Quantitative Trading as a Mathematical Science
QuantCon Singapore 2016
Haksun Li [email protected]
www.numericalmethod.com
Abstract
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Quantitative trading is distinguishable from other trading methodologies like technical analysis and analystsโ 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 HamiltonโJacobiโBellman 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
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What is Quantitative Trading?
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Quantitative Trading?
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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
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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
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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 โworksโ, 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)
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The Quantitative Trading Research Process
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NM Quantitative Trading Research Process
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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
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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
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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 โ Evaluation/Justification
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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/public/repository/svn-algoquant/show/core/src/main/java/com/numericalmethod/algoquant/execution/performance
Step 4 - Trading
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Put in capitals incrementally. Install safety measures. Monitor the performance. Regime change detection.
Mathematical Analysis of Moving Average Crossover
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Moving Average Crossover as a Quantitative Trading Strategy
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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 (๐) and faster (๐). Monitor the crossovers.
๐ต๐ก = 1๐โ ๐๐กโ๐๐โ1๐=0 โ 1
๐โ ๐๐กโ๐๐โ1๐=0 , ๐ > ๐
Long when ๐ต๐ก โฅ 0. Short when ๐ต๐ก < 0.
How to Choose ๐ and ๐? It is an art, not a science (so far). They should be related to the length of market cycles. Different assets have different ๐ and ๐. Popular choices: (250, 5) (250 , 20) (20 , 5) (20 , 1) (250 , 1)
Two Simplifications
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Reduce the two dimensional problem to a one dimensional problem. Choose ๐ = 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)
๐ต๐ก โฅ 0 iff ๐๐ก โฅ โ ๐๐กโ๐๐โ1๐=0
1๐
๐ ๐ก โฅ โโ ๐โ ๐+1๐โ1
๐ ๐กโ๐๐โ2๐=1 (by taking log)
๐ต๐ก < 0 iff ๐๐ก < โ ๐๐กโ๐๐โ1๐=0
1๐
๐ ๐ก < โโ ๐โ ๐+1๐โ1
๐ ๐กโ๐๐โ2๐=1 (by taking log)
What is ๐? ๐ = 2 ๐ = โ
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 ๐๐ก
Zt = 0 DOWN TREND
๐๐ฟ ๐ฅ
=๐2
๐ผ2๐ฅ๐ผ2โ1
ฮ ๐ผ2๐โ๐2๐ฅ
Zt = 1 UP TREND ๐๐ ๐ฅ
=๐1
๐ผ1๐ฅ๐ผ1โ1
ฮ ๐ผ1๐โ๐1๐ฅ
q p
1-q
1-p
Knight-Satchell-Tran Process ๐ ๐ก = ๐๐ + ๐๐ก๐๐ก โ 1 โ ๐๐ก ๐ฟ๐ก ๐๐: long term mean of returns, e.g., 0 ๐๐ก, ๐ฟ๐ก: positive and negative shocks, non-negative, i.i.d
๐๐ ๐ฅ = ๐1๐ผ1๐ฅ๐ผ1โ1
ฮ ๐ผ1๐โ๐1๐ฅ
๐๐ฟ ๐ฅ = ๐2๐ผ2๐ฅ๐ผ2โ1
ฮ ๐ผ2๐โ๐2๐ฅ
Step 3 โ Evaluation/Justification Assume the long term mean is 0, ๐๐ = 0. When ๐ = 2, ๐ต๐ก โฅ 0 โก ๐ ๐ก โฅ 0 โก ๐๐ก = 1 ๐ต๐ก < 0 โก ๐ ๐ก < 0 โก ๐๐ก = 0
GMA(2, 1) โ Naรฏve 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.
Naรฏve 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.
๐ Period Returns ๐ ๐ ๐ = โ ๐ ๐ก ร ๐ผ ๐ต๐กโ1โฅ0
๐๐ก=1
Sell at this time point
๐
๐ต๐ < 0
0 1
hold
Holding Time Distribution ๐ ๐ = ๐ = ๐ ๐ต๐ < 0,๐ต๐โ1 โฅ 0, โฆ ,๐ต1 โฅ 0,๐ต0 โฅ 0 = ๐ ๐๐ = 0,๐๐โ1 = 1, โฆ ,๐1 = 1,๐0 = 1 = ๐ ๐๐ = 0,๐๐โ1 = 1, โฆ ,๐1 = 1|๐0 = 1 ๐ ๐0 = 1
= ๏ฟฝฮ ๐๐โ1 1 โ ๐ , T โฅ1
1 โ ฮ , T=0
Stationary state probability: ฮ = 1โ๐
2โ๐โ๐
Conditional Returns Distribution (1)
ฮฆ๐ ๐ ๐|๐=๐ ๐ = E ๐ ๐ โ ๐ ๐กร๐ผ ๐ต๐กโ1โฅ0๐๐ก=1 ๐ |๐ = ๐
= E ๐ ๐ โ ๐ ๐กร๐ผ ๐ต๐กโ1โฅ0๐๐ก=1 ๐ |๐ต๐ < 0,๐ต๐โ1 โฅ 0, โฆ ,๐ต0 โฅ 0
= E ๐ ๐ โ ๐ ๐ก๐๐ก=1 ๐ |๐๐ = 0,๐๐โ1 = 1, โฆ ,๐1 = 1
= E ๐ ๐ ๐1+โฏ+๐๐โ1โ๐ฟ๐ ๐
= ๏ฟฝฮฆ๐๐โ1 ๐ ฮฆ๐ฟ โ๐ , T โฅ1ฮฆ๐ฟ โ๐ , T =0
Unconditional Returns Distribution (2) ฮฆ๐ ๐ ๐ ๐ =
โ E ๐ ๐ โ ๐ ๐กร๐ผ ๐ต๐กโ1โฅ0๐๐ก=1 ๐ |๐ = ๐ ๐ ๐ = ๐โ
๐=0 =โ ฮ ๐๐โ1 1 โ ๐ ฮฆ๐
๐โ1 ๐ ฮฆ๐ฟ โ๐ โ๐=1 + 1 โ ฮ ฮฆ๐ฟ โ๐
= 1 โ ฮ ฮฆ๐ฟ โ๐ + ฮ 1 โ ๐ ฮฆ๐ฟ โ๐ 1โ๐ฮฆ๐ ๐
Expected Returns E ๐ ๐ ๐ = โ๐ฮฆ๐ ๐ ๐
โฒ 0
= 11โ๐
ฮ ๐๐๐ โ 1 โ ๐ ๐๐ฟ
When is the expected return positive? ๐๐ โฅ
1โ๐ฮ ๐
๐๐ฟ, shock impact
๐๐ โซ ๐๐ฟ, shock impact ฮ ๐ โฅ 1 โ ๐, if ๐๐ โ ๐๐ฟ, persistence
GMA(โ,1) Rule
๐๐ก โฅ โ ๐๐กโ๐๐โ1๐=0
1๐
ln๐๐ก โฅ1๐โ ln๐๐กโ๐๐โ1๐=0
ln๐๐ก โฅ ๐1
GMA(โ,1) Expected Returns ฮฆ๐ ๐ ๐ ๐ =
1 โ ฮ ๐ ฮฆ๐ฟ ๐ + ฮฆ๐ฟ โ๐ +1 โ ๐ 1 โ ฮ ฮฆ๐ ๐ + ฮฆ๐ โ๐
E ๐ ๐ ๐ = โ 1 โ ๐ 1 โ ฮ ๐๐ + ๐๐ฟ
MA Using the Whole History An investor will always expect to lose money using
GMA(โ,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 ๐ and ๐?
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
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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.
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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, ๐, of that the market is going up given all the available information.
When ๐ 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 stockโs prices: BULL and
BEAR.
๐๐๐ = ๐๐ ๐๐ผ๐๐๐ + ๐๐๐ต๐ , ๐ก โค ๐ โค ๐ < โ The trading period is between time ๐ก,๐ . ๐ผ๐ = 1,2 are the two Markov states that indicates
the BULL and BEAR markets.
๐1 > 0
๐2 < 0
๐ = โ๐1 ๐1๐2 โ๐2
, the generator matrix for the Markov
chain. When ๐ = 0, expected return is
E0,๐ก ๐ ๐ก =
E๐ก ๐๐ ๐1โ๐ก โ ๐๐๐๐๐๐
1โ๐พ๐ 1+๐พ๐
๐ผ ๐๐<๐๐๐ ๐๐+1โ๐๐โ
๐=1
We are long between ๐๐ and ๐๐ and the return is determined by the price change discounted by the commissions.
We are flat between ๐๐ and ๐๐+1 and the money grows at the risk free rate.
Value function:
J0 ๐,๐ผ, ๐ก,ฮ0 =
E๐ก๐ ๐1 โ ๐ก +
โ log ๐๐๐๐๐๐
+ ๐ผ ๐๐<๐ log 1โ๐พ๐ 1+๐พ๐
+ ๐ ๐๐+1 โ ๐๐โ๐=1
J1 ๐,๐ผ, ๐ก,ฮ1 =
E๐กlog
๐๐1๐
+ ๐ ๐2 โ ๐1 + log 1 โ ๐พ๐ +
โ log ๐๐๐๐๐๐
+ ๐ผ ๐๐<๐ log 1โ๐พ๐ 1+๐พ๐
+ ๐ ๐๐+1 โ ๐๐โ๐=2
Find an optimal trading sequence (the stopping times) so that the value functions are maximized.
๐๐ ๐, ๐ก = supฮ๐
๐ฝ๐ ๐, ๐, ๐ก,ฮ๐
๐๐: the maximum amount of expected returns
๏ฟฝ๐0 ๐, ๐ก = sup
๐1๐ธ๐ก ๐ ๐1 โ ๐ก โ log 1 + ๐พ๐ + ๐1 ๐๐1 , ๐1
๐1 ๐, ๐ก = sup๐1
๐ธ๐ก log๐๐1๐๐ก
+ log 1 โ ๐พ๐ + ๐0 ๐๐1 , ๐1
Hamilton-Jacobi-Bellman Equations
๏ฟฝmin โโ๐0 โ ๐,๐0 โ ๐1 + log 1 + ๐พ๐ = 0
min โโ๐1 โ ๐ ๐ ,๐1 โ ๐0 โ log 1 โ ๐พ๐ = 0
with terminal conditions: ๏ฟฝ ๐0 ๐,๐ = 0๐1 ๐,๐ = log 1 โ ๐พ๐
โ = ๐๐ก + 12
๐1โ๐2 ๐ 1โ๐๐
2๐๐๐ + โ ๐1 + ๐2 ๐ + ๐2 ๐๐
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)
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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)
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Assume a risk free asset ๐๐ก, which satisfies ๐๐๐ก = ๐๐๐ก๐๐ก
Assume two assets,๐ด๐ก and ๐ต๐ก. Assume ๐ต๐กfollows a geometric Brownian
motion. ๐๐ต๐ก = ๐๐ต๐ก๐๐ก + ๐๐ต๐ก๐๐ง๐ก
๐ฅ๐กis the spread between the two assets. ๐ฅ๐ก = log๐ด๐ก โ log๐ต๐ก
๐๐๐ก๐๐ก
= โ๐ก๐๐ด๐ก๐ด๐ก
+ โ๐ก๏ฟฝ๐๐ต๐ก๐ต๐ก
+ ๐๐๐ก๐๐ก
= โ๐ก ๐ ๐ โ ๐ฅ๐ก + 12๐2 + ๐๐๐ + ๐ ๐๐ก +
โ๐ก๐๐๐๐ก max
โ๐ก๐ธ ๐๐๐พ , s.t.,
๐ 0 = ๐ฃ0, x 0 = ๐ฅ0 ๐๐ฅ๐ก = ๐ ๐ โ ๐ฅ๐ก ๐๐ก + ๐๐๐๐ก ๐๐๐ก = โ๐ก๐๐ฅ๐ก = โ๐ก๐ ๐ โ ๐ฅ๐ก ๐๐ก + โ๐ก๐๐๐๐ก
โ ๐ก โ = ๐๐ก1โ๐พ
โ ๐๐2
๐ฅ๐ก โ ๐ + 2๐ผ ๐ก ๐ฅ๐ก + ๐ฝ ๐ก
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.
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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 ๐๐ก, we
define H-variation ๐๐ ๐ป,๐ = sup
๐โ ๐ ๐ก๐ โ ๐ ๐ก๐โ1๐ฟ๐=1
It can be shown that for any H, there exists a sequence ๐๐โ , ๐๐ ๐=0,1,โฆ,๐ such that ๐๐ ๐=0,1,โฆ,๐ are Markovian and ๐๐โ are defined by ๐๐ก on intervals ๐๐โ1, ๐๐ . And they satisfy the equality: ๐๐ ๐ป,๐ = โ ๐ ๐๐โ โ ๐ ๐๐โ1โ๐
๐=1
๐๐ ๐ป,๐ is the number of KAGI-inversion in the T-interval.
H-volatility:
๐๐ ๐ป,๐ = ๐๐ ๐ป,๐๐๐ ๐ป,๐
For an no-arbitrage Wiener process, we have lim
๐โโ๐๐ ๐ป,๐๐ = ๐พ๐ป = 2๐ป
Define a trading strategy such that the position of X is: ๐พ๏ฟฝ๐ก๐พ ๐ป,๐ =
โ sign ๐ ๐๐โ1 โ ๐ ๐๐โ1โ ๐ ๐๐โ1,๐๐ ๐ก๐๐ ๐ป,๐๐=1
The trend following P&L is:
๐๏ฟฝ๐ก๐พ ๐ป,๐ = โซ ๐พ๏ฟฝ๐ข๐พ ๐ป,๐ ๐๐ ๐ข๐ก0
= ๐๐ ๐ป,๐ โ 2๐ป ๐๐ ๐ป,๐ + ๐ The expected income per trade is:
๐๐ก๐พ ๐ป,๐ = โซ ๐พ๐ข๐พ ๐ป,๐ ๐๐ ๐ข๐ก0
๐ฆ๐ก๐พ ๐ป,๐ = ๐๐ก๐พ ๐ป,๐๐๐ก๐พ ๐ป,๐
lim๐โโ
E๐ฆ๐ก๐พ ๐ป,๐ = ๐พ โ 2 ๐ป
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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.
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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:
๐,๐,๐, ๐ cash, inventory, mid price, spread
Objective:
max๐ผ
E ๐ ๐๐ โ ๐พ โซ ๐ ๐๐ก ๐๐ก๐0
๐๐ = 0, e.g., donโt hold position overnight ๐: utility function ๐๐: terminal wealth ๐พ: penalty for holding inventory
Liquidation function (how much we get by selling everything):
๐ฟ ๐ฅ,๐ฆ,๐, ๐ = ๐ฅ โ ๐ โ๐ฆ,๐, ๐ = ๐ฅ + ๐ฆ๐ โ ๐ฆ ๐ 2โ ๐
Equivalent problem (get rid of ๐๐ = 0):
max๐ผ
E ๐ ๐ฟ ๐๐ ,๐๐ ,๐๐ , ๐๐ โ ๐พ โซ ๐ ๐๐ก ๐๐ก๐0
Value function:
๐ฃ ๐ก, ๐ง, ๐ = sup๐ผ E๐ก,๐ง,๐
๐ ๐ฟ ๐๐ , ๐๐ โ ๐พ โซ ๐ ๐๐ข ๐๐ข๐๐ก
๐ง = ๐ฅ,๐ฆ,๐
This is a mixed regular/impulse control problem in a regime switching jump-diffusion model.
Quasi-Variational Inequality
min โ๐๐๐๐กโ sup โ๐,๐๐ฃ + ๐พ๐,๐ฃ โโณ๐ฃ = 0
Terminal condition:
๐ฃ ๐, ๐ฅ, ๐ฆ,๐, ๐ = ๐ ๐ฟ ๐ฅ,๐ฆ, ๐, ๐
For each state ๐, we have
min
โ๐๐๐๐๐กโ ๐ซ๐ฃ๐ โ โ ๐๐๐ ๐ก ๐ฃ๐ ๐ก, ๐ฅ,๐ฆ,๐ โ ๐ฃ๐ ๐ก, ๐ฅ,๐ฆ, ๐๐
๐=1
โ sup ๐๐๐ ๐๐ ๐ฃ๐ ๐ก, ๐ฅ โ ๐๐๐ ๐๐,๐ ๐๐ ,๐ฆ + ๐๐ ,๐ โ ๐ฃ๐ ๐ก, ๐ฅ, ๐ฆ,๐โ sup ๐๐๐ ๐๐ ๐ฃ๐ ๐ก, ๐ฅ + ๐๐๐ ๐๐,๐ ๐๐ ,๐ฆ โ ๐๐ ,๐ โ ๐ฃ๐ ๐ก, ๐ฅ,๐ฆ, ๐
+๐พ๐,๐ฃ๐ ๐ก, ๐ฅ,๐ฆ,๐ โ sup ๐ฃ๐ ๐ก, ๐ฅ โ ๐๐ ๐, ๐ ,๐ฆ + ๐, ๐
= 0
๐ฃ๐ ๐, ๐ฅ, ๐ฆ,๐ = ๐ ๐ฟ๐ ๐ฅ,๐ฆ,๐
Assumptions:
๐ ๐ฅ = ๐ฅ; we care about only how much money made.
๐๐ก ๐ก is a martingale; we know nothing about where the market will move.
Solution:
๐ฃ๐ ๐ก, ๐ฅ,๐ฆ, ๐ = ๐ฅ + ๐ฆ๐ + ๐๐ ๐ก, ๐ฆ
๐๐ ๐ก,๐ฆ is the solution to the system of integro-differential equations (IDE):
min
โ๐๐๐๐๐ก
โ โ ๐๐๐ ๐ก ๐๐ ๐ก,๐ฆ โ ๐๐ ๐ก,๐ฆ๐๐=1
โ sup ๐๐๐ ๐๐ ๐๐ ๐ก,๐ฆ + ๐๐ โ ๐๐ ๐ก,๐ฆ + ๐๐ฟ2โ ๐ฟ1๐๐=๐๐ก๐+ ๐๐
โ sup ๐๐๐ ๐๐ ๐๐ ๐ก, ๐ฆ โ ๐๐ โ ๐๐ ๐ก, ๐ฆ + ๐๐ฟ2โ ๐ฟ1๐๐=๐๐ก๐โ ๐๐
+๐พ๐ ๐ฆ ,๐๐ ๐ก,๐ฆ โ sup ๐๐ ๐ก,๐ฆ + ๐ โ ๐๐ฟ
2๐ โ ๐
= 0
๐๐ ๐, ๐ฆ = โ ๐ฆ ๐๐ฟ2โ ๐
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
โข Linear algebra
โข Calculus
๏ฟฝ ๐ ๐ฅ๐
๐
= ๐น ๐ โ ๐น ๐
โข Unconstrained optimization
โข Statistics
โข Differential Equations
local minimum global minimum
Financial Mathematics
Advanced Mathematics
Foundation Mathematics
โข Parallelization
โข Cointegration
โข Optimization (LP, QP, SQP, SDP, SOCP, IP, GA)
โข Digital Signal Processing
โข Time Series Analysis
Applications
โข Optimal Trading Strategies
โข Portfolio Optimization
โข Extreme Value Theory
โข Trading Signals
โข Asset Allocation
โข 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).