Danny Pandhare

Binomial Process, Binomial Tracking Portfolio & Value of the Derivative is one of the most challenging problem covered in FIN622
The theory & equation shown/explained below helps to find the relation of a derivative with an underlying asset, which has ability to use the underlying asset to perfectly track the derivative's future/present cash flow.

Step1: Binomial Process (Binomial Tree)
Lay out the tree for derivative and underlying asset on paper with a "up” state & “down” state.
UU
_U
C -UD/DU
_D
DD

Step2: Tracking Portfolio
The first of these equations deals with the “up” values.
The equation is ΔSu + Β(1+rf) = Vu,
where:
Δ is the number of units of the underlying asset
Su is the value of the underlying asset in the “up” state
Β is the number of dollars in the risk free security
rf is the risk free rate
Vu is the value of the derivative in the “up” state.
Similarly there is a down node governed by the equation ΔSd + Β(1+rf) = Vd.
where:
Sd is the value of the underlying asset in the “down” state
Vd is the value of the derivative in the “down” state.

Step3: Value of the Derivative
V = ΔS + Β

Consider following Example for 2-Period Binomial Valuation:
Underlying asset
Current value = $100
U = $160
D = $40
UU = $256
UD = $64
DU = $64
DD = $16

Derivatives is as follows
UU = $156
UD = $0
DU = $0
DD = $0

Risk free rate = 15.4%

Solving the following equation we get
ΔSu + Β(1+rf) = Vu; Δ256 + Β(1+rf) = 156
ΔSd + Β(1+rf) = Vd; Δ64 + Β(1+rf) = 0
Δ=0.8125 & B=-45.06
V=0.8125*160-45.06=84.94

Solving the next set of equation we get
ΔSu + Β(1+rf) = Vu; Δ64 + Β(1+rf) = 0
ΔSd + Β(1+rf) = Vd; Δ16 + Β(1+rf) = 0
Δ=0 & B=0
V=0

Solving the last set of equation to find the present value
ΔSu + Β(1+rf) = Vu; Δ160 + Β(1+rf) = 84.94
ΔSd + Β(1+rf) = Vd; Δ40 + Β(1+rf) = 0
Δ=0.7078 & B=-24.53
V=0.7078*100-24.53=46.25

The step by step approach shown above calulates derivative's future/present cash flows: Vc=46.25, Vu=84.94, Vd=0