Large tick assets: implicit spread and optimal tick value Khalil Dayri 1 and Mathieu Rosenbaum 2 1 Antares Technologies 2 University Pierre and Marie Curie (Paris 6) 15 February 2013 Khalil Dayri and Mathieu Rosenbaum Implicit spread and optimal tick value 1
Outline Tick value, tick size and spread 1 Tick value, tick size and spread 2 3 4 Khalil Dayri and Mathieu Rosenbaum Implicit spread and optimal tick value 2
Outline Tick value, tick size and spread 1 Tick value, tick size and spread 2 3 4 Khalil Dayri and Mathieu Rosenbaum Implicit spread and optimal tick value 3
Tick value Definitions The market fixes a price grid on which traders can place their prices. The smallest interval between two prices is called the tick value, measured in the currency of the asset. The market may change the tick value. Also, in some markets, the spacing of the grid can depend on the price. For example, stocks trading on Euronext Paris have a price dependent tick scheme. Stocks priced 0 to 9.999 have a tick value of 0.001 but all stocks above 10 have a tick of 0.005. Here we will consider time periods so that for a given security, the tick grid is evenly spaced. Khalil Dayri and Mathieu Rosenbaum Implicit spread and optimal tick value 4
Tick size Tick value, tick size and spread Notion of tick size When it comes to actual trading, the tick value is given little consideration. What is important is the tick size. A trader considers that an asset has a small tick size when he feels it to be negligible, in other words, when he is not averse to price variations of the order of one single tick. Khalil Dayri and Mathieu Rosenbaum Implicit spread and optimal tick value 5
Tick size (2) Tick value vs tick size The trader s perception of the tick size is qualitative and empirical, and depends on many parameters such as the tick value, the price, the usual amounts traded in the asset, and even his own trading strategy. The tick value is not a good measure of the perceived size of the tick. For instance, every trader considers that the ESX future contract has a much greater tick size than the DAX index future contract, though the tick values are of the same orders. Khalil Dayri and Mathieu Rosenbaum Implicit spread and optimal tick value 6
Large tick asset and spread What is a large tick asset? From Eisler, Bouchaud and Kockelkoren : Large tick stocks are such that the bid-ask spread is almost always equal to one tick, while small tick stocks have spreads that are typically a few ticks. This leads to the following questions : Khalil Dayri and Mathieu Rosenbaum Implicit spread and optimal tick value 7
Large tick asset and spread (2) Issues For small tick assets, the spread is a good measure for the tick size. In the case where the spread is almost always equal to one tick, how to quantify the tick size? Many studies have pointed out special relationships between the spread and some market quantities. However, these studies reach a limit when discussing large tick assets since the spread is artificially bounded from below. How to extend these studies in the large tick case? What happens to the relevant market quantities when the tick value is changed and what is the optimal tick value? Khalil Dayri and Mathieu Rosenbaum Implicit spread and optimal tick value 8
Spread theory for small tick assets Madhavan, Richardson, Roomans economic model p i : ex post true or efficient price after the i-th trade (all transactions have the same volume), ε i : sign of the i-th trade. The MRR model is defined by : p i+1 p i = ξ i + θε i, with ξ i an independent centered shock component (new information,... ) with variance v 2. Khalil Dayri and Mathieu Rosenbaum Implicit spread and optimal tick value 9
Spread theory for small tick assets (2) MRR model (2) Market makers cannot guess the surprise of the next trade. So, they post (pre trade) bid and ask prices a i and b i given by a i = p i + θ + φ, b i = p i θ φ, with φ an extra compensation claimed by market makers, covering processing costs and the shock component risk. The above rule ensures no ex post regrets for market makers (if φ = 0 the traded price is in average the right one). If φ = 0, the ex post average cost of a market order with respect to the efficient price a i p i+1 or p i+1 b i is equal to 0. Khalil Dayri and Mathieu Rosenbaum Implicit spread and optimal tick value 10
Spread theory for small tick assets (3) MRR model (3) We can compute several quantities : The spread S = a b = 2(θ + φ). The volatility per trade of the efficient price σ1 2 = E[(p i+1 p i ) 2 ] = θ 2 + v 2 θ 2 (the news component being negligible, see Wyart et al.). Therefore : S 2σ 1 + 2φ. Khalil Dayri and Mathieu Rosenbaum Implicit spread and optimal tick value 11
The Wyart et al. approach Market making strategy Market makers are patient traders who prefer to send limit orders and wait to be executed, thus avoiding to cross the spread but taking on volatility risk. Market takers are impatient traders who prefer to send market orders and get immediate execution, thus avoiding volatility risk but crossing the spread in the process. Wyart et al. consider a simple market making strategy and show that its average P&L per trade is S 2 c 2 σ 1, with c depending on the assets but of order 1 2. This P&L corresponds to the average cost of a market order. Khalil Dayri and Mathieu Rosenbaum Implicit spread and optimal tick value 12
The Wyart et al. approach (2) Market maker vs market taker Wyart et al. argue that on electronic market, any agent can choose between market orders and limit orders. So the market should stabilize so that both types of orders have the same average (ex post) cost, that is zero. In particular, market makers do not make profit (if so another market maker comes with a slightly tighter spread). Therefore : S cσ 1. This relationship is very well satisfied on market data. Khalil Dayri and Mathieu Rosenbaum Implicit spread and optimal tick value 13
Outline Tick value, tick size and spread 1 Tick value, tick size and spread 2 3 4 Khalil Dayri and Mathieu Rosenbaum Implicit spread and optimal tick value 14
Model with uncertainty zones Properties of this model Model for transaction prices and durations, based on an efficient semi-martingale type price. Essentially one important scalar parameter : η. Reproduces almost all the stylized facts of (ultra) high frequency and low frequency data. Originally built in the purpose of high frequency statistical estimation and hedging. Khalil Dayri and Mathieu Rosenbaum Implicit spread and optimal tick value 15
Aversion for price changes Aversion for price changes In an idealistic framework, transactions would occur when the efficient price crosses the tick grid. In practice, uncertainty about the efficient price and aversion for price changes of market participants. The price changes only when market participants are convinced that the efficient price is sufficiently far from the last traded price. We introduce a parameter η quantifying this aversion for price changes. Khalil Dayri and Mathieu Rosenbaum Implicit spread and optimal tick value 16
Model with uncertainty zones (simplified) Model with uncertainty zones : notation Efficient price : X t. α : tick size. t i : time of the i-th transaction with price change. P ti : transaction price at time t i. Uncertainty zones : U k = [0, ) (d k, u k ) with d k = (k + 1/2 η)α and u k = (k + 1/2 + η)α. Khalil Dayri and Mathieu Rosenbaum Implicit spread and optimal tick value 17
Model with uncertainty zones (simplified) (2) Model with uncertainty zones : dynamics d log X u = a u du + σ u dw u. t i : i-th exit time of an uncertainty zone : t i+1 = inf { t > t i, X t = X (α) t i ± α( 1 2 + η)}, with X (α) t i the value of X ti rounded to the nearest multiple of α. P ti = X (α) t i. Khalil Dayri and Mathieu Rosenbaum Implicit spread and optimal tick value 18
Model with uncertainty zones 101.5 101 100.5 Price 100 99.5 2ηα α 99 0 50 100 150 200 250 300 Time Khalil Dayri and Mathieu Rosenbaum Implicit spread and optimal tick value 19
Estimation of η Estimation of η The parameter η can be very easily estimated. Let N α,t = card{t i, t i t} and N N (c) α,t α,t = I {(Pti P ti 1 )(P ti 1 P ti 2 )>0}, i=2 We define N N (a) α,t α,t = I {(Pti P ti 1 )(P ti 1 P ti 2 )<0}. i=2 ˆη t = N(c) α,t 2N (a) α,t. Khalil Dayri and Mathieu Rosenbaum Implicit spread and optimal tick value 20
Bund and DAX, estimation of η, October 2010 Day η (Bund) η (FDAX) Day η (Bund) η (FDAX) 1 Oct. 0.18 0.41 18 Oct. 0.16 0.33 5 Oct. 0.15 0.37 19 Oct. 0.13 0.37 6 Oct. 0.15 0.37 20 Oct. 0.13 0.33 7 Oct. 0.15 0.38 21 Oct. 0.15 0.33 8 Oct. 0.15 0.41 22 Oct. 0.11 0.33 11 Oct. 0.14 0.36 25 Oct. 0.12 0.31 12 Oct. 0.14 0.36 26 Oct. 0.14 0.31 13 Oct. 0.14 0.32 27 Oct. 0.14 0.32 14 Oct. 0.16 0.35 28 Oct. 0.14 0.32 15 Oct. 0.16 0.35 29 Oct. 0.14 0.34 Khalil Dayri and Mathieu Rosenbaum Implicit spread and optimal tick value 21
Buy only, sell only and buy/sell areas The market order areas We assume for simplicity that the bid-ask spread is constant equal to α and that the efficient price X t satisfies X t = σw t, with W a Brownian motion. For given bid-ask quotes, the model enables to define in term of the efficient price buy only, sell only and buy/sell areas. We call them respectively ask zone, bid zone and buy/sell zone. Khalil Dayri and Mathieu Rosenbaum Implicit spread and optimal tick value 22
Ask Zone, Bid Zone and Buy/Sell Zone 102 101.5 101 ask=101 Ask Zone Price 100.5 α = Spread 2ηα = Buy/Sell Zone 100 bid=100 Bid Zone 99.5 99 0 100 200 300 400 500 600 700 Time Khalil Dayri and Mathieu Rosenbaum Implicit spread and optimal tick value 23
Some intuitions Intuitions about η The size of the buy/sell zone is 2ηα. If η is small, there is a lot of mean reversion in the price and the buy/sell zone is very small : the tick size is very large. If η is close to 1/2, the last traded price can be seen as a sampled Brownian motion, there is no microstructure effects and the buy/sell zone is equal to one tick : the tick size is, in some sense, optimal. 2ηα can be seen as a kind of implicit spread. So, if M denotes the number of trades over the considered period, can we extend the relationship : S 2 σ to ηα M σ M? Khalil Dayri and Mathieu Rosenbaum Implicit spread and optimal tick value 24
Outline Tick value, tick size and spread 1 Tick value, tick size and spread 2 3 4 Khalil Dayri and Mathieu Rosenbaum Implicit spread and optimal tick value 25
Setup Tick value, tick size and spread The assets We want to investigate the relationship for large tick assets. ηα σ M + φ We consider Futures on : the DAX index (DAX), the Euro-Stoxx 50 index (ESX), the Dow Jones index (DJ), SP500 index (SP), 10-years Euro-Bund (Bund), 5-years Euro-Bobl (Bobl), 2-years Euro-Schatz (Schatz), 5-Year U.S. Treasury Note Futures (BUS5), EUR/USD futures (EURO), Light Sweet Crude Oil Futures (CL). Khalil Dayri and Mathieu Rosenbaum Implicit spread and optimal tick value 26
Cloud (ηα M, σ), for each day, for each asset 1500 1000 Dax DJ EURO BUS5 CL Bobl Bund Schatz Eurostoxx SP CL Dax EURO σ SP Bund 500 Eurostoxx DJ BUS5 Bobl Schatz 0 0 100 200 300 400 500 600 700 800 900 1000 ηα M For each asset : linear relationship, same slope, different intercepts. Khalil Dayri and Mathieu Rosenbaum Implicit spread and optimal tick value 27
Regression design Linear regression We consider the relationship ηα σ M + φ for large tick assets. φ includes operational costs/profits related to the inventory control so we take φ = k S. Daily regression : σ = p 1 ηα M + p 2 S M + p 3. Khalil Dayri and Mathieu Rosenbaum Implicit spread and optimal tick value 28
Daily regression 1.8 1.6 1.4 p1 1.2 1 0.8 0.6 Dax EURO DJ BUS5 CL Bobl Bund Schatz Eurostoxx SP 0.1 p2 0.05 0 Dax EURO DJ BUS5 CL Bobl Bund Schatz Eurostoxx SP 50 0 p3 50 100 150 200 Dax EURO DJ BUS5 CL Bobl Bund Schatz Eurostoxx SP Khalil Dayri and Mathieu Rosenbaum Implicit spread and optimal tick value 29
Checking that the constant is equal to zero 1400 1200 1000 Dax DJ EURO BUS5 CL Bobl Bund Schatz Eurostoxx SP Dax EURO σ p2s M 800 600 CL 400 200 DJ SP Bund BUS5 Bobl Eurostoxx Schatz 0 0 200 400 600 800 1000 1200 p 1 ηα M Khalil Dayri and Mathieu Rosenbaum Implicit spread and optimal tick value 30
Cost analysis Market orders cost In our setup, it can be easily shown that the average ex post cost of a market order is α/2 ηα. Therefore, since the average P&L per trade of the market makers is equal to the average cost of a market order, we exactly derive ηα = c σ + φ. M Thus limit orders are profitable whereas market orders are costly. However, many market participants try to make profit from this, therefore, the individual gains remain small. Khalil Dayri and Mathieu Rosenbaum Implicit spread and optimal tick value 31
Explanation of microstructure effects Signature plot Assume we have price observations P t at times t = i/n, n N, i = 0,..., n, where t = 1 represents for example one trading day. The signature plot is the function which to k = 1,..., n associates RV n (k) = n/k 1 i=0 (P k(i+1)/n P ki/n ) 2. If P t is a continuous semi-martingale, as soon as (n/k) is large enough, RV n (k) stabilizes. A distinctive feature of high frequency data, particularly of large tick assets, is the decreasing behavior of this signature plot. Khalil Dayri and Mathieu Rosenbaum Implicit spread and optimal tick value 32
Explanation of microstructure effects (2) Bund Signature plot 1 2 3 4 5 6 Oct06 Nov06 Feb07 2 4 6 8 10 Dyadic subsampling (calendar time) Khalil Dayri and Mathieu Rosenbaum Implicit spread and optimal tick value 33
Explanation of microstructure effects (3) Modeling the signature plot Many models aim at reproducing this decreasing shape. However, there are only few agent based explanations for this phenomenon. Our approach enables us to provide a very simple one. Khalil Dayri and Mathieu Rosenbaum Implicit spread and optimal tick value 34
Explanation of microstructure effects (4) Explaining the signature plot Recall that the ex post expected cost of a market order is α/2 ηα. This does explain why for large tick assets with average spread close to one tick, the parameter η is systematically smaller than 1/2, which means the signature plot is decreasing. Otherwise we would be in a situation where the cost of market orders is negative and market makers lose money. To avoid that, market makers would naturally increase the spread, which they can always do. Khalil Dayri and Mathieu Rosenbaum Implicit spread and optimal tick value 35
Outline Tick value, tick size and spread 1 Tick value, tick size and spread 2 3 4 Khalil Dayri and Mathieu Rosenbaum Implicit spread and optimal tick value 36
Changing the tick value Consequences of a change Market platforms face the question of choosing a tick value A tick value that is too small encourages free-riding, where market participants jump marginally ahead of market makers and others who suffer the time and expense of determining at which level they should place their bid and ask quotes. Free-riding discourages market makers and tends to suppress liquidity. It creates messy order books and forces people to make absurd judgments about prices. One has certainly no rational basis for assessing the price of, say, Microsoft, down to the level of fractions of a penny. Khalil Dayri and Mathieu Rosenbaum Implicit spread and optimal tick value 37
Changing the tick value (2) Consequences of a change (2) At the same time, a tick value that is too large creates needless frictions or sloppiness in pricing. One may not have a rational basis for pricing Microsoft in fractions of a penny, but certainly has a rational basis for pricing it in multiples of a dollar. It is usually acknowledged that it is not possible to have an a priori idea of what is the right tick value. Thus, a market designer could only determine, after the fact, whether his chosen tick value has the desired effect, usually adjudged on the basis of price formation, spread, and liquidity. Therefore, it is commonly thought that tick values have to be determined by trial and error. Khalil Dayri and Mathieu Rosenbaum Implicit spread and optimal tick value 38
Changing the tick value (3) Consequences of a change (3) What happens to η if one changes the tick value? How to obtain η close to 1/2? The volatility, p 1, p 2 and the daily traded volume should be invariant after a change of the tick value, however, the number of trades M should not. Khalil Dayri and Mathieu Rosenbaum Implicit spread and optimal tick value 39
Changing the tick value (4) The η equation σ, p 1, p 2 being constant, we get p 1 ηα M + p 2 α M = p 1 η 0 α 0 M0 + p 2 α 0 M0. Assuming the cumulative latent order book is linear : available volume up to price p is equal to c (p p ref ) : α0 η = η 0 α + p 2 α0 p 1 α p 2. p 1 Assuming the cumulative latent order book is concave : available volume up to price p is equal to c (p p ref ) 1/2 : η = η 0 ( α 0 α )3/4 + p 2 p 1 ( α 0 α )3/4 p 2 p 1. Khalil Dayri and Mathieu Rosenbaum Implicit spread and optimal tick value 40
Testing on the Bobl futures 0.5 0.45 α=5 α = 10 linear concave 0.4 0.35 15 6 2009 0.3 η 0.25 0.2 0.15 0.1 0.05 0 0 5 10 15 20 25 day Khalil Dayri and Mathieu Rosenbaum Implicit spread and optimal tick value 41
s New tick values Futures Tick Value β = 1 β = 1/2 BUS5 7.8125 $ 2.7 $ 3.8 $ DJ 5.00 $ 1.6 $ 2.3 $ EURO 12.50 $ 3.1 $ 5.0 $ SP 12.50 $ 0.3 $ 0.9 $ Bobl 1 5.00 e 1.8 e 2.6 e Bobl 2 10.00 e 1.6 e 2.8 e Bund 10.00 e 1.6 e 2.9 e DAX 12.50 e 4.9 e 6.7 e ESX 10.00 e 1.3 e 2.6 e Schatz 5.00 e 0.8 e 1.5 e CL 10.00 $ 3.1 $ 4.6 $ Khalil Dayri and Mathieu Rosenbaum Implicit spread and optimal tick value 42