8.2 Optimal strategies without market impact
8.2.3 Calibration and test
8.2 Optimal strategies without market impact 143 Parameter Estimate
κ 9.55×10−5 µ 2.99×10−2 σCIR 9.23×10−4 r −2.23×10−7 σGBM 2.51×10−4
λ 6.73×10−4
Table 8.2: Model parameter estimates computed from order book data (AAPL, February 17, 2010, 10am-11am).
144 Optimal ’Peg-Cross’ strategies
α Optimal boundary GBM Optimal boundary CIR
8×10−5 1,1 ticks 1,0 ticks
16×10−5 2,2 ticks 2,0 ticks
24×10−5 3,2 ticks 3,1 ticks
31×10−5 4,2 ticks 4,0 ticks
Table 8.3: Optimal boundaries for different values of α computed from order book data (AAPL, February 17, 2010, 10am-11am).
Brownian motion (GBM) and CIR model, using the formulae from theorems 8.2 and 8.4. The results are displayed in table 8.3. Note that the tick size is 1 cent.
When the spreadsis below or equal to the optimal boundary displayed in table 8.3, theorems 8.2 and 8.4 say that it is optimal to cross the spread and cancel the outstanding limit order.
Finally, we will see how the theoretical optimal boundaries perform when tested against real order book data. To carry out this test, we test how the optimal boundaries computed from the 10am-11am data perform, when tested with orderbook data of the same stock, one hour later. For this delayed data, we compute the average costs for six different strategies associated to buying one single share:
’Cross’ The costs associated to crossing immediately.
’n ticks’ The costs associated to pegging on the best bid until the limit order is executed and crossing as soon as the spread size is ≤nticks, for n= 1, ...,5.
’Peg’ The costs associated to pegging on the best bid until the limit order is executed.
Figure 8.1 displays the costs associated to the six strategies for a risk-neutral trader (i.e. with α= 0). Since we normalized the price of the best bid to zero in our model, we will compute the minimum costs of the six strategies from the data and set it to zero. The bars show the average additional cost in US dollar of buying one share using the corresponding strategy, compared to the strategy with minimum costs. For the risk-neutral trader, we see that the minimum cost strategy is pegging on the best bid. This corresponds to our analytical result: when there is no penalization for holding the shares, the best strategy is ’Peg’, i.e. the trader waits until the limit order is executed, irrespective of the current spread. The second best strategy is ’1 tick’, the third-best ’2 ticks’, and so on, and finally ’Cross’ has the highest additional costs compared to ’Peg’.
Figure 8.2 displays the costs associated to the six strategies for different risk-averse traders (i.e.
with different levels of risk-aversionα). We see that forα = 8×10−5, the optimal strategy is ’1 tick’, which corresponds to our analytic results, for both GBM and CIR process. Similary, for α = 16×10−5, the optimal strategy obtained from analytic and empirical results match and consist in crossing as soon as the spread is lower or equal to 2 ticks. Forα= 24×10−5, however, the empirical result suggests an optimal crossing boundary of 2 ticks, whereas the analytical result in this section suggests that it should be equal to 3 ticks. Yet, the additional costs of following strategy ’3 ticks’ instead of the optimal empirical strategy are rather low. Similary, for α= 24×10−5 the optimal empirical and analytical strategies do not correspond (’3 ticks’
vs. ’4 ticks’), but the corresponding additional cost is rather low.
8.2 Optimal strategies without market impact 145
Figure 8.1: Costs associated to different strategies for risk-neutral trader (α= 0) computed from order book data (AAPL, February 17, 2010, 11am-12am).
Comparing the optimal boundaries of GBM and CIR model in table 8.3, we note that - for this special case of parameters - there is not much difference between the two models: both models have roughly the same optimal boundaries, with the GBM model giving slightly higher optimal boundaries than the CIR model. Comparing this to the empiricial results in figure 8.2, we see that the CIR model performs slightly better in practice, but the difference is negligible.
The mean benefit of using the best versus the second best strategy is in the range of 1/10 cent to 5/10 cent. However, one needs to keep in mind that this applies each time a single share is bought. For a large trader engaged in high frequency trading, these fractions of a cent may accumulate to a considerable amount.
Even though the theoretical analysis of the model does not correspond exactly to the optimal strategies found empirically, it is surprising that such a simple model performs so well. Clearly, our model has a lot of shortcomings:
We model the spread as a diffusion process, however, in reality it is a discrete process, due to the discrete tick size. This leads to results which are hard to interpret, such as an optimal crossing barrier of 3.2 ticks (see table 8.3).
The time-to-fill is modelled as a simple Poisson process. However, as we saw in chapter 5, it depends heavily on spread and volume-imbalance.
The best bid is assumed to be constant. A more realistic model would reproduce volatility and drift of the best bid, which will influence the price of the limit order placed on the best bid.
These obvious shortcomings had to be introduced for the sake of mathematical tractability.
Despite all this, our the analytical results are very close to the empirical observations.
For practical purposes, I would propose a simple finite-state (or denumerable-state) Markov chain. The state space should at least include best quotes and volume on the best quotes. This model then automatically reflects the discrete tick stucture. Moreover, transition probabilities can easily be estimated from historical data, and will reproduce both movements of best bid/ ask as well as state-dependent time-to-fill. This framework can also easily be extended to include queuing priorities of limit orders. Using a standard dynamic programming algorithm, one can then numerically solve the peg/cross problem with a finite time horizon.
146 Optimal ’Peg-Cross’ strategies
(a)α= 8×10−5 (b) α= 16×10−5
(c) α= 24×10−5 (d) α= 31×10−5
Figure 8.2: Costs associated to different strategies for averse trader, with varying level of risk-aversion computed from order book data (AAPL, February 17, 2010, 11am-12am).