Question

A stock priced at $61 has three-month calls and puts with an exercise price of $55 available. The calls have a premium of $4.94, and the puts cost $1.48. The risk-free rate is 3.7%. If the put options are mispriced, what is the profit per option assuming no transaction costs?

Answer 1

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	$   4.940
Strike Price	$   55.00
Int rate	3.70%
Maturity Period in Year	0.2500
Vp	$   1.480
Stock Price	$   61.00

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
= $ 55 * e^-0.037 * 0.25
= $ 55 * e^-0.0093
= $ 55 * 0.9908
= $ 54.49

Vc + PV of Strike Price
= $ 4.94 + $ 54.4936
= $ 59.4336

Vp + Stock Price
= $ 1.48 + $ 61
= $ 62.48

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

If (VC + PV of Strike Price) < ( Vp + Stock Price )

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
= $ 1.48 + $ 61 - $ 4.94
= $ 57.54

Invest Amount in bank

Maturity Value of amount invested :
= Amount invested * e ^ rt
r - Int rate per anum
t - Time in Years
= $ 57.54 * e ^ 0.037 * 0.25
= $ 57.54 * e ^ 0.0093
= $ 57.54 * 1.0093
= $ 58.0747

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 $ 55

Arbitrage gain on Maturity date = Mayurity Value of Investemnt - Strike Price
= $ 58.0747 - $ 55
= $ 3.0747

Arbitrage gain in Today's Value:
= Arbitrage gain on maturity * e ^-rt
= $ 3.0747 * e^ - 0.037 * 0.25
= $ 3.0747 * e^ - 0.0093
= $ 3.0747 * 0.9908
= $ 3.0464