# Papers and algorithms on bidding schemes for best order execution?

I'm building an automated option trading bot that executes common options multi-leg strategies (straddles, spreads) and I want to learn the best way to execute my orders.

The naive execution is to just submit a order at the full size at the mid of the bid/ask of all of your spreads. But that order is rarely best execution and if it is filled right away, you know that you could've gotten a better price.

So I'd like to seek out some papers/algorithms that can come up with the best execution for both discretionary orders and must-filled hedging orders.

It is possible to think of a strategy for splitting orders for one large sell over time as a function $x_t$ which describes how much to sell at each timestep $t$. If the instantaneous trading rate $\dot{x_t}$ is too large, i.e. too much is sold at once you get immediate impact which is bad. If selling takes too much time, there is the risk of negative price movements, so both should be balanced.

Solution: Gatheral and Schied have suggested an approach to this problem to optimize trade execution under market impact. The idea is to model two components of market impact into the SDE, permanent impact and short term impact. The result is an optimization problem of the form

$$minimize\quad E\left[\int_0^T (\dot{x}^2_t+ k^2x^2+\lambda x_t S_t)dt\right]$$

where $k$ and $\lambda$ are parameters for immediate and permanent impact, and $\dot{x}$ is the instantaneous speed of selling the asset.

References:

• One nice point of their formulation is that they are able to derive a closed form solution for the optimal strategy, so this can be evaluated quickly.
• They do not include trading costs, so I don't know how that impacts your optimal strategy.

With respect to what you need, you have to consider different aspects of optimal trading:

• the Almgren-Chriss framework (cited by Anna, since Jim and Alex -amongst others- extended it) focus on obtaining an optimal trading rate, it is nice but not really what you need. You can nevertheless use it to plan / schedule your trading during the day.
• but what you need is to obtain prices at which your algorithm need to post orders; you have Avellaneda-Stoikov like frameworks. I believe the most advanced form is in the two Guéant-Fernandez-L papers:
• Guéant, O., Lehalle, C.-A., Fernandez-Tapia, J., Sep. 2012. Dealing with the inventory risk: a solution to the market making problem. Mathematics and Financial Economics. URL http://arxiv.org/abs/1105.3115
• Guéant, O., Lehalle, C.-A., Fernandez-Tapia, J., 2012. Optimal Execution with Limit Orders. SIAM Journal on Financial Mathematics 13 (1), 740-764. http://arxiv.org/abs/1106.3279
• Then you have literature on hedging with transaction costs, one typical paper is D. Possamaï, N. Touzi and M. Soner, Homogenization and asymptotics for small transaction costs: the multidimensional case. arXiv:1212.6275. Another paper of interest for you is Stoikov-Saglam's one, since it simultenaously make the market on the option and the underlying stock: Option Market Making Under Inventory Risk.

Is it enough to solve your problem? unfortunately not! you will need a lot of empirical work to put all this together since there is not "off the shelf" solution by now. I would just attract you attention to a paper recently accepted by Quantitative Finance: Realtime market microstructure analysis: online transaction cost analysis - by Azencott, Beri, Gadhyan, Joseph, L, Rowley (2013) http://arxiv.org/abs/1302.6363 . In this paper we develop an approach to conduct empirical analysis of market microstructure in real-time.

• +1 for "you will need a lot of empirical work", down to earth (and correct) view. Feb 9, 2014 at 7:55
• I liked yours too @Anna , it would be great to undelete it since the references are adequate. Feb 9, 2014 at 20:32

I can speak from experience that options with next to no volume and ridiculously large spreads have market makers that accept nothing short of 5% effective spreads, right below liquidation value for deep in the money, and quickly nothing for out of the money.

Also, the parameters should be expected to move against your fund flows very quickly. I've found it difficult to enter and exit large positions.

If one takes the approach of walking orders from the bid to the ask, the bid will become quickly crowded out, and the parameters move even faster. The same is true for walking from the ask.

If one tries to make a good estimation of the parameters, they seem to move more slowly.

Parameters don't move quite as much when trading the position instead of entering and exiting.

This is probably common knowledge, but I had to experience to believe it.