Explain this arbitrage opportunity based on put/call parity

Discuss all general (i.e. non-personal) investing questions and issues, investing news, and theory.
Post Reply
Topic Author
sharukh
Posts: 593
Joined: Mon Jun 20, 2016 10:19 am

Explain this arbitrage opportunity based on put/call parity

Post by sharukh »

Hi,

The formula for options on futures is:
Futures price - Call price + Put price - strike = zero

I pick the same thing for different dates based on same underlying future
ES June future

Please explain if this is an arbitrage opportunity or what I am missing. It might be zero coupon interest for some difference.

Even if the transaction is made at bid, it will credit $2 (*50, multiplier)

Image
comeinvest
Posts: 1910
Joined: Mon Mar 12, 2012 6:57 pm

Re: Explain this arbitrage opportunity based on put/call parity

Post by comeinvest »

Are all options in your example settled into the same futures contract? No they are not.
Topic Author
sharukh
Posts: 593
Joined: Mon Jun 20, 2016 10:19 am

Re: Explain this arbitrage opportunity based on put/call parity

Post by sharukh »

comeinvest wrote: Sat Mar 18, 2023 2:21 pm Are all options in your example settled into the same futures contract? No they are not.
Yes, I made sure of that.

I pick the same thing for different dates based on same underlying future
ES June future

June 16 options are of American style
March 24 options are of European style

Note: there is one more European option for June 16 that settle to September future. I didn't pick it.
comeinvest
Posts: 1910
Joined: Mon Mar 12, 2012 6:57 pm

Re: Explain this arbitrage opportunity based on put/call parity

Post by comeinvest »

sharukh wrote: Sat Mar 18, 2023 6:35 pm
comeinvest wrote: Sat Mar 18, 2023 2:21 pm Are all options in your example settled into the same futures contract? No they are not.
Yes, I made sure of that.

I pick the same thing for different dates based on same underlying future
ES June future

June 16 options are of American style
March 24 options are of European style

Note: there is one more European option for June 16 that settle to September future. I didn't pick it.
I think American exercise style FOPs have a different pricing formula. Different from equity index options which are always suboptimal to exercise early, it can make sense for the owner of the long position to exercise FOPs early. Do some reading.

I think your formula is also somehow missing the risk-free rate.
Topic Author
sharukh
Posts: 593
Joined: Mon Jun 20, 2016 10:19 am

Re: Explain this arbitrage opportunity based on put/call parity

Post by sharukh »

comeinvest wrote: Sat Mar 18, 2023 11:21 pm
sharukh wrote: Sat Mar 18, 2023 6:35 pm
comeinvest wrote: Sat Mar 18, 2023 2:21 pm Are all options in your example settled into the same futures contract? No they are not.
Yes, I made sure of that.

I pick the same thing for different dates based on same underlying future
ES June future

June 16 options are of American style
March 24 options are of European style

Note: there is one more European option for June 16 that settle to September future. I didn't pick it.
I think American exercise style FOPs have a different pricing formula. Different from equity index options which are always suboptimal to exercise early, it can make sense for the owner of the long position to exercise FOPs early. Do some reading.

I think your formula is also somehow missing the risk-free rate.
Thank you. Yes the difference is due to American style.

American call on a futures contract should carry higher price than a European call on a futures contract.

For calls, early exercise may be worthwhile even if the underlying does not make any cash payments. For example, if the holder of a deep-in-the-money American call option exercises it and establishes a futures position, he earns interest on the futures margin account.

https://analystnotes.com/cfa-study-note ... racts.html


https://www.cmegroup.com/education/cour ... arity.html
For options on futures, the put call parity formula don't need explicit interest rates as it is already embedded in futures price
Topic Author
sharukh
Posts: 593
Joined: Mon Jun 20, 2016 10:19 am

Re: Explain this arbitrage opportunity based on put/call parity

Post by sharukh »

sharukh wrote: Sun Mar 19, 2023 7:07 am
comeinvest wrote: Sat Mar 18, 2023 11:21 pm
sharukh wrote: Sat Mar 18, 2023 6:35 pm
comeinvest wrote: Sat Mar 18, 2023 2:21 pm Are all options in your example settled into the same futures contract? No they are not.
Yes, I made sure of that.

I pick the same thing for different dates based on same underlying future
ES June future

June 16 options are of American style
March 24 options are of European style

Note: there is one more European option for June 16 that settle to September future. I didn't pick it.
I think American exercise style FOPs have a different pricing formula. Different from equity index options which are always suboptimal to exercise early, it can make sense for the owner of the long position to exercise FOPs early. Do some reading.

I think your formula is also somehow missing the risk-free rate.
Thank you. Yes the difference is due to American style.

American call on a futures contract should carry higher price than a European call on a futures contract.

For calls, early exercise may be worthwhile even if the underlying does not make any cash payments. For example, if the holder of a deep-in-the-money American call option exercises it and establishes a futures position, he earns interest on the futures margin account.

https://analystnotes.com/cfa-study-note ... racts.html


https://www.cmegroup.com/education/cour ... arity.html
For options on futures, the put call parity formula don't need explicit interest rates as it is already embedded in futures price
On a second thought. I am buying American option and selling European option. So the total buy price should be positive.

So the posted transaction looks even more profitable from arbitration pov.

Will need to read up more.
Topic Author
sharukh
Posts: 593
Joined: Mon Jun 20, 2016 10:19 am

Re: Explain this arbitrage opportunity based on put/call parity

Post by sharukh »

sharukh wrote: Sun Mar 19, 2023 9:32 am
sharukh wrote: Sun Mar 19, 2023 7:07 am
comeinvest wrote: Sat Mar 18, 2023 11:21 pm
sharukh wrote: Sat Mar 18, 2023 6:35 pm
comeinvest wrote: Sat Mar 18, 2023 2:21 pm Are all options in your example settled into the same futures contract? No they are not.
Yes, I made sure of that.

I pick the same thing for different dates based on same underlying future
ES June future

June 16 options are of American style
March 24 options are of European style

Note: there is one more European option for June 16 that settle to September future. I didn't pick it.
I think American exercise style FOPs have a different pricing formula. Different from equity index options which are always suboptimal to exercise early, it can make sense for the owner of the long position to exercise FOPs early. Do some reading.

I think your formula is also somehow missing the risk-free rate.
Thank you. Yes the difference is due to American style.

American call on a futures contract should carry higher price than a European call on a futures contract.

For calls, early exercise may be worthwhile even if the underlying does not make any cash payments. For example, if the holder of a deep-in-the-money American call option exercises it and establishes a futures position, he earns interest on the futures margin account.

https://analystnotes.com/cfa-study-note ... racts.html


https://www.cmegroup.com/education/cour ... arity.html
For options on futures, the put call parity formula don't need explicit interest rates as it is already embedded in futures price
On a second thought. I am buying American option and selling European option. So the total buy price should be positive.

So the posted transaction looks even more profitable from arbitration pov.

Will need to read up more.


Concluding this thread.

I was able to get the correct answer from book:
Options, Futures, and Other Derivatives 10th Edition by John Hull

the put-call parity formula is different for options on futures compared to options on equity.

The put-call parity relationship for options on equity is:
Call Option Price - Put Option Price = Stock Price - Strike Price / (1 + r)^t
where r is the risk-free interest rate, t is the time to expiration, and (1 + r)^t is the discount factor.

For options on futures, the put-call parity relationship is:
Call Option Price - Put Option Price = (Futures Price - Strike Price) x e^(-r x t)
where r is the risk-free interest rate, t is the time to expiration, and e^(-r x t) is the discount factor.

The difference in the put-call parity formula arises due to the fact that options on futures are based on a different underlying asset (futures contracts) compared to options on equity (stocks). Options on futures are settled by delivery of the underlying futures contract, while options on equity are settled by delivery of the underlying stock. The pricing of options on futures takes into account the cost of carry, which is the cost of holding the underlying futures contract until expiration, while options on equity do not consider this factor.


Its unfortunate that CME website have a wrong formulae for the put call parity
https://www.cmegroup.com/education/cour ... arity.html
Post Reply