Option Pricing Theory
2022, May 05
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Option Pricing Theory
Contents are subject to copyright.
About
- Theory notebook.
- In the next link you will find the whole Mathematica notebook of the work presented. Gitlab Repo.
Project status And Requirements
Table of contents
- Basic securities,
- Call an Put options.
- Brownian Motion Process.
- Stochastic Differential Equation.
Contributing
Contribution is welcomed on component (or project if there is no component for that project).
To contribute to
- Fork this repository.
- Create a branch:
git checkout -b <branch_name>. - Make your changes and commit them:
git commit -m '<commit_message>' - Push to the original branch:
git push origin <project_name>/<location> - Create the pull request.
License
Option Pricing Notes
Stocks:
- Issued by firms to finance operations.
- Represent ownership of the firm.
- Price known today, but not in the future.
- May or may not pay dividends.
Bond:
- Price known today.
- Future pay offs known at fixed dates.
- Otherwise, the price movement is random.
- Final payoff at maturity: face vale/nominal value/principal.
- Intermediate payoffs: coupons.
- Exposed to default/credit risk.
Derivatives (Basic Securities):
- Sell for a price/value/premium today.
- Future value derived from the value of underlying securities.
- Tared at exchanges - standardized contract, no credit risk.
- or, over-the-counter(OTC)- a network of dealers and institutions, can be non-standard, some credit risk.
Forward Contract:
- An agreement to buy (long) or sell (short) a given underlying asset S:
- At predetermined future date T (maturity).
- At a predetermined price F (forward price).
- F is chosen so that the contract has zero value today.
Call and Put options
Vanilla options:
- Call option: a right to buy the underlying.
- Put option: a right to sell the underlying.
- European option: the right can be exercised only at maturity.
- American option: can be exercised at any time before maturity.
Exotic options:
- Asian options: the payoff depends of the average underlying asset price.
- Lookback options: the payoff depends on the maximum r minimum of the underlying asset price.
- Barrier option: the payoff depends on whether the underlying crossed a barrier or not
- Basket options: the payoff depends on the value of several underlying assets.
Brownian motion process
History:
- Brown, 1800’s.
- Bachelier, 1900.
- Einstein, 1905, 1906.
- Wiener, Levy 1920.b4s, 30.b4s.
- Ito, 1940.b4s.
- Samuelson, 1960.b4s.
- Merton, Black, Scholes, 1970.b4s.
Short introduction to the Merton-Black-Scholes model:
- Risk-free asset : \(B(t)=e^{rt}\)
- Stock has a log normal distribution: \(log S(t)=log S(0)+ (\mu-\frac{1}{2} \sigma^2) t+\sigma \sqrt{t}\ z(t)\)
- Where z (t) is a standard normal random variable. Thus, \(S(t) = S(0) e^{(\mu-\frac{1}{2} \sigma^2)t+\sigma \sqrt{t}\ z(t)}\)
- And it can be shown that: \(ES(t) = S(0)*e^{(\mu t)}\) and \( \frac{1}{2} Var [log \frac{S(t)}{S(0)}]=\sigma^{2}\)
To see the whole Mathematica notebook click on the Gitlab project site.