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There is a vast literature on the extensions of the Merton model. As introduced in Section 1.4, Davis and Norman [13] extend the Merton model into an incomplete financial market with proportional transaction costs. Apart from that, the Merton model is also extended into other two mainstream directions, namely, non-traded assets and asset liquidation.

1.5.1 Non-traded assets

Whilst active trading in financial assets are allowed in the Merton problem, in other contexts dynamic trading is not possible. Svensson and Werner [54] were the first to consider the problem of pricing non-traded assets in the Merton model. In their model, an agent endowed with units of an asset can sell the asset, but may not make purchases. In the simplest case the agent is endowed with a single unit of an indivisible asset which cannot be traded and the problem reduces to an optimal sale problem for an asset. Evans et al [16], see also Henderson and Hobson [28, 30], consider an agent with power utility function who owns an indivisible, non-traded asset and wishes to choose the optimal time to sell the asset in order to maximise the expected utility of terminal wealth

sup ⌧

sup ⇡2A⌧

EU(X⌧,⌧),

where ⌧ is a stopping time. Their results show that the optimal criterion for the sale

of the asset is to sell the first time the value of the non-traded asset exceeds a certain proportion of the agent’s trading wealth and this critical threshold is governed by a transcendental equation. Henderson and Hobson [26] also study the problem in the context of real options, where the investor with power utility function, has a claim on units of non-traded assets correlated with the risky asset and wishes to find

sup ⇡t,0tTE

in a finite horizon. Monoyios [44] considers a similar problem in which the investor has an exponential utility function and the non-traded asset is the underlying of an European option. An explicit solution is obtained for put option by perturbation techniques. Miao and Wang [42] consider a risk averse entrepreneur with exponential utility function who is endowed with a non-traded investment project and has access to another correlated risky asset. The project generates cash flow governed by an arithmetic Brownian motion and the price of the risky asset follows a geometric Brownian motion. By a similar approach in [26], they show that risk aversion delays investment and lowers the value of the project.

1.5.2 Asset liquidation

In a separate strand of literature, Merton’s model is generalised into the context of asset liquidation. Rogers and Singh [49] model illiquidity of a portfolio in the way that investors with large trading volumes have to pay an inflated price. In contrast, Bank and Baum [6] consider a large trader whose trading behaviour impacts the market price, governed by a family of semi-martingales. Under the assumption that there exists a universal martingale measure for all price processes, they prove the no arbitrage condition in the financial market comprising such a large trader. Henderson and Hobson [29] consider the problem of a risk averse investor who wishes to liquidate a portfolio of infinitely divisible American style options. Longstaff[40] models the illiquidity of portfolio by constraining trading strategies to be of bounded variation so that a trader cannot extricate himself from a position immediately. Schied and Schöneborn [50] introduce two price impacts in order to model illiquidity, one permanent price impact which accumulates over time and one temporary impact which is only affected by the instantaneous change in the number of shares.

1.5.3 Incomplete financial market with transaction costs

Another extension of the Merton’s model involves incomplete financial market setting where perfect hedging is no longer possible. Constantinides and Magill [12] (see also Constantinides [11]) were the first to introduce proportional transaction costs to the Merton model and considered an investor whose aim is to maximise the expected utility of consumption over an infinite horizon under power utility. They conjectured the existence of a ‘no-transaction’ region, and that it is optimal to keep the proportion of wealth invested in the risky asset within some interval.

[12] and their work is a landmark in the study of proportional transaction cost problems. Motivated by Davis and Norman’s work, Shreve and Soner [52] studied the same problem but with an approach via viscosity solutions. They recover the results from Davis and Norman [13] without imposing all of the conditions of [13]. These approaches remain the main methods for solving portfolio optimisation problems with transaction costs, although recently a different technique based on shadow prices has been proposed, see Guasoni and Muhle-Karbe [22] for a users’ guide.

In related work, Duffie and Sun [15], Liu [39] and Korn [38] study the problem when there are fixed (as opposed to proportional) transaction costs. Liu modelled the fixed transaction cost by a constant brokerage feeF >0and the holdings in stock evolves

as

dXt= (rXt Ct)dt F 1(dLt>0)+1(dUt>0) ,

Liu used a dynamic programming approach, deriving an ordinary differential equation to characterise the value function and solving it numerically. He found that if there is only a fixed transaction cost, the optimal trading strategy is to trade to a certain target amount as soon as the fraction of wealth in stock goes outside a certain range. Korn [38] solved a similar problem by an impulse control and optimal stopping approach. He proved the Bellman principle and solved for the reward function by an iteration procedure under the assumption that the value function is finite.

In the literature of multi-asset problems with transaction costs, however, there are relatively limited results. In the multi-asset case, and on the computational side, Muthuraman and Kumar [45] use a process of policy improvement to construct a numer- ical solution for the value function and the associated no-transaction region, and Collings and Haussman [10] derive a numerical solution via a Markov chain approximation, for which they prove the convergence. On the theoretical front Akianet al[2] show that the value function is the unique viscosity solution of the HJB equation (and provide some numerical results in the two-asset case) under the assumption that the price processes of the risky assets are uncorrelated. Chen and Dai [8] identify and prove the shape of the no-transaction region in the two-asset case. Explicit solutions of the general problem remain very rare. One situation when an explicit solution is possible is the rather special case of uncorrelated risky assets, and an agent with constant absolute risk aversion. In that case the problem decouples into a family of optimisation problems, one for each risky asset, see Liu [39]. Another setting for which some progress has been made is the problem with small transaction costs, see Atkinson et al [3], Whalley and Willmott [57], and for a more recent analysis Soner and Touzi [53]. Whalley and Willmott use an expansion

method to provide asymptotic formulae for the optimal strategy.