BSM model (1)
After reviewing the content of Chapter 11 of Financial Engineering, I think it is necessary to sort out the relevant formula derivation and model establishment in this chapter. Check the relevant information and summarize it as follows.
Preliminary knowledge
option pricing
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Concept comparison
(1) European options VS American options
European options cannot be exercised in advance, while American options can be exercised in advance (loss time value). However, it should be noted that the price of American options is not necessarily higher than the price of European options, and it is necessary to consider whether early exercise is reasonable.
(2) Call options VS puts options
(3) Asset options with dividends VS non-dividend asset options -
Factors affecting option prices
(1) Market price of the underlying asset (when exercised, S T ) S_{T}) ST) and option exercise price (K)
(2) Option remaining period (T-t)
(3) Underlying asset price change rate ( σ \sigma σ)
(4) Risk-free interest rate (r)
(5) Return on the underlying asset (Dividend and interest payment) -
Reasonable circumstances for early exercise of American options
(1) Non-dividend assets are bullish: early exercise is unwise.
(2) Bearish non-dividend assets: Generally speaking, early exercise is beneficial only when the degree of real value is high and the interest rate is high.
(3) Bullish assets with dividends: Generally speaking, early exercise is beneficial only when the degree of real value is high, the dividends are high, and the interest rates are low.
(4) Bearish assets with dividends: early exercise is less likely. -
Option term structure
There are three theories for the formation of the volatility term structure:
(1) The price movement process is not stable: this statement is It means that changes in fundamentals during the validity period will cause permanent changes in the expected distribution of underlying asset prices. If the market expects that the underlying asset will undergo significant changes in a certain period, the implied volatility of options before and after the event will also be different.
(2) The theory of non-uniform volatility: This theory holds that the actual volatility is expected to be different on different days, especially the difference between the day when important events occur and other days is more obvious. Volatility should therefore be a function of the number of events that occur during the life of the option and their importance.
(3) Volatility mean reversion theory: In a given market, volatility cannot remain at extreme levels for a long time, but will return to its long-term equilibrium level. We can also assume that realized volatility is a relatively stable level over the long term. When volatility levels exceed equilibrium levels, volatility returns to normal levels rather than sustaining this difference.
stochastic process
- Ordinary Brownian motion
Brownian motion, also known as Wiener process, is a random process if it satisfies the following properties:
( 1) Independent increment
For any t>s, B(t)-B(s) is independent of the previous process B(u).
(2) Normal increment
B(t)-B(s) satisfies the normal distribution with mean 0 and variance t-s. That is, B(t)-B(s)~ N(0,t-s).
(3) Continuous path
B(t), t≥0 is a continuous function about t. Fixed a path, B(t)->B(s) satisfies convergence according to probability.
The Brownian motion hypothesis is the core hypothesis of modern capital market theory. Modern capital market theory believes that securities and futures prices have random characteristics. The so-called randomness here refers to the memorylessness of data, that is, past data does not form the basis for predicting future data. At the same time, there will be no strikingly similar repetitions. The mathematical definition of random phenomena is: in individual experiments, the results show uncertainty; in a large number of repeated experiments, the results have statistical regularity. The Wiener process of Brownian motion, one of the models describing stock price behavior, is a special form of Markov random process; and the Markov process is a special type of random process. Stochastic process is a probability model based on probability space and is considered to be the dynamics of probability theory, that is, its research object is a random phenomenon that evolves over time. So random behavior is a behavior with statistical regularity. Stock price behavior models are often expressed in terms of the well-known Wiener process. It is tempting to assume that stock prices follow a generalized Wiener process, that is, that they have constant expected drift and variance rates. The Wiener process states that only the current value of a variable is relevant to future predictions, while the past history of the variable and the way the variable evolves from the past to the present are not relevant to future predictions. The Markovian property of stock prices is consistent with weak-form market efficiency, that is, the current price of a stock already contains all information, including of course all past price records. But when people began to use fractal theory to study financial markets, they found that its operation did not follow Brownian motion, but obeyed the more general geometric Brownian motion (from Baidu Encyclopedia) - Geometric Brownian Motion
Geometric Brownian motion (GBM) (also called exponential Brownian motion) is a continuous-time stochastic process in which the logarithm of a random variable follows Brownian motion.
Definition: The stochastic process St is said to follow geometric Brownian motion if it satisfies the following stochastic differential equation (SDE):
dS t _{t} t= m m μS t _{t} tdt+ σ \sigma σS t _{t} tdW t _{t} t
Here Wt is a Wiener process, or Brownian motion, while the drift percentage μ and fluctuation percentage σ are constants. - Ito process
S t _{t} tThe drift term and variance themselves may be random processes. - Hikiri Ito
dx t _{t} t =a