Question

The stock price is $100. There are two European options that expire in 1 year with an exercise price of $110. The call option premium is $3 and the put option premium is $12.5. The risk free rate is 6% compounded annually. Is Put-Call Parity violated? If so, you must show the appropriate strategy to capture that profit. You must show a full arbitrage table with payoffs today and in the future. Round to 4 decimals

SHOW YOUR WORK TO GET CREDIT FOR ANY CALCULATIONS

Answer 1

Call Option:

Holder of call option will have right to buy underlying asset at the agreed price ( Strike Price). As he is receiving right, he needs to pay premium to writer of call option.
Holder of calloption will exercise the right, when expected future spot price > Strike Price. Then writer of option has obligation to sell at the strike Price.
Holder will go for call option if he is bullish.

If the Future SPot Price > Strike Price - In the Money
If the Future SPot Price = Strike Price - At the Money
If the Future SPot Price < Strike Price - Out of the Money

Put Option:

Holder of Put option will have right to sell underlying asset at the agreed price ( Strike Price). As he is receiving right, he needs to pay premium to writer of Put option.
Holder of put option will exercise the right, when expected future spot price < Strike Price. Then writer of option has obligation to buy at the strike Price.
Holder will go for put option, if he is bearish.

If the Future SPot Price < Strike Price - In the Money
If the Future SPot Price = Strike Price - At the Money
If the Future SPot Price > Strike Price - Out of the Money

Put Call Parity Theorm:

It shows the long term equilibrium relation between Value of call with certain exercise price, Value of put with same exercise price, excercise price, exercise date and stock price today.

Vc + PV of Strike Price = Vp + Stock price

Vc = Value of call
Vp = Value of Put

Particulars	Values
Vc	$   3.000
Strike Price	$ 110.00
Int rate	6.00%
Maturity Period in Year	1.0000
Vp	$ 12.500
Stock Price	$ 100.00

According to Put call parity Theorm,
Vc + PV of Strike Price = Vp + Stock price

Vc = Value of Call
Vp = Value of Put

Computation of PV of Strike Price
PV of Strike Price = Strike Price * e^-rt
e - Exponential factor
r - Int Rate per anum
t - Time in Years
= $ 110 * e^-0.06 * 1
= $ 110 * e^-0.06
= $ 110 * 0.9418
= $ 103.59

Vc + PV of Strike Price
= $ 3 + $ 103.5941
= $ 106.5941

Vp + Stock Price
= $ 12.5 + $ 100
= $ 112.5

As Vc + PV of strike Price is not equal to Vp + Stock price, Hence arbutrage gain exists.
Arbitrage strategy:

Hold a Call Option
Short sell a share
Write a Put Option

Initial Cash Inflow:
= Premium on Put Option + Stock Price - Premium on Call option
= $ 12.5 + $ 100 - $ 3
= $ 109.5

Invest Amount in bank

Maturity Value of amount invested :
= Amount invested * e ^ rt
r - Int rate per anum
t - Time in Years
= $ 109.5 * e ^ 0.06 * 1
= $ 109.5 * e ^ 0.06
= $ 109.5 * 1.0618
= $ 116.2711

Amount required to buy the stock and clear the short position
If the Stock price on Maturity Date is More Than Strike Price, Put potion will be lapsed. Being Holder of call option, stock will be purchased at strike Price.
If the Stock price on Maturity Date is less than Strike Price, Call potion will be lapsed. Holder of put option, We will exercise his right and We need to buy the stock at strike price.

i.e in any case, we would be able to purchase at strike price i.e $ 110

Arbitrage gain on Maturity date = Mayurity Value of Investemnt - Strike Price
= $ 116.2711 - $ 110
= $ 6.2711

Arbitrage gain in Today's Value:
= Arbitrage gain on maturity * e ^-rt
= $ 6.2711 * e^ - 0.06 * 1
= $ 6.2711 * e^ - 0.06
= $ 6.2711 * 0.9418
= $ 5.9059