Stock Price Behavior
Stock Price Model
Assumptions: (based on empricial analysis of data):
- stock returns are normally distributed
- return today is uncorrelated with return yesterday
where is the average return and is some noise with an average of zero
Let , then the revised stock price model is
or
This is called an Ito process, the mean and variance of which are not constant. Instead, the change in the stock price has mean and variance .
Itoβs Lemma
Suppose is a function of and , i.e. , where
then
The stock price process is
for a function of and
Example: the price of a forward contract
therefore
Example:
therefore
that is
and thus
This works because the average, and the variance, , do not depend on .
This means, based on our assumptions on the stock price, the log of the stock price is normally distributed,
and thus the stock price is lognormally distributed (definition).
Monte Carlo Simulation
We can sample random paths for the stock price by sampling values for
If the stock price is lognormally distributed, suppose , , , then
Popular Processes
Geometric Brownian motion
Properties:
Markove Ito process:
Properties:
Ornstein-Uhlenbeck process:
Properties:
Black-Scholes Model
Risk Neutral Valuation
Construct the Portfolio
Assume the stock price is . Let denote some derivative whose value depends on (for example, a call option). We set up a portfolio consisting of
The value of the portfolio, , is given by
The change in the value of the portfolio in time is given by
Because the portfolio has no risk (), the return on the portfolio must be the risk-free rate. Hence
Combining the two equations for , we have
which produces the Black-Scholes differential equation:
The solution to the equation is
Remark:
- Any security whose price is dependent on the stock price satisfies the differential equation.
- The particular security being valued is determined by the boundary conditions of the differential equation. For example, in a forward contract the boundary condition is , when .
Β
Risk-Neutral Analysis
Assume the expected return from the stock price is the risk-free rate and is log-normally distributed,
The expected payoff on a call option is
where
The price of a call-option and a put option is the risk-free discounted payoff
where
Greeks
The value of a portfolio of derivatives dependent on an asset is a function of the asset price , its volatility , and time . Denote the portfolio value by
(ignoring terms of higher order than and assume constant volatility and interest rates)
Delta
Delta () is the rate of change of the option price w.r.t. the underlying asset price.
Variation of Delta with Time to Maturity, for example, , , .
Delta Hedging, for example, for a shorting call
At time :
- Option delta =
- Buy shares stock
- Trading Revenus =
- Interest Income =
Cash Balance = Previous Cash Balance + Trading Revenue + Interest Income
Theta
Theta () is the rate of change of the value (of a derivative or portfolio of derivatives) w.r.t. the passage of time.
The theta of a call or put is usually negative. This means that, if time passes with the price of the underlying asset and its volatility remaining the same, the value of a long option declines.
Gamma
Gamma () is the rate of change of delta () w.r.t. the price of the underlying asset.
Gamma can address the Delta hedging errors caused by curvature.
- If gamma is small, delta changes slowly, and adjustments to keep a portfolio delta neutral need to be made relatively infrequently.
- If gamma is large (in absolute value), however, delta is very sensitive to changes in the underlying. In this case, it is risky to leave a delta-neutral portfolio unchanged for long periods of time.
Vega
Vega is the rate of change of the value of a derivatives portfolio w.r.t. volatility. Vega tends to be greatest for options that are close to the money.
Rho
Rho is the rate of change of the value of a derivative w.r.t. the interest rate.
Greeks for European Options
Suppose the underlying asset provides a yield at rate (e.g. stock dividends)
γ
€ | European Call | European Put |
Delta | ||
Gamma | ||
Theta | ||
Vega | ||
Rho |
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