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September 06, 2007, at 10:55 PM by 24.148.11.217 -

Changed lines 32-33 from:

{$$ d< e^{(r-\delta)h}< u $$}. Otherwise we can arbitrage on exchanging stock and bonds, ignoring options altogether.

to:

{$$ d< e^{(r-\delta)h}< u. $$}
Otherwise we can arbitrage by exchanging stock and bonds, ignoring options altogether.

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September 06, 2007, at 10:20 PM by 24.148.11.217 -

Added lines 1-36:

[[MFE topic list -> Actuary.MFE]]

![-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 on 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 {$B>0$}.

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