Binomial

MFE topic list

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 {$B>0$}.