of positive (negative) shocks  the persistence of these shocks  Shocks are modeled by gamma processes.  Asymmetry  Fat tails  Persistence is modeled by a Markov switching process.
  • 24. Knight-Satchell-Tran 𝑍𝑡 Zt = 0 DOWN TREND 𝑓𝛿 𝑥 = 𝜆2 𝛼2 𝑥 𝛼2−1 Γ 𝛼2 𝑒−𝜆2 𝑥 Zt = 1 UP TREND 𝑓𝜀 𝑥 = 𝜆1 𝛼1 𝑥 𝛼1−1 Γ 𝛼1 𝑒−𝜆1 𝑥 q p 1-q 1-p
  • 25. Knight-Satchell-Tran Process  𝑅𝑡 = 𝜇𝑙 + 𝑍𝑡 𝜀𝑡 − 1 − 𝑍𝑡 𝛿𝑡  𝜇𝑙: long term mean of returns, e.g., 0  𝜀𝑡, 𝛿𝑡: positive and negative shocks, non-negative, i.i.d  𝑓𝜀 𝑥 = 𝜆1 𝛼1 𝑥 𝛼1−1 Γ 𝛼1 𝑒−𝜆1 𝑥  𝑓𝛿 𝑥 = 𝜆2 𝛼2 𝑥 𝛼2−1 Γ 𝛼2 𝑒−𝜆2 𝑥
  • 26. Step 3 – Evaluation/Justification  Assume the long term mean is 0, 𝜇𝑙 = 0.  When 𝑛 = 2,  𝐵𝑡 ≥ 0 ≡ 𝑅𝑡 ≥ 0 ≡ 𝑍𝑡 = 1  𝐵𝑡 < 0 ≡ 𝑅𝑡 < 0 ≡ 𝑍𝑡 = 0
  • 27. GMA(2, 1) – Naïve MA Trading Rule  Buy when the asset return in the present period is positive.  Sell when the asset return in the present period is negative.
  • 28. Naïve MA Conditions  The expected value of the positive shocks to asset return >> the expected value of negative shocks.  The positive shocks persistency >> that of negative shocks.
  • 29. 𝑇 Period Returns  𝑅𝑅 𝑇 = ∑ 𝑅𝑡 × 𝐼 𝐵𝑡−1≥0 𝑇 𝑡=1 Sell at this time point 𝑇 𝐵 𝑇 < 0 0 1 hold
  • 30. Holding Time Distribution  𝑃 𝑁 = 𝑇  = 𝑃 𝐵 𝑇 < 0, 𝐵 𝑇−1 ≥ 0, … , 𝐵1 ≥ 0, 𝐵0 ≥ 0  = 𝑃 𝑍 𝑇 = 0, 𝑍 𝑇−1 = 1, … , 𝑍1 = 1, 𝑍0 = 1  = 𝑃 𝑍 𝑇 = 0, 𝑍 𝑇−1 = 1, … , 𝑍1 = 1|𝑍0 = 1 𝑃 𝑍0 = 1  = � Π𝑝 𝑇−1 1 − 𝑝 , T ≥1 1 − Π, T=0  Stationary state probability:  Π = 1−𝑞 2−𝑝−𝑞
  • 31. Conditional Returns Distribution (1)  Φ 𝑅𝑅 𝑇|𝑁=𝑇 𝑠 = E 𝑒 𝑖 ∑ 𝑅 𝑡×𝐼 𝐵 𝑡−1≥0 𝑇 𝑡=1 𝑠 |𝑁 = 𝑇  = E 𝑒 𝑖 ∑ 𝑅 𝑡×𝐼 𝐵 𝑡−1≥0 𝑇 𝑡=1 𝑠 |𝐵 𝑇 < 0, 𝐵 𝑇−1 ≥ 0, … , 𝐵0 ≥ 0  = E 𝑒 𝑖 ∑ 𝑅 𝑡 𝑇 𝑡=1 𝑠 |𝑍 𝑇 = 0, 𝑍 𝑇−1 = 1, … , 𝑍1 = 1  = E 𝑒 𝑖 𝜀1+⋯+𝜀 𝑇−1−𝛿 𝑇 𝑠  = � Φ 𝜀 𝑇−1 𝑠 Φ 𝛿 −𝑠 , T ≥1 Φ 𝛿 −𝑠 , T =0
  • 32. Unconditional Returns Distribution (2)  Φ 𝑅𝑅 𝑇 𝑠 = ∑ E 𝑒 𝑖 ∑ 𝑅 𝑡×𝐼 𝐵 𝑡−1≥0 𝑇 𝑡=1 𝑠 |𝑁 = 𝑇 𝑃 𝑁 = 𝑇∞ 𝑇=0  = ∑ Π𝑝 𝑇−1 1 − 𝑝 Φ 𝜀 𝑇−1 𝑠 Φ 𝛿 −𝑠∞ 𝑇=1 + 1 − Π Φ 𝛿 −𝑠  = 1 − Π Φ 𝛿 −𝑠 + Π 1 − 𝑝 Φ 𝛿 −𝑠 1−𝑝Φ 𝜀 𝑠
  • 33. Expected Returns  E 𝑅𝑅 𝑇 = −𝑖Φ 𝑅𝑅 𝑇 ′ 0  = 1 1−𝑝 Π𝑝𝜇 𝜀 − 1 − 𝑝 𝜇 𝛿  When is the expected return positive?  𝜇 𝜀 ≥ 1−𝑝 Π𝑝 𝜇 𝛿, shock impact  𝜇 𝜀 ≫ 𝜇 𝛿, shock impact  Π𝑝 ≥ 1 − 𝑝, if 𝜇 𝜀 ≈ 𝜇 𝛿, persistence
  • 34. GMA(∞,1) Rule  𝑃𝑡 ≥ ∏ 𝑃𝑡−𝑗 𝑛−1 𝑗=0 1 𝑛  ln 𝑃𝑡 ≥ 1 𝑛 ∑ ln 𝑃𝑡−𝑗 𝑛−1 𝑗=0  ln 𝑃𝑡 ≥ 𝜇1
  • 35. GMA(∞,1) Expected Returns  Φ 𝑅𝑅 𝑇 𝑠 = 1 − Π 𝑞 Φ 𝛿 𝑠 + Φ 𝛿 −𝑠 + 1 − 𝑝 1 − Π Φ 𝜀 𝑠 + Φ 𝜀 −𝑠  E 𝑅𝑅 𝑇 = − 1 − 𝑝 1 − Π 𝜇 𝜀 + 𝜇 𝛿
  • 36. MA Using the Whole History  An investor will always expect to lose money using GMA(∞,1)!  An investor loses the least amount of money when the return process is a random walk.
  • 37. Optimal MA Parameters  So, what are the optimal 𝑛 and 𝑚?
  • 39. Step 2 : ARMA(1, 1) no systematic winner optimal order
  • 40. Step 2 : ARIMA(0, d, 0)
  • 41. Live Results of Quantitative Trading Strategies 41
  • 42. Unique Guiding Principle  What Others Do:  Start with a trading strategy.  Find the data that the strategy works.  Result:  Paper P&L looks good.  Live P&L depends on luck.  Trading strategies are results of a non-scientific, a pure data snooping process.  What We Do:  Start with simple assumptions about the market.  Compute the optimal trading strategy given the assumptions.  Result:  Can mathematically prove that no other strategy will work better in the same market conditions.  Trading strategies are results of a scientific process. 42
  • 43. Optimal Trend Following (TREND)  We make assumptions that the market is a two (or three) state model. The market state is either up, down, (or sideway).  In each state, we assume a random walk with positive, negative, or zero drift.  We use math to compute what the best thing to do is in each of the states.  We estimate the conditional probability, 𝑝, of that the market is going up given all the available information.  When 𝑝 is big enough, i.e., most certainly that the market is going up, we buy. 0.00% 20.00% 40.00% 60.00% 80.00% 100.00% 120.00% 140.00% 160.00% trading period 2015/1/2 - 2016/5/2 assets traded Hang Seng china enterprises index futures annualized return 107.00% max drawdown 6.61% Sharpe ratio 4.79 Result:
  • 44. Optimal Trend Following (Math)  Two state Markov model for a stock’s prices: BULL and BEAR.  𝑑𝑆𝑟 = 𝑆𝑟 𝜇 𝛼 𝑟 𝑑𝑑 + 𝜎𝜎𝐵𝑟 , 𝑡 ≤ 𝑟 ≤ 𝑇 < ∞  The trading period is between time 𝑡, 𝑇 .  𝛼 𝑟 = 1,2 are the two Markov states that indicates the BULL and BEAR markets.  𝜇1 > 0  𝜇2 < 0  𝑄 = −𝜆1 𝜆1 𝜆2 −𝜆2 , the generator matrix for the Markov chain.  When 𝑖 = 0, expected return is  E0,𝑡 𝑅𝑡 = E 𝑡 𝑒 𝜌 𝜏1−𝑡 ∏ 𝑆 𝜈 𝑛 𝑆 𝜏 𝑛 1−𝐾𝑠 1+𝐾 𝑏 𝐼 𝜏 𝑛<𝑇 𝑒 𝜌 𝜏 𝑛+1−𝜐 𝑛∞ 𝑛=1  We are long between 𝜏 𝑛 and 𝜈 𝑛 and the return is determined by the price change discounted by the commissions.  We are flat between 𝜈 𝑛 and 𝜏 𝑛+1 and the money grows at the risk free rate.  Value function:  J0 𝑆, 𝛼, 𝑡, Λ0 = E 𝑡 𝜌 𝜏1 − 𝑡 + ∑ log 𝑆 𝜈 𝑛 𝑆 𝜏 𝑛 + 𝐼 𝜏 𝑛<𝑇 log 1−𝐾𝑠 1+𝐾 𝑏 + 𝜌 𝜏 𝑛+1 − 𝜐 𝑛 ∞ 𝑛=1  J1 𝑆, 𝛼, 𝑡, Λ1 = E 𝑡 log 𝑆 𝜈1 𝑆 + 𝜌 𝜏2 − 𝜈1 + log 1 − 𝐾𝑠 + ∑ log 𝑆 𝜈 𝑛 𝑆 𝜏 𝑛 + 𝐼 𝜏 𝑛<𝑇 log 1−𝐾𝑠 1+𝐾 𝑏 + 𝜌 𝜏 𝑛+1 − 𝜐 𝑛 ∞ 𝑛=2  Find an optimal trading sequence (the stopping times) so that the value functions are maximized.  𝑉𝑖 𝑝, 𝑡 = sup Λ 𝑖 𝐽𝑖 𝑆, 𝑝, 𝑡, Λ 𝑖  𝑉𝑖: the maximum amount of expected returns  � 𝑉0 𝑝, 𝑡 = sup 𝜏1 𝐸𝑡 𝜌 𝜏1 − 𝑡 − log 1 + 𝐾𝑏 + 𝑉1 𝑝𝜏1 , 𝜏1 𝑉1 𝑝, 𝑡 = sup 𝜈1 𝐸𝑡 log 𝑆 𝜈1 𝑆𝑡 + log 1 − 𝐾𝑠 + 𝑉0 𝑝 𝜈1 , 𝜈1  Hamilton-Jacobi-Bellman Equations  � min −ℒ𝑉0 − 𝜌, 𝑉0 − 𝑉1 + log 1 + 𝐾𝑏 = 0 min −ℒ𝑉1 − 𝑓 𝜌 , 𝑉1 − 𝑉0 − log 1 − 𝐾𝑠 = 0  with terminal conditions: � 𝑉0 𝑝, 𝑇 = 0 𝑉1 𝑝, 𝑇 = log 1 − 𝐾𝑠  ℒ = 𝜕𝑡 + 1 2 𝜇1−𝜇2 𝑝 1−𝑝 𝜎 2 𝜕 𝑝𝑝 + − 𝜆1 + 𝜆2 𝑝 + 𝜆2 𝜕 𝑝  Based on: M Dai, Q Zhang, QJ Zhu, "Trend following trading under a regime switching model," SIAM Journal on Financial Mathematics, 2010.
  • 45. Optimal Mean Reversion (MR) 45  Basket construction problem:  Select the right financial instruments.  Obtain the optimal weights for the selected financial instruments.  Basket trading problem:  Given the portfolio can be modelled as a mean reverting OU process, dynamic spread trading is a stochastic optimal control problem.  Given a fixed amount of capital, dynamically allocate capital over a risky mean reverting portfolio and a risk-free asset over a finite time horizon to maximize a general constant relative risk aversion (CRRA) utility function of the terminal wealth .  Allocate capital amongst several mean reverting portfolios.  Based on: Mudchanatongsuk, S., Primbs, J.A., Wong, " Optimal Pairs Trading: A Stochastic Control Approach," Dept. of Manage. Sci. & Eng., Stanford Univ., CA.
  • 46. Optimal Mean Reversion (Math) 46  Assume a risk free asset 𝑀𝑡, which satisfies  𝑑𝑀𝑡 = 𝑟𝑀𝑡 𝑑𝑑  Assume two assets,𝐴 𝑡 and 𝐵𝑡.  Assume 𝐵𝑡follows a geometric Brownian motion.  𝑑𝐵𝑡 = 𝜇𝐵𝑡 𝑑𝑑 + 𝜎𝐵𝑡 𝑑𝑧𝑡  𝑥 𝑡is the spread between the two assets.  𝑥 𝑡 = log 𝐴 𝑡 − log 𝐵𝑡  𝑑𝑉𝑡 𝑉𝑡 = ℎ 𝑡 𝑑𝐴 𝑡 𝐴 𝑡 + ℎ 𝑡 � 𝑑𝐵𝑡 𝐵𝑡 + 𝑑𝑀𝑡 𝑀𝑡  = ℎ 𝑡 𝑘 𝜃 − 𝑥 𝑡 + 1 2 𝜂2 + 𝜌𝜌𝜌 + 𝑟 𝑑𝑑 + ℎ 𝑡 𝜂𝜂𝜔 𝑡  max ℎ 𝑡 𝐸 𝑉𝑇 𝛾 , s.t.,  𝑉 0 = 𝑣0, x 0 = 𝑥0  𝑑𝑥𝑡 = 𝑘 𝜃 − 𝑥 𝑡 𝑑𝑑 + 𝜂𝜂𝜔 𝑡  𝑑𝑉𝑡 = ℎ 𝑡 𝑑𝑥 𝑡 = ℎ 𝑡 𝑘 𝜃 − 𝑥 𝑡 𝑑𝑑 + ℎ 𝑡 𝜂𝜂𝜔 𝑡  ℎ 𝑡 ∗ = 𝑉𝑡 1−𝛾 − 𝑘 𝜂2 𝑥 𝑡 − 𝜃 + 2𝛼 𝑡 𝑥 𝑡 + 𝛽 𝑡 covariance selection
  • 47. Intraday Volatility Trading (VOL)  In mid or high frequency trading, or within a medium or short time interval, prices tend to oscillate.  If there are enough oscillations before prices move in a direction, arbitrage exists. 47 profit region loss region -20.00% 0.00% 20.00% 40.00% 60.00% 80.00% 100.00% 120.00% 140.00% trading period 2014/2/27 - 2015/2/27 assets traded Hang Seng china enterprises index futures annualized return 122.32% max drawdown 10.24% Sharpe ratio 18.45 Live Result:
  • 48. Intraday Volatility Trading (Math)  For a continuous price process 𝑋𝑡, we define H-variation  𝑉𝑇 𝐻, 𝑋 = sup 𝑇 ∑ 𝑋 𝑡𝑙 − 𝑋 𝑡𝑙−1 𝐿 𝑙=1  It can be shown that for any H, there exists a sequence 𝜏 𝑛 ∗ , 𝜏 𝑛 𝑛=0,1,…,𝑁 such that 𝜏 𝑛 𝑛=0,1,…,𝑁 are Markovian and 𝜏 𝑛 ∗ are defined by 𝑋𝑡 on intervals 𝜏 𝑛−1, 𝜏 𝑛 . And they satisfy the equality:  𝑉𝑇 𝐻, 𝑋 = ∑ 𝑋 𝜏 𝑛 ∗ − 𝑋 𝜏 𝑛−1 ∗𝑁 𝑙=1  𝑁 𝑇 𝐻, 𝑋 is the number of KAGI- inversion in the T-interval.  H-volatility:  𝜂 𝑇 𝐻, 𝑋 = 𝑉 𝑇 𝐻,𝑋 𝑁 𝑇 𝐻,𝑋  For an no-arbitrage Wiener process, we have  lim 𝑇→∞ 𝜂 𝑇 𝐻, 𝜎𝜎 = 𝐾𝐾 = 2𝐻  Define a trading strategy such that the position of X is:  𝛾�𝑡 𝐾 𝐻, 𝑋 = ∑ sign 𝜒 𝜏 𝑛−1 − 𝜒 𝜏 𝑛−1 ∗ 𝜒 𝜏 𝑛−1,𝜏 𝑛 𝑡 𝑁 𝑇 𝐻,𝑋 𝑛=1  The trend following P&L is:  𝑌�𝑡 𝐾 𝐻, 𝑋 = ∫ 𝛾� 𝑢 𝐾 𝐻, 𝑋 𝑑𝑑 𝑢 𝑡 0  = 𝜂 𝑇 𝐻, 𝑋 − 2𝐻 𝑁 𝑇 𝐻, 𝑋 + 𝜀  The expected income per trade is:  𝑌𝑡 𝐾 𝐻, 𝑋 = ∫ 𝛾𝑢 𝐾 𝐻, 𝑋 𝑑𝑑 𝑢 𝑡 0  𝑦𝑡 𝐾 𝐻, 𝑋 = 𝑌𝑡 𝐾 𝐻,𝑋 𝑁𝑡 𝐾 𝐻,𝑋  lim 𝑇→∞ E 𝑦𝑡 𝐾 𝐻, 𝑋 = 𝐾 − 2 𝐻 48  Based on: SV Pastukhov, "On some probabilistic-statistical methods in technical analysis," Theory of Probability & Its Applications, SIAM, 2005.
  • 49. Optimal Market Making (MM)  We optimally place limit and market orders depending on the current inventory and spread. 49 the best market making strategy: trading period 2015/7/16 - 2016/3/1 assets traded rebar + iron ore commodity futures annualized return 65% max drawdown 0.90% Sharpe ratio 16.71 Live Result: 0.00% 5.00% 10.00% 15.00% 20.00% 25.00% 30.00% 35.00% 40.00%
  • 50. Optimal Market Making (Math)  State variable:  𝑋, 𝑌, 𝑃, 𝑆  cash, inventory, mid price, spread  Objective:  max 𝛼 E 𝑈 𝑋 𝑇 − 𝛾 ∫ 𝑔 𝑌𝑡 𝑑𝑑 𝑇 0  𝑌𝑇 = 0, e.g., don’t hold position overnight  𝑈: utility function  𝑋 𝑇: terminal wealth  𝛾: penalty for holding inventory  Liquidation function (how much we get by selling everything):  𝐿 𝑥, 𝑦, 𝑝, 𝑠 = 𝑥 − 𝑐 −𝑦, 𝑝, 𝑠 = 𝑥 + 𝑦𝑦 − 𝑦 𝑠 2 − 𝜀  Equivalent problem (get rid of 𝑌𝑇 = 0):  max 𝛼 E 𝑈 𝐿 𝑋 𝑇, 𝑌𝑇, 𝑃 𝑇, 𝑆 𝑇 − 𝛾 ∫ 𝑔 𝑌𝑡 𝑑𝑑 𝑇 0  Value function:  𝑣 𝑡, 𝑧, 𝑠 = sup 𝛼 E 𝑡,𝑧,𝑠 𝑈 𝐿 𝑍 𝑇, 𝑆 𝑇 − 𝛾 ∫ 𝑔 𝑌𝑢 𝑑𝑑 𝑇 𝑡  𝑧 = 𝑥, 𝑦, 𝑝  This is a mixed regular/impulse control problem in a regime switching jump-diffusion model.  Quasi-Variational Inequality  min − 𝜕𝜕 𝜕𝜕 − sup ℒ 𝑞,𝑙 𝑣 + 𝛾𝛾, 𝑣 − ℳ𝑣 = 0  Terminal condition:  𝑣 𝑇, 𝑥, 𝑦, 𝑝, 𝑠 = 𝑈 𝐿 𝑥, 𝑦, 𝑝, 𝑠  For each state 𝑖, we have  min − 𝜕𝑣 𝑖 𝜕𝜕 − 𝒫𝑣𝑖 − ∑ 𝑟𝑖𝑖 𝑡 𝑣𝑗 𝑡, 𝑥, 𝑦, 𝑝 − 𝑣𝑖 𝑡, 𝑥, 𝑦, 𝑝𝑚 𝑗=1 − sup 𝜆𝑖 𝑏 𝑞 𝑏 𝑣𝑖 𝑡, 𝑥 − 𝜋𝑖 𝑏 𝑞 𝑏 , 𝑝 𝑙 𝑏 , 𝑦 + 𝑙 𝑏 , 𝑝 − 𝑣𝑖 𝑡, 𝑥, 𝑦, 𝑝 − sup 𝜆𝑖 𝑎 𝑞 𝑎 𝑣𝑖 𝑡, 𝑥 + 𝜋𝑖 𝑎 𝑞 𝑎 , 𝑝 𝑙 𝑎 , 𝑦 − 𝑙 𝑎 , 𝑝 − 𝑣𝑖 𝑡, 𝑥, 𝑦, 𝑝 +𝛾𝛾, 𝑣𝑖 𝑡, 𝑥, 𝑦, 𝑝 − sup 𝑣𝑖 𝑡, 𝑥 − 𝑐𝑖 𝑒, 𝑝 , 𝑦 + 𝑒, 𝑝 = 0  𝑣𝑖 𝑇, 𝑥, 𝑦, 𝑝 = 𝑈 𝐿𝑖 𝑥, 𝑦, 𝑝  Assumptions:  𝑈 𝑥 = 𝑥; we care about only how much money made.  𝑃𝑡 𝑡 is a martingale; we know nothing about where the market will move.  Solution:  𝑣𝑖 𝑡, 𝑥, 𝑦, 𝑝 = 𝑥 + 𝑦𝑦 + 𝜙𝑖 𝑡, 𝑦  𝜙𝑖 𝑡, 𝑦 is the solution to the system of integro-differential equations (IDE):  min − 𝜕𝜙 𝑖 𝜕𝜕 − ∑ 𝑟𝑖𝑖 𝑡 𝜙𝑗 𝑡, 𝑦 − 𝜙𝑖 𝑡, 𝑦𝑚 𝑗=1 − sup 𝜆𝑖 𝑏 𝑞 𝑏 𝜙𝑖 𝑡, 𝑦 + 𝑙 𝑏 − 𝜙𝑖 𝑡, 𝑦 + 𝑖𝑖 2 − 𝛿1 𝑞 𝑏=𝑃𝑡 𝑏+ 𝑙 𝑏 − sup 𝜆𝑖 𝑎 𝑞 𝑎 𝜙𝑖 𝑡, 𝑦 − 𝑙 𝑎 − 𝜙𝑖 𝑡, 𝑦 + 𝑖𝑖 2 − 𝛿1 𝑞 𝑎=𝑃𝑡 𝑎− 𝑙 𝑎 +𝛾𝛾 𝑦 , 𝜙𝑖 𝑡, 𝑦 − sup 𝜙𝑖 𝑡, 𝑦 + 𝑒 − 𝑖𝑖 2 𝑒 − 𝜀 = 0  𝜙𝑖 𝑇, 𝑦 = − 𝑦 𝑖𝑖 2 − 𝜀 50  Based on: F Guilbaud, H Pham, "Optimal high-frequency trading with limit and market orders," Quantitative Finance, 2013.
  • 52. FinMath Infrastructure Support  All these mathematics and simulations are possible only with a finmath technology that serves as the modeling infrastructure. 52 • Linear algebra • Calculus � 𝑓 𝑥 𝑏 𝑎 = 𝐹 𝑏 − 𝐹 𝑎 • Unconstrained optimization • Statistics • Differential Equations local minimumglobal minimum Financial Mathematics Advanced Mathematics Foundation Mathematics • Parallelization • Cointegration • Optimization (LP, QP, SQP, SDP, SOCP, IP, GA) • Digital Signal Processing• Time Series Analysis Applications • Optimal Trading Strategies • Portfolio Optimization • Extreme Value Theory • Trading Signals • Asset Allocation • Risk Management
  • 53. The Essential Skills 53  Financial intuitions, market understanding, creativity.  Mathematics.  Computer programming.
  • 54. An Emerging Field 54  It is a financial industry where mathematics and computer science meet.  It is an arms race to build  more reliable and faster execution platforms (computer science);  more comprehensive and accurate prediction models (mathematics).