Frequently Asked Questions In Quantitative Finance

264 Frequently Asked Questions In Quantitative Finance

S

uS

δt vS

Probability of rise = p

Figure 4-1:The model.

Suppose that we know the value of the option at the

timet+δt. For example this time may be the expiration

of the option. Now construct a portfolio at timetcon-

sisting of one option and a short position in a quantity

of the underlying. At timetthis portfolio has value

=V−S,

where the option valueVis for the moment unknown.

At timet+δtthe option takes one of two values, de-

pending on whether the asset rises or falls

V+ or V−.