 Replicating portfolio in a binomial market

Let $c_0$ be the initial monetary balance in the bank account, negative for borrowing. Matching successor financial payoffs gives the linear systemSubtracting determines the stock holding $\Delta$ in the header; substitution determines $c_0$. The initial cost is $V_0=\Delta S_0+c_0$. For the European call option with $S_u=120$, $S_d=80$, strike price 100 and payoffs 20 and zero,The debt becomes 40 at maturity: half a share minus 40 yields $60-40=20$ in the up state and $40-40=0$ in the down state. Holding these positions until maturity is self-financing. Repeating this two-equation construction at earlier nodes gives backward option pricing and the unique price in the complete market.

The discrete replication method was developed in Cox, Ross and Rubinstein's 1979 paper, whose Sections 2–3 explain borrowing, statewise matching and why physical probability drops out of relative option valuation. Their symbols for physical and pricing probabilities differ from the $p,q$ convention here. ^[ref]