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Marsden
Topic Author
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Joined: August 20th, 2001, 5:42 pm
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### Re: "Option Pricing, Risk Premium, and Arbitrage: An Argument for Volatility-Modified Risk-Neutral Prices."

If I understand correctly, bearish, this would just in effect be an "invest in everything" index (assuming market prices are all efficient)? I guess that's the part that you say is mysterious to you ... although it makes sense to me in a way that I would be at a loss to explain.

I guess it would more accurately be an "invest in everything, in proportion to each thing's market value" index.

Sraffa had a notion of a "commodity of constant marginal utility," which I guess would take any utility effects out of pricing, if everything were denominated in it. (I sort of think there would still tend to be risk premium effects, because planning tends to improve utility -- $100,000 that you know for years your uncle was going to give you on your 40th birthday [40th rather than 21st, for obvious reasons] tends to get more efficiently spent than$100,000 that you find in a paper bag in a parking lot -- and I'm not sure how a time zero "commodity of constant marginal utility" could account for planning; also, planning isn't free, so it would make sense I think to concentrate your planning on the most likely outcomes).

A little difficult to wrap my head around your "natural numeraire" idea. If I'm more risk averse than average, so I weight my investments to fixed income of one sort or another relative to the natural numeraire ... what does that look like? If the value of all investments drops (... by some currency measure), then I'm relatively a winner; if the value skyrockets, then I'm relatively a loser. What does that say about what price would be appropriate for buying (concave upward, relative to some currency measure) or selling (concave downward) some sort of insurance for my portfolio?

I guess the expected rate of return, measured in the natural numeraire, is constant, so I should be able to shape my portfolio however I like at no cost.

Am I missing something obvious?

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

### Re: "Option Pricing, Risk Premium, and Arbitrage: An Argument for Volatility-Modified Risk-Neutral Prices."

This is even simpler to carry out in a binomial model, but using the BSM environment and standard notation, define the risk premium as $\lambda = \frac{\mu - r}{\sigma}$. Then you can create a dynamic portfolio with a constant fraction of its value given by $\frac{\lambda}{\sigma}$ invested in the stock and the remaining balance in the bond. The value process for this portfolio, initialized at a value of 1, is $A_t = e^{(r+\frac{1}{2} \lambda ^2) t + \sigma W_t}$. It is straightforward to show that each of the stock price and the bond price is a P-martingale once divided by this value process. This is the sense in which A is the “natural numeraire”; it turns the natural measure into a martingale measure. Seemingly like magic, calculating $E \frac{(S_T - K)^+}{A_T}$ yields the standard BSM formula for the value of a call. This may not be as well known as it ought to be.

Marsden
Topic Author
Posts: 248
Joined: August 20th, 2001, 5:42 pm
Location: Maryland

### Re: "Option Pricing, Risk Premium, and Arbitrage: An Argument for Volatility-Modified Risk-Neutral Prices."

I'll have to take a moment to parse your comment, bearish.

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

### Re: "Option Pricing, Risk Premium, and Arbitrage: An Argument for Volatility-Modified Risk-Neutral Prices."

Sorry, the volatility in the process for A should of course be $\lambda$.

Marsden
Topic Author
Posts: 248
Joined: August 20th, 2001, 5:42 pm
Location: Maryland

### Re: "Option Pricing, Risk Premium, and Arbitrage: An Argument for Volatility-Modified Risk-Neutral Prices."

Bearish, when you write "with a constant fraction of its value given by (fraction) invested in the stock," does that envision instantaneous and costless rebalancing?

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

### Re: "Option Pricing, Risk Premium, and Arbitrage: An Argument for Volatility-Modified Risk-Neutral Prices."

Yes

Alan
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### Re: "Option Pricing, Risk Premium, and Arbitrage: An Argument for Volatility-Modified Risk-Neutral Prices."

I wonder if it is such magic. Generally (in the absence of arbitrage), there is a stochastic discount factor process $m_t$ such that the time-0 value of payoffs $V_T$ is given by $V_0 = E[m_T V_T]$. In the case of the BSM model, $m_t = 1/A_t$.

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

### Re: "Option Pricing, Risk Premium, and Arbitrage: An Argument for Volatility-Modified Risk-Neutral Prices."

I agree, I don’t think it is magic. But it is universal, at least in complete markets. You can get there via a martingale representation theorem or a separation of the Radon-Nikodym derivative. The aspect that is still somewhat mysterious to me is that the natural numeraire is equal to the growth optimal portfolio (which suffers from a terrible acronym!). Superficially, the properties of maximizing expected log utility and being the numeraire that makes all market prices martingales are not obviously equivalent.

DavidJN
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Joined: July 14th, 2002, 3:00 am

### Re: "Option Pricing, Risk Premium, and Arbitrage: An Argument for Volatility-Modified Risk-Neutral Prices."

Speaking of numeraires and the growth optimal portfolio, you will definitely profit from reading "The numeraire portfolio" by John B. Long Jr., Journal of Financial Economics, July 1990. Directly relevant to the topic at hand, an easier and more intuitive read than Harrison or Pliska, and imho one of the more unusual, powerful, and least understood finance papers ever written. Peter Carr made creative use of Long's results in his late work on extracting return expectations from option prices.

Marsden
Topic Author
Posts: 248
Joined: August 20th, 2001, 5:42 pm
Location: Maryland

### Re: "Option Pricing, Risk Premium, and Arbitrage: An Argument for Volatility-Modified Risk-Neutral Prices."

There's a Harrison-Pliska paper, "Martingales and Stochastic Integrals in the Theory of Continuous Trading," from 1981 that is far more readable to me than the 1979 Harrison-Kreps paper (Pliska was a full-on mathematician). The Harrison-Pliska paper announces up front that its model assumes "a frictionless security market with continuous trading," which immediately diminishes my interest.

And, just now looking through the Harrison-Kreps paper I find, "we are assuming frictionless markets (no transaction costs and unrestricted short sales)," although they avoid the term "continuous" in that. They sure don't spare the term later on, although mostly to point to continuous preferences for agents.

Haven't been able to track down the Long article without fee on the interwebs, but I'll keep looking.

Alan
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### Re: "Option Pricing, Risk Premium, and Arbitrage: An Argument for Volatility-Modified Risk-Neutral Prices."

Haven't been able to track down the Long article without fee on the interwebs, but I'll keep looking.
I thought I had a copy but haven't been able to find it. I suggest that someone with a copy post it here for a few days for discussion and then take it down. IMO, that's "fair use".

p.s. Oh wait -- you will have to ask the admin to take it down.
Last edited by Alan on July 4th, 2022, 2:05 pm, edited 1 time in total.

osmium76
Posts: 3
Joined: November 15th, 2021, 8:02 pm

### Re: "Option Pricing, Risk Premium, and Arbitrage: An Argument for Volatility-Modified Risk-Neutral Prices."

I believe this is the right one?

Harrison-Pliska, Martingales and Stochastic Integrals in the Theory of Continuous Trading (1981)

Found it on the Northwestern/Kellogg site.

Alan
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Location: California
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### Re: "Option Pricing, Risk Premium, and Arbitrage: An Argument for Volatility-Modified Risk-Neutral Prices."

No, we're looking for J. B.  Long, "The Numeraire Portfolio"

osmium76
Posts: 3
Joined: November 15th, 2021, 8:02 pm

### Re: "Option Pricing, Risk Premium, and Arbitrage: An Argument for Volatility-Modified Risk-Neutral Prices."

Right, sorry.

Here is a similar paper by Long and Hentschel (2002) on SSRN - could be similar enough to the one by Long alone in the JFE (1990):

Numeraire Portfolio Tests of the Size and Source of Gains from International Diversification