We get the auction data of all the artists from the site d' Artprice. Artists have been broken down into three categories as they are usually by sales houses: contemporary artists whose works were mostly produced after the World War II, modern artists who begin with Edgar Degas and the impressionist movement, and finally the old masters who put together all the other artists.
We select artists whose at least one artwork has already reached the million-dollar mark. For each work of the artist, the ratio between the hammer price and the low estimate is calculated, or zero if the auction did not reach the reserve price.
auction basquiat


The idea is to consider for each artwork awarded, the hammer price ratio by the low estimate, and select for each artist the most recent auctions depending on the number of records available over the period. This methodology makes all sales homogeneous and eliminates the influence of most bias factors such as the type, style and dimensions of the work, the exchange rate and inflation, place of sale or commission rates. As the majority sales houses start auctions between 60% and 80% of the low estimate, this solution allows to build up remarkably homogeneous distributions for the same artist and also between the various artists.
The distribution thus consists of two parts: the first one is a measure of Dirac located in zero and whose weight is the rate of Unsold, and the second part is a Weibull law with two parameters d and s whose weight is: 1 - Unsold. The density of the Weibull distribution is written:
and the correponding repartition function :
The parameter d is called dispersion and measures the artist's performance during public auctions. The paramleter s is called sharpness and its inverse measures prices volatility when they exceed the reserve price. The estimation of the parameters of the distribution law is performed using the maximum log-likelihood method.


When a seller hesitates to put a valuable artwork in auction for fear of a bad sale, the sales house may offer a guaranteed price, usually above the high estimate, backed up by an option on auction. The theoretical initial value of the option is equal to the guaranteed price minus the reserve price. The seller is assured of receiving the guaranteed price if the hammer price exceeds the reserve price. This mechanism encourages the seller to propose a lower reserve price, and therefore increases the expectation of the sales house to receive a commission on the sale of the work.
The strike K of the option is calculated as the break even, rounded up to $10K, between the initial theoretical value of the option and the present value of the option evaluated by applying the Weibull distribution law of the artist, plus the commission for issuing warrants. If rp and k stand for the reserve price and the strike divided by the low estimate, the value k is the solution of the equation:
where r is the risk-free rate of the currency in which the sale price is expressed and T is the time to expiry until the auction date. The numerical resolution of the equation is performed through trisection and Newton-Raphson algorithms, and using the trapezoidal method for the integral calculation.
As an artist's rating is not tradable on financial markets, the probabilistic universe built to evaluate the option's price is not complete and therefore there is no neutral risk universe in which it would be possible to replicate the option's return with an asset and cash. Strictly speaking, the above formula should include a market risk premium, corresponding to the correlation between the artist's rating and the general index of financial markets. It is assumed that in general this correlation is zero, and therefore the market risk premium is zero, which justifies the validity of the formula. The quantity: K * (1 - fees) represents the price guaranteed to the seller by the sales house. The bank is then responsible for securitizing the option call warrants with a parity equal to the inverse of the number of securities issued, and with a quotity which represents the minimum number of warrants bought by each investor. The issue price of each warrant is therefore equal to:
warrant price
The Greeks are calculated according to the variations of the different parameters of the pricing formula: the risk-free rate r for the rho, the maturity T of the option for theta, the performance d for delta and gamma, and volatility, inverse of sharpness s, for vega and volga.
The bank is also responsible for marketing warrants to its customers and facilitating the secondary market by proposing a purchase price and an offer price of the warrant until expiration, which matches the auction date of the artwork by the sales house. The first investors interested in buying call warrants are potential bidders of the artwork who will have interest in raising the highest possible auctions to increase leverage, and make the deal even more attractive to other investors.


Weibull's distribution law associated with the results of each artist's auctions can be represented in a volatility/performance graph by a point whose corresponding X-axis opposite the sharpness parameter s and the ordinate to the dispersion parameter d. Art investors aim to maximize performance pc of their collection by minimizing the volatility vc. That means the goal is to determine the optimal weights of each artist in the collection for a given performance to minimize the risks.
By solving this constrained minimization problem, we show using the calculated values performances and volatilities for each artist, based on the results of the auctions contained in the database, that the efficient border is a branch of hyperbole that verifies approximately the equation:
The singular point of the hyperbole represents a performance of 1 (corresponding to an average of hammer prices equal to the low estimate) and a volatility of 20%. The points of the graph volatility/performance corresponding to Weibull's distribution laws of the main artists, and the efficient border curve, are shown in the graph below:
efficient border
Artists closest to the efficient border represent the best investments for collectors. Among contemporary artists, the best ones are currently KAWS, Georges Condo, A.R. Penck and Keith Haring. Among the modern ones, we find Edgar Degas, Vincent Van Gogh, the American realistic painter Edward Hopper, Edvard Munch and Emil Nolde. Among the old masters are Nicolas Poussin, Jacques-Louis David and Jean-François Millet. Among contemporary French painters, Pierre Soulages and Robert Combas present the best performances.
The point on the graph and the rate of unsold allow to determine which artists are likely to have artworks backed up by option on auction. The first condition is that the low estimate must exceed the million dollars mark. For the seller, the work must present a significant risk of not being sold or poorly sold. For investors in call warrants, the point of the graph representing the artist must have good volatility and performance, and are as close as possible to the efficient border.
Banksy and Edgar Degas are examples of artists who fit this profile. The shape of the density curves of Weibull distribution laws is characterized by strong flattening and thick tails distribution. However, the works of Pierre Soulages or Vincent Van Gogh present unsold rates too low, so do not pose any risk to their owners. In the opposite, investors in call warrants would not be interested in investing in warrants about artworks by Matisse or Andy Wahrol because the distribution profiles present too low performance in relation to their volatility.
van gogh