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 FORUMS > General Forum < refresh >
 Topic Title: Black-Scholes: What basis used for time to expiration? Created On Fri May 23, 08 10:53 AM Topic View: Branch View Threaded (All Messages) Threaded (Single Messages) Linear

Max79
Junior Member

Posts: 2
Joined: May 2008

Fri May 23, 08 10:53 AM

Hello Guys,

I know that the Black Sholes formula for a cap depends on d1 with:
d1= (log(S/K) +sigma^2/2 * T) / (sigma * sqr(T))
But do you know what is the market practice for the basis of the time to expiration T (ACT/360, ACT/365, ...) in d1 formula ?

Best regards,
Max
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RiskUser
Senior Member

Posts: 291
Joined: Jul 2006

Fri May 23, 08 11:19 AM

Day count conventions such as ACT/360 are primarily used for interest flow type calculations. This would sugest that you stick with ACT/365 for all currencies when calculating d1?

Quote

Originally posted by: Max79
Hello Guys,

I know that the Black Sholes formula for a cap depends on d1 with:
d1= (log(S/K) +sigma^2/2 * T) / (sigma * sqr(T))
But do you know what is the market practice for the basis of the time to expiration T (ACT/360, ACT/365, ...) in d1 formula ?

Best regards,
Max

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BrightDay
Member

Posts: 103
Joined: Aug 2003

Fri May 23, 08 11:26 AM

Actual/Actual is the one that makes most sense to me.

But it is slower to compute than a ACT/365.25 and the difference is small, so I would use the second...

In the past I've seen people using ACT/365 as well, but on long maturities (e.g. 30y to 50y) the extra days become noticeable....
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gjlipman
Senior Member

Posts: 1102
Joined: May 2002

Fri May 23, 08 11:41 AM

Ideally you should use the same basis that was used to estimate or imply the volatility. So first try and find out what assumption was made to generate your vols.
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BrightDay
Member

Posts: 103
Joined: Aug 2003

Fri May 23, 08 11:54 AM

Quote

Ideally you should use the same basis that was used to estimate or imply the volatility. So first try and find out what assumption was made to generate your vols.

I don't agree with you: the volatility is the volatility of the forward rate underlying your caplet, so with regards to your answer, the day count convention used to price the caplet should be the same as the day count convention your volatility quote refers to (just to make sure we are talking about the same thing; given that:

caplet value = Black76 Call value*(d/Basis)/(1+F*d/Basis)

In other words, I'm saying that d/Basis should be the same as the day basis used for volatility. This is generally not relevant, since in most market one would rely on a broker to have a market quote of the volatility without much need to estimate it).

It seems to me that Max79 is asking something different: he is not asking for d/Basis but the value of "T" that enters the computation of the Black76 call value, and to my knowledge has nothing to do with the accrual conventions.
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Max79
Junior Member

Posts: 2
Joined: May 2008

Fri May 23, 08 12:46 PM

Yes I confirm what Brightday says: my question is on the basis convention used to compute the time to expiration "T" in the term d1 of the Black-Scholes formula, for a cap.
Traders told me that it is ACT/365, but I need a confirmation to make sure that it is a well known market practice (and not just what a trader needs).
I search on ISDA conventions, web-sites, Hull book, ... but I didn't find the answer.

So in your opinion, it is ACT/365 or ACT/ACT in case of years with 366 days?

Regards,
Maxime
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BrightDay
Member

Posts: 103
Joined: Aug 2003

Fri May 23, 08 01:06 PM

Quote

I search on ISDA conventions, web-sites, Hull book, ... but I didn't find the answer.

I know what you mean, I went through the same and couldn't find any definitive answer myself.

Quote

Traders told me that it is ACT/365, but I need a confirmation to make sure that it is a well known market practice (and not just what a trader needs).

To me there are two important things:
1. the computation needs to be fast and simple
2. if I'm looking at a caplet with expiry on (T+2)+50years, the time in years should be around 50

With ACT/365 you end up with a period that is about 13 days longer than it should be.
Hardly a big error, but it can make a little difference. With a cap this probably isn't really noticeable: nobody would trade single caplets or a forward starting cap that much in the future, and since the vols quoted on the market are flat vols, the error would be diluited through the whole cap length. In this situation the different interpolation assumptions used for the bootstrapping of the yield curve can cause even greater differences, so I think that for caps&floors you can probably get away with ACT/365.

ACT/365.25 can still be incorrect, but it will generally be incorrect by at most 1 day, so i like it more on long terms. The problem that if an error of 1day in one year (with ACT/365.25) will be worse than the error of 12 days in 50 years :-), so maybe ACT/ACT may be a better idea.
What did other people do? In my working experience I have seen both ACT/365 and ACT/365.25 but not ACT/ACT yet. What are other people in the forum doing?

I think the problem becomes worse with swaptions. In a 20y10y swaption those extra 5 days can change your volatility quote by a few bps so getting it wrong could be an issue... mn... i maybe we can try to reprice a few brokers' screens (ICAP,BGC, TRAD, TULLET) and see if they all use ACT/365 or there are different streams of thought....

Quote

So in your opinion, it is ACT/365 or ACT/ACT in case of years with 366 days?

In my opinion ACT/ACT is better, but I'd settle (as I have done) for either ACT/365 or ACT/365.25

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unkpath
Senior Member

Posts: 288
Joined: Jan 2004

Fri May 23, 08 01:18 PM

this question and thread are irrelevant.

the only thing that counts is that we all agree on the standard deviation or variance from the
date the deal is struck to the date the option expires. so, yes it absolutely depends on the
time basis that was used to compute the volatility.
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BrightDay
Member

Posts: 103
Joined: Aug 2003

Fri May 23, 08 01:32 PM

Quote

the only thing that counts is that we all agree on the standard deviation or variance from the
date the deal is struck to the date the option expires. so, yes it absolutely depends on the
time basis that was used to compute the volatility.

Thank you for making this cristal clear now... There is still a detail that i don't understand, so allow me to add some details to it if we can pin-point where I'm wrong...

I want to price a 3m-caplet with expiry in 50y for a cap denominated in EUR.

The formula is:

caplet value = Black76 Call value*(d/Basis)/(1+F*d/Basis)

Where "d/Basis" in this case is ACT/360. Since I am in a Black76 scenario (and not in a Black-Scholes) the variance is not the variance from the date the deal is struck to the date the option expiries but it is the variance of the rate between 50y and 50y3m. I would say that the volatility is a data coming from the market (in terms of flat volatility),so every one agrees that the day count convention this quote refers to is ACT/360.

What you are saying is that you would insert for "T" in the "Black76 call value" the time from today to 50y computed with ACT/360 (hence a number similar to 50.7). I have two problems about this:
1. The option expiries in 50 years, and we are inserting in Black76 an expiry for 50y8m; this sounds outright wrong to me!
2. I cannot see any relationship between ACT/360 used for the rate accrual and volatility estimation, and the time to expiry to insert in the black formula. Can you elaborate please on this second point, for me to see your reasons?

Now I compute a similar caplet but for GBP where the daycount convention is ACT/365. Same reasonings, but now we insert 50 in the black call premium...

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gjlipman
Senior Member

Posts: 1102
Joined: May 2002

Sat May 24, 08 10:45 AM

I'm not an expert on interest rate caps, but I think I know what is confusing matters.

The first thing to ask yourself when pricing an option using Black76 is: what is the underlying. As you correctly point out, the underlying is the rate at tiem 50 years from 50 years to 50years and 3 months. And in converting this to a dollar amount, yes, you should use the day count convention that that rate is expressed in (which may depend on the market). Even if you were pricing a swap for that period, you'd still need to know this. (I'm pretty sure this is the act/360 term in your last post).

Buying an option, you get the advantage of uncertainty in this underlying. And the sigma x sqrt(t) term in the Black Scholes is all about quantifying that uncertainty. And, in this option, you have 50 years of uncertainty (not 3 months of uncertainty). Now this term is never used in d1, except multiplied by the volatility, so different people can use a different daycount method in expressing the 50 years and the volatility, and as long as they are consistent, all is fine.
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BrightDay
Member

Posts: 103
Joined: Aug 2003

Sun May 25, 08 02:24 AM

Quote

The first thing to ask yourself when pricing an option using Black76 is: what is the underlying. As you correctly point out, the underlying is the rate at tiem 50 years from 50 years to 50years and 3 months. And in converting this to a dollar amount, yes, you should use the day count convention that that rate is expressed in (which may depend on the market). Even if you were pricing a swap for that period, you'd still need to know this. (I'm pretty sure this is the act/360 term in your last post). Buying an option, you get the advantage of uncertainty in this underlying. And the sigma x sqrt(t) term in the Black Scholes is all about quantifying that uncertainty. And, in this option, you have 50 years of uncertainty (not 3 months of uncertainty).

Yep...
Quote

Now this term is never used in d1, except multiplied by the volatility, so different people can use a different daycount method in expressing the 50 years and the volatility, and as long as they are consistent, all is fine.

This is what I understood being Maxime's original question. Only, since cap/floor volatilities are quoted on the market as alternative to their price, saying "as long as they are consistent" isn't sufficient, since the market must be agreeing on how to compute this number of days: what is the convention that you have in your pricing formulas for caps and even swaptions?

This isn't a theorical question:the issue is simply what do people use in practice? I wonder if we coud try a poll....

From the thread so far we have:

AJSKJ: ACT/365
BrightDay: (in the order of preference): ACT/ACT, ACT/365.25, ACT/365
Unkpath: same as *IBOR basis

Anyone wants to contribute their experience?

Edited: Sun May 25, 08 at 02:32 AM by BrightDay
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gjlipman
Senior Member

Posts: 1102
Joined: May 2002

Sun May 25, 08 09:43 AM

So, if I'm not misunderstanding you, you're agreeing that it is the daycount convention for the volatility that we're after (because people are already quoting volatility, we need to know what daycount convention this volatility is based on).

I suspect this may differ in different countries/markets - you might want to clarify which you are after.
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jfuqua
Senior Member

Posts: 1255
Joined: Jul 2002

Sun May 25, 08 08:16 PM

Are you also considering the adjustments pre- and post-calculation for the time between Trade and Spot Settle, Expire and Expire Settle and days that interest accrues----all of these can be substantial [e.g. JGBs]. Also you may want to consider the adjustment for days of volatility---i.e. the weighted volatility by day of the week [e.g Sat. and Sunday maybe <1 and Monday and Friday >1].
If American then you also want to get from your numerics the 'Fugit' [see Mark Garman's article] or the effective days to Expire---i.e. when Early Exericse will occur.

Edited: Sun May 25, 08 at 08:16 PM by jfuqua
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BrightDay
Member

Posts: 103
Joined: Aug 2003

Sun May 25, 08 09:03 PM

There is something I don't understand here... Maxine's question is very precise. Why do people keep on adding things on the side, just to confuse the topic we are talking about?

Quote

Are you also considering the adjustments pre- and post-calculation for the time between Trade and Spot Settle, Expire and Expire Settle and days that interest accrues----all of these can be substantial [e.g. JGBs]

Of course these are very substantial, and I clearly consider them in my pricing. I'm not considering them in the thread because: Max79 didn't ask about them and because differently from "T" in Black76 formula they are very well documented in books and ISDA documents.

Quote

Also you may want to consider the adjustment for days of volatility

No i don't want to consider this: it's not relevant to the original question and....

Quote

---i.e. the weighted volatility by day of the week [e.g Sat. and Sunday maybe <1 and Monday and Friday >1].

What we are talking here is implied volatility, so once all the market conventions are given the number in a one-to-one mapping with the caps and floor prices! Weighting volatility if you are computing realised volatility is a fascinating topic, but has nothing to do with what we are discussing here.

Quote

If American then you also want to get from your numerics the 'Fugit' [see Mark Garman's article] or the effective days to Expire---i.e. when Early Exericse will occur.

Seems out of topic here... even if theorically possible I have never seen on the market a cap whose caplets have American style exercise.... Have you ever traded one? And even if they are traded, they are not the commonly traded cap and floors, to which the original question refers to. If you are looking at non standard instruments, then any possible convention about anything can apply, but Max79's question only makes sense in case we are talking about mainstream, standard caps and floor contracts.

I apologise for my bad personality but this thread is really winding me up: it started with a simple and precise question and it seems to me that most of the contributors (maybe myself included) either:
1. Don't understand the question or the instrument they are commenting about
2. Haven't read the original question and keep on talking about something else, clearly not caring about getting to the bottom of a problem before opening others

P.S. Max79: Let me know if you consider my postings either abusive or written in a style that is not helping clearing this issue and i'll step out immediately from the discussion

Edited: Sun May 25, 08 at 09:13 PM by BrightDay
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BrightDay
Member

Posts: 103
Joined: Aug 2003

Sun May 25, 08 09:12 PM

Quote

Originally posted by: gjlipman
So, if I'm not misunderstanding you, you're agreeing that it is the daycount convention for the volatility that we're after (because people are already quoting volatility, we need to know what daycount convention this volatility is based on).

I suspect this may differ in different countries/markets - you might want to clarify which you are after.

I don't know how people want to call this... maybe they call it "daycount convention for the volatility", i don't know... Often names are ambiguous while mathematical formulas are not. A caplet (according to the market convention) is priced once its black76 vol is given, as:

caplet value = Black76 Call value*(d/Basis)/(1+F*d/Basis)

According to Max79's first posting he is looking for how "T" is computed in:

d1= (log(S/K) +sigma^2/2 * T) / (sigma * sqr(T))

I think Max79's question cannot be any clearer regardless how we call it...

Quote

I suspect this may differ in different countries/markets - you might want to clarify which you are after

From my experience it is common through all currencies markets (apart from the Mexican market because of it's 13 months year that begs for custom treatment everywhere)

Edited: Tue May 27, 08 at 09:12 AM by BrightDay
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