Mathematical Finance Programming in TI-BASIC

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1 Financial background
2 TI-BASIC programs
- 2.1 Ito's lemma
- 2.2 Black-Scholes Equation

[edit] Financial background

Quantitative (mathematical) finance is based on stochastic calculus, and more specifically on the Ito's Lemma.

[edit] TI-BASIC programs

[edit] 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:

$d S = f (S, t) d t + g (S, t) d W$

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 $S$ and $t$ :

: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 $ln(S)$ :

dsto(ln(S),S,t,ds(S))
>> { m - sigma^2/2 , sigma }

this tell us that:

$d \ln(S) = \left(m-{1\over 2}\sigma^2\right) dt + \sigma dW, \;{\rm when}\; dS = m S dt + \sigma S dW$

[edit] Black-Scholes Equation

Now we can try to prove the Black-Scholes equation.

Define a portfolio with an option and $Δ$ shares of $S$ :

V(S,t) - Delta * S → Pi

and apply Ito's lemma to obtain $d Π$ :

dsto(Pi, S, t, ds(S)) → dPi

we now want to nullify the stochastic part of $d Π$ by chosing an appropriate value for $Δ$ :

solve( dPi[2]=0, Delta)
>> Delta = d(V(S,t), S) or sigma S = 0

we now know that the correct value for $Δ$ is:

$\Delta = {\partial V(S,t)\over \partial S}$

On another side, we have:

$d\Pi = r \Pi dt = r ( V(S,t)-\Delta\cdot S) dt$

which leads us to the equation:

$(\partial_t V + {1\over 2}\sigma^2 S^2 \partial^2_S V) dt = r ( V(S,t)-\Delta S) dt$

At first we need to replace $Δ$ by its value into $d Π$ , 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]!

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