On Continuous time

Hello! This week I want to write about Stochastic Calculus. Why? Because it is interesting and useful for economic models. What I want to do is a brief introduction for those who know nothing about it, but are eager to learn, as well as point you towards relevant literature which might help.

To begin with, let me give a little bit of intuition on the Binomial Asset Pricing, as well as a random walk, which both are basic elements for continuous time models.

Suppose an asset bought today can have up and low returns tomorrow with a given probability. This way, the value of the portfolio would be $X_1$:

$$ X_1 = n S_1 + (1+r)(X_0 – n S_0),$$

where $n$ is the number of stocks bought, $r$ is the risk free rate (money market), $X_0$ represents the initial wealth, and $S_i$ represents the price of the asset in time $0$ and $1$ respectively ($i \in \{0,1\}$). It turns out that the initial wealth happens to be also the no-arbitrage price of the option at time zero. What option? It is what is called a European call option, which confers the owner the right to buy one share of the stock at time one for the strike price of $K$, so it is betting on the stock to go up. It is similar to what you do when you buy a house: you make a downpayment and guarantee that the price you will pay for the house remains static, even if the price of the house per se goes up.

There is a lot of theory there, but now I move on to random walks. Imagine you have a frog standing in $F_0 = 0$. Now the frog can go up or down with one leap, so that $F_1 \in \{ 1, -1 \}.$ Then one can measure the expected value in leap $N$, $E[F_N].$ Hopefully it is easy to see that that value is zero. Why? Because each jump has an expected value of zero. What is interesting, though, is what the variance is. Because each jump has a variance of $1$, and each leap is independent, then the variance of $N$ jumps is equal to $N.$

With these simple concepts, we have all the basic ingredients for Brownian motions, which I will write about on the next blog. For now, I send you a picture of the place I will hike this Saturday! 🙂

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