TOC
One-step binomial tree
Example:
- A non-dividend paying stock currently trading for $20
- Its value in 3 months can be $22 or $18 with unknown probabilities
- We hold a call with maturing at time T
Payoff of the call option
- For :
- For :
Discount the option payoff to determine the value of the option. But we cannot use the risk-free rate as discount rate directly because the option is far riskier (i.e. has a larger market beta) than the stock
Construct the replication portfolio
Idea: Combine the stock & the call option into a portfolio so that the portfolio is risk-free (that is, have the same payoffs for an up- & a down- move)
Buy stocks and short one call, then the portfolio payoff is:
- For :
- For :
since , thus
therefore, the portfolio payoff is always equal to 4.5 (risk-free, i.e. no matter the price go up or down)
Determining the option value
Risk-free payoffs should be discounted at the risk-free rate, thus
the discounted value shoule be equal to the present value of the portfolio, which is
therefore:
All derivation is based on only one assumption: No arbitrage
The General Case
Form a portfolio consisting of stocks & one short option
The portfolio payoff is:
thus , thus
- Delta means change in the option value for a unit change in the stock price
- Delta varies from node to node, it depends on the current stock price and is bounded between zero and one
- Highly in-the-money call: Delta is close to one (almost sure to exercise, prepare the stock)
- Highly out-of-the-money call: Delta is close to zero (almost sure not to exercise, no need for stock)
Determing the option value:
therefore,
Substitute into the equation get
Define , then
While using the binomial option pricing, the expected return of both option and underlying asset is the risk-free rate.
Risk-neutral valuation
We can interpret p and (1-p) as pseudo probabilities:
- p is the probability of an up-move
- (1-p) is the probability of a downward-move
- But in reality, we do not know the (real world) probability of price go up or go down
The option value is then the expected payoff discounted at the risk-free rate:
and
Binomial trees illustrate the general result known as risk-neutral valuation:
The value of a derivative is the expected payoff under the risk-neutral probability measure discounted at the risk free rate
The probability of up & down movements in the real world are irrelevant because
- We value the option in terms of the underlying asset price
- The real world probabilities of up & down movements are already incorporated in the underlying asset price
Example
Multiple-step Binomial Tree
Value the trees in the last step, and then move back
Example
European and American put options
- Binomial trees can be used to value any type of option
- Risk-neutral probabilities & risk-free rate do not depend on the option type
- Only the option payoff is different
European put options, Example
Assume , , each time step over one month
American put options, Example
Binomial trees are extremely flexible:
- Allow for any possible stock price distribution
- Allow for early exercise
For American options:
- The option holder can exercise the option prior to maturity when the intrinsic value exceeds the option premium
Analysis Steps
- At maturity (), decides whether to exercise or to let the option expire
- One time step before , compute the option value & the exercise value
- Pick the higher one the value of the option at this node
- Continue in this way until
Example
Assume , , each time step over one month
Girsanov’s theorem
When we move from the risk-neutral world to the real world:
- The expected rate of return on the stock changes
- The volatility remains the same
General case:
- When we move from a world with one set of preferences to a world with another set of preferences
- Changing the probability measure
- The expected rate of return on the stock changes
- The volatility remains the same
Define: - Standard deviation of the underlying asset return over
Risk-neutral world
Real world
Other underlying assets
Options on other assets can be valued in the same way. But the definition of changes slightly
general definition for :
is defined as follow:
- for a non-dividend paying stock
- for a stock index where is the index dividend yield
- for a currency where is the foreign risk-free rate
- 1 for a futures contract
Since different assets have different payouts, and hence the payout of the replication portfolio will be different
Approaching the continuous-time limit
By increasing the number of time steps, while keeping fixed, one can derive the Black & Scholes formula. Intuitively, this occurs because with n time-steps, the probability of reaching a specific option payoff follows the binomial distribution.
With steps, the option value is hence:
- Black & Scholes only holds for European Options
- Binomial option pricing is applicable to both European option and American option
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