Question

Zuti has a capital investment project that could start immediately. The project will require a machine costing $2.4 million. The total discounted value now of the cash inflows from the project will be either $2.6 million or $1.9 million with equal probability. The risk-free rate is 3%.
Instead of starting immediately the project could be delayed until one year from now to gain more market information. Its total discounted cash inflows at that time will be known as either $2.6 million, or $1.9 million, with certainty.
(i) What is the present value of the option to delay?
(ii) The supplier of the machine has offered to deliver it (if required) in one year’s time at a price of only $2 million, if Zuti pays a non-refundable deposit now. What is the maximum the firm should pay as a deposit now? What type of real option does this represent for Zuti? Identify the specific components of the option contract.

Answer 1

As per the given information , the real option given to Zuti is an option to delay (which is a type of call option) and the value of this option can be derived using the Black-Scholes Option Pricing Model, as below -

where,

wherein,

P_a = underlying asset value, which is the present value of future cash flows arising from the project [here, since there are two possible cash inflows at equal probability, P_a = $2.6M*50%+$1.9M*50% = $2.25M]

P_e = exercise price, which is the amount paid when the call option is exercised [here, cost of asset, P_e = $2.40M]

r = risk-free rate [here, 3%]

s = volatility of cash flows, which is the risk attached to the project or underlying asset, measured by the standard deviation [here, 50%, since there is equal probability]

t = time in years, that is left before the opportunity to exercise ends [here, 1 year]

N(d) = area under the normal curve upto d

e = 2.71828, the exponential constant

ln = the natural logarithm

P_ee^(-rt) = the PV of the exercise price calculated by using the continuous discounting factors.

Thus, we first compute the values for N(d₁) and N(d₂) as below -

d₁ = [ln($2,250,000/$2,400,000)+(0.30+0.50*(0.50)²)*1]/(0.50*1)] = [ln(0.9375)+0.425]/0.50 = [-0.0280+0.425]/0.50 = 0.794 = 0.79

d₂ = 0.794- (0.50*1) = 0.294 = 0.29

N(d₁) = 0.50+0.2852 = 0.7852 ; N(d₂) = 0.50+0.1141 = 0.6141

Then, value of call option can be computed as

Value of call option = [$2,250,000*0.7852]-[$2,400,000*0.6141*2.7183^(-0.03*1)] = $1,766,700 - $1,473,840*0.9705 = $1,766,700 - $1,430,362 = $336,338

In the case where supplier of the machine has offered to deliver it in one year’s time at a price of only $2 million, the P_e changes to $ 2 million and the same is substitued in the above formula and computations are made as below -

We first compute the values for N(d₁) and N(d₂) as below -

d₁ = [ln($2,000,000/$2,400,000)+(0.30+0.50*(0.50)²)*1]/(0.50*1)] = [ln(0.8333)+0.425]/0.50 = [-0.0792+0.425]/0.50 = 0.6916 = 0.69

d₂ = 0.6916- (0.50*1) = 0.1916 = 0.19

N(d₁) = 0.50+0.2549 = 0.7549 ; N(d₂) = 0.50+0.0753 = 0.5753

Then, value of call option can be computed as

Value of call option = [$2,000,000*0.7549]-[$2,400,000*0.5753*2.7183^(-0.03*1)] = $1,509,800 - $1,380,720*0.9705 = $1,509,800 - $1,339,989 = $169,811

Thus, Zuti should pay a maximum security deposit of $169,811.