Replication & Price

Replicating the Payoff

No-arbitrage pricing starts by matching the derivative payoff with traded assets.

highlighted = computed this step

Replicating portfolio

Look for a portfolio of delta shares plus money in the bond that pays $15.00 in the up state and $0.00 in the down state.

ΔSu+B=$15.00,ΔSd+B=$0.00\Delta S_u+B=\$15.00,\quad \Delta S_d+B=\$0.00
Replicating call valueThe derivative value is recomputed from the one-period tree.updownTodayS0 $100.00V0 $5.00Up stateSu $120.00Vu $15.00Down stateSd $90.00Vd $0.00

Share count

Subtracting the down equation from the up equation gives delta equal to 1/2 share.

Δ=$15.00−$0.00$120.00−$90.00=1/2\Delta=\frac{\$15.00-\$0.00}{\$120.00-\$90.00}=1/2

Bond position

The bond position is $-45.00, so the replicating portfolio borrows $45.00.

B=$0.00−1/2⋅$90.00=$−45.00B=\$0.00 - 1/2\cdot \$90.00=\$-45.00