Question

 

A trader observes that a) A put on stock X which does not pay dividends with strike price
$20 and exercisable in one year trades at $3 and b) A call on the same stock with the same strike
price and also exercisable in one year trades at $25. If the stock trades today at $30/share and the
risk-free rate is 10%, design a profitable arbitrage strategy for the trader.

Answer 1

Particulars	Values
Vc	$ 25.000
Strike Price	$   20.00
Int rate	10.00%
Maturity Period in Years	1.0000
Vp	$   3.000
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
= $ 20 * e^-0.1 * 1
= $ 20 * e^-0.1
= $ 20 * 0.9048
= $ 18.1

Vc + PV of Strike Price
= $ 25 + $ 18.0967
= $ 43.0967

Vp + Stock Price
= $ 3 + $ 30
= $ 33

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
= $ 3 + $ 30 - $ 25
= $ 8

Borrow the amount required from Bank

Maturity Value of Loan :
= Amount borrowed * e ^ rt
r - Int rate per anum
t - Time in Years
= $ 8 * e ^ 0.1 * 1
= $ 8 * e ^ 0.1
= $ 8 * 1.1052
= $ 8.8414

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 $ 20

Arbitrage gain on Maturity date = Sale Proceeds - Maturity value of Loan
= $ 20 - $ 8.84
= $ 11.16

Arbitrage gain in Today's Value:
= Arbitrage gain on maturity * e ^-rt
= $ 11.16 * e^ - 0.1 * 1
= $ 11.16 * e^ - 0.1
= $ 11.16 * 0.9048
= $ 10.1