Algorithmic And High – Frequency Trading – Alvaro Cartea

Then, the agent’s perfonnance criterion is Hv (t, X, S, q) = lEt,x,S,q [ X!f + Qf (S” -aQf)- ¢.f (Q) 2 du], ‘-y-/ ——-  Terminal Cash Tcnninal Execution Inventory Penalty (6.20) and the value function H(t, x, S, q) = sup Hv(t, x, S, q) . vEA The DPP implies that the value function should satisfy the HJB equation 0= (at+½a.2ass)H -¢q2 + sup {(v (S – f(v)) ax – g(v) as – vaq) H}, (6.21) subject to the terminal condition H(T, x, S, q) = x +Sq – a q2 .

We use the simplifying assumption that permanent and temporary price im­ pact functions are linear in the speed of trading, i.e. f (v) = k v and g(v) = b v for finite constants k 2: 0 and b 2: 0. The first order condition allows us to obtain the optimal speed of trading in feedback control form as * 1 (Sa,r – bas-aq)H v =21c aH (6.22) Upon substituting the optimal feedback control into the DPE, it reduces to _ (a 2 a ) H ,,1, 1 [(Sax – bas -aq)H] 0 – t + 2 (}’ ss – ‘+’ q + 4 k axH By inspecting the terminal condition H(T, x, S, q) = x +Sq – a q2, it suggests the ansatz H(t, x, S, q) = .T +Sq+ h(t, S, q) , (6.23) where h, with terminal condition h(T, S, q) = -a q2, is yet to be determined.

The first term of the ansatz is the accumulated cash of the strategy, the second is the marked-to-market book value (at midprice) of the remaining inventory, and h is the extra value stemming from optimally liquidating the rest of the shares. Using this ansatz in the equation above and simplifying, we find the following non-linear PDE for h: o =(at+ ½(}’2 ass) h -¢q2 + A [b (q +ash)+ aq hJ Since the above PDE contains no explicit dependence on S and the terminal condition is independent of S, it follows that ash(t, S, q) = 0, and we can write h(t, S, q) = h(t, q) (with a slight abuse of notation).

The design of trading algorithms requires sophisticated mathematical models, a solid anal- of financial data, and a deep understanding of how markets and exchanges function. In this textbook the authors develop models for algorithmic trading in contexts such as: executing large orders, market making, targeting VWAP and other schedules, trading pairs or collection of assets, and executing in dark pools. These models are grounded on how the exchanges work, whether the algorithm is trading with better informed traders (adverse selection), and the type of information available to market participants at both ultra-high and low frequency.

Algorithmic and High-Frequency Trading is the first book that combines sophisticated mathematical modelling, empirical facts and financial economics, taking the reader from basic ideas to the cutting edge of research and practice. If you need to understand how modern electronic markets operate, what information provides a trading edge. and how other market participants may affect the profitability of the algorithms, then this is the book for you. AL VAR o c ARTE A is a Reader in Financial Mathematics at University College London.

Before joining UCL he was Associate Professor of Finance at Universidad Carlos III, Madrid-Spain (2009-2012) and from 2002 until 2009 he was a Lecturer (with tenure) in the School of Economics, Mathematics and Statistics at Birkbeck – University of London. He was previously JP Morgan Lecturer in Financial Mathematics at Exeter College, University of Oxford. s EB As TI AN J A IM u NG AL is an Associate Professor and Chair, Graduate Studies in the Department of Statistical Sciences at the University of Toronto where he teaches in the PhD and Masters of Mathematical Finance programs.

He consults for major banks and hedge funds focusing on implementing advance derivative valuation engines and algorith­ mic trading strategics. He is also an associate editor for the SIAM Journal on Financial Mathematics, the International Journal of Theoretical and Applied Finance, the journal Risks and the Argo newsletter. Jaimungal is the Vice Chair for the Financial Engineering & Mathematics activity group of SIAM and his research is widely published in academic and practitioner journals.

His recent interests include High-Frequency and Algorithmic trading, applied stochastic control, mean-field games, real options, and commodity models and derivative pricing . .r o sf: PEN AL v A is an Associate Professor at the Universidad Carlos Ill in Madrid where he teaches in the PhD and Master in Finance programmes, as well as at the undergraduate level. He is currently working on information models and market microstructure and his research has been published in Econometrica and other top academic journals.

ALGORITHMIC AND HIGH-FREQUENCY TRADING ALVARO CARTEA University College London SEBASTIAN JAIMUNGAL University of Toronto JOSE PENALVA Universidad Carlos III de Madrid CAMBRIDGE UNIVERSITY PRESS CAMBRIDGE UNIVERSITY PRESS University Printing Honse, Cambridge CB2 8BS, United Kingdom Cambridge University Press is part of the University of Cambridge.

This is a short excerpt from the opening of “” by Unknown, quoted for review and introduction purposes. All rights belong to the copyright holders.

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