Actuary » Binomial

The binomial model

Basic setup

Key: replicating the purchase of a call option by buying a fractional (\Delta) shares of stock and borrowing B dollars.

Assume at time T the stock will have two values, u\cdot S and d\cdot S. Let the values of a corresponding call option be C_u and C_d. h denotes the length of one time period. Then a cash flow table shows the replicating portfolio will satisfy

\Delta \cdot dS \cdot d^{\delta h} + B\cdot e^{rh} = C_d,

\Delta \cdot uS \cdot d^{\delta h} + B\cdot e^{rh} = C_u.

Solving gives

\Delta = \left( \frac{C_u-C_d}{S(u-d)} \right) e^{-\delta h}

B = \left( \frac{uC_d-dC_u}{u-d} \right) e^{-r h}

So the call option price is

C = \Delta S + B.

The put option price is obtained entirely the same way by replacing C with P.

If the price of an option is mispriced, we can arbitrage by buying low, selling high, using the fact that the buying a call option is equivalent to buying \Delta shares and borrowing B dollars.

Risk-neutral pricing

If we write

p^* = \frac{ e^{(r-\delta)h} -d}{u-d},

which is called the risk neutral probability, then the formula simplifies to

C = e^{-rh} \left( p^*C_u + (1-p^*) D_d \right)

as if C is a kind of (discounted) expected value. [In fact, a calculation also shows p^* is related to the forward price:

F_{t, t+h} = p^*uS + (1-p^*)dS.

]

Properties of \Delta and B

The factor e^{(r-\delta)h} satisfies

d< e^{(r-\delta)h}< u.

Otherwise we can arbitrage by exchanging stock and bonds, ignoring options altogether.

For a call option, 0<\Delta\leq 1 and B<0.
For a put option, -1\leq \Delta <0 and B>0.

James’s Page

The binomial model

Basic setup

Risk-neutral pricing

Properties of \Delta and B