Arbitrage Opportunity
Arbitrage opportunity means an opportunity of making profit without any risk. In this situation trader doesn't bear any risk other than default risk and regulatory risk.
In this transaction trader doesn't invest his own funds and whatever fund required to effect these transactions, he borrow that fund.
Concept applied :- Put-Call parity theory.
Put-Call parity theory.
As per Put-Call parity theory, if two assets are expected to have same value on a future date, then they should have same cost today.
Example
Let Mr. A has two portfolios.
(I) Consists of buying 1 BTC at $10,000 and 1 European Put option of 1 BTC at a strike price of $12,000 having 1 year of maturity.
(II) Consists of buying 1 European Call Option of 1 BTC at a strike price of $12,000 having 1 year of maturity and 1 Bond carrying Risk free return, which will be equal to the strike price i.e $12,000 at the time of expiration of the option.
Analysis
Case 1:- At the time of expiration of Option, spot price of BTC is less than strike price ( Let 1 BTC = $9,000 )
then,
Value of (I)st portfolio
= Value of Put Option + Spot price of BTC
= $( 12,000 - 9,000 ) + $9,000
= $12,000
Value of (II)nd portfolio
= Value of Call Option + Value of Bond
= $0 + $12,000
= $12,000
Case 2:- At the time of expiration of Option, spot price of BTC is equal to the strike price ( Let 1 BTC = $12,000 )
then,
Value of (I)st portfolio
= Value of Put Option + Spot price of BTC
= $0 + $12,000
= $12,000
Value of (II)nd portfolio
= Value of Call Option + Value of Bond
= $0 + $12,000
= $12,000
Case 3:- At the time of expiration of Option, spot price of BTC is higher than strike price ( Let 1 BTC = $12,500 )
then,
Value of (I)st portfolio
= Value of Put Option + Spot price of BTC
= $0 + $12,500
= $12,500
Value of (II)nd portfolio
= Value of Call Option + Value of Bond
= $(12,500-12,000) + $12,000
= $12,500
Observation
At the time of expiration of the Option, whatever be the price of BTC, value of (I)st and (II)nd portfolio are equal.
Now, as per Put-Call parity theory, if two assets are expected to have same value on future date, then they should have same cost today.
Therefore
Cost of (I)st portfolio = Cost of (II)nd portfolio.
=> Spot price + Put premium = Call premium+ Value of Bond
=> Spot price + Put premium = Call premium+ Present Value of Strike Price
**Conclusion
**
Spot price + Put premium = Call premium+ Present Value of Strike Price
When the Arbitrage opportunity in Options exist ?
By Put-Call parity theory (PCPT), we have
Spot price + Put premium = Call premium+ Present Value of Strike Price
When,
(a) LHS of PCPT is not equal to the RHS of PCPT, then there Arbitrage opportunity exists.
(b) LHS of PCPT = RHS of PCPT, then there is no Arbitrage opportunity exists.
Profit = Future value of difference between LHS and RHS.
Rule for Arbitrage
(i) When LHS < RHS, then LHS is cheaper than RHS, so Purc(I)hase Put and Sale Call.
(ii) When LHS > RHS, then RHS is cheaper than LHS, so Purchase Call and Sale Put.
(iii) Whatever fund required, borrow that fund.
Example
Given,
Spot price of BTC. $14,400
Strike price of Option. $15,000
Put premium. $450
Call premium. $270
Time of expiration. 2 month
Risk free rate of return. 6% p.a cc
Now
LHS of PCPT
= Spot price + Put premium
= $14,400 + $450
= $14,850
RHS of PCPT
= Call premium + Present Value of Strike Price
= $270 + $(15,000e^-0.01)
= $270 + $14,850
= $15,120
Since LHS of PCPT is lower than RHS of PCPT, it means LHS is cheaper than RHS, so Purchase Put and Sell Call
Profit
= Future value of difference between LHS of PCPT and RHS of PCPT
= $(15,120-14,850)e^0.01
= $273
Steps for Arbitrage
Purchase Put. -$450
Sell Call. $270
Net -$180
Purchase 1BTC. -$14,400
Amount required to borrow. -$14,580
Loan ammount $14,580
Amount required to repayment of loan at the time of expiration
= $14,580e^0.01
= $14,727
Analysis
At the time of expiration
Case I :- Spot price of BTC is less than strike price.
then, Call will not be exercised, so sell 1BTC and repay the loan.
Profit
= Sale at Spot price + Receipt of Put - Repayment of the loan
= $14,900 + $(15,000-14900) - $14,727
= $273
Case II :- Spot price of BTC = Strike Price
then, neither Put nor Call will be exercised, so sell 1BTC and repay the loan
Profit
= Sale at Spot price - Repayment of the loan
= $15,000 - $14,727
= $273
Case III :- Spot price of BTC is higher than strike price ( Let 1 BTC = $15,100 )
then, Put will be exercised, so sell BTC and pay to Call Option holder and repay the loan.
Profit
= Sale at Spot price - Payment to the Call Option holder - Repayment of the loan
= $15,100 - $(15,100-15,000) - $14,727
= $273
Observation
In all cases, profit is equal to $273, which is future value of the difference between the costs today.
Thank you.