Chapter 4: Ten Different Ways to Derive Black–Scholes 271
S
Long option
Short stock
Hedged portfolio
Figure 4-2:How our portfolio depends onS.
Our portfolio has a dependence onSas shown in
Figure 4-2.
We are only concerned with small movements in the
stock over a small time period, so zoom in on the cur-
rent stock position. Locally the curve is approximately
a parabola, see Figure 4-3.
Now think about how our net worth will change from
now to a timeδtlater. There are three reasons for our
total wealth to change over that period.
1.The option price curve changes
2.There is an interest payment on the money in the bank
3.The stock moves
The option curve falls by the time value, the theta mul-
tiplied by the time step:
×δt.
