# 金融代写|期权定价理论代写Option Pricing Theory代考|MATH150

## 金融代写|期权定价理论代写Option Pricing Theory代考|The No Trade Region in Continuous Time for an Infinite

The extension of the asset allocation results under proportional transaction costs to continuous time is not a trivial exercise, and rigorous proofs with useful results exist only for special cases. This is surprising in view of the fact that the frictionless case for simple diffusion asset dynamics was derived by Merton as early as 1969 and has been extended since that time to several more complex cases. The key result is that the optimal consumption and portfolio policy keeps a constant proportion of risky to riskless asset at all times, the so-called Merton line, that depends on the risk premium and volatility of the risky asset as well as the risk aversion of the investor. Needless to say, this solution is infeasible in the presence of transaction costs since it requires continuous rebalancing of the portfolio.

Constantinides (1986) was the first to formulate and solve the asset allocation problem under proportional transaction costs and simple diffusion asset dynamics in continuous time, for a trader who maximizes the discounted flow of a constant relative risk aversion (CRRA) utility function of consumption $\frac{c_t^\gamma}{\gamma}$ over an infinite horizon. Nonetheless, as he states (p. 846), “results on the existence and form of the optimal consumption and investment policy have not been derived” under his formulation. For this reason he assumed a “simple” portfolio revision policy, in which the NT zone is as in the discrete time case covered in his 1979 paper: for CRRA utility investors it is a compact cone or wedge in $(x, y)$ space, in which the investor or trader refrains from trading as long as the portfolio stays within the NT zone and restructures the portfolio propor-tions to the nearest boundary when they go outside the zone. Further, he assumed that all consumption was a constant proportion of the riskless asset $x_t$, or $c_t=\beta x_t$, and both consumption and transaction costs are charged to the riskless asset. These assumptions resulted in an elegant solution that is sketched below.

## 金融代写|期权定价理论代写Option Pricing Theory代考|What Happens When the Horizon Is Finite?

When the upper limit of integration in (3.12) is a finite horizon $T$, as it would happen when the investor is a trader acting on behalf of an institution, the PDE (3.14) acquires an extra term $V_t$ that prevents the existence of a closed-form solution. Although to our knowledge there is no exact solution for this problem, there are two approximations that have appeared in the literature. Liu and Loewenstein (2002) formulated and solved the problem of the maximization of expected CRRA utility of terminal wealth for lognormal dynamics of the risky asset return and an exponentially distributed horizon, as in the case where a single Poisson event (“death”) takes place and terminates the portfolio. They also subsequently extended their analysis for a sequence of such events, in which case the terminal time has an Erlang distribution and tends to a constant limit as the number of events increases.

This approach produced some interesting and elegant results, such as the fact that the optimal strategy is clearly horizon-dependent and may not include any holdings of the risky asset when the horizon is short and/ or the transaction cost rate is large. Unfortunately, the approximation of the solution for the fixed finite horizon by a single Poisson event is poor when the horizon is relatively short, of the order of a couple of years or so, as can be realistically expected in the case of option traders. For the Erlangdistributed horizon the numerical work consists of recursive simultaneous solutions of several PDEs; it is computationally extensive and does not converge for several realistic parameter configurations. The same is true a fortiori when the risky asset dynamics are enriched by including jump components in the diffusion. ${ }^8$

# 期权理论代考

## 金融代写|期权定价理论代写Option Pricing Theory代考|The No Trade Region in Continuous Time for an Infinite

Constantinides (1986) 是第一个制定和解决连续时间比例交易成本和简单扩散资产动态下的资产配置问题，对于最大化恒定相对风险厌恶 (CRRA) 消费效用函数的贴现流量的交易者C吨CC在无限的地平线上。尽管如此，正如他所说（第 846 页），根据他的表述，“尚未得出关于最优消费和投资政策的存在和形式的结果”。出于这个原因，他假设了一个“简单”的投资组合修正政策，其中 NT 区域与他 1979 年论文中涵盖的离散时间案例一样：对于 CRRA 公用事业投资者来说，它是一个紧凑的锥形或楔形(X,是)空间，只要投资组合留在 NT 区域内，投资者或交易者就不会进行交易，并且当他们离开该区域时，将投资组合比例重组到最近的边界。此外，他假设所有消费都是无风险资产的固定比例X吨， 或者C吨=bX吨，并且消费和交易成本都计入无风险资产。这些假设导致了一个优雅的解决方案，如下所示。

## 金融代写|期权定价理论代写Option Pricing Theory代考|What Happens When the Horizon Is Finite?

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