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lynn
Topic Author
Posts: 35
Joined: September 13th, 2002, 8:02 am

### Commodity Option on Futures - deriving the formula

(i) Basic Black-Black Scholes - assumption no dividend paidC = SN(d1) -Xe^(-rt)N(d2)(ii) assumption with dividend paying on stockC = Se^(-qt)N(d1) -Xe^(-rt)N(d2), where q is the dividend yield(iii) for commodity option on futures, we note the text book is substituting the above formula where S=F(future prices) & q=r, therefore the formula readsC = e^(-rt)[FN(d1) -XN(d2)]First question, why is q equated to r?Second question, we have also seen in hull's latest edition that futures style option the formula can be derived as C = [FN(d1) -XN(d2)]; can we understand what happen to e^(-rt)?Thanks for the help!

spursfan
Posts: 893
Joined: October 7th, 2001, 3:43 pm

### Commodity Option on Futures - deriving the formula

the correct link is that F = S *exp((r-q)*T) and so in equation (iii) the first term F * exp(-r*T) and S * exp(-q*T) are equivalent

daveangel
Posts: 17031
Joined: October 20th, 2003, 4:05 pm

### Commodity Option on Futures - deriving the formula

the term r-q is the cost of carry. thisis zero on futures on contracts. second , if the option is settled futures style (daily MTM rather than an upfront premium) then there is no carry benefit on the premium either.
knowledge comes, wisdom lingers

choewl
Posts: 8
Joined: February 15th, 2007, 7:40 am

### Commodity Option on Futures - deriving the formula

As you mentioned in the 2nd point, if it is daily mark-to-market ("mtm") the cost of carry is practically zero. Would like to confirm with you this is refering to mtm of the "option on futures" or mtm of the "underlying futures" on daily basis?

choewl
Posts: 8
Joined: February 15th, 2007, 7:40 am

### Commodity Option on Futures - deriving the formula

As you mentioned in the 2nd point, if it is daily mark-to-market ("mtm") the cost of carry is practically zero. Would like to confirm with you this is refering to mtm of the "option on futures" or mtm of the "underlying futures" on daily basis?

choewl
Posts: 8
Joined: February 15th, 2007, 7:40 am

### Commodity Option on Futures - deriving the formula

As you mentioned in the 2nd point, if it is daily mark-to-market ("mtm") the cost of carry is practically zero. Would like to confirm with you this is refering to mtm of the "option on futures" or mtm of the "underlying futures" on daily basis?

daveangel
Posts: 17031
Joined: October 20th, 2003, 4:05 pm

### Commodity Option on Futures - deriving the formula

yes - not all options on futures are settled daily. for some contracts the premium is exchanged upfront in which case there is a cost of carry. to clarify - all exchange traded (and some OTC) options have daily variation margin applied to them - some options contracts require that the buyer pay the premium upfront and others do not.
Last edited by daveangel on September 22nd, 2008, 10:00 pm, edited 1 time in total.
knowledge comes, wisdom lingers

cosmologist
Posts: 640
Joined: January 24th, 2005, 8:08 am

### Commodity Option on Futures - deriving the formula

commodity options done OTC are required to pay premiums and it is marked-to-marketed daily. this is true for NYMEX options. We pay premiums when we buy and get get MTM on a dily basis. The price used is Black model. The basic idea is probably from discounting. You don't wait for the pay-off at the end date. No matter whether the option is American or European, every bodyy marks-you-to-the market ,at the end of the day. SSo the basic idea is every option is a sum of large number of options with 1 day to expiry. The professor Spursfan will be in a better position to throw some light. Dave can tell us what he notices in his part of the world.Is the new edition out Mr. Spusfan? Is it out with the Stochastic vol treatment

spursfan
Posts: 893
Joined: October 7th, 2001, 3:43 pm

### Commodity Option on Futures - deriving the formula

thanks for your interest in the new edition - still writing new code to cover lots of Levy models including Heston

cosmologist
Posts: 640
Joined: January 24th, 2005, 8:08 am

### Commodity Option on Futures - deriving the formula

So do we agree that exp(-rt) should not be there.I am still waiting for the new edition.You probably can answer the Edgeworth question the best. Do you think,Edgeworth for smile is of any use now-a-days?regards

spursfan
Posts: 893
Joined: October 7th, 2001, 3:43 pm

### Commodity Option on Futures - deriving the formula

I code up stuff that interests me - and Gram-Charlier and Edgeworth expansions provide an easy way to see the impact of skewness and kurtosis on option prices - and compare them with skewness and kurtosis from some of the Levy processes -

gjlipman
Posts: 1102
Joined: May 20th, 2002, 9:13 pm

### Commodity Option on Futures - deriving the formula

I only have v6 of Hull, but there, in section 14.8 Black's Model For Valuing Futures Options, he says c=e^{-rT}[FN(d1) - KN(d2)] - I can't believe it would have changed between editions, so can only imagine it is a typo?Certainly, in the usual case where the premium was paid up front, you'd expect there to be an e^{-rT} term.

choewl
Posts: 8
Joined: February 15th, 2007, 7:40 am

### Commodity Option on Futures - deriving the formula

Greeting,I need to really get this to my headOption on Futures valuation can be with a discount factor and without. When do you use with a discount factor:-- for contracts where the premium is exchanged upfront - in which case there is a cost of carry. When do you use without a discount factor:-- is for option on futures; where there are daily margining on the futures.My question is will the option then also be subjected to daily margining. Also, is there premium paid here?Do shade some light in dark tunnel

gjlipman
Posts: 1102
Joined: May 20th, 2002, 9:13 pm

### Commodity Option on Futures - deriving the formula

Margining is a real nightmare, as different people mean different things by it. All exchange traded commodities options require margins, in the sense that you put up collateral, but that collateral is earning you interest, so doesn't technically need to be taken into account in valuing the option (at least it is second order).So, if the only margining required for an option is putting up collateral, for which you receive interest, or there is no margining at all (eg in the case of an OTC option on a future), you do use a discount factor.There are a few examples in commodities options where you have to make margin payments that don't go into your collateral account, ie you don't get the value from them. These are (correct me if I'm wrong) typically the options where the premium isn't paid up front. IPE have some options on futures that use this approach (though these might all be American anyway, in which case you shouldn't be using Blacks formula). In this case, I don't think you would use a discount factor.My logic for this is as follows: imagine you've got a long dated european in the money option on an underlying with zero volatility, premium deferred. So, if we don't incorporate discount factors, it's value is F-K the whole time, we never make variation margin payments, and never get any money at the end, which makes sense. If we did incorporate discount factors, the value would be increasing each day, and we'd be making payments each day, to not receive anything in the end, which wouldn't make sense.

daveangel
Posts: 17031
Joined: October 20th, 2003, 4:05 pm

### Commodity Option on Futures - deriving the formula

everything that is exchange traded is margined (initial and variation) so at the expiry of the option you do not need to make any further payments. say you sell a call option on a future and there is no exchange of premium. lets call the premium X. lets say you hedge this option with a position in the future. the exchange will margin your position to the tune of M for which you will have to pony up today. this margin earns interest at the risk free rate. lets assume that the premium is exchanged upfront then the exchange will ask for M still (the risk of the position is the same) but you will need to only pony up (M - X*pv factor) - so the premium has to be discounted when paid upfront.
knowledge comes, wisdom lingers

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