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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