Have you ever wondered how the put prices relate to the call prices? If you happen to know the call price on an asset, would that help you to get some idea of the price of a put on the same asset?
Do put prices have anything at all to do with call prices? Of course, they do. The put and the call prices are related by a condition called the put-call parity. We shall see how.
Put-call parity
To get an intuitive understanding about the put-call parity, we could think of it in the following way. I buy the asset on spot, paying S. I buy a put at X, paying P, so my downside below X is taken care of (if S < X, I will exercise the put). I sell a call at X, earning C, so if S > X, the call holder will exercise on me, so my upside beyond X is gone. This gives me X on T with certainty. This means that the portfolio of S+P-C is nothing but a zero-coupon bond which pays X on date T.
What happens if the above equation does not hold good ? It gives rise to arbitrage opportunities. The put-call parity basically explains the relationship between put, call, stock and bond prices.
It is expressed as:
S + P – C = X / (1+r) T
Where:
S: Current index level
X: Exercise price of option
T: Time to expiration
C: Price of call option
P: Price of put option
R: risk-free rate of interest
The above expression shows that the value of a European call with a certain exercise price and exercise date can be deduced from the value of a European put with the same exercise price and date and vice versa. It basically means that the payoff from holding a call plus an amount of cash equal to X / (1+r) T is the same as that of holding a put option plus the index.
Case 1
Suppose Nifty stands at 1265, the risk-free rate of interest is 12% per annum, the price of a three month Nifty 1260 call is Rs.96.50 and the price of a three month Nifty 1260 put is Rs.60. In this case we can see that
S + P – C (not equal to) X / (1+r) T
1325 > 1321.30
What does this mean? If we think of index plus put as portfolio A and the call plus cash as portfolio B, clearly portfolio A is overpriced relative to portfolio B. What would be the arbitrage strategy in this case? Sell the securities in portfolio A and buy those in portfolio B.
This involves shorting the index and a put on the index and buying a call. How would one short the index? One way to do it would be to actually sell off all 50 Nifty stocks in the proportions in which they exist in the index.
Another easier way to do this would be to sell units of Index funds instead of the actual index stocks. This would achieve a similar outcome. This entire set of transactions generates an up-front cash-flow of (1265 + 60 - 96.50) = Rs.1228.50. When invested at the riskfree rate of 12%, this amount grows to Rs.1265.35.
At expiration, if the index is higher than 1260, you will exercise the call. If the index is lower than 1260, the buyer of the put will exercise on you. In either case, the investor ends up buying the index at Rs.1260. Hence the net profit on the entire transaction is Rs.5.35 (i.e. 1265.35-1260).
How do we actually do this?
1. Sell off all 50 index shares on the cash market in the proportion in which they exist in the index. This can be done using a single keystroke using the NEAT software.
2. Sell a three month Nifty 1260 put.
3. Buy a three month Nifty 1260 call.
4. You will receive the money for the stocks and the put sold and have to make delivery of the 50
shares.
5. Invest this money at the riskless interest rate. In three months Rs.1228.50 will grow to Rs.1265.35.
6. On the exercise date at 3:15 PM, if the Nifty is above 1260, exercise the call. If the Nifty is below 1260, the put will be exercised on you.
7. Either way, you end up buying the index at Rs.1260.
8. The riskless profit on the transaction works out to be Rs.5.35.
Case 2:
Suppose Nifty stands at 1265, the risk-free rate of interest is 12% per annum, the price of a three month Nifty 1260 call is Rs.96 and the price of a three month Nifty 1260 put is 51.50. In this case, we can see that
S + P – C (not equal to) X / (1+r) T
1316.50 < 1320.80
What does this mean? If we think of index plus put as portfolio A and the call plus cash as portfolio B, clearly portfolio B is overpriced relative to portfolio A. What would be the arbitrage strategy in this case? Buy the securities in portfolio A and sell those in portfolio B.
This involves buying the index and a put on the index and selling a call. How would one buy the index? One way to do it would be to actually buy all 50 Nifty stocks in the proportions in which they exist in the index. An easier way to do this would be to buy units of Index funds instead of the actual index stocks.
This would achieve a similar outcome. This entire set of transactions involves an initial investment of Rs.1220.50(i.e. 1265 - 51.50 + 96) When financed at the riskfree rate of 12%, the repayment required at the end of three months is Rs.1257.
At expiration if the index is lower than 1260, you will exercise the put. If the index is higher than 1260, the buyer of the call will exercise on you. In either case, the investor ends up buying the index at Rs.1260. Hence the net profit on the entire transaction is Rs.3 (1260 - 1257).
How do we actually do this?
1. Buy all 50 index shares on the cash market in the proportion in which they exist in the index. This can be done using a single keystroke using the NEAT software.
2. Buy a three month Nifty 1260 put.
3. Sell a three month Nifty 1260 call.
4. You will have to pay for the shares and the put, and will receive the call premium. The entire set of transactions will require an
initial outflow of Rs.1221.20.
5. Finance this money at the riskless interest rate. The repayment at the end of three months works out to Rs.1257.
6. On the exercise date at 3:15 PM, if the Nifty is below 1260, exercise the put. If the Nifty is above 1260, the call will be exercised
on you.
7. Either way, you end up selling the index at Rs.1260.
8. The riskless profit on the transaction works out to be Rs.3.
Do put prices have anything at all to do with call prices? Of course, they do. The put and the call prices are related by a condition called the put-call parity. We shall see how.
Put-call parity
To get an intuitive understanding about the put-call parity, we could think of it in the following way. I buy the asset on spot, paying S. I buy a put at X, paying P, so my downside below X is taken care of (if S < X, I will exercise the put). I sell a call at X, earning C, so if S > X, the call holder will exercise on me, so my upside beyond X is gone. This gives me X on T with certainty. This means that the portfolio of S+P-C is nothing but a zero-coupon bond which pays X on date T.
What happens if the above equation does not hold good ? It gives rise to arbitrage opportunities. The put-call parity basically explains the relationship between put, call, stock and bond prices.
It is expressed as:
S + P – C = X / (1+r) T
Where:
S: Current index level
X: Exercise price of option
T: Time to expiration
C: Price of call option
P: Price of put option
R: risk-free rate of interest
The above expression shows that the value of a European call with a certain exercise price and exercise date can be deduced from the value of a European put with the same exercise price and date and vice versa. It basically means that the payoff from holding a call plus an amount of cash equal to X / (1+r) T is the same as that of holding a put option plus the index.
Case 1
Suppose Nifty stands at 1265, the risk-free rate of interest is 12% per annum, the price of a three month Nifty 1260 call is Rs.96.50 and the price of a three month Nifty 1260 put is Rs.60. In this case we can see that
S + P – C (not equal to) X / (1+r) T
1325 > 1321.30
What does this mean? If we think of index plus put as portfolio A and the call plus cash as portfolio B, clearly portfolio A is overpriced relative to portfolio B. What would be the arbitrage strategy in this case? Sell the securities in portfolio A and buy those in portfolio B.
This involves shorting the index and a put on the index and buying a call. How would one short the index? One way to do it would be to actually sell off all 50 Nifty stocks in the proportions in which they exist in the index.
Another easier way to do this would be to sell units of Index funds instead of the actual index stocks. This would achieve a similar outcome. This entire set of transactions generates an up-front cash-flow of (1265 + 60 - 96.50) = Rs.1228.50. When invested at the riskfree rate of 12%, this amount grows to Rs.1265.35.
At expiration, if the index is higher than 1260, you will exercise the call. If the index is lower than 1260, the buyer of the put will exercise on you. In either case, the investor ends up buying the index at Rs.1260. Hence the net profit on the entire transaction is Rs.5.35 (i.e. 1265.35-1260).
How do we actually do this?
1. Sell off all 50 index shares on the cash market in the proportion in which they exist in the index. This can be done using a single keystroke using the NEAT software.
2. Sell a three month Nifty 1260 put.
3. Buy a three month Nifty 1260 call.
4. You will receive the money for the stocks and the put sold and have to make delivery of the 50
shares.
5. Invest this money at the riskless interest rate. In three months Rs.1228.50 will grow to Rs.1265.35.
6. On the exercise date at 3:15 PM, if the Nifty is above 1260, exercise the call. If the Nifty is below 1260, the put will be exercised on you.
7. Either way, you end up buying the index at Rs.1260.
8. The riskless profit on the transaction works out to be Rs.5.35.
Case 2:
Suppose Nifty stands at 1265, the risk-free rate of interest is 12% per annum, the price of a three month Nifty 1260 call is Rs.96 and the price of a three month Nifty 1260 put is 51.50. In this case, we can see that
S + P – C (not equal to) X / (1+r) T
1316.50 < 1320.80
What does this mean? If we think of index plus put as portfolio A and the call plus cash as portfolio B, clearly portfolio B is overpriced relative to portfolio A. What would be the arbitrage strategy in this case? Buy the securities in portfolio A and sell those in portfolio B.
This involves buying the index and a put on the index and selling a call. How would one buy the index? One way to do it would be to actually buy all 50 Nifty stocks in the proportions in which they exist in the index. An easier way to do this would be to buy units of Index funds instead of the actual index stocks.
This would achieve a similar outcome. This entire set of transactions involves an initial investment of Rs.1220.50(i.e. 1265 - 51.50 + 96) When financed at the riskfree rate of 12%, the repayment required at the end of three months is Rs.1257.
At expiration if the index is lower than 1260, you will exercise the put. If the index is higher than 1260, the buyer of the call will exercise on you. In either case, the investor ends up buying the index at Rs.1260. Hence the net profit on the entire transaction is Rs.3 (1260 - 1257).
How do we actually do this?
1. Buy all 50 index shares on the cash market in the proportion in which they exist in the index. This can be done using a single keystroke using the NEAT software.
2. Buy a three month Nifty 1260 put.
3. Sell a three month Nifty 1260 call.
4. You will have to pay for the shares and the put, and will receive the call premium. The entire set of transactions will require an
initial outflow of Rs.1221.20.
5. Finance this money at the riskless interest rate. The repayment at the end of three months works out to Rs.1257.
6. On the exercise date at 3:15 PM, if the Nifty is below 1260, exercise the put. If the Nifty is above 1260, the call will be exercised
on you.
7. Either way, you end up selling the index at Rs.1260.
8. The riskless profit on the transaction works out to be Rs.3.