Assumptions and notations

We need the following assumptions and notation before establishing the DF structure pricing models of options on a non-dividend -paying stock.

Assumption 3.1

(i) The basic assumptions in 2.1, the assumption 2.1 and assumption 2.2 all come into existence.

(ii) There are no dividends during the life of the derivative.

(iii) The risk-free rate of interest, r, is constant.

(iv)There are no riskless arbitrage opportunities.

(v) Security trading is continuous.

All the following discussions are under Assumption 3.1. We will use the following notation:

t—current time.

S(t)—market price of the stock at t.

X—strike price of option on S(t).

T—time of expiration of option.

r—risk-free rate of interest to maturity T.

S(t)e^r(T-t)—forward value of S(t)(E(S(T)) ,the expected value in a risk-neutral world).

X_s(t,T)—DF stochastic structure of X on S(t)e^r(T-t).

C_s—value of call option to buy one share.

P_s—value of put option to sell one share.

If S(t)εP( µ(t), σ²(t)) and XS(t,T)εP(X, D[S(t)e^r(T-t)](T-t)), we have the DF structure models of options pricing (DF model for short) as "DF structure models of call options pricing".

Prof. Feng Dai, Prof. Zifu Qin

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Summary: Index