Question

ABC Inc stock sells for $30 per share. Put and call options of ABC with strike price

of $34 and expiration of 6 months are available. Both these options are priced at $7.

The annual risk free rate is 4%. If the put option is priced correctly, is the call option

priced right? If not, what should its price be and how can you take advantage of this

arbitrage opportunity?

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	$   7.000
Strike Price	$   34.00
Int rate	4.00%
Maturity Period in Year	0.5000
Vp	$   7.000
Stock Price	$   30.00

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
= $ 34 * e^-0.04 * 0.5
= $ 34 * e^-0.02
= $ 34 * 0.9802
= $ 33.33

Vc + PV of Strike Price
= $ 7 + $ 33.3268
= $ 40.3268

Vp + Stock Price
= $ 7 + $ 30
= $ 37

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

Arbitrage Strategy
If (VC + PV of Strike Price) > ( Vp + Stock Price )

Hold a Put Option
Buy a stock
Write a call Option

Initial Outflow:
= Premium on Put Option + Stock Price - Premium on Call option
= $ 7 + $ 30 - $ 7
= $ 30

Borrow the amount required from Bank

Maturity Value of Loan :
= Amount borrowed * e ^ rt
r - Int rate per anum
t - Time in Years
= $ 30 * e ^ 0.04 * 0.5
= $ 30 * e ^ 0.02
= $ 30 * 1.0202
= $ 30.606

Sale Proceeds on Maturity:
If the Stock price on Maturity Date is More Than Strike Price, Put potion will be lapsed. Holder of call option will exercise his right. We need to sell at strike price.
If the Stock price on Maturity Date is less than Strike Price, Call potion will be lapsed. Being Holder of put option, We will exercise his right and sell the stock at strike price.

i.e in any case, we would be able to sell at strike price i.e $ 34

Arbitrage gain on Maturity date = Sale Proceeds - Maturity value of Loan
= $ 34 - $ 30.61
= $ 3.39

Arbitrage gain in Today's Value:
= Arbitrage gain on maturity * e ^-rt
= $ 3.39 * e^ - 0.04 * 0.5
= $ 3.39 * e^ - 0.02
= $ 3.39 * 0.9802
= $ 3.33

Correct Call Price:

Particulars	Values
Strike Price	$   34.00
Int rate	4.00%
Maturity Period in Year	0.5000
Vp	$      7.00
Stock Price	$   30.00

Vc = Value of Call
Vp = Value of Put

Vc = Vp + Stock Price - PV of Strike Price

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
= $ 34 * e^-0.04 * 0.5
= $ 34 * e^-0.02
= $ 34 * 0.9802
= $ 33.33

Vc = Vp + Stock Price - PV of Strike Price
= $ 7 + $ 30 - $ 33.33
= $ 3.67
Correct price of call option is$ 3.67