Optimal Liquidation of Derivative Portfolios

Henderson, Vicky; Hobson, David

doi:10.1111/j.1467-9965.2011.00477.x

Cited by 7 publications

(6 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In addition to the classical paper [7], for an introduction to non-cooperative games and Nash equilibria we refer to [3,6,13,14].…”

mentioning

confidence: 99%

A Bidding Game in a Continuum Limit Order Book

Bressan¹,

Facchi²

2013

SIAM J. Control Optim.

View full text Add to dashboard Cite

Abstract. The paper is concerned with a continuum model of the limit order book, viewed as a noncooperative game for n players. An external buyer asks for a random amount X > 0 of a given asset. This amount will be bought at the lowest available price, as long as the price does not exceed a given upper bound P . One or more sellers offer various quantities of the asset at different prices, competing to fulfill the incoming order, whose size is not known a priori.The first part of the paper deals with solutions to the measure-valued optimal pricing problem for a single player, proving an existence result and deriving necessary and sufficient conditions for optimality. The second part is devoted to Nash equilibria. For a general class of random variables X and an arbitrary number of players, the existence and uniqueness of the corresponding Nash equilibrium is proved, explicitly determining the pricing strategy of each player. For a different class of random variables, it is shown that no Nash equilibrium can exist.The paper also describes the asymptotic limit as the total number of players approaches infinity, and provides formulas for the price impact produced by an incoming order.Key words. measure-valued optimization, optimality conditions, Nash equilibrium, bidding game, optimal pricing strategy, limit order book, price impact AMS subject classifications. 49K21, 49J21, 91A06, 91A13, 91A60.1. Introduction. This paper is concerned with a continuum model of the limit order book in a stock market, viewed as a noncooperative game for n players. Our main goal is to study the existence and uniqueness of a Nash equilibrium, determining the optimal bidding strategies of the various agents who submit limit orders.We consider a one-sided limit order book. In our basic setting, we assume that an external buyer asks for a random amount of X > 0 of shares of a certain asset. This external agent will buy the amount X at the lowest available price, as long as this price does not exceed a given upper bound P . One or more sellers offer various quantities of this asset at different prices, competing to fulfill the incoming order, whose size is not known a priori.Having observed the prices asked by his competitors, each seller must determine an optimal strategy, maximizing his expected payoff. Of course, when other sellers are present, asking a higher price for a stock reduces the probability of selling it.In our model we assume that the i-th player owns an amount κ i of stock. He can put all of it on sale at a given price, or offer different portions at different prices. In general, his strategy will thus be described by a measure µ i on IR + , where µ i ([0, p]) denotes the total amount of shares put on sale by the i-th player at a price ≤ p.In practice, it is clear that prices can take only a discrete set of values. However, by studying a continuum model where strategies are described by Radon measures one obtains clear-cut results on existence or non-existence of Nash equilibria, and clean, explicit solution formulas. In ge...

show abstract

“…In addition to the classical paper [7], for an introduction to non-cooperative games and Nash equilibria we refer to [3,6,13,14].…”

mentioning

confidence: 99%

A Bidding Game in a Continuum Limit Order Book

Bressan¹,

Facchi²

2013

SIAM J. Control Optim.

View full text Add to dashboard Cite

show abstract

“…If at some time it is not the first-to-be-exercised, thresholds for all options exercised before it, including the first-to-be-exercised, are reduced since the portfolio remaining after their exercise now includes the new option. 44 We know for a single grant of identical options, as the number of options becomes infinite, the individual exercise thresholds will limit to the option strike K, see Henderson and Hobson (2011).…”

Section: Larger Portfoliosmentioning

confidence: 99%

Executive Stock Options: Portfolio Effects

Henderson

Whalley

Sun³

2012

SSRN Journal

Self Cite

View full text Add to dashboard Cite

Since executives typically receive new grants of stock options (ESOs) each year, longerserving executives often have portfolios of ESOs with differing strikes and maturities. Valuation models for stand-alone ESO grants have shown that trading restrictions, which force executives to bear unhedgeable risk until the options are exercised, induce earlier exercise and hence a lower cost to shareholders than in a risk-neutral setting. However, since unhedgeable risk varies non-linearly with portfolio size and composition, the executive's exercise strategy and thus also the shareholder cost of an option held as part of a portfolio depend also on the remainder of the executive's ESO portfolio. We show that such portfolio effects matter -lowering both the moneyness required for exercise and the shareholder cost of most options in a portfolio. In contrast to a risk-neutral setting, both exercise thresholds and costs depend on an option's position in the optimal exercise order. An option's cost varies with the maturities and, nonlinearly, with the strike prices of other options in the executive's portfolio. The model explains several empirical findings in the literature -which options are attractive to exercise first, how exercise can be induced by a new grant, and the prevalence of early exercise.

show abstract

“…Numerical solutions in a parametric model can be straightforwardly obtained using the sequential representation of U in (5.2). Finally, one can also consider more complex models of contract accumulation/liquidation following the methods in [14].…”

mentioning

confidence: 99%

Optimal Timing to Purchase Options

Leung¹,

Ludkovski²

2011

SIAM J. Finan. Math.

View full text Add to dashboard Cite

We study the optimal timing of derivative purchases in incomplete markets. In our model, an investor attempts to maximize the spread between her model price and the offered market price through optimally timing her purchase. Both the investor and the market value the options by risk-neutral expectations but under different equivalent martingale measures representing different market views. The structure of the resulting optimal stopping problem depends on the interaction between the respective market price of risk and the option payoff. In particular, a crucial role is played by the delayed purchase premium that is related to the stochastic bracket between the market price and the buyer's risk premia. Explicit characterization of the purchase timing is given for two representative classes of Markovian models: (i) defaultable equity models with local intensity; (ii) diffusion stochastic volatility models. Several numerical examples are presented to illustrate the results. Our model is also applicable to the optimal rolling of long-dated options and sequential buying and selling of options.Comment: 24 pages, 6 figures; http://dx.doi.org/10.1137/10080938

show abstract

Optimal Liquidation of Derivative Portfolios

Cited by 7 publications

References 22 publications

A Bidding Game in a Continuum Limit Order Book

A Bidding Game in a Continuum Limit Order Book

Executive Stock Options: Portfolio Effects

Optimal Timing to Purchase Options

Contact Info

Product

Resources

About