Eric Philips

Solving for the number of dollars in the risk-free security.

Finding the amount of money borrowed at the risk free rate is an import step towards valuing a derivative, the amount borrow is represented by B.

B = Number of dollars in the risk-free security

Typical inputs into the problem would include the value of both the derivative and it's underlying asset in a 'good' condition (up) and a 'bad' condition (down). These conditions represent the probable changes in value of the derivative and asset depending on changes in the economy.

rf = Risk-free rate

Su = Value of the underlying asset at the up node
Sd = Value of the underlying asset at the down node

Vu = Value of the derivative at the up node
Vd = Value of the derivative at the down node

The number of dollars that in the risk-free asset is represented by this equation.

B = $\frac{S_d V_u - S_u V_d} {(1+r_f)(S_d - S_u)}$

The good and bad values of the underlying asset and derivative could be given; if not the value might be given purely as a % increase or decrease of the original value one period in the future. To calculate the up node value add 1 to the % increase and multiply by the starting value, for the down node subtract the % decrease from 1 before multiplying by the start value. Be sure to pay attention to the numbers given; it's critical to keep the underlying security information separate from the derivative information. Additionally, make sure your up and down values for each asset makes sense.

For example:

An underlying asset is worth 100 dollars and will be 20% more one year from now if the economy is good, however it will be worth 30% less if the economy sours.

Su = (1+20%)*$100 = $120
Sd = (1-30%)*$100 = $70

If the derivative is worth $200 in an up state and $130 in the down state and the current risk free rate is 6.65%:

Vu = $200
Vd = $130
rf = 6.65%

B = $\frac{(70 * 200) - (120 *130)} {(1+.0665)(70 - 120)}$

B = $\frac{-1,600} {(1.0665)(-50)}$

B = $30 invested in the risk free asset

The value of understanding the calculation of B comes from it's use in many other formulas, for example:
Subtract B from the product of $\Delta$ (the number of units required in underlying security) and S (the cost one share) will give you the no-arbitrage value of the derivative
V = $\Delta$S + B