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MATHEMATICAL MODELS IN FINANCE:TRADING STRATEGIES
Paul Johnson
School of Mathematics
July 2019
Paul Johnson Mathematical Models in Finance: Trading Strategies
TRADING STRATEGIES
TODAY’S LECTURE
Background
A Mathematical Model for Stock Price
Simulating and Testing Trading Strategies
Can it Work on Real Data?
Paul Johnson Mathematical Models in Finance: Trading Strategies
BACKGROUND
I’m Paul Johnson, a Senior Lecturer inMathematical Finance
Worked in the department for over 10yearsResearch in numerical solutions tonon-linear PDEs arising in financeApplications in Finance, Mining,Revenue Management systems,Renewable EnergyTeach Mathematical Finance in theundergraduate program andComputational Finance at postgraduatelevel.
Paul Johnson Mathematical Models in Finance: Trading Strategies
BACKGROUND
I’m Paul Johnson, a Senior Lecturer inMathematical FinanceWorked in the department for over 10years
Research in numerical solutions tonon-linear PDEs arising in financeApplications in Finance, Mining,Revenue Management systems,Renewable EnergyTeach Mathematical Finance in theundergraduate program andComputational Finance at postgraduatelevel.
Paul Johnson Mathematical Models in Finance: Trading Strategies
BACKGROUND
I’m Paul Johnson, a Senior Lecturer inMathematical FinanceWorked in the department for over 10yearsResearch in numerical solutions tonon-linear PDEs arising in finance
Applications in Finance, Mining,Revenue Management systems,Renewable EnergyTeach Mathematical Finance in theundergraduate program andComputational Finance at postgraduatelevel.
Paul Johnson Mathematical Models in Finance: Trading Strategies
BACKGROUND
I’m Paul Johnson, a Senior Lecturer inMathematical FinanceWorked in the department for over 10yearsResearch in numerical solutions tonon-linear PDEs arising in financeApplications in Finance, Mining,Revenue Management systems,Renewable Energy
Teach Mathematical Finance in theundergraduate program andComputational Finance at postgraduatelevel.
Paul Johnson Mathematical Models in Finance: Trading Strategies
BACKGROUND
I’m Paul Johnson, a Senior Lecturer inMathematical FinanceWorked in the department for over 10yearsResearch in numerical solutions tonon-linear PDEs arising in financeApplications in Finance, Mining,Revenue Management systems,Renewable EnergyTeach Mathematical Finance in theundergraduate program andComputational Finance at postgraduatelevel.
Paul Johnson Mathematical Models in Finance: Trading Strategies
TRADING STRATEGIES
Algorithmic Trading and technological solutions are rapidlyadvancing in the financial sectorAn Algorithmic Trade executes a series of trades (to buy orsell a stock) according to a predefined strategy
So who comes up with the strategy?We need a model ... a strategy ... and a way to test it ...
We need Mathematics!!!
Paul Johnson Mathematical Models in Finance: Trading Strategies
TRADING STRATEGIES
Algorithmic Trading and technological solutions are rapidlyadvancing in the financial sectorAn Algorithmic Trade executes a series of trades (to buy orsell a stock) according to a predefined strategy
So who comes up with the strategy?We need a model ... a strategy ... and a way to test it ...
We need Mathematics!!!
Paul Johnson Mathematical Models in Finance: Trading Strategies
TRADING STRATEGIES
Algorithmic Trading and technological solutions are rapidlyadvancing in the financial sectorAn Algorithmic Trade executes a series of trades (to buy orsell a stock) according to a predefined strategy
So who comes up with the strategy?We need a model ... a strategy ... and a way to test it ...
We need Mathematics!!!
Paul Johnson Mathematical Models in Finance: Trading Strategies
TRADING STRATEGIES
Algorithmic Trading and technological solutions are rapidlyadvancing in the financial sectorAn Algorithmic Trade executes a series of trades (to buy orsell a stock) according to a predefined strategy
So who comes up with the strategy?We need a model ... a strategy ... and a way to test it ...
We need Mathematics!!!
Paul Johnson Mathematical Models in Finance: Trading Strategies
TRADING STRATEGIES
Algorithmic Trading and technological solutions are rapidlyadvancing in the financial sectorAn Algorithmic Trade executes a series of trades (to buy orsell a stock) according to a predefined strategy
So who comes up with the strategy?We need a model ... a strategy ... and a way to test it ...
We need Mathematics!!!
Paul Johnson Mathematical Models in Finance: Trading Strategies
TRADING STRATEGIES
Algorithmic Trading and technological solutions are rapidlyadvancing in the financial sectorAn Algorithmic Trade executes a series of trades (to buy orsell a stock) according to a predefined strategy
So who comes up with the strategy?We need a model ... a strategy ... and a way to test it ...
We need Mathematics!!!
Paul Johnson Mathematical Models in Finance: Trading Strategies
TRADING STRATEGIES
Algorithmic Trading and technological solutions are rapidlyadvancing in the financial sectorAn Algorithmic Trade executes a series of trades (to buy orsell a stock) according to a predefined strategy
So who comes up with the strategy?
We need a model ... a strategy ... and a way to test it ...
We need Mathematics!!!
Paul Johnson Mathematical Models in Finance: Trading Strategies
TRADING STRATEGIES
Algorithmic Trading and technological solutions are rapidlyadvancing in the financial sectorAn Algorithmic Trade executes a series of trades (to buy orsell a stock) according to a predefined strategy
So who comes up with the strategy?We need a model ... a strategy ... and a way to test it ...
We need Mathematics!!!
Paul Johnson Mathematical Models in Finance: Trading Strategies
TRADING STRATEGIES
Algorithmic Trading and technological solutions are rapidlyadvancing in the financial sectorAn Algorithmic Trade executes a series of trades (to buy orsell a stock) according to a predefined strategy
So who comes up with the strategy?We need a model ... a strategy ... and a way to test it ...
We need Mathematics!!!
Paul Johnson Mathematical Models in Finance: Trading Strategies
MODELLING THE REAL WORLD
1 The Mathematical ModelHow does stock price change?How do we capture our uncertainty about predicting thefuture?
2 The Simulation EnvironmentHow do we execute a trade in mathematics?How do we keep track of our money/stock?
3 The Real WorldHow realistic is our model?How likely is it to cause huge losses?
Paul Johnson Mathematical Models in Finance: Trading Strategies
MODELLING THE REAL WORLD
1 The Mathematical ModelHow does stock price change?How do we capture our uncertainty about predicting thefuture?
2 The Simulation EnvironmentHow do we execute a trade in mathematics?How do we keep track of our money/stock?
3 The Real WorldHow realistic is our model?How likely is it to cause huge losses?
Paul Johnson Mathematical Models in Finance: Trading Strategies
MODELLING THE REAL WORLD
1 The Mathematical ModelHow does stock price change?How do we capture our uncertainty about predicting thefuture?
2 The Simulation EnvironmentHow do we execute a trade in mathematics?How do we keep track of our money/stock?
3 The Real WorldHow realistic is our model?How likely is it to cause huge losses?
Paul Johnson Mathematical Models in Finance: Trading Strategies
A MATHEMATICAL MODEL FOR STOCK PRICES
A Brownian motion model for stocks is one of the most simpleand popular models in finance. Consider St is the price of thestock at time t , it looks like this:
dS = µStdt + σStdW
which means:
change in stock price = deterministic trend + random component
Paul Johnson Mathematical Models in Finance: Trading Strategies
SIMULATED TRADING ACCOUNT
The total value Wt of our trading account is:-
Wt = ∆tSt + Bt
where:
∆t ≥ 0 is the number of stocks we own at time tSt ≥ 0 is the price to buy/sell stock at time tBt ≥ 0 is the amount of money we have at time t
Paul Johnson Mathematical Models in Finance: Trading Strategies
SIMULATED TRADING ACCOUNT
The total value Wt of our trading account is:-
Wt = ∆tSt + Bt
where:∆t ≥ 0 is the number of stocks we own at time t
St ≥ 0 is the price to buy/sell stock at time tBt ≥ 0 is the amount of money we have at time t
Paul Johnson Mathematical Models in Finance: Trading Strategies
SIMULATED TRADING ACCOUNT
The total value Wt of our trading account is:-
Wt = ∆tSt + Bt
where:∆t ≥ 0 is the number of stocks we own at time tSt ≥ 0 is the price to buy/sell stock at time t
Bt ≥ 0 is the amount of money we have at time t
Paul Johnson Mathematical Models in Finance: Trading Strategies
SIMULATED TRADING ACCOUNT
The total value Wt of our trading account is:-
Wt = ∆tSt + Bt
where:∆t ≥ 0 is the number of stocks we own at time tSt ≥ 0 is the price to buy/sell stock at time tBt ≥ 0 is the amount of money we have at time t
Paul Johnson Mathematical Models in Finance: Trading Strategies
EXECUTING A TRADE
To buy a stock, increase ∆:-
∆t+1 = ∆t + 1
and we pay S out of the bank account
Bt+1 = Bt − St+1
To sell a stock, decrease ∆:-
∆t+1 = ∆t − 1
and we deposit S into the bank account
Bt+1 = Bt + St+1
Paul Johnson Mathematical Models in Finance: Trading Strategies
EXECUTING A TRADE
To buy a stock, increase ∆:-
∆t+1 = ∆t + 1
and we pay S out of the bank account
Bt+1 = Bt − St+1
To sell a stock, decrease ∆:-
∆t+1 = ∆t − 1
and we deposit S into the bank account
Bt+1 = Bt + St+1
Paul Johnson Mathematical Models in Finance: Trading Strategies
A TRADING STRATEGY
A trading strategy is a function T that instructs a change in theamount of stock we hold. So
∆t+1 = ∆t + T (S,W ,∆,B)
We are interested in defining thefunction Tand examing the expected profit of thestrategy
expected profit = E [WT ] − W0
Paul Johnson Mathematical Models in Finance: Trading Strategies
A TRADING STRATEGY
A trading strategy is a function T that instructs a change in theamount of stock we hold. So
∆t+1 = ∆t + T (S,W ,∆,B)
We are interested in defining thefunction Tand examing the expected profit of thestrategy
expected profit = E [WT ] − W0
Paul Johnson Mathematical Models in Finance: Trading Strategies
A TRADING STRATEGY
A trading strategy is a function T that instructs a change in theamount of stock we hold. So
∆t+1 = ∆t + T (S,W ,∆,B)
We are interested in defining thefunction Tand examing the expected profit of thestrategy
expected profit = E [WT ] − W0
Paul Johnson Mathematical Models in Finance: Trading Strategies
EXAMPLE STRATEGIES
BUY UNCONDITIONAL
T (S,W ,∆,B) = 1
BUY LOW – SELL HIGH
T (S,W ,∆,B) =
1 if St < αS00 if αS0 ≤ St ≤ βS0−1 if St > βS0
with α < 1 and β > 1.
Click on the binder icon to see these strategies in action:
Paul Johnson Mathematical Models in Finance: Trading Strategies
EXAMPLE STRATEGIES
BUY UNCONDITIONAL
T (S,W ,∆,B) = 1
BUY LOW – SELL HIGH
T (S,W ,∆,B) =
1 if St < αS00 if αS0 ≤ St ≤ βS0−1 if St > βS0
with α < 1 and β > 1.
Click on the binder icon to see these strategies in action:
Paul Johnson Mathematical Models in Finance: Trading Strategies
EXAMPLE STRATEGIES
BUY UNCONDITIONAL
T (S,W ,∆,B) = 1
BUY LOW – SELL HIGH
T (S,W ,∆,B) =
1 if St < αS00 if αS0 ≤ St ≤ βS0−1 if St > βS0
with α < 1 and β > 1.
Click on the binder icon to see these strategies in action:
Paul Johnson Mathematical Models in Finance: Trading Strategies
REAL WORLD SCENARIOS
HOLD ON ...
This is just a mathematical model.
Strategies that prove effective herecould be catastrophic in the real worldClick the icon to see performance in thereal world!
Paul Johnson Mathematical Models in Finance: Trading Strategies
REAL WORLD SCENARIOS
HOLD ON ...
This is just a mathematical model.Strategies that prove effective herecould be catastrophic in the real world
Click the icon to see performance in thereal world!
Paul Johnson Mathematical Models in Finance: Trading Strategies
REAL WORLD SCENARIOS
HOLD ON ...
This is just a mathematical model.Strategies that prove effective herecould be catastrophic in the real worldClick the icon to see performance in thereal world!
Paul Johnson Mathematical Models in Finance: Trading Strategies