InvestmentMath

In this post, I present the so-called martingale method in a complete market setup where the time scale is discrete and the probability space is discrete as well. In the absence of arbitrage, there is a strictly positive stochastic discount factor (SDF) and a risk neutral measure can be defined. The market is said to be complete if any consumption stream which is adapted to the underlying filtration can be traded using the available financial instruments. If this is the case, the SDF and the risk neutral measure are both unique. A portfolio optimization problem without labour income is defined. Despite the dynamic context, the maximization problem is essentially static. The gradient of the utility function has to be proportional to the SDF, and the optimal plan is found by varying the coefficient of proportionality to satisfy the intertemporal budget constraint. The optimal consumption plan is thus in principal easily derived. The corresponding portfolio policy has to be identified in a second step however.

This post explains how to use the IPython/Jupyter notebooks which are provided in this blog through Github. The notebooks can easily be uploaded on a personal Sagemathcloud account. This avoids having to install a local implementation of Jupyter.

This post collects exercises for the ENSAE course. See this post for the solutions.

In the cross section, portfolio returns are a weighted sum of securities’ returns. This opens up the door to diversification gains when returns are imperfectly correlated. Optimal diversification is reached when the independent underlying factors are weighted proportionately to their prospective Sharpe ratios. It is perhaps less well known that the same principle holds across periods. When returns are independent from one period to another (the benchmark case), efficiency requires that the risk budget be allocated proportionately to the prospective Sharpe ratio of the period. This leads to constant volatility strategies, when the portfolio’s Sharpe ratio is constant over time. Constant volatility strategies can be contrasted with policies which hold the weight of the risky asset constant. The latter are efficient when the incentive to invest is proportional to volatility. Choosing between the two types of policies requires an empirical investigation of the link between volatility and the prospective Sharpe ratio.

This is the solution of the exam for the 2015 ensae course.

This is the exam for the 2015 ensae course.

In this post, I introduce the notion of market completeness in the context of a two period model. I also review the notion of a stochastic discount factor (SDF). The absence of arbitrage coupled with market completeness imply the existence of a unique strictly positive SDF. In this context, the investor chooses wealth/consumption in each state of the world so as to equalize marginal rates of substitution across states and the ratio of the values the SDF takes in the corresponding states. In other words, the marginal value of wealth/consumption in each state is proportional to the SDF, the constant of proportionality being the same for all states . For a given level of initial wealth and a given utility function, the SDF determines the end wealth/consumption distribution.

In this post, I look at continuous time models. Price dynamics are described and wealth dynamics are derived. I then look at dynamic programming, describing the Hamilton Jacobi Bellman partial differential equation. I apply this to the case of constant investment opportunities and CRRA utility (including log utility). The results described in discrete time are shown to hold here as well. I then look at cases where investment opportunities fluctuate, reflecting the dynamics of a scalar Markovian process. In the first example, the driving variable is the (stochastic) cash rate. This model allows to discuss the role of bonds as hedging instruments within portfolios. In the second example, the cash rate is constant but the equity premium fluctuates and the equity index is more volatile in the short term than in the long term. This makes the optimal equity weight dependent on the investment horizon.

In this post, I briskly review stochastic calculus. I concentrate on continuous martingales. Continuous martingales have trajectories with infinite first order variation and finite quadratic variation. This forces the development of a specific integration theory. I introduce continuous semimartingales, for which I spell out the Ito formula. I then consider the geometric Brownian motion and the Ornstein Uhlenbeck process.

In this short post, I sketch the derivation of the dynamic programming principle in discrete time.