Currency derivatives

[Winod Dhamnekar]

On December 9 of a particular year, a January Swiss Franc call option with an exercise price of 46 had a price of 1.63. The January 46 put was at 0.14.The spot rate was 47.28.All prices are in cents per Swiss Franc. The option expired on January 13.The U.S. risk free rate was 7.1% while the Swiss risk-free rate was 3.6%. Find out the following:-
a. Determine the intrinsic value of the call
b. Determine the lower bound of the call
c. Determine the time value of the call
d. Determine the intrinsic value of the put
e. Determine the lower bound of the put
f. Determine the time value of the put
g. Determine whether the put-call parity holds.

My answer:

To solve the problem, we will use the following formulas and definitions:

1. Intrinsic Value of a Call Option:
[math]\text{Intrinsic Value (Call)} = \max(0, S - K)[/math] where [imath]S[/imath] is the spot rate and [imath]K[/imath] is the exercise price.

2. Intrinsic Value of a Put Option:
[math]\text{Intrinsic Value (Put)} = \max(0, K - S)[/math]
3. Lower Bound of a Call Option:
[math]\text{Lower Bound (Call)} = S - K e^{-r_d T}[/math] where [imath]r_d[/imath] is the domestic risk-free rate, and [imath]T[/imath] is the time to expiration in years.

4. Lower Bound of a Put Option:
[math]\text{Lower Bound (Put)} = K e^{-r_d T} - S[/math]
5. Time Value of an Option:
[math]\text{Time Value} = \text{Option Price} - \text{Intrinsic Value}[/math]
6. Put-Call Parity:
[math]C + K e^{-r_f T} = P + S[/math] where [imath]C[/imath] is the call price, [imath]P[/imath] is the put price, [imath]r_f[/imath] is the foreign risk-free rate, and [imath]T[/imath] is the time to expiration in years.

Given Data:
- Call option price [imath]C = 1.63[/imath] cents
- Put option price [imath]P = 0.14[/imath] cents
- Spot rate [imath]S = 47.28[/imath] cents
- Exercise price [imath]K = 46[/imath] cents
- U.S. risk-free rate [imath]r_d = 7.1\% = 0.071[/imath]
- Swiss risk-free rate [imath]r_f = 3.6\% = 0.036[/imath]
- Time to expiration [imath]T = \frac{35}{365}[/imath] years (from December 9 to January 13)

Calculations:

a. Intrinsic Value of the Call
[math]\text{Intrinsic Value (Call)} = \max(0, 47.28 - 46) = 1.28 \text{ cents}[/math]
b. Lower Bound of the Call
[math]T = \frac{35}{365} \approx 0.09589 \text{ years}[/math]
[math]\text{Lower Bound (Call)} = 47.28 - 46 e^{-0.071 \times 0.09589} \approx 47.28 - 46 e^{-0.0068} \approx 47.28 - 45.73 \approx 1.55 \text{ cents}[/math]
c. Time Value of the Call
[math]\text{Time Value (Call)} = 1.63 - 1.28 = 0.35 \text{ cents}[/math]
d. Intrinsic Value of the Put
[math]\text{Intrinsic Value (Put)} = \max(0, 46 - 47.28) = 0 \text{ cents}[/math]
e. Lower Bound of the Put
[math]\text{Lower Bound (Put)} = 46 e^{-0.071 \times 0.09589} - 47.28 \approx 45.73 - 47.28 \approx -1.55 \text{ cents (not valid, so it is 0)}[/math]
f. Time Value of the Put
[math]\text{Time Value (Put)} = 0.14 - 0 = 0.14 \text{ cents}[/math]
g. Put-Call Parity
[math]C + K e^{-r_f T} = P + S[/math]Calculating [imath]K e^{-r_f T}[/imath]:
[math]K e^{-r_f T} = 46 e^{-0.036 \times 0.09589} \approx 46 e^{-0.00345} \approx 46 \times 0.99655 \approx 45.83[/math]Now substituting into the parity equation:
[math]1.63 + 45.83 \approx 0.14 + 47.28[/math][math]47.46 \approx 47.42 \text{ (approximately holds)}[/math]
Summary of Results:
- Intrinsic Value of the Call: 1.28 cents
- Lower Bound of the Call: 1.55 cents
- Time Value of the Call: 0.35 cents
- Intrinsic Value of the Put: 0 cents
- Lower Bound of the Put: 0 cents
- Time Value of the Put: 0.14 cents
- Put-Call Parity: Approximately holds

This analysis shows that the options are priced in a way that is consistent with the theoretical framework of options pricing.