Expose the II price variance and MM Lucas weight a parameters

[sbenthall]

We should get clarity about what the different populations we will be using for FIRE Shark are.

It sounds like we will be exploring:

• A very basic uniform population that can replicate Lucas Asset Pricing results, synced up with the institutional investors
• ...
• The full Distribution of Wealth realistic population model (But ... how big? How many agents simulated?)

Anything in between?

@alanlujan91 @sbenthall @llorracc

[sbenthall]

A homogeneous population is good enough for SPARK SHARK.

The full Distribution of Wealth population can wait until FIRE SHARK.

[sbenthall]

@mesalas I think all we need is for the institutional investors to vary systematically around the consumer population preferences.

Maybe the wideness of that distribution around the consumer preferences could be a parameter?

If that distribution of II preferences is narrow, then it probably keeps the price tracked tightly to the LAP. If that distribution is wide, then there may be more equilibria price points in between their price points, and it could lead to more departures from the LAP (and more falling into the STRANGE regime, etc.)

[llorracc]

A disciplined way of controlling the variation in the target prices of the institutional investors would be to assume that there is some distribution of beliefs among those investors about one of the parameters that goes into the LAP price/dividend formula. For example if some institutional investors believed that consumers' relative risk aversion was 2.5, some believed it was 3, and some believed it was 3.5, then each of those kinds of institutions would have a different view of the target.

Or, say, you could assume that there were many institutional investors among whom the distribution of believed-CRRA was uniform over the interval [2.5,3.5].

[sbenthall]

The relationship between the preference parameters (beta and rho) and the LAP is not linear. These are some example plots:

If we were to distribute along beta or rho linearly, we would potentially get curvature in the distribution of II asset prices.

Based on my understanding so far, we will be expecting II to have price impact roughly proportional to their liquidity, and so this curvature might depress (or raise) the equilibrium price in ways that distort the results due to a mathematical artifact.

What if the prices themselves were linearly distributed?

[llorracc]

Seb,

It’s pretty linear in rho in the interval [2.5-3.5] that I proposed. (For whatever [image: \beta] you are using).

When P=f(x), you can only make prices linearly distributed by making x distributed in some nonlinear way that undoes the nonlinearity of the f function.

On Thu, Apr 20, 2023 at 12:52 PM Sebastian Benthall < @.***> wrote:

The relationship between the preference parameters (beta and rho) and the LAP is not linear. These are some example plots:

[image: image] https://user-images.githubusercontent.com/68752/233432916-ec3105ea-1577-42d5-9d6a-047b06b6c1b7.png

[image: image] https://user-images.githubusercontent.com/68752/233432943-c0c1f3a6-8a38-415a-b335-6a352930dd69.png

If we were to distribute along beta or rho linearly, we would potentially get curvature in the distribution of II asset prices.

Based on my understanding so far, we will be expecting II to have price impact roughly proportional to their liquidity, and so this curvature might depress (or raise) the equilibrium price in ways that distort the results due to a mathematical artifact.

What if the prices themselves were linearly distributed?

— Reply to this email directly, view it on GitHub https://github.com/sbenthall/SHARKFin/issues/180#issuecomment-1516651838, or unsubscribe https://github.com/notifications/unsubscribe-auth/AAKCK7ZQJGZOFSOUXEZER2DXCFSWNANCNFSM6AAAAAASUYS72A . You are receiving this because you were mentioned.Message ID: @.***>

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• Chris Carroll

[sbenthall]

What I am suggesting is that the variance of this distribution over preferences be itself variable, i.e. part of the parameter sweep. So it would be different intervals depending on the value of that parameter.

Yes, that's right, it would require nonlinearity of x.

Given the number of LAP assumptions we are violating, essentially we are just picking a constant of approximately correct size. The relationship with the preference parameters won't matter much because the consumers are homogeneous. So what we are really looking at is how II divergence effects equilibrium prices.

[sbenthall]

To put it another way: if we vary both beta and rho, we will have two dimensions over which to sweep. If we vary just the price spread, we will have just one. Which simplifies both the calculation and interpretation.

[sbenthall]

In the paper it would be good to write up this about the effect of rho and beta on the II price selection.

[sbenthall]

Also expose the MM Lucas weight.

[sbenthall]

Based on the meeting today, I think the plan is that we'll continue with what @mesalas is doing: each II draws an idiosyncratic IID normal shock and adds it to the LAP for the consumer population, and uses this as the target price each day. The variance of that shock will be exposed as a parameter.

[mesalas]

The ammps configuration generator now takes the market makers Lucas pricing weight and the stdev of the II idiosyncratic valuation as input parameters

[sbenthall]

Cool. I guess we just put that into the parameter grid for the simulation, and we don't need command line arguments for that?

[mesalas]

What are we using for CRRA and discount factors?

[sbenthall]

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