DF structure models of call options pricing

From Definition 2.1, Definition 2.5 and Theorem 2.2, we have

3.2.1. The price of call option at time t is

(4)

Click to enlarge image

When the call option is brought forward to execute at any time τε[t,T], the price of underlying stock, S(τ), becomes a constant to the option contract, thus D[S(τ)]=0. According to (4) and definition 2.4, the current value of the option is

i.e. At this time, the intrinsic value of the call option is max {S(τ)-X, 0}, thus

C_s(τ)≥max{S(τ)-X, 0} (5)

3.2.2. The price of put option at time t is

(6)

Click to enlarge image

when the put option is brought forward to execute at any time τε[t, T], the price of underlying stock, S(τ), becomes a constant to the option contract, thus D[S(τ)]=0. According to (6) and definition 2.4, the current value of the option is

i.e. . At this time, the intrinsic value of the put option is max{X-S(τ), 0}, thus,

P_s(τ)≤max{X-S(τ), 0} (7)

From (4) and (6), we could see that the price of option is made up of two parts. One is the market price of stock, and the other is the fluctuation of stock price. According to (5), we know call option should not be executed before the time to expiration. Otherwise, the current option value will be encashed by the intrinsic value, and the intrinsic value may be lower at the same time. And from (7), put option should be executed before the time to expiration since the current option value will be encashed by the intrinsic value, and the intrinsic value may be higher at the same time.

When Xe^-r(T-t)-S(t)>α, t≤T, thus the put option should be executed at the same time, where α is, including the transactions costs and tax, all the expenses of trade options.

Noting:

the expression (4) can, for short, be written as:

(8)

the expression (6) can be written as:

(9)

Prof. Feng Dai, Prof. Zifu Qin