Mathematical Finance Programming In Ti-basic

Quantitative (mathematical) Finance is based on Stochastic Calculus , and more specifically on the Ito's Lemma .

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:

dS = f(S,t) dt + g(S,t) dW

is defined by a set of two terms:

{ f(S,t), g(S,t) }

For an exponential brownian motion, we define:

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,x)+ds[2]^2---d(d(f,x),x)/2" class="copylinks" target="_blank">, ds[2 ---d(f,x)}

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
ight) dt + \sigma dW, \;{
m when}\; dS = m S dt + \sigma S dW

Black-Scholes Equation

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

Define a portfolio with an option and

\Delta

shares of

S

S → Pi

and apply Ito's lemma to obtain

d \Pi

:

dsto(Pi, S, t, ds(S)) → dPi

we now want to nullify the stochastic part of

d \Pi

by chosing an appropriate value for

\Delta

:

solve( dPi {Link without Title} =0, Delta)
>> Delta = d(V(S,t), S) or sigma S = 0

we now know that the correct value for

\Delta

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

\Delta

by its value into

d\Pi

, and then equalize with

DPi

defsto( r(V(S,t) - Delta S) ) sol &rarr BS

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Mathematical Finance Programming In Ti-basic