In this below, we explain Derivative pricing on 1-step Binomial tree.
Functions to load: BinP1()
Usage: BinP1(S, Su, Sd, Du, Dd, r, delta, T)
S: spot stock price
Su, Du: upper leg stock price and derivative payoff
Sd, Dd: lower leg stock price and derivative payoff
r: continuously compounding interest rate
delta: continuously compounding dividend yield
Returning value: : a vector of 3-dim
: option price
: number of shares needed for perfect hedge of one unit option
: EMM probability for upper leg
Given a constant interest rate , and dividend yield , one step binomial tree of the stock model assumes that, (1) today’s stock price is ; (2) next step stock price goes up to or down to . One step period is given by .
We consider a derivative on this 1-step binomial tree with -payoff is either or for up and down case respectively. Then, to price a derivative, one shall first compute the equivalent martingale measure (EMM) by
The derivative price is the discounted average with the above EMM,
It is also possible to replicate the derivative with the position computed below, i.e. a portfolio of shares of stocks and units of zero coupon bonds. In fact, is computed by
Example: Stock A with no dividend: today’s price is , and in year the price can be either dollars with probability , or dollars with probability . The risk free continuously compounded rate .
- Find the risk-neutral probability
- Find the price of European call with maturity 12 months and strike .
- Find the number of shares needed for the replication of 1 unit above option.
Click here for the answer.