Derivative pricing on 1-Step Binomial Tree

This function with an example is available in R-fiddle (here). See also its upper level page (here).

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)
Arguments:
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
T: maturity
Returning value: {val}: a vector of 3-dim
{val[1]}: option price
{val[2]}: number of shares needed for perfect hedge of one unit option
{val[3]}: EMM probability for upper leg

Given a constant interest rate {r>0}, and dividend yield {\delta \ge 0}, one step binomial tree of the stock model assumes that, (1) today’s stock price is {S}; (2) next step stock price goes up to {S_{u} = u S} or down to {S_{d} = d S}. One step period is given by {T>0}.

We consider a derivative on this 1-step binomial tree with {T}-payoff is either {D_{u}} or {D_{d}} for up and down case respectively. Then, to price a derivative, one shall first compute the equivalent martingale measure (EMM) by

\displaystyle q = \frac{e^{(r-\delta)T}- d}{u - d}.

The derivative price is the discounted average with the above EMM,

\displaystyle D_{0} = e^{-rT} (D_{u} q + D_{d}(1-q)).

It is also possible to replicate the derivative with the position {(\Delta, \beta)} computed below, i.e. a portfolio of {\Delta} shares of stocks and {\beta} units of zero coupon bonds. In fact, {\Delta} is computed by

\displaystyle \Delta = \frac{D_{u} - D_{d}}{S (u-d)},

and {\beta = (D_{0} - \Delta S)/e^{-rT}}.
Example: Stock A with no dividend: today’s price is {S= 80}, and in {1} year the price can be either {90} dollars with probability {1/3}, or {78} dollars with probability {2/3}. The risk free continuously compounded rate {r= .05}.

  1. Find the risk-neutral probability
  2. Find the price of European call with maturity 12 months and strike {\$81}.
  3. Find the number of shares needed for the replication of 1 unit above option.

Click here for the answer.

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