Please provide steps to justify the below.
1) Can we use a constant as a numeraire?
Related Question: Scaling Stock Price and Strike etc. by a Constant
The rest of standard Geometric Brownian Motion and Black Scholes assumptions apply.
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2$\begingroup$ If you prefer thinking in millions of dollars rather than in dollars, you can. $\endgroup$– M. JeunesseAug 16, 2016 at 6:13
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$\begingroup$ Let we can do it. Please continue $\endgroup$– user16651Aug 16, 2016 at 6:50
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$\begingroup$ i am not wholly sure what the question. Is it "can we do martingale pricing with $S_t/N_t$ a martingale when $N_t$ is a constant? " The answer to that is no! $\endgroup$– Mark JoshiAug 17, 2016 at 6:16
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$\begingroup$ @MarkJoshi Thanks for your comment. I think I need to make this two questions. 1) Can a constant be a numeraire? 2) What is the impact on option prices when the underlying price and strike are scaled by a constant? $\endgroup$– texmexAug 17, 2016 at 11:09
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1$\begingroup$ @MarkJoshi What's the difference between choosing a constant as your numeraire, and choosing a hypothetical instrument with zero drift and zero volatility? $\endgroup$– willAug 17, 2016 at 13:13
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4 Answers
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Either $r=0$ in which $B_t$ is constant and is a valid numeraire (as is any multiple of it.)
or $ r \neq 0$ in which case an asset of constant value would give an arbitrage since we could take $$ B_t - N_t $$ with $B_0 = N_0$ and get a riskless profit. (or the opposite if $r<0.$) and so it would be a very flawed model.
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A Numeraire must be a tradeable asset. If you can find a constant tradeable asset, then yes a constant can be used as a numeraire.
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$\begingroup$ Can we assume that there is a traded asset with constant price. Surely we can buy and sell something at the same value (though it defeats the purpose of a trade, which is to buy and sell for a profit ?? ) One instance where I can think of such a constant price asset being used is to fill it as part of a bigger trade to make up for part of the value ?? Please let me know if this would be a valid assumption. $\endgroup$– texmexAug 17, 2016 at 4:22
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1$\begingroup$ This is the right answer. Furthermore such a tradeable asset will not exist, unless the interest rate is zero. The accepted answer does not make any sense to me. $\endgroup$– KiwiakosAug 17, 2016 at 6:26
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$\begingroup$ @user249613 Even a dollar tomorrow is not worth the same as a dollar today unless interest rates are exactly zero. As Kiwiakos points out the asset will not exist. Another way to think about it is that option values will be the same regardless of the numeraire (redo the BS derivation using the stock as a numeraire to demonstrate this). The only way for the price to coincide when using a constant numeraire is if the constant asset itself is the bond; that is, a bond with zero interest rates. $\endgroup$– user9403Aug 17, 2016 at 11:33
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$\begingroup$ @user903 Thanks for the very helpful answer. I see the issue with my question and perhaps the need for two subquestions .. I have edited the question now to reflect that. $\endgroup$– texmexAug 17, 2016 at 11:34
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Actually, all investments, retirement accounts, mutual fund accounts, utility bills, supermarket price listings are reported or stated in the Constant Numeraire, which may also be called Dollar-kept-under-the-mattress Numeraire
It is the most widely (indeed the only) Numeraire used in real life.
How nice it would be if my retirement account or mutual fund account reported my accumulated wealth in the Bank Account Numeraire. Or atleast in the Inflation Numeraire. Even better, in the Nominal GDP Numeraire.
However, all reporting is necessarily required to be made in the Constant "Dollar-under-the-mattress" Numeraire
For ease of derivatives pricing, we change the numeraire from Constant Numeraire to Bank Account numeraire, or T-forward measure or whatever Numeraire but always convert the computed price back to Constant "Dollar-under-the-mattress" Numeraire because that is the value that mutual funds, retirement funds, investments need to report.
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Use a constant to scale the numeraire and S and K would scale the same, volatility (of relative returns) would remain unchanged, S/K and ln(S/K) would remain unchanged and of course time to strike and the interest rate; now look in the Black formula and you see that the call price would scale like S and K as you would expect.
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$\begingroup$ Thanks for your answer. So can we conclude that, all option prices remain unchanged by a change of constant numeraire and by changing the strike accordingly (That is scaling the price and strike by a constant). Are there any examples of options where this might not hold? $\endgroup$– texmexAug 16, 2016 at 14:06
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$\begingroup$ No, that would if nothing else violate the future = call - put arbitrage $\endgroup$ Aug 16, 2016 at 14:08
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$\begingroup$ Sorry meant to add earlier, not just vanilla options, but including exotics, American, Bermudean etc.. $\endgroup$– texmexAug 16, 2016 at 14:41
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$\begingroup$ In the practical cases the option price should scale with the units of underlying it is written on. Otherwise you would have to factor in and specify how to handle events like stock-splits. But for the general case I guess you could have derivatives where the price would vary non-trivially with the size of the underlying. $\endgroup$ Aug 16, 2016 at 14:56
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