SERVING THE QUANTITATIVE FINANCE COMMUNITY

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dsantanu
Topic Author
Posts: 19
Joined: August 29th, 2009, 5:41 am

Hi,Can constants ( 1 ) be treated as "tradable" asset. That is, under T-forward measure, is {1/B(t,T)}_t a maringale ? I think 1 is not self financing.Can some one please point me to a heuristic and a rigourous argument?Thanks

ZhuLiAn
Posts: 220
Joined: June 9th, 2011, 7:21 am

Can a constant be a Numeraire? What about the constant zero?

dsantanu
Topic Author
Posts: 19
Joined: August 29th, 2009, 5:41 am

by definition Numeraire is positive. so zero ruled out for Numeraire. I think no harm in assuming zero is a tradable! lets restrict to non zero constants.it seems 1 is not a tradable.If you put 1 dollar under your matress it better not be called tradable, but can't see why!

ZhuLiAn
Posts: 220
Joined: June 9th, 2011, 7:21 am

So the tradability cannot be extended continuously in the volatility space?

VivienB
Posts: 130
Joined: August 6th, 2012, 3:32 pm

HiIf 1 is a tradable asset, then 1 / C is a martingale under the cash measure. as dC(t) = r(t)C(t) dt, d(1/C)(t) = -r(t) / C(t) dt, 1/C is a martingale leads to r(t) = 0 for all t >= 0. Reciprocally, if r(t) = 0 for all t, then 1 is the value of the cash, then it is a tradable asset. Thus, 1 is a tradable asset <=> r(t) = 0 for all t >=0.

spv205
Posts: 478
Joined: July 14th, 2002, 3:00 am

isn't it enough to consider a discount bond DF_t which has value 1 at time t then you have arbitrage...?

VivienB
Posts: 130
Joined: August 6th, 2012, 3:32 pm

You can indeed build many arbitrage to show that constants can not be tradable when rates are not 0, what is strongly linked with the fact that the asset divided by the numeraire is not a martingale.By the way, answer of the initial question is "constants can be treadable asset if and only if rates are 0".

dsantanu
Topic Author
Posts: 19
Joined: August 29th, 2009, 5:41 am

vivienb's argument is simple and clear. thanks to all of you.

spacemonkey
Posts: 443
Joined: August 14th, 2002, 3:17 am

Is £10 tradeable? Are you kidding? EDIT: And someone thinks the answer is no?!?
Last edited by spacemonkey on January 12th, 2013, 11:00 pm, edited 1 time in total.

Alan
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Joined: December 19th, 2001, 4:01 am
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The previous discussion seems ok to me.If you define a "tradeable" as "any asset that does not offer an arbitrage opportunity",then a £10 banknote is, indeed, not a tradeable in a world with a positive money market rate.

spacemonkey
Posts: 443
Joined: August 14th, 2002, 3:17 am

Think what you are saying man!Call me crazy, but I prefer to use the definition 'thing that can be traded'. Doesn't have anything to do with martingales or whatnot. If your theory suggests cash isn't tradeable, you'd better rethink the theory. As it happens, £10 is exactly as tradeable as a dividend paying stock, and you can happily call both tradeable. Obviously, because any definition that excluded these would just be silly.
Last edited by spacemonkey on January 12th, 2013, 11:00 pm, edited 1 time in total.

Alan
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Well, I'm prepared to admit maybe that definition I offered was silly, and maybe I am confused.So, let me ask: is there any issue of arbitrage opps (or any issue at all) in a theory with both cash and non-zero money market rates?For example, could Black and Scholes, with no problem, have started by saying: consider a simple market with 3 tradeable assets: a stock (following GBM), a riskless money market account (paying r > 0) and cash.Now, we want to price an option on the stock. ? (honest question!)
Last edited by Alan on January 12th, 2013, 11:00 pm, edited 1 time in total.

spacemonkey
Posts: 443
Joined: August 14th, 2002, 3:17 am

QuoteOriginally posted by: AlanWell, I'm prepared to admit maybe that definition I offered was silly, and maybe I am confused.So, let me ask: is there any issue of arbitrage opps (or any issue at all) in a theory with both cash and non-zero money market rates?For example, could Black and Scholes, with no problem, have started by saying: consider a simple market with 3 tradeable assets: a stock (following GBM), a riskless money market account (paying r > 0) and cash.Now, we want to price an option on the stock. ? (honest question!)As long as you accept that cash pays dividends (interest), then everything is fine. It's analogous to a stock that pays dividends, which also isn't a martingale under the R.N. measure.This really is just terminology, you're still better off working with the non-dividend paying bond and forward processes.
Last edited by spacemonkey on January 13th, 2013, 11:00 pm, edited 1 time in total.

bearish
Posts: 5389
Joined: February 3rd, 2011, 2:19 pm

QuoteOriginally posted by: spacemonkeyQuoteOriginally posted by: AlanWell, I'm prepared to admit maybe that definition I offered was silly, and maybe I am confused.So, let me ask: is there any issue of arbitrage opps (or any issue at all) in a theory with both cash and non-zero money market rates?For example, could Black and Scholes, with no problem, have started by saying: consider a simple market with 3 tradeable assets: a stock (following GBM), a riskless money market account (paying r > 0) and cash.Now, we want to price an option on the stock. ? (honest question!)As long as you accept that cash pays dividends (interest), then everything is fine. It's analogous to a stock that pays dividends, which also isn't a martingale under the R.N. measure.This really is just terminology, you're still better off working with the non-dividend paying bond and forward processes.But, as we all know, actual cash does not pay any dividends/interest. So, to justify the rational holding of actual cash (as opposed to money market accounts and the like), you have to invoke some kind of convenience yield accruing to its holders, e.g. because you can use it to pay for coffee and bus fare and the like. In the world of negative interest rates (Denmark, Switzerland), the opposite problem arises -- you need to be able to explain why people (and institutions) choose to buy government securities at negative yields rather than keeping their money in cash. The cost of secure storage is one candidate, but only goes so far. When all is said and done, the whole cash bit is so messy that it is no wonder that most finance researchers choose to ignore it...

Alan
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Joined: December 19th, 2001, 4:01 am
Location: California
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Good, bearish, that was helpful to me. Of course, I raised the issue of "real-world" cash as a kind of a follow-up question, so let me see if I can paraphrase spacemonkey's resolution of the original question:"are constants tradeable?"In a world with dividends, a tradeable should be thought of as the price process or the total returnprocess of anything that can be traded, as long as it remains positive. Since the money market account,in a world with non-zero interest rate r(t) can always be thought of as having a price process of 1 and paying dividends at the rate r(t), then "1" is certainly a tradeable. Then a separate issue is the role of actual cash (see bearish's remarks).

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