TOC
Review of stochastic processes
Stochastic Processes
- Assumptions: Stock prices evolve randomly over time
- Stock prices incorporate expectations about the future.
- Hence, stock prices only change with news
- News cannot be forecasted & arrives in a random fashion over time
- Stochastic process
- can be classified as discrete variable or continuous variable
- Examples:
- Discrete variable: Each day a stock price increases by $1 with probability 30%, stays the same with probability 50% and decrease by $1 with probability 20%
- Continuous variable: Each day a stock price change is drawn from a normal distribution with a specific mean & variance.
Markov processes
Markov processes depend only on the current level of the random variable. Markov property of prices is consistent with weakly efficiency markets.
Weak-form of market efficiency
- The spot price impounds all the information contained in past prices
- Historical prices contain no information about future prices
- Technical trading is therefore futile (useless)
- Competition ensures that the markets are weakly efficient
A good example of a Markov process is the random walk:
The wiener process
Wiener process is a continuous-time stochastic process.
Let donote a normal distribution with expectation & variance
A random variable z follows a Wiener process if:
- , where
- for any two non-overlapping time periods are independent
where is the change in over a small time interval
The Wiener process is also called as Brownian motion and it is a particular type of a Markov process.
Expected value and variance of
Thus
The standard deviation is ⇒ uncertainty is proportional to the square root of time
The generalized wiener process
In a generalized Wiener process, the drift rate and the variance rate can be set to any chosen constants (with the variance > 0)
- Drift rate: Mean change per unit of time for a stochastic process
- Variance rate: Variance per unit of time for a stochastic process
- Basic Wiener process: Drift=0, Variance rate = 1
The generalized Wiener process can be written as:
where is the mean change in per time unit, and is the variance of the change per time unit.
When ,
Process
Generalized Wiener process:
process: Obtained when are deterministic functions of and
- Both the expected drift rate () & variance rate () are liable to change over time
- The process is a particular type of Markov process (The change in at time depends on the value of at , not on its history)
Generalized wiener process is inappropriate for stocks
- The stock price cannot become negative due to limited liability
- The expected % change (i.e. return) should remain constant over time - not the actual change
- The variance of the actual chagne should depend on the stock price level (i.e. greater price greater variance)
Geometric Brownian motion (GBM)
For stock price, the Geometric Brownian motion (GBM), makes more sense:
where is the stock’s expected rate of return and is the volatility (i.e. standard deviation) of the stock price. Thus the discrete-time equivalent is:
We can sample random paths for the stock price by sampling values for
Suppose , and week ( years):
A stock price path can be simulated by sampling repeatedly from
For example, assume the stock price at the beginning of week 1 is $100, then the stock price at the beginning of week 2 is $100+
To calculate during week 1, we sample from , so is the stock price at the beginning of week 3
Ito’s lemma
The stock options price is a function of the underlying stock’s price & time.
Assume that follows an process:
lemma shows that a function follows the process (a equation of partial differential)
If a stock price follows tht GBM () then follows:
Example 1
Apply lemma when the stock price follows a GBM &
that is , which means
Prices are lognormally distributed and continuously compounded returns are normally distributed
Example 2
Apply lemma to the price of a forward contract written on a non-dividend paying stock:
Black-Scholes model
Basic idea:
- The price of an option is the discounted expected payoff but it’s difficult to determine the correct discount rate
- Identical to that underlying the binomial tree
- Form a riskless protfolio (long in stocks & short in the option)
- Set the portfolio return over a short period of time equal to the risk-free rate
- This yields a partial differential equation (p.d.e.)
Derivation:
Assume that stock price follows the Geometric Brownian Motion
Additional assumptions:
- No transaction costs
- Continuous trading
- Markets do not jump
- Volatility is constant and known
From lemma, the option value (funciton of stock price and time ) follows the process:
The discrete version of these equations is
The replication portfolio: Buy units of the stock () and short one option ()
We can set up a riskless portfolio based on the truth (assumptions):
- The stock price & the option price are affected by the same underlying source of uncertainty (i.e. stock price movements)
- Over a short period of time the stock price and the option price are perfectly correlated
- can be chosen so that the gain (loss) from the stock position is offset by the loss (gain) from the option position
The value of this portfolio is: and the change in value of the portfolio over the time interval is:
Substituting (1) and (2) into we get:
terms are responsible for the uncertainty in portfolio value. And thus we can choose to cancel out the uncertainty and
A risk-free portfolio must pay the risk-free rate:
thus
since , thus
Re-arranging yields the p.d.e. :
Note: As and change, the also changes. To keep the portfolio riskless, we need to frequently rebalance it.
The Black-Scholes p.d.e. & risk-neutral valuation
- Any security whose price is dependent on the stock price and time (i.e. price is ) satisfies the former p.d.e.
- The p.d.e. is independent of risk preferences
- We have the risk free rate and not the expected return on the stock (dividends, etc)
- Any set of risk preferences can be used → we can treat all investors as if they are risk-neutral investors
- Risk neutral valuation
- Risk neutral valuation:
- We can treat all investors as if they are risk-neutral
- The expected return on the stock and the portfolio is the risk-free rate
- The price of the option is the expected payoff under the risk-neutral probability measure discounted at the risk-free rate.
Risk-neutral valuation: European call price
Under risk-neutral valuation, all assets have a risk-neutral drift ():
Consider a discounted call price , through ’s lemma:
From ,
If the underlying asset pays no dividends, their solutions are
- Call:
- Put:
with and
We can interpret ( ) as the risk-neutral probability of the call (put) option ending up in-the-money
- ( ) becomes a real-world probability when swapping for
Note: function
is the probability that a normally distributed random variable with a mean of 0 and a standard deviation of 1 is less than
No closed-form solutions for . An efficient approximation
where
Black-Scholes Example
Assume having a non-dividend paying stock current price of which is 30 and volatility is 25%. Calculate the price of a European call written on the stock with strike price and time to maturity months. The continuously-compounded risk-free rate is 2%
- Calculate and
- Look up and using a distribution table: ,
- Plug into BS formula
Properties of the Black-Scholes formula
Call:
Put:
with and
As becomes very large, tends to be
- The call option moves more and more into-the-money
- We can be relatively certain that the call will be exercised
- The call becomes similar to a forward contract ()
- Note also that and
As becomes vary large, tends to be zero
- The put option moves more and more out-of-the-money
- We can be relatively certain that it will not be exercised
- Note also that and
As becomes large, the values of calls and puts increase
- Holders profit from the upside, but do not suffer from the downside.
- In fact, we can derivate that and
- The amount of unit change is determined by vega ()
As the time-to-maturity increases, the call option value converges to the stock value and the put option value converges to zero
Dividends & the Black-Scholes Model
Dividends change the stock price distribution at maturtiy: In an efficient market, a $5 dividend per share drops the stock price by $5.
Dividend payouts can be modeled in two ways:
- Continuous payouts occuring at a rate of
- Discrete payouts occuring at specific points in time
To handle the effect of dividends, we decrease the initial stock price by the value of the dividends:
- Continuous payouts:
- Discrete payouts:
Thus, if there are dividends, the BS formula bacome:
- Call:
- Put:
with and
Example:
Value European index options ⇒ Set to current index level and to average dividend yield
Value European currency options ⇒ Set to the FX rate and to the foreign interest rate
Settings:
- Stock: , and it pays a $2 dividend after ten months
- European call: and
- Continuously-compounded risk-free rate: (p.a.)
The present value of the dividend:
new stock price is 30-1.97=28.03
Calculate and :
Look up and in the normal distribution tables: ,
Plug into BS formula:
Probability of ending up ITM
Assume no dividends. Using ’s lemma and assuming that and , we have shown that:
Hence:
Risk-neutral probability of European put ending up ITM
We can interpret as the risk-neutral probability of the put option ending up in-the-money (ITM). becomes a real-world probability when we replace by
Risk-neutral probability of European call ending up ITM
We can interpret as the risk-neutral probability of the call option ending up in-the-money (ITM). becomes a real-world probability when we replace by
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