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School of Finance and Economics
University of Technology, Sydney
25762 Synthetic Financial Products
Autumn 2009
Case Study: A Reset Strike Option
Background
In this case study we are going to investigate an exotic option on a non-代写留学生论文dividend-paying
stock, known as a reset strike option. We begin by introducing some notation:
Symbol Interpretation
S Stock price
¾ Stock price volatility
r Risk-free interest rate (with continuous compounding)
K Option strike
T Option maturity
¿ Strike reset date
CR(S) Price of a reset strike call if the initial stock price is S
PR(S) Price of a reset strike put if the initial stock price is S
C(S) Price of a vanilla call if the initial stock price is S
P(S) Price of a vanilla put if the initial stock price is S
A reset strike option is a European contingent claim with the feature that its strike price
may be modi¯ed at the reset date ¿ 2 [0; T]. The way in which the strike price is modi¯ed
depends on the stock price at the reset date and the °avour of the option. In particular,
if the contract is a call, then the reset strike price is minfS¿ ;Kg. On the other hand,
if it is a put, then the reset strike price is maxfS¿ ;Kg. We therefore have the following
expressions for the payo®s of the reset strike call and the reset strike put, respectively:
max
½
ST ¡ minfS¿ ;Kg; 0
¾
and max
½
maxfS¿ ;Kg ¡ ST ; 0
¾
:
Now solve the following problems:
Problems
1. Prove that CR(S) ¸ C(S) and PR(S) ¸ P(S), for all values S.
2. For what value of ¿ do the inequalities above become equalities?
3. Using risk-neutral pricing, it is possible to prove that
CR(S) = SM(a1; y1; ½) ¡ Ke¡rTM(a2; y2; ½) ¡ Se¡r(T¡¿)N(¡a1)N(z2)
+ SN(¡a1)N(z1)
1
and
PR(S) = Se¡r(T¡¿)N(a1)N(¡z2) ¡ SN(a1)N(¡z1) + Ke¡rTM(¡a2;¡y2; ½)
¡ SM(¡a1;¡y1; ½);
where
a1 :=
ln(S=K) + (r + ¾2=2)¿
¾
p
¿
; a2 := a1 ¡ ¾
p
¿ ;
z1 :=
(r + ¾2=2)(T ¡ ¿ )
¾
p
T ¡ ¿
; a2 := a1 ¡ ¾
p
T ¡ ¿ ;
y1 :=
ln(S=K) + (r + ¾2=2)T
¾
p
T
; y2 := a1 ¡ ¾
p
T;
and
½ :=
q
¿=T :
In the expressions above,
N(x) := P(Z · x)
is the cumulative distribution function of a standard normal random variable Z,
while
M(x1; x2; ½) := P(Z1 · x1;Z2 · x2)
is the joint cumulative distribution function of two standard normal random vari-
ables Z1 and Z2, whose correlation is ½. The Excel workbook Reset Strike
Option.xls, which can be downloaded from UTSOnline, contains an implemen-
tation of this function.1 In particular, we have
bivar(x1; x2; ½) :=M(x1; x2; ½):
Now, using the parameter values r = 5%, ¾ = 30%, K = 100, T = 1 year and
¿ = 6 months, plot the pricing functions of the reset strike call and the reset strike
put against the stock price, for S 2 [50; 150].
4. Use the data r = 5%, ¾ = 30%, S = 100, K = 100, T = 1 year and ¿ = 6 months,
to price the reset strike call and the 本论文由英语论文网提供整理，提供论文代写，英语论文代写，代写论文，代写英语论文，代写留学生论文，代写英文论文，留学生论文代写相关核心关键词搜索。