TI-Basic Z80 Programming/Mathematical Finance Programming

TI-BASIC is a simple programming language used on Texas Instruments graphing calculators. This module shows you how to program some standard financial calculations: TI-BASIC programs Ito's lemma Let's begin defining a stochastic process through its Ito's definition: :defsto(t,x) :Func :{t,x} :EndFunc So for our TI-calculator, a diffusion process formally defined by: is defined by a set of two terms: { f(S,t), g(S,t) } For an exponential brownian motion, we define: defsto(m*s, sigma*s) → ds(s) Now we want to use Ito's lemma on functions of and : :dsto(f,x,t,ds) :Func :{d(f,t)+ds[1]*d(f,x)+ds[2]^2*d(d(f,x),x)/2 , ds[2]*d(f,x)} :EndFunc This can now be used to apply Ito's lemma to : dsto(ln(S),S,t,ds(S)) >> { m - sigma^2/2 , sigma } this tell us that: Black-Scholes Equation Now we can try to prove the Black-Scholes equation. Define a portfolio with an option and shares of : V(S,t) - Delta * S → Pi and apply Ito's lemma to obtain : dsto(Pi, S, t, ds(S)) → dPi we now want to nullify the stochastic part of by chosing an appropriate value for : solve( dPi[2]=0, Delta) >> Delta = d(V(S,t), S) or sigma S = 0 we now know that the correct value for is: On another side, we have: which leads us to the equation: At first we need to replace by its value into , and then equalize with solve( dPi[2]=0, Delta) | sigma > 0 and S > 9 → sol >> Delta = d(V(S,t), S) dPi | sol → dPi >> {sigma^2 d^2(V(S,t), S^2) S^2 /2 + d(V(S,t), t) , 0 } dPi = defsto( r(V(S,t) - Delta S) ) | sol → BS >> { BS_equation , true } and now we've got the Black-Schole Differential Equation into the variable BS_equation[1]!