Bid-Ask Spreads, Conic Finance and the MINVAR Function

Conic Finance

This post recaps the interesting concept of conic finance in the context of incorporating bid-ask spreads onto the idealized models of markets that are originally neutral to risks. It draws from original papers on this topic by Dilip Madan, Alexander Cherny, Ernst Eberlein, Juan Jose Vicente Alvarez, Peter Carr and others found at An Introduction to Conic Finance and its Applications.

Ideas from this area are illustrated with an elementary example of an asset that follows a binomial process. At first the bid-ask spread is evaluated to be zero under a risk-neutral market assumption. Then the example illustrates how a conic market distortion can create a non-zero spread.

Complete Risk Neutral Market

A fundamental concept in financial theory is the assumption of a complete market and the existence of a risk neutral measure. A risk neutral market is indifferent to either buying or selling at any size since all cashflows can be hedged perfectly and no residual risk is left.

Let’s for example consider being long an asset X in a market with a risk neutral probability for a one period move of (1/3)  to $120 and (2/3) to $90.

The risk neutral expectation holding this position is

E[X_{long}] = 1/3*\$ 120 +2/3*\$90 = \$100

The corresponding short position in X represents itself as

with expected outcome of

E[X_{short}] = 2/3*(-\$ 90) +1/3*(-\$ 120) = -\$ 100

A risk neutral market does not have any preference between being long or short. It will pay (bid) $100 for owning the asset and charge (ask) $100 for assuming the liability.

The transition probabilities imply cumulative probabilities given by the following step functions

P_{long}(X \le 120) = 1
P_{long}(X < 120) = 2/3
P_{long}(X \le 90) = 2/3
P_{long}(X < 90 ) = 0
P_{short}(X \le -90) = 1
P_{short}(X < -90) = 1/3
P_{short}(X \le -120) = 1/3
P_{short}(X < -120) = 0

as depict in this graph

We also have
P_{long}(X \le x) = 1 - P_{short}(X \le -x)
P_{short}(X \le x) = 1 - P_{long}(X \le -x)

Bidding and Asking

Generally real markets have a bid-ask spread. The ask price a is the price one must pay to buy an asset or a product from the market. The bid price b is the price one receives selling an asset to the market. Generally we observe a positive bid-ask spread

s = a-b \ge 0

In a market that depicts a non zero bid-ask spread long assets and liabilities (short assets) are valued differently. Unwinding a position of a long asset entails selling the asset to the market for the bid price, thus the bid price reflects the value of a long position. Unwinding liabilities, or a position in a short asset, requires buying the asset from the market for paying the asking price, which reflects the liability of a short position.

The MINVAR Distortion Function

As seen above the risk neutral market distribution does imply a zero bid-ask spread. In order to introduce a difference in valuation between long and short positions the distribution is distorted away from the risk neutral mean. The result of the distortion is that greater weights are given towards losses for the long position and larger liabilities for the short position. The leads to a lower expected value of the long position, thus a lower bid price, and a greater expected liability for the short position, thus a higher ask price.

At this point we regard the selection of the distortion function as a model input. In this framework the distortion function maps from risk neutral probabilities in the range from to 1 to ‘real’ market probabilities also in the range from 0 to 1, necessarily mapping the points 0 and 1 onto itself.

One interesting choice in the possible class of distortion function is the ‘minvar’ function \psi

\psi_n(y) = 1 - (1-y)^{1+n}

as introduced by A. Cherny and D. Madan in New Measures for Performance Evaluation..

For n=0 the function is the trivial identity \psi_0(y) = y , for larger value of n it is a monotonic increasing concave function, with each larger order of n function dominating the previous orders.

As Cherny and Madan pointing out the minvar function has a nice interpretation. It reflects the probability distribution of the minimum over n+1 successive drawings. Concretely for a given probability function y(x)=P(X<x) , the minvar distortion \psi_n(y(x)) reflects the probability that the minimum of n+1 independent samples of X is below x. By construction, the term (1-y)^{(1+n)} is the probability that all n+1 samples are greater than x. That means the minimum of the samples is greater than x . The complementary probability that the minimum is smaller than x is then given by the difference to 1.

As a caveat the minvar function is only one possible choice for a distortion function and the discussion above linking it to the distribution of the minimum is not needed in the present context.

Distorted Cumulative Probabilities

Taking the MINVAR function for n=1 we have

\psi_1 (y) = 1 - (1-y)^2

In the following we use \psi_1 to directly distort the risk neutral probabilities from the section above. Alternatively one could first removed the mean price of $100, and applied the distortion to the net P&L process. However for the sake of better comparisons between the risk neutral and distorted probabilities and to clearly distinguish the long and short position the mean is left in. With this we get

\psi_1 (P_{long}(X \le 120)) = 1
\psi_1 (P_{long}(X < 120)) = 8/9
\psi_1 (P_{long}(X \le 90)) = 8/9
\psi_1 (P_{long}(X < 90 )) = 0
\psi_1 (P_{short}(X \le -90)) = 1
\psi_1 (P_{short}(X < -90)) = 5/9
\psi_1 (P_{short}(X \le -120)) = 5/9
\psi_1 (P_{short}(X < -120)) = 0

Bid Price: Distorted Asset

The price process for the long asset becomes

with the expectation

E_{\psi_1}[X_{long}] = 1/9* \$ 120+8/9 * \$ 90 = \$ 93.33

Bidding prices up to the calculated amount contain all possible trades with expected positive net cash flows, or positive alpha trades. On the other hand competitive market forces maximize the bid price up to this acceptable limit.

The formal calculation of the bid price in the collection of reference papers under the aforementioned link is given as the expectation integral over the distorted asset distribution

b = \int_{-\infty}^{\infty} x d \psi ( F_{X} (x) )

Ask Price: Distorted Liability

The distorted short position involves according to

with expectation

E_{\psi_1}[X_{short}] = 4/9*(- \$ 90)+5/9*(- \$120) = - \$ 106.66

Here the market is asking to be compensated for taking on the trade. Again any in absolute terms larger bid price contains the possible positive alpha positions the market is willing to accept while competitive forces limit the asking price to the calculated amount.

Consequently the asking price is given by the (negative) expectation integral over the distorted liability distribution

a = - \int_{-\infty}^{\infty} x d \psi ( F_{-X} (x) )

or in terms of the long asset distribution F_{X}(-x) = 1 - F_{-X}(x) , after a change of signs resulting in an ask price of

a = \int_{-\infty}^{\infty} x d \psi ( 1-F_{X} (x) )

In “Markets, Profits, Capital, Leverage and Return”, Carr, Madan and Alvarez further express bid and ask expectations in terms of the inverse x = G(u) of the distribution u = F_X(x) as

b = \int_{0}^{1} G(u) d \psi ( u )
a = \int_{0}^{1} G(u) d \psi ( 1-u )

They identity the capital reserve requirement for holding a position in X  with the size of the bid-ask spread, since it is the charge to be paid for unwinding in the presence of otherwise prefect market hedges. Similarly the difference between the mid and the risk-neutral price is the average profit that is disseminated between market participants by trading. Note that the simple binomial state example in this post is not rich enough to feature non-equal mid and risk neutral expected prices.

In summary we find the distorted expectation of the long asset to be lower than the (absolute) price of the short asset. In accordance with the discussion above we identify the bid-ask prices from the asset and the liability valuated on the distorted market. In our simple example the bid and ask prices are centered around the risk-neutral mid price.

Ask: $106.66 (liability)
Bid: $93.33 (asset)
Mid: $100
Spread: $13.33

This concludes this note on the concept of conic finance illustrated on an elementary example in the context of bid-ask spreads. The methodology has a rich area of application to markets such as expected profits, capital requirements, leveraging, hedging etc, which can be found under the reference link to An Introduction to Conic Finance and its Applications.

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