1 Self-financing process: from discrete-time tocontinuous-time

Let us first look at the discrete time version of a self-financing process. Denotet0 < t1 < ... < tN , and the stock price Sti

, and wealth Ztiat time ti. Assume at

time ti, we hold ∆ti stock until ti+1, then Wti −∆tiSti is in the bank account.We change the positions only at time ti, i = 0, 1, ..., N − 1. So the profit/lossduring [ti, ti+1]

r (Zti −∆tiSti) (ti+1 − ti) + ∆ti

(Sti+1 − Sti

).

The accumulative profit/loss in [0, T ] is then

ZT − Z0 =N−1∑

i=0

r (Zti−∆ti

Sti) (ti+1 − ti) + ∆ti

(Sti+1 − Sti

)

→∫ T

0

r (Z −∆tSt) dt +∫ T

0

∆tdSt, as maxi|ti+1 − ti| → 0

=∫ T

0

r (Z −∆tSt) dt +∫ T

0

∆t (µStdt + σStdBt)

=∫ T

0

[rZ + (µ− r)∆tSt] dt +∫ T

0

σ∆tStdBt.

In differential form,

dZt = [rZ + (µ− r)∆tSt] dt + σ∆tStdBt.

This is the governing equation for a self-financing wealth process Zt : Note that1) Zt is continuous;2) The change of wealth is due to the change of asset prices, instead of the

change of the positions we take.

2 On the linkage between PDE and expectation

f(St, t) = Et

[(X − ST )+

]

= Et

[Et+δt (X − ST )+

]

= Et [f(St+δt, t + δt)] ,

where E is the risk-neutral expectation. So,

0 = Et [f(St+δt, t + δt)− f(St, t)]

= Et

[∫ t+δt

t

(∂f

∂t+

12σ2S2 ∂2f

∂S2+ rS

∂f

∂S

)dt +

∫ t+δt

t

σS∂f

∂SdBt

]

1

= Et

[∫ t+δt

t

(∂f

∂t+

12σ2S2 ∂2f

∂S2+ rS

∂f

∂S

)dt

].

It follows

Et

[1δt

∫ t+δt

t

(∂f

∂t+

12σ2S2 ∂2f

∂S2+ rS

∂f

∂S

)dt

]= 0

which yields by letting δt → 0

∂f

∂t+

12σ2S2 ∂2f

∂S2+ rS

∂f

∂S= 0. (1)

Assume constant r. And consider

V (St, t) = Et

[e−r(T−t) (X − ST )+

]

= e−r(T−t)f(St, t)

or f(S, t) = er(T−t)V (S, t). Then

∂f

∂S= er(T−t) ∂V

∂S,

∂2f

∂S2= er(T−t) ∂

2V

∂S2,

∂f

∂t= er(T−t) ∂V

∂t− rer(T−t)V.

Substituting to the equation (1), we get the Black-Scholes equations.

2