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# Financehasevolvedintoadisciplineofappliedmathematics infinancecamewithfewmathematicalexpressionsandequations comprisedofmathematicalexpressionsandequations providingadetailedbackgroundforeachofthem topicavailable withpython finance pricingwhenitcomestomodelcalibration evaluationofintegrals systemsof matplotlib importnumpyasnp importmatplotlib matplotlibinlinethroughoutthisdiscussion iscomprisedofatrigonometrictermandalinearterm deff returnnp xthemainfocusistheapproximationofthisfunctionoveragivenintervalbyregression andinterpolation exactlytheapproximationshallachieve figure displaysthefunctionoverthefixedintervaldefinedviathelinspacefunction withstop plt plt plt plt examplefunctionplotregressionregressionisaratherefficienttoolwhenitcomestofunctionapproximation suitedtoapproximateone-dimensionalfunctionsbutalsoworkswellinhigher dimensions easilyimplementedandquicklyexecuted whereyif consideredindependentobservationsandtheyidependentobservations statisticalsense minimizationproblemofregression

Chapter�9.�Mathematical�Tools

The�mathematicians�are�the�priests�of�the�modern�world.

Regression�and�interpolation�are�among�the�most�often�used�numerical�techniques�in finance.

Convex�optimization

Python�provides�with�SymPy�a�powerful�tool�for�symbolic�mathematics,�e.g.,�to�solve (systems�of)�equations.

Approximation

The�main�focus�is�the�approximation�of�this�function�over�a�given�interval�by�regression and�interpolation.�First,�let�us�generate�a�plot�of�the�function�to�get�a�better�view�of�what exactly�the�approximation�shall�achieve.�The�interval�of�interest�shall�be�[–2,2].

Figure�9-1�displays�the�function�over�the�fixed�interval�defined�via�the�linspace�function.

Figure�9-1.�Example�function�plot

Regression

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