InvestmentMath

The category ‘ENSAE’ groups posts that correspond to a course tought at ENSAE on portfolio choice. This post gives contains references and warnings.

I review the impact on the future price level of random noise in the return process. This effect operates through the non-linearity of compounding. I look at how an increased level of volatility changes the price trajectories. Normally distributed (say) return variability lowers the median of the price distribution and skews it towards the upside. Over time, the skew increases as compounding operates. The majority of price trajectories are depressed by volatility. The mean price level is however unaffected as the lower trajectories are made up by a few ‘explosive’ ones. This will be important to bear in mind when discussing portfolio choice and the effect of portfolio rebalancing.

This post introduces discrete time price dynamics in a log-normal setup. This framework allows an easy translation of some of the continuous time results discussed on this site.

In this post, I shed some light on the volatility of the price process of a zero coupon equity contract. The aim is to understand how the volatility of prospective expected returns impacts the volatility of the price process. I derive under a specific condition¹ a simple accounting relationship linking price volatility on one hand and cash flow as well as discount factor volatility on the other hand. There is some empirical presumption that markets exhibit excess volatility, i.e. that market prices are more volatile than warranted by cash flow volatility. Under the afore mentionned condition, this requires that discount rate shocks be negatively correlated with cash flow shocks².

This post introduces zero coupon contracts, contracts which promise a single payment at a fixed date \(T\). Truthful to this introductory post, the price process is derived backwards. The terminal pay-off is modeled as a random variable usually not known before maturity. The price process is then defined as adapted to the filtration, with a drift which is the instantaneous expected return and converging to the terminal pay-off at maturity. This is an example of a very simple backward stochastic differential equation. The data of the zero coupon contract is the terminal pay-off and the drift (expected return) process. A key component of the solution is the volatility of the price process, which can be related to the data of the problem.

This post introduces Gaussian processes, i.e. processes with Gaussian finite dimensional multivariate distributions. Amongst Gaussian processes, the Ornstein Uhlenbeck process is the only Markovian covariance stationary example. It plays a key role in applications thanks to its tractability.

The martingale representation problem in its simplest form is the following. Given a filtration generated by a martingale \(M\) and given another martingale \(N\) adapted to the filtration, can we express \(N\) as a stochastic integral with \(M\) as the integrator? The martingale \(N\) is generally closed, i.e. it can be expressed as the conditional expectation of a terminal variable \(N_{T}\). In this case, the integrand \(H_{t}\) of the stochastic integral representation is heuristically the sensitivity of \(N_{T}\) to the shock \(dM_{t}\).The Brownian filtration is the most important example where a Martingale Representation Theorem holds.

This post introduces the Ito rule of stochastic calculus. This rule provides the semimartingale decomposition of a nonlinear (this being the non trivial case) function of given semimartingales. Its simplest incarnation is the product rule of stochastic calculus which gives the semimartingale decomposition of the product of two given semimartingales.

The previous posts introduced stochastic integration of an integrand \(H\) against an integrator \(M\) which was chosen to be a continuous martingale. Continuous semimartingales are now defined as continuous processes that can be decomposed into a continuous martingale and a continuous finite variation process. Because ‘standard’ integration theory allows to integrate against finite variation processes, one can easily extend stochastic integration from continuous martingale integrators to continuous semimartingale integrators.

As explained in this post, stochastic integration has initially been developed under the condition \(E\left[\int_{0}^{T}H^{2}_{s}d[M]_{s}\right]<\infty\). This condition has then been replaced by the more general condition \(P\left(\int_{0}^{T}H^{2}_{s}d[M]_{s}<\infty \right)=1\). This turns out to be the most general integrability condition under which stochastic integration can be defined. This post introduces the idea behind this generalization. The process whereby stochastic integration is thus generalized is called localization. In words, the above condition means that the set of trajectories for which integrated variance remains bounded should have probability one.