TD - European average rate option- Quiz & Solution

Question
Consider the EUROPEAN AVERAGE RATE option with expiry T whose
value is governed by the partial differential equation
Vt + (log S)VI +
1
2
σ
2S
2VSS + rSVS − rV = 0
where the independent variable I is defined by
I =
Z t
0
log S(τ ) dτ.
(a) By considering definition of I, and the quantity
Q =
ÃXn
i=1
S(ti)
! 1
n
in the limit n → ∞ or otherwise, explain what kind of average is being
used in the average rate option.
(b) If the payoff of the option is a function of I only, show that solutions of
the form
V = F(θ,t)
θ =
I + (T − t)log S
T
exist provided F satisfies
Ft + a(t)Fθθ + b(t)Fθ − rF = 0 −→ (1)
where a(t) and b(t) are functions that should be determined.
(c) Explain briefly why (1) is easier to solve than the original problem. If
the payoff of the option depends on S as well as I, does this method
still work?