= Replicating portfolio in a binomial market
{title2=$\Delta=(V_u-V_d)/(S_u-S_d),\quad c_0=(V_d-\Delta S_d)/R$}
Let $c_0$ be the initial monetary balance in the <bank account>, negative for borrowing. Matching successor <financial payoffs> gives the <linear system>
$$
\Delta S_u+Rc_0=V_u,\qquad \Delta S_d+Rc_0=V_d.
$$
Subtracting 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,
$$
\Delta=\frac{20-0}{120-80}=\frac12,\qquad c_0=\frac{0-(1/2)80}{21/20}=-\frac{800}{21},\qquad V_0=50-\frac{800}{21}=\frac{250}{21}.
$$
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>.
\Image[/binomial-option-replication.png]
{title=A half-share position and a debt of 800/21 reproduce the call payoff in both states}
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. https://bpb-us-w2.wpmucdn.com/u.osu.edu/dist/7/36891/files/2017/07/CRR79-1yy8av8.pdf[Cox, Ross and Rubinstein, Option Pricing: A Simplified Approach (1979), Sections 2–3]{ref}
Back to article page
