Author: Chen
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Gambler’s Ruin
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in MathematicsRead more: Gambler’s RuinThis post discusses the gambler’s ruin concept. Stopping Problems and Boundary Problems There are two classes of problems we can solve when discussing gambler’s ruin: stopping problems and boundary problems. Stopping problems are concerned with the termination of a process, either by ruin or conditionally. They include questions like “What is the probability of ruin?”, “What…
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Binomial Models
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in MathematicsRead more: Binomial ModelsA binomial model is made up of nodes that split into two separate branches. Suppose the branches in the model above represent log returns rt and the nodes represent asset values St. We can represent the model using the equations below: If we let X = log(ST/S0), we can rewrite the equation above as This…
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Forecasting
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in MathematicsRead more: ForecastingForecasts in time series models are predictions of future observations conditioned on information that is known at the time of forecast. Theorem (cf. Granger) Granger states that the optimal forecast, ft,h, of a value is the conditional expectation of that value provided the cost function is symmetric and convex. In other words, provided the cost…
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Model Selection
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in MathematicsRead more: Model SelectionAlternatives to the Random Walk Model Besides the random walk model, we have other types of models, including autoregressive, moving average, and ARMA models. Autoregressive models have terms that depend on previous lagged returns (i.e. rt-k). Moving average models have terms that depend on previous lagged innovations (i.e. previous random variables zt-k). ARMA models generate…
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Simple vs Logarithmic Returns
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in MathematicsRead more: Simple vs Logarithmic ReturnsMost financial reports give us simple returns, but financial modelling uses log returns because they are consistent with the formulation of the model. We can convert from one to the other. Let R be the simple return and rt be the log return, where rt = log(Pt/Pt-1). R = er – 1 m = E[R]…
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Testing the Random Walk Model
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in MathematicsRead more: Testing the Random Walk ModelThe Efficient Market Hypothesis (EMH) EMH states that markets are efficient in the sense that investors take into account all available information when making an investment decision. Therefore, the only reason prices change is due to randomness. In other words, randomness of prices is a sign of markets operating efficiently. The random walk model The…
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From AR(1) to MA
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in MathematicsRead more: From AR(1) to MAIn this post, we’ll learn to convert a AR(1) model to a MA model. The moving-average model specifies that the output variable depends linearly on the current and various past values of a stochastic (imperfectly predictable) term. https://en.wikipedia.org/wiki/Moving-average_model So, a MA model should only depend on zt-k, where k >= 0. Converting a AR(1) Model…
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Solving the AR(1) Model – Finding its Mean, Variance and Covariance
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in MathematicsRead more: Solving the AR(1) Model – Finding its Mean, Variance and CovarianceWe discussed the AR(p) model previously when discussing linear time series. In this post, we’ll learn to solve the AR(1) model, where Rt = c0 + c1Rt−1 + σ*zt Finding the Mean of the AR(1) model E[Rt] = c0 + c1E[Rt−1] + σ*E[zt]= c0 +c1E[Rt−1] + σ*0= c0 +c1E[Rt−1] Since AR(1) is stationary, E[Rt] = E[Rt−1]…
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What is Stationarity? Definition, Examples and Worked Solutions
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in MathematicsRead more: What is Stationarity? Definition, Examples and Worked SolutionsStationarity Stationarity means the probability distribution of a stochastic process does not change over time. Examples: Probability on the dice is the same every day on every roll Probability on the roulette wheel does not change over time Probability on a deck of cards is the same on every draw In mathematics and statistics, a…
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Linear Time Series
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in MathematicsRead more: Linear Time SeriesWhat is a time series? Time series are used to model processes that are discrete. These processes can be genuinely discrete or they can be continuous, but we observe them periodically. For instance, we can use time series to model corporation cuμlative income. The income of a corporation is continuous, but we may only observe…