The Martingale Method in Continuous Time

This post describes the martingale method in continuous time. It should be read after this one. It illustrates the method on a simple example within the Black and Scholes framework where an investor maximizes the utility of terminal wealth. In this context, the martingale method allows to spell out how optimal terminal wealth depends on the unique stochastic discount factor, or alternatively, how it is obtained as a transformation of the stock return. The transformation hinges on the shape of the utility function. The case of a CRRA utility function is fully spelled out. The results obtained through dynamic programming here are recovered although the martingale method does not easily uncover the trading policy that generates the optimal terminal wealth.

Constrained Optimization

This post collects constrained optimzation results for reference.

Solution of the Exam (2016)

This is the solution of the exam for the 2016 ensae course.

Exam (2016)

This is the exam for the 2016 ensae course.

Solutions to Exercises

This post collects the solutions to the exercises of the ENSAE course.

Complete Markets, Discrete Time

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.

Using Jupyter notebooks

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.

Allocating risk across periods

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.

Solution of the Exam (2015)

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