Transcript Put-Call Option Interest Rate Parity

Put-Call Option Interest Rate Parity
Objective
Determine the international parity relationship between Call, Put,
and Forward prices
Outline
• Two arbitrage portfolios
• Derivation of parity conditions
• Exemplification
Two arbitrage portfolios
Consider:
C: the premium of a call option on the Sfr
P: the premium of a put option on the Sfr
X: the strike price of call and put options
niSfr : Sfr nominal interest rate
ni$ : $ nominal interest rate
s = $/Sfr : spot exchange rate
f = forward rate
Two arbitrage portfolios
e = $/Sfr
Portfolio I
today
Lend Sfr1/(1+niSFr)
Buy a put option on Sfr,
struck at X
Borrow $X/(1+ni$)
Portfolio II
Two arbitrage portfolios
s = $/Sfr
today
Portfolio I
Portfolio II
Lend SFR 1/(1+niSFR)
Buy a call option on
SFR1, struck at X
Buy a put option on SFR,
struck at X
Borrow $X/(1+ni$)
Two arbitrage portfolios
s = $/Sfr
today
Portfolio I
Portfolio II
Lend SFR 1/(1+niSFR)
Buy a call option on
SFR1, struck at X
Buy a put option on SFR,
struck at X
Borrow $X/(1+ni$)
scenario 1:
s1 > X
$payoff :
$payoff :
Two arbitrage portfolios
s = $/Sfr
today
Portfolio I
Portfolio II
Lend SFR 1/(1+niSFR)
Buy a call option on
SFR1, struck at $X
Buy a put option on SFR,
struck at $X
Borrow $X/(1+ni$)
scenario 1:
s>X
$payoff :
s–X
$payoff :
s-X
Two arbitrage portfolios
s = $/Sfr
today
Portfolio I
Portfolio II
Lend SFR 1/(1+niSFR)
Buy a call option on
SFR1, struck at X
Buy a put option on SFR,
struck at X
Borrow $X/(1+ni$)
scenario 1:
s>X
$payoff :
s–X
$payoff :
s-X
scenario 2:
s<X
$payoff:
$payoff :
Two arbitrage portfolios
s = $/Sfr
today
Portfolio I
Portfolio II
Lend SFR 1/(1+niSFR)
Buy a call option on
SFR1, struck at X
Buy a put option on SFR,
struck at X
Borrow $X/(1+ni$)
scenario 1:
s>X
$payoff :
s–X
$payoff :
s-X
scenario 2:
s<X
$payoff:
(s - X) - (s - X) = 0
$payoff :
0
Parity conditions derived
It follows that C = s0/(1+niSfr) - X/(1+ni$) +P
According to interest rate parity we know that s0/(1+niSfr) = f1/(1+ni$)
Hence, if we are in Canada,
C = (f1 - X)/(1+ni$) + P
In general,
C = (f1 - X)/(1+nih) + P
Exemplification
e0 = C$0.7143/Sfr
ni$ = 3.5%
niSfr = 4.4%
One-year forward = C$0.70814/Sfr
A call on the Sfr struck at C$0.701/Sfr, expiring in one year sells at
C$0.035/Sfr
A put on the Sfr struck at C$0.701/Sfr, expiring in one year sells at
C$0.023/Sfr
Note
C$0.70814/C$0.7143 = (1.035)/(1.044)
IRP holds
C$0.035 > C$(0.70814-0.701)/(1.035) + C$0.023
Arbitrage opportunity
Another two arbitrage portfolios
today
Portfolio I
Portfolio II
a. Lend SFR 84,000/(1.044)
b. Buy a put option on SFR84,000, struck at
C$0.701
c. Borrow C$84000(0.701)/(1.035)
Sell a call option on
SFR84,000, struck at
C$0.701/SFR
Cash flow:
Cash flow:
- C$84,000(0.7143)/(1.044)
- C$1,932
+ C$84,000(0.701)/(1.035)
= -C$2,511.25
+C$2,940
scenario 1:
e1=C$0.72/Sfr
payoff ($):
C$84,000(0.72)
-C$84,000(0.701)
scenario 2:
e1=C$0.7/SFR
payoff ($):
+ C$84,000(0.7)
+ C$84,000(0.701-0.7)
- C$84,000(0.701)
=0
payoff (in $):
-C$84,000(0.72-0.701)
0
Another two arbitrage portfolios
today
Portfolio I
Portfolio II
a. Lend SFR 84,000/(1.044)
b. Buy a put option on SFR84,000, struck at
C$0.701
c. Borrow C$84000(0.701)/(1.035)
Sell a call option on
SFR84,000, struck at
C$0.701/SFR
Cash flow:
Cash flow:
- C$84,000(0.7143)/(1.044)
- C$1,932
+ C$84,000(0.701)/(1.035)
= -C$2,511.25
+C$2,940
scenario 1:
e1=C$0.72/SFR
payoff ($):
C$84,000(0.72)
-C$84,000(0.701)
scenario 2:
e1=C$0.7/SFR
payoff ($):
+ C$84,000(0.7)
+ C$84,000(0.701-0.7)
- C$84,000(0.701)
=0
payoff (in $):
-C$84,000(0.72-0.701)
0
Another two arbitrage portfolios
today
Portfolio I
Portfolio II
a. Lend SFR 84,000/(1.044)
b. Buy a put option on SFR84,000, struck at
C$0.701
c. Borrow C$84000(0.701)/(1.035)
Sell a call option on
SFR84,000, struck at
C$0.701/SFR
Cash flow:
Cash flow:
- C$84,000(0.7143)/(1.044)
- C$1,932
+ C$84,000(0.701)/(1.035)
= -C$2,511.25
+C$2,940
scenario 1:
e1=C$0.72/Sfr
payoff ($):
C$84,000(0.72)
-C$84,000(0.701)
scenario 2:
e1=C$0.7/SFR
payoff ($):
+ C$84,000(0.7)
+ C$84,000(0.701-0.7)
- C$84,000(0.701)
=0
payoff (in $):
-C$84,000(0.72-0.701)
0
Another two arbitrage portfolios
today
Portfolio I
Portfolio II
a. Lend SFR 84,000/(1.044)
b. Buy a put option on SFR84,000, struck at
C$0.701
c. Borrow C$84000(0.701)/(1.035)
Sell a call option on
SFR84,000, struck at
C$0.701/SFR
Cash flow:
Cash flow:
- C$84,000(0.7143)/(1.044)
- C$1,932
+ C$84,000(0.701)/(1.035)
= -C$2,511.25
+C$2,940
scenario 1:
e1=C$0.72/Sfr
payoff ($):
C$84,000(0.72)
-C$84,000(0.701)
scenario 2:
e1=C$0.7/SFR
payoff ($):
+ C$84,000(0.7)
+ C$84,000(0.701-0.7)
- C$84,000(0.701)
=0
payoff (in $):
-C$84,000(0.72-0.701)
0
Analysis
At expiration, the combined payoff from the two portfolios is always
zero.
However, buying the first portfolio and shorting the second one has
produced an arbitrage profit of (C$2,940- C$2,511.25)=C$428.75
up-front.