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January 13th, 2017. The Martingale Method in Continuous Time. This post describes the martingale method in continuous time. It should be read after this one. Are recovered although the martingale method does not easily uncover the trading policy that generates the optimal terminal wealth. December 16th, 2016. This post collects constrained optimzation results for reference. December 15th, 2016. Solution of the Exam (2016). This is the solution of the exam for the 2016 ensae course. November 29th, 2016.

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InvestmentMath | investmentmath.com Reviews

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January 13th, 2017. The Martingale Method in Continuous Time. This post describes the martingale method in continuous time. It should be read after this one. Are recovered although the martingale method does not easily uncover the trading policy that generates the optimal terminal wealth. December 16th, 2016. This post collects constrained optimzation results for reference. December 15th, 2016. Solution of the Exam (2016). This is the solution of the exam for the 2016 ensae course. November 29th, 2016.

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InvestmentMath | investmentmath.com Reviews

https://investmentmath.com

January 13th, 2017. The Martingale Method in Continuous Time. This post describes the martingale method in continuous time. It should be read after this one. Are recovered although the martingale method does not easily uncover the trading policy that generates the optimal terminal wealth. December 16th, 2016. This post collects constrained optimzation results for reference. December 15th, 2016. Solution of the Exam (2016). This is the solution of the exam for the 2016 ensae course. November 29th, 2016.

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investmentmath.com

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Allocating risk across periods

http://www.investmentmath.com/finance/2015/03/11/time-diversification.html

Allocating risk across periods. March 11th, 2015. In this paragraph, I consider an investor who has access to a risky asset and to cash. The risky asset provides a Sharpe ratio which I take to be constant ( lambda 0 ). In a first step. I assume that its return distribution is lognormal, with volatility ( sigma {t} ). I assume volatility changes with time but is deterministic. I also assume that the cash rate (r {f} ). Where the shocks ( epsilon {t}) ). The portfolio return between date (t ). To date (T ).

2

Dynamic Models (Continuous Time)

http://www.investmentmath.com/ensae/2014/10/27/cont.html

Dynamic Models (Continuous Time). October 27th, 2014. Decisions are taken at specific instants which are themselves decision variables. The underlying time scale is continuous. Realism therefore suggests exploring continuous time. Continuous time leads to some simplifications. Discrete time mathematics are harder than it looks. As an example, continuous time calculus often provides closed form expressions where discrete time does not. Example of a simplification: log normal markets. If I own (n {t} ).

3

Exam (2015)

http://www.investmentmath.com/ensae/2015/01/23/exam.html

January 23rd, 2015. This is the exam for the 2015 ensae course. Part 1: Efficient frontier, without risk free asset ( ( cal{E} ). The context is that of static optimization, i.e. decisions are taken in period (0 ). And results are observed in period (1 ). There are (N ). Risky assets ( (i=1, ldots,N ). With stochastic rates of return ( tilde{r} {i}) {i=1, ldots,N} ). Expected rates of return are noted ( mu {i}) {i=1, ldots,N} ). And the covariance matrix is ( Sigma ). As a proportion of wealth (W {0} ).

4

Foreword to the ENSAE Course in Portf. Choice

http://www.investmentmath.com/ensae/2014/10/06/warning.html

Foreword to the ENSAE Course in Portf. Choice. October 6th, 2014. The category ‘ENSAE’ groups posts that correspond to a course tought at ENSAE on portfolio choice. This post gives contains references and warnings. Kerry A. Back,. Asset pricing and portfolio choice theory. Oxford University Press, 2010. John H. Cochrane,. Princeton University Press, 2005. Gabrielle Demange and Guy Laroque,. Finance and the Economic of Uncertainty. Working paper, Copenhagen Business School, 2013. Richard F. Bass,.

5

Complete Markets, Static Case

http://www.investmentmath.com/ensae/2014/11/16/complete.html

Complete Markets, Static Case. November 16th, 2014. Discrete probability space, static case. Consider the static portfolio problem, i.e. there are two dates. Investment is carried out in period (0 ). And consumption takes place in period (1 ). I now assume that the probability space is discrete, so that the securities payoff distributions are described by a vector of outcomes with components ( tilde{x} {i}( omega {k}) ). For state ( omega {k} ). There are (K ). Satisfying: [ sum {k=1} {K}q {k}=1. ].

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January 13th, 2017. The Martingale Method in Continuous Time. This post describes the martingale method in continuous time. It should be read after this one. Are recovered although the martingale method does not easily uncover the trading policy that generates the optimal terminal wealth. December 16th, 2016. This post collects constrained optimzation results for reference. December 15th, 2016. Solution of the Exam (2016). This is the solution of the exam for the 2016 ensae course. November 29th, 2016.

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