Chapter 2 Option Pricing Theory in Investment Decisions
Answer 1
(a)
This is a straightforward application of the Black and Scholes option pricing model. Each of the input components is stated in the question:
Current price = Present Value of the Project = $24m + $4m = $28 million
Exercise price = capital expenditure = $24 million
Exercise date = 2 years (or 500 trading days)
Risk free rate = 5%
Volatility = 25%
[4 marks]
Using the formula as specified:
[4 marks]
Using the normal distribution tables:
N(d1) = 0.5 + 0.3159 = 0·8159 (using 0.90 as an approximation for d1)
N(d2) = 0.5 + 0.2054 = 0·7054 (using 0.54 as an approximation for d2)
Using the derived values for N(d1) and N(d2) the value of the call option on the value represented by this project is as follows:
c = 0.8159 × 28 – 0.7054 × 24 × e-0.05 × 2 = $7.53 million
[3 marks]
This implies that at the current time the project has a value equal to its net present value plus the value of the call option to delay, i.e., $11·53 million. The additional value arises because the delay option allows the company to avoid the downside element of risk.
[1 mark]
(b)
Overall value of the project
The overall value of the project is the NPV of $4 million plus the value of the option of $7.53 million – that is, $11.53 million. This indicates the considerable payoff from having the option to delay the commencement of the project. [2 marks]
The Black-Scholes model was used in part (a) to determine the value of the real option. The payoff of an option to delay is the same as the payoff of a call option, hence the use of the Black-Scholes model, although this model is more often used for the valuation of financial options, is a good approximation for the value of real options. And it allows companies to determine how valuable having such options available actually are. [2 marks]
(c)
Limitations of the Black-Scholes model
Volatility
One of the main limitations of using the Black-Scholes model for valuing real options is the estimation of volatility. Real options and their underlying assets are not traded, therefore it is very difficult to establish the volatility of the value. In addition this volatility is assumed to be constant throughout the life of the option. [1 mark]
Risk free rate
Likewise, the model assumes knowledge of the risk-free rate of interest, and also assumes the risk-free rate will be constant throughout the option's life, which may not be the case over a longer time period such as two years. [1 mark]
Style of option
The Black-Scholes model assumes that the option is a European style option – that is, it can only be exercised at the maturity date. Where the option can be exercised at any point up to the maturity date (that is, an American style option), the results of the Black-Scholes model are invalid. [1 mark]
Which model should be used if the government made its announcement at any time during the two year time period?
If this were the case then the option could be exercised at any point during the next two years – that is, it would become an American style option. As mentioned above, the Black-Scholes model ceases to be accurate in such circumstances. As a result, the binomial option pricing approach would have to be used. [1 mark]
Answer 2
(a)
Financial impact of option to delay
First of all we calculate the PV of the project without the option to delay.
(b)
Implications of the results
The option to delay the project gives management time to consider and monitor the potential investment before committing to its execution. This extra time will allow management to assess the popularity of similar launches and also to monitor competition. The success of the film will be heavily reliant on the marketing campaign launched by the film’s promoters prior to its release – management will be able to monitor the extent of this campaign before committing to an expensive (and potentially unsuccessful) project.
However the calculations of the value of the option to delay are subject to several limiting assumptions, primarily the volatility of the cash flows. The value of the option to delay ($9.127m) is not an exact figure but rather an indication of how much management would value the opportunity to delay. The result shows that management should not dismiss the project immediately, despite the current negative NPV.
There may be other options embedded within the project. The technology used to develop the game may be used for other projects in the future (option to redeploy). Alternatively the project could lead to follow-on projects if the film is successful enough to generate sequels.
(c)
The value of the option depends on the following variables.
(i) The price of the security
A decrease in the price of the security will mean that a call option becomes less valuable. Exercising the option will mean purchasing a security that has a lower value.
(ii) The exercise price of the option
A decrease in the exercise price will mean that a call option becomes more valuable; the profit that can be made from exercising the option will have increased.
(iii) Risk free rate of return
A decrease in the risk free rate will mean that a call option becomes less valuable. The purchase of an option rather than the underlying security will mean that the option holder has spare cash available which can be invested at the risk free rate of return. A decrease in that rate will mean that it becomes less worthwhile to have spare cash available, and hence to have an option rather than having to buy the underlying security.
(iv) Time to expiry of the option
A decrease in the time of expiry will mean that a call option becomes less valuable, as the time premium element of the option price has been decreased.
(v) Volatility of the security price
A decrease in volatility will mean that a call option becomes less valuable. A decrease in volatility will decrease the chance that the security price will be above the exercise price when the option expires.
A2-1