Faculty of Engineering & Computing 2014/15 Financial Mathematics (329MSE)

329MSE Financial Mathematics CW sheet 4
Instructions
1. This is an individual assignment, not group work.
2. The assignment contributes 25% to your coursework mark.
3. Please submit online as one pdf file using the naming convention as in “YavorskyiTaras-
329MSE-CW4.pdf”
4. The deadline is 23:55, 11 March 2015. Work submitted later than that without evidence of a
deferral/extension will receive zero marks.
5. In the head of your script please include details usually required on a coursework cover sheet:
Module Code and Title, Assignment number, Full Name, Student ID, Time taken.
6. When solving exercises, be maximally clear. Print or use legible handwriting. Use standard
notations. Describe explicitly the steps followed to produce your answer.
7. Perform anyl calculations to at least 7 significant digits (not counting leading zeros).
8. Do your work yourself.
(Questions overleaf)
T. Yavorskii/C. von Ferber February 18, 2015
Faculty of Engineering & Computing 2014/15 Financial Mathematics (329MSE)
Questions
1. In the Black-Scholes framework compute
@C
@E
and @2C
@E2 . (1)
Then, use Put-Call parity to compute
@P
@E
and @2P
@E2 . (2)
Assume that the underlying asset S gains continuous dvidend payments with rate  according
to the Black-Scholes formula
C = Se-(T-t)N(d1) – Ee-r(T-t)N(d2), (3)
where
d1 =
ln( S
E ) + (r –  + 2
2 )(T – t)

p
T – t
; (4)
d2 =
ln( S
E ) + (r –  – 2
2 )(T – t)

p
T – t
(5)
= d1 – 
p
T – t (6)
and use the fact that
Se-(T-t)N0(d1) = Ee-r(T-t)N0(d2). (7)
2. Show that the price of a plain vanilla European call option is a convex function of the strike
E (Exercise price) of the option, i.e. show that
@2C
@E2  0. (8)
Recall Eq. (7) and differentiate the Black Scholes Formula Eq. (3) with respect to E.
3. A straddle is a an options strategy in which you hold a position in both a call and a put with
the same strike (exercise) price E and same date of expiry.
Within the Black-Scholes framework, for what strike (excercise price E) is the value of a
straddle minimal?
In other words, given S, T, , r find the strike E such that P(E) + C(E) is minimal. Start
with Put-Call parity in the form
P(E) = C(E) + Ee-rT – Se-T (9)
and re-write as
P(E) + C(E) = 2C(E) + Ee-rT – Se-T (10)
to derive the minimal value of the straddle.
T. Yavorskii/C. von Ferber February 18, 2015