**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