TD - Profit or loss on options and the Black-Scholes equation

Question
(a) On the last day of 1999, two investors (Asif and Barbara) decided to
spend money in buying options. On this day, the share price of BOOTS
was 495p, the share price of BP AMOCO was 495p and the share price
of SHELL was 438p. The expiry date for all of the options that they
bought was April 13th. Options were available as follows
Option Strike Price Available for
BOOTS PUTS 460p 20p
BOOTS CALLS 460p 60p
BP PUTS 500p 37p
BP CALLS 500p 34p
SHELL PUTS 460p 45p
SHELL CALLS 460p 20p
Asif bought 1000 BOOTS calls, 1000 BP calls and 500 SHELL puts,
and Barbara bought 1000 BOOTS puts, 2000 BP puts and 2000 SHELL
calls.
Assuming that there was no bid/ask spread and no dealing charges,
and that on April 13th 2000 the share prices were given by:
BOOTS 555p
BP AMOCO 410p
SHELL 500p
determine how much Asif and Barbara paid for their options, and what
the total profit or loss was for each investor after expiry.
(b) Now YOU MAY ASSUME that small charges df in the function f(S,t)
are related to small changes in S and t by Taylor’s theorem and that
the asset prices S of a share follows the lognormal random walk
dS = rSdt + σSdX
where X is a random variable, r and σ are constants, and dX2 → dt
as dt → 0.
By considering a portfolio Π = V − ∆S (where ∆ is to be determined),
show that the fair value V of an option satisfies the Black-Scholes equation
Vt +
1
2
σ
2S
2VSS + rSVS − rV = 0.