When a Constant Can Serve as a Derivatives Numeraire
Summary
The document discusses whether a constant-valued quantity can be used as a numeraire in a standard Geometric Brownian Motion and Black–Scholes setting. One answer distinguishes the zero-interest-rate case, where the bank account is constant and can serve as a numeraire, from a nonzero rate, where a constant-traded asset would create an arbitrage under the model. Another answer emphasizes that a numeraire must be tradeable, so a constant is admissible only if such an asset exists.
The responses also distinguish practical reporting units from pricing numeraires: market values are commonly reported in currency units, while derivatives calculations may use a bank account or forward-based numeraire before converting prices back. A scaling argument notes that scaling the underlying and strike together leaves their ratio and relative volatility unchanged, so the option price scales accordingly. These points are conditional on the assumed market model and do not establish that a constant-valued tradeable asset exists in a given market.
Key ideas
- A numeraire must be a tradeable asset in the pricing framework.
- With zero interest rates, the bank account is constant and may serve as a numeraire.
- At a nonzero interest rate, a constant-valued tradeable asset would imply arbitrage under the stated model.
- Scaling the underlying and strike by the same constant preserves their ratio and relative volatility.
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Full text
# Using a Constant as a Numeraire # Using a Constant as a Numeraire 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. ## Answer by Mark Joshi (score 7) https://quant.stackexchange.com/a/29701 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. ## Answer by user9403 (score 6) https://quant.stackexchange.com/a/29665 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. ## Answer by bhutes (score 2) https://quant.stackexchange.com/a/45314 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. ## Answer by Mats Lind (score 0) https://quant.stackexchange.com/a/29666 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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