Adding a lecture on Monte Carlo and European option pricing #115
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Thanks for writing such a beautiful and valuable lecture @jstac . I've learned a lot from it.
I've included my reviews below.
PS. A possible new exercise could be added where we introduce the more realistic distributional dynamics for the option price with the persistence and transitory parameters. We could ask students to investigate how the changes in these parameters could affect the distributional dynamics and therefore the option price.
| Suppose that, after analyzing the data, we guess that $S$ is well | ||
| represented by a lognormal distribution with parameters $\mu, \sigma$ . | ||
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| * $S$ has the same distribution as $\exp(\mu + \sigma Z)$ where $Z$ is standard normal. |
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Maybe change where $Z$ is standard normal. -> where $Z$ is standard normal $N(\mu, \sigma)$.
| * $S$ has the same distribution as $\exp(\mu + \sigma Z)$ where $Z$ is standard normal. | ||
| * we write this statement as $S \sim LN(\mu, \sigma)$. | ||
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| Any good reference on statistics (such as |
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Probably statistics -> lognormal distribution.
| * we write this statement as $S \sim LN(\mu, \sigma)$. | ||
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| Any good reference on statistics (such as | ||
| [Wikipedia](https://en.wikipedia.org/wiki/Log-normal_distribution)) will tell |
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[Wikipedia] --> [the one from Wikipedia]?
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| $$ | ||
| \mathop{\mathrm{Var}} S | ||
| = [\exp(\sigma^2) - 1] \exp(2\mu + \sigma^2) |
lectures/monte_carlo.md
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| But now suppose that we study the distribution of $S$ more carefully. | ||
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| We decide that the share price depends on three variables, $X_1$, $X_2, and $X_3$ (for example, sales, inflation, etc.). |
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$X_2, -> $X_2$,
Maybe add interest rates to the variable series sales, inflation, since we mentioned 3 variables before.
lectures/monte_carlo.md
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| Now that our model is more complicated, we cannot easily determine the | ||
| distribution of $S_n$. | ||
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| So to compute the price $P_0$ of the option, we use Monte Carlo |
lectures/monte_carlo.md
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| So to compute the price $P_0$ of the option, we use Monte Carlo | ||
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| WE average over realizations $S_n^1, \ldots, S_n^M$ of $S_n$ and appealing to |
lectures/monte_carlo.md
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| ## ExerciseS |
lectures/monte_carlo.md
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| s = np.full(M, np.log(S0)) | ||
| h = np.full(M, h0) | ||
| for t in range(n): | ||
| Z = np.random.randn((2, M)) |
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Z = np.random.randn((2, M)) -> Z = np.random.randn(2, M)
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| Notice that this version is faster than the one using a Python loop. | ||
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| Now let's try with larger $M$ to get a more accurate calculation. |
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with larger $M$ -> with a larger $M$
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The commit above updates the exercise and solution environment to exercise 1. |
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Added to toc in 51ea510 |
Addresses #76
TODO