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M ATHEMATICAL M ODELS IN F INANCE :T RADING S TRATEGIESPaul JohnsonSchool of MathematicsJuly 2019Paul JohnsonMathematical Models in Finance: Trading Strategies
T RADING S TRATEGIEST ODAY ’ S L ECTUREBackgroundA Mathematical Model for Stock PriceSimulating and Testing Trading StrategiesCan it Work on Real Data?Paul JohnsonMathematical Models in Finance: Trading Strategies
BACKGROUNDI’m Paul Johnson, a Senior Lecturer inMathematical FinancePaul JohnsonMathematical Models in Finance: Trading Strategies
BACKGROUNDI’m Paul Johnson, a Senior Lecturer inMathematical FinanceWorked in the department for over 10yearsPaul JohnsonMathematical Models in Finance: Trading Strategies
BACKGROUNDI’m Paul Johnson, a Senior Lecturer inMathematical FinanceWorked in the department for over 10yearsResearch in numerical solutions tonon-linear PDEs arising in financePaul JohnsonMathematical Models in Finance: Trading Strategies
BACKGROUNDI’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 EnergyPaul JohnsonMathematical Models in Finance: Trading Strategies
BACKGROUNDI’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 JohnsonMathematical Models in Finance: Trading Strategies
T RADING S TRATEGIESAlgorithmic 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 strategyPaul JohnsonMathematical Models in Finance: Trading Strategies
T RADING S TRATEGIESAlgorithmic 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 strategyPaul JohnsonMathematical Models in Finance: Trading Strategies
T RADING S TRATEGIESAlgorithmic 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 strategyPaul JohnsonMathematical Models in Finance: Trading Strategies
T RADING S TRATEGIESAlgorithmic 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 strategyPaul JohnsonMathematical Models in Finance: Trading Strategies
T RADING S TRATEGIESAlgorithmic 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 strategyPaul JohnsonMathematical Models in Finance: Trading Strategies
T RADING S TRATEGIESAlgorithmic 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 strategyPaul JohnsonMathematical Models in Finance: Trading Strategies
T RADING S TRATEGIESAlgorithmic 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 strategySo who comes up with the strategy?Paul JohnsonMathematical Models in Finance: Trading Strategies
T RADING S TRATEGIESAlgorithmic 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 strategySo who comes up with the strategy?We need a model . a strategy . and a way to test it .Paul JohnsonMathematical Models in Finance: Trading Strategies
T RADING S TRATEGIESAlgorithmic 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 strategySo who comes up with the strategy?We need a model . a strategy . and a way to test it .We need Mathematics!!!Paul JohnsonMathematical Models in Finance: Trading Strategies
M ODELLING THE R EAL W ORLD1The Mathematical ModelHow does stock price change?How do we capture our uncertainty about predicting thefuture?Paul JohnsonMathematical Models in Finance: Trading Strategies
M ODELLING THE R EAL W ORLD1The Mathematical ModelHow does stock price change?How do we capture our uncertainty about predicting thefuture?2The Simulation EnvironmentHow do we execute a trade in mathematics?How do we keep track of our money/stock?Paul JohnsonMathematical Models in Finance: Trading Strategies
M ODELLING THE R EAL W ORLD1The Mathematical ModelHow does stock price change?How do we capture our uncertainty about predicting thefuture?2The Simulation EnvironmentHow do we execute a trade in mathematics?How do we keep track of our money/stock?3The Real WorldHow realistic is our model?How likely is it to cause huge losses?Paul JohnsonMathematical Models in Finance: Trading Strategies
A M ATHEMATICAL M ODEL FOR S TOCK P RICESA 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 µSt dt σSt dWwhich means:change in stock price deterministic trend random componentPaul JohnsonMathematical Models in Finance: Trading Strategies
S IMULATED T RADING ACCOUNTThe total value Wt of our trading account is:Wt t St Btwhere:Paul JohnsonMathematical Models in Finance: Trading Strategies
S IMULATED T RADING ACCOUNTThe total value Wt of our trading account is:Wt t St Btwhere: t 0 is the number of stocks we own at time tPaul JohnsonMathematical Models in Finance: Trading Strategies
S IMULATED T RADING ACCOUNTThe total value Wt of our trading account is:Wt t St Btwhere: t 0 is the number of stocks we own at time tSt 0 is the price to buy/sell stock at time tPaul JohnsonMathematical Models in Finance: Trading Strategies
S IMULATED T RADING ACCOUNTThe total value Wt of our trading account is:Wt t St Btwhere: 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 tPaul JohnsonMathematical Models in Finance: Trading Strategies
E XECUTING A TRADETo buy a stock, increase : t 1 t 1and we pay S out of the bank accountBt 1 Bt St 1Paul JohnsonMathematical Models in Finance: Trading Strategies
E XECUTING A TRADETo buy a stock, increase : t 1 t 1and we pay S out of the bank accountBt 1 Bt St 1To sell a stock, decrease : t 1 t 1and we deposit S into the bank accountBt 1 Bt St 1Paul JohnsonMathematical Models in Finance: Trading Strategies
A T RADING S TRATEGYA trading strategy is a function T that instructs a change in theamount of stock we hold. So t 1 t T (S, W , , B)Paul JohnsonMathematical Models in Finance: Trading Strategies
A T RADING S TRATEGYA 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 thestrategyPaul JohnsonMathematical Models in Finance: Trading Strategies
A T RADING S TRATEGYA 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 thestrategyexpected profit E[WT ] W0Paul JohnsonMathematical Models in Finance: Trading Strategies
E XAMPLE S TRATEGIESB UY UNCONDITIONALT (S, W , , B) 1Paul JohnsonMathematical Models in Finance: Trading Strategies
E XAMPLE S TRATEGIESB UY UNCONDITIONALT (S, W , , B) 1B UY LOW – SELL HIGH 10T (S, W , , B) 1 if St αS0 if αS0 St βS0 if St βS0with α 1 and β 1.Paul JohnsonMathematical Models in Finance: Trading Strategies
E XAMPLE S TRATEGIESB UY UNCONDITIONALT (S, W , , B) 1B UY LOW – SELL HIGH 10T (S, W , , B) 1 if St αS0 if αS0 St βS0 if St βS0with α 1 and β 1.Click on the binder icon to see these strategies in action:Paul JohnsonMathematical Models in Finance: Trading Strategies
R EAL W ORLD S CENARIOSH OLD ON .This is just a mathematical model.Paul JohnsonMathematical Models in Finance: Trading Strategies
R EAL W ORLD S CENARIOSH OLD ON .This is just a mathematical model.Strategies that prove effective herecould be catastrophic in the real worldPaul JohnsonMathematical Models in Finance: Trading Strategies
R EAL W ORLD S CENARIOSH OLD 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 JohnsonMathematical Models in Finance: Trading Strategies
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 simple and popular models in finance. Consider St is the price of the stock at time t, it looks like this: dS Stdt StdW which means: change in stock price deterministic trend random component Paul Johnson Mathematical Models
