Derivative pricing on 1-Step Binomial Tree

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In this below, we explain Derivative pricing on 1-step Binomial tree.

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.