# Value of the investment opportunity

Investment opportunity is the identification of additional, more valuable, opportunities in the future that result from a current opportunity or operation. (http://financialdictionary.thefreedictionary.com) Following this further the value of the investment opportunity of underlying asset is consider as the value of the Call option to buy a non-traded real asset (not a stock or share).

According to Pike, author of Corporate finance and investment “Options are contractual arrangements giving the owner the right, but not the obligation, to buy or sell something at a given price, at some time in the future.” (Pike & Neale, 2003) In addition to this, options are one of the investment opportunities available with the investors. It is a kind of derivative securities that provides the buyer with the option to buy or sell an underlying asset at a certain date for a specified price.

The option price is under the influence of many factors, which are the current price of underlying asset, the exercise price of stock, the time of expiration date, the variance of underlying asset the risk free interest rate and the dividend payments. To simplify the formular Black-Scholes had eliminated the dividend factor out of the consideration, but in the real option price applied in the real investment project, dividend can not be eliminated out of the calculation of the option price. To make the equation more correct and include the dividend, in 1973 Merton had introduce the Black-Scholes extended model, which also involve the dividend to compute the option price (Bahaguna, 2000). By this time, when the Chicago Board Option Exchange (CBOE) began listing call options, trading of standardized option contracts on a national exchange started.

A call option gives it holder the right to purchase an asset for a specified price, call the exercise or strike price, on or before some specified expiration date. Black and Scholes and Merton derived a formula for a value of call option. Scholes and Merton shared the 1997 Nobel Prize in Economics for their accomplishment.

Now widely used by options market participants, the Black-Scholes pricing formula for a call option is

C = S * N (d1) - K * (e ^ -rt) * N (d2)

ln (S / K) + (r + (sigma) ^ 2 / 2) * t

d1 = --------------------------------------

sigma * sqrt(t)

d2 = d1 - sigma * sqrt(t)

Where:

C = theoretical call premium

S = current stock price

N = cumulative standard normal distribution

t = time until option expiration

r = risk-free interest rate

K = option strike price

e = the constant 2.7183..

Sigma = standard deviation of stock returns (usually written as lower-case 's')

ln() = natural logarithm of the argument

sqrt() = square root of the argument

^ means exponentiation (i.e., 2 ^ 3 = 8)

In order to understand the model itself, we divide it into two parts. The first part, SN(d1), derives the expected benefit from acquiring a stock outright. This is found by multiplying stock price [S] by the change in the call premium with respect to a change in the underlying stock price [N(d1)]. The second part of the model, K(e^-rt)N(d2), gives the present value of paying the exercise price on the expiration day. The fair market value of the call option is then calculated by taking the difference between these two parts.

N(d) = the probability that a random draw from a standard normal distribution will be less than (d).

ln = natural logarithm function

e = 2.71828, the base of the natural log function

σ = √ σ2 = 0.2236

d1 = [in (350/500)] + [(0.07- (1/20) + (0.05/2)) 20] / 0.2236 √20

= [-0.356675 +0.9 / 0.9999696]

= 0.5433542

And;

d2 = 0.5433542 - (0.2236 √20)

= 0.5433251 - 0.9999696

= -0.456615

For N(d1), d1 falls in the range of 0.54 and 0.55 with the difference of 0.01 and the cumulative normal distribution falls between 0.7088 and 0.7054 with the difference of 0.0034

So therefore:

N(d1) = 0.5434 - 0.54 = 0.0034

=(0.0034 * 0.0034) / 0.01 = 0.00115

= 0.00115 + 0.7054

N(d1) = 0.70655

Similarly, N(d2 ), d2 falls amid -0.45 and -0.46 with a difference of -0.01 and therefore the cumulative normal distribution stuck between 0.3264 and 0.3228 respectively with a difference of 0.0036

N(d2 ) = -0.456615 - (-0.45) = -0.006615

= (-0.006615 * 0.0036) / -0.01 = -0.0024

= -0.0024 + 0.3264

N(d2 ) = 0.324

Applying the calculated N(d1) = 0.70653 and N(d2 ) = 0.324

C= Se-dTN(d1) - Ke-rTN(d2)

= [(350*e (- (1/20)*20)) * 0.70655] - [(500*e (-0.07*20)) * 0.324]

= 51.025119

Therefore;

The value of the investment opportunity is £51.025 million.

### The Way To Test The Robustness Of The Valuation

The sensitivity analysis, one of the common method, provides a way to test the robustness of the valuation. Besides that it is performed using the option Greeks such as Delta, Gamma, Vega, Theta, and Rho (Mun, 2002). This sensitivity analysis shows how the option value changes when there is a change in the following variables,

1. Stock Price
2. Volatility
3. Risk free interest rate
4. Time

The Delta, Gamma, Theta, Vega and Rho are also known as the hedge parameters because they are used in hedging as well as helping the traders to adopt postures regarding the type of risk. These option Greeks shows a fairly comprehensive view of how the option changes when there is change in above variables and thus helps in providing a test of sensitivity of the valuation. The analysis also helps in finding out which of the variable have the most impact in the real option pricing (Chriss, 1997).

### Delta

Delta which is also known as the hedge ratio is used to represent the relationship between the changes in the market price of an option when there is a change in the stock price (S). It is the option's sensitivity to small changes in the value of underlying asset (Lumby & Jones, 2003)

The Call Delta for the Black-Scholes dividend model is given by the formula:

The delta can take a value between 0 and +1 for call options and 0 and -1 for put options. There are diverse case scenarios where the value of Delta represents different meanings that involve changes in the inherent value of the option (Lumby & Jones, 2003).

And if the call delta value is

* Greater than 0.5 then it is said to be In-The-Money (ITM) that is the options intrinsic value is positive.

* Lesser than 0.5 then it is said to be Out-of-the-Money (OTM) that is the options intrinsic value is zero.

* Equal to 0.5 then it is said to be At-The-Money (ATM).

From the preliminary financial analysis it was given the value of the underlying asset (S) = £ 350m

Then the call delta value if S = £ 350m is

= e-(0.05 * 20) * 0.70655

= 0.3679 * 0.70655

Δ = 0.2599

### Here Is Less Than 0.5 So It Is Out Of The Money.

To see the options sensitivity to small changes in the value of underlying asset lets take some prices which is above and below the given underlying asset of £ 350m. The table below shows the various delta value for change in the value of S and their corresponding option prices. The values are calculated using the formulas under the Black-Scholes dividend model,

### Sensitivity Table:

 Underlying Asset price (S) (£ in million) Call Delta Value Call Option price (£0000) (£ in million) 250 0.2230 29.40 300 0.2397 38.51 350 0.2599 51.025 400 0.2765 64.43 450 0.289 78.59

From the above table it can seen that the call delta value for all the values of underlying asset is less than 0.5 and therefore they are out of the money. Also the value of the call option increases with increase in the underlying asset price.

### Gamma

Gamma represents the rate of change in delta with respect to changes in the stock price.

It is represented by the second derivative of the option price with respect to the underlying stock price.

It measures the degree to which the delta moves as the share price moves. The gamma takes a value between 0 and 1. If the gamma value is higher then, it represents that the value of delta value is more sensitive to the changes in the underlying stock price (Lumby & Jones, 2003). Also traders used de Gamma analysis to corroborate if there was any error choosing the Delta Hedging strategy. The Call Gamma for the BS dividend model is given by the formula (Chriss, 1997)

The Gamma's sensitivity to the various underlying asset price is given in the following table:

 Underlying asset price (S) in millions Call Gamma 250 0.00058 300 0.00045 350 0.00036 400 0.00029 450 0.00024

Since the gamma values for the underlying asset price £250m and £300m is higher compared to others, so the delta is more sensitive in these values.

### Vega

Vega represents the relationship between stock volatility and option value. The Vega parameter is determined by the first derivate of the option price with respect to the volatility (Lumby, Jones 2003).

In order to find a significance factor for the behaviour of the price I necessary to set the volatility as constant, therefore it is a linear relationship. It will so measure the sensitivity of the option price with respect to changes in the volatility. A higher volatility stock is expected to have higher fluctuations in option price on average than one with lower volatility.

The Call Vega for the BS dividend model is given by the formula (Chriss, 1997):

Let's take different volatilities to measure the sensitivity of the option price. The Call Vega for the given volatility of 22.36% (√0.05) is 198.2 and the option price is 51.025.

To find the option price at 30% the formula used is

Where c(σ)is the option value at volatility σ, Δσ is the small change in volatility, Ѵ is the value of Call Vega. So for the volatility of 30% the difference or the change (Δσ) is 7.64%

c(σ + Δσ) = c(σ) + (Ѵ *Δσ)

c(30) = c(22.36) + (198.2 * 0.0764)

c(30) = 51.025 + (15.142)

c(30) = 66.17

The option value at 30% is 66.17

And similarly the option value at 15% is 36.44

### Sensitivity Table:

 Volatility (%) Call option value (£ in millions) 15 36.44 22.36 51.025 30 66.17

Therefore it can be seen that as the volatility increases the option value also increases.

### Theta (Θ)

Theta represents the relationship between the time to maturity and the option price. It measures the option's sensitivity with respect to changes in the time. It shows how the option price changes at each period of time. As the time passes the option's value will be changing even if the underlying asset price remains the same. A negative theta means that an option's value decreases with time and positive theta means that the option's value increases with time and the theta of the European call option is always negative. It means if all the other variables are kept constant then the value of the option will naturally decrease as time passes. So because of this reason theta is known as the time decay of the option (Chriss, 1997).

The Call Theta for the Black Scholes dividend model is given by

Θ(CALL) = -S N'(d1) σ e-yT - r K e-rT N(d2) + y S N(d1) e-yT

2√T

WhereN'(d1)

The given time of expiration is 20 years, then call theta is

Θ(CALL)= -S N'(d1) σ e-yT - r K e-rT N(d2) + y S N(d1) e-yT

2√T

Θ(CALL) = [-350 * e-(0.5434)^2/2 * 0.2236 * e-0.05*20] - [0.07 * 500 * e-0.07*20 * 0.324]

√(2p)

+ [ 0.05 * 350 * 0.7066 * e-0.05*20 ]

= -9.90 -2.796 + 4.55

Θ = -8.146

Let's take different times to measure the sensitivity of the option price. The Call Theta for the given expiration period of 20 years is -8.146 and the option price is 51.025.

To find the option value for a change in time, the formula used is

### c(t + Δt) = c(t) + (Θ * Δt)

Where C(t) is option value at time ‘t' and Δt is the small change in time measured in years.

Here C(20) = 51.025

Therefore for an expiration period of 25 years the option value is

C(t + Δt) = c(t) + (Θ * Δt), here Δt = 5yr

C(25) = c(20) + (-8.146 * 5)

= 51.025 + (-40.73)

C(25) = £ 10.295m

Also similarly the option value for 15years is £ 91.755m

### Sensitivity Table:

 Time of expiration (years) Call option value (£ in million) 15 91.755 20 51.025 25 10.295

Hence it can be seen that as the time of expiration increases the value of the option decreases.

### Rho (ρ)

The Rho represents the relationship between the option value and the risk free rate. It measures the options sensitivity to changes in the risk free rate. As the risk free rate changes the option value also changes. Rho is always said to be positive for European Call options and negative for European put options. So when the interest rate increases the value of the European call option rise and the value of European put option falls (Chriss, 1997).

The Call Rho for Black Scholes dividend model is given by

= K T e-rT N(d2)

The given risk free rate is 7%, so the Call Rho is

= 500 * 20 * e-0.07*20 *0.324

### = 798.974

Now lets take risk free rates to measure the sensitivity of the option price. The Call Rho for the given risk free rate of 7% is 798.974.

To find the option value for a change in risk free rate, the formula used is

### C(r + Δr) = c(r) + (P * Δr)

For a risk free rate of 10% the option value is

C(7% + 3%) = c(7%) + (798.974 * 0.03)

= 51.025 + 23.97

C(10%) = 74.99

Also similarly the option value for risk free rate of 4% is 27.06

### Sensitivity Table:

Risk free rate (%)

Call option value (£ in million)

4

27.06

### 7

51.025

10

74.99

From the table it can be seen that the as the option value increases the risk free rate increases.

Therefore from the above sensitivity analysis using the Greeks such as Delta, Gamma, Vega, Theta and Rho we can measure how the option price changes with respect to the changes in the underlying asset price, volatility, time and risk free rate. Hence this provides a good way in testing the robustness of the valuation.

Nevertheless, throughout the Greek methods we could perceive that the major assumption of the changing nature of the volatility has not been covered entirely, and conjectures are still remaining (Belgrade & Benhamou, 2004).

So now the uncertain of whether over or under estimate the volatility value will be analyse. First of all, the deterministic function of the price of the stock is use to generate an error function. This new function will help to create a link between the volatility and other traceable parameter.

The stock price is related with the annualized risk-free interest rate, the volatility and the time. The term stands in for all sources of uncertainty in the price of a stock.

This expression shows the tracking error function, where stands in for the variability in the option price in function of the time and stock and represent the option price with the “actual” or realized volatility.

Using the self-financing relationship between and we obtain:

And now with the help of the Ito's lemma on we find:

If the BS partial differentiation equation is introduce the above to de last expression the volatility will be now in terms of.

Then we get the difference between the actual volatility and the BS approximation:

So we obtain the ordinary differential expression for tracking the error in volatility:

If necessary the equation can be integrated, so the error function can be easy incorporated in the above Greek analyses correcting errors associated with the uncertain in the volatility.

The final expression combines the Gamma parameter with the volatilities how it is shows in the below equation:

Finally the volatility has to be overestimated if the value for Gamma is positive and underestimate its value if Gamma is negative in order to obtain a positive tracking error. (Belgrade & Benhamou, 2004)

After all it is necessary to comment that diverse authors agree that the robustness of the Black-Scholes methods for pricing options it does not replicate the phenomenon because of the multiples assumptions it does during it develop, however is the “most powerful and simple model to understand what influences the price of an option and estimate its manufacturing cost.” (Bossu, 2005)

Some assumptions that can affect the robustness of the method are listed below.

• The price of the underlying stock not always follows a lognormal distribution with σ(standard deviation) and μ(mean) remaining constant.
• Short selling is allowed without any limits, meaning that people will never have a limited stock to purchase.
• In any of the methods the cost of the transaction is a variable that can affect the valuing of the option.
• The constant behaviour of the free-risk rate.
• No arbitrage opportunity on the market.

### Significance Of Patent Protection To The Value Of This Opportunity

Patent is a property right which is granted by an independence state to the inventor of a new, non-obvious and useful invention from preventing others from producing or copying the inventor's invention. Furthermore Patents protect inventors by creating a significant barrier to competitors in the market commercializing technology the company own rights for a certain period of time. In other word, patenting inventions make drug companies to protect their R&D investment and to secure competitive advantage. A patent is one of the few assets that can increase in value over time

The graph below shows the number of patents issued in drug and plant in the USA as investors are more likely to put money in patented drugs R&D. In addition to this speciality pharmaceutical spending accounted for nearly %10 of total pharmacy spending, up from %5.6 in 2003, according to “Drug Trend Report 2006; Personalizing Healthcare” from Medco Health Solutions.

Source: http://www.luzzatto.com available form: 13/08/09

In the same way, a patent also increases the value of the business because it is considered a valuable asset by banks and potential investors of the business.For example, IBM receives thousands of patents every year that provides IBM approximately 20 years of exclusive use of the invention with little competition. (National Inventor Fraud Center, Inc., 2001)

There are many ways to financially benefit from a patent. The patent may be sold for remuneration, or the company owns the patent may license it to other companies for a percentage of the sale price. A patent is valuable asset that keeps the copycats off the market. For instance, Bristol-Myers Squibb used court cases to delay generic versions of cancer drug Taxol and BuSpar. Generic Taxol was delayed for years. Keeping the copycats off the market made Bristol hundreds of millions of dollars. (Herper, 2002)

The patent is provided for only a particular period of time, which makes the company to work hard on their new invention and develop new drugs as quickly as possible. When a pharmaceutical company takes a long period of time in the invention of a new drug, it will directly affect the profit of the company. Since they need to pay for their researchers who are all involved in that mission of new drug invention.) Patent protection is important to the innovative pharmaceutical industry. Patents are required by innovative companies for the guaranteed period of market exclusivity so as to sustain drug prices, recoup research and development (R&D) expenditures and finance the development of new products. In a Pharmaceutical company the combination of research and development cost and long development time will determine cost of a drug and it is very essential for a pharmaceutical company for a patent protection since, patent covers healing molecule of the drug, the drug manufacturing process. This patent is mainly offered to R&D companies to regain their investment on the research and development investment and make profit on their invented drug. The patent is more effective for pharmaceutical industry rather than other industry, since a slight change in the disease curing drug will turn into a harmful drug. To reduce the risk employed in consumers the Pharmaceutical industries should get patent on their inventions. (Thakkar & Argaval, 1997)

Therefore, patent protection is quite significant to the value of the investment opportunity. However, decision makers need to consider the costs and benefits of patent protection. In spite of the obvious benefits of patent protection, there are some costs points to consider. For instance, what is the projected commercial value of the invention, what are the projected out-of-pocket expenses for registering the patent, in addition to legal fees, what advertising, marketing or even re-tooling expenses will be incurred? (Sherman, 2009) And also need to consider the timing issues. To grant a patent usually takes 24-30 months, and how long the patent is granted.

The presence of patents makes the new entrants difficult to copy the existing product. However, when patent protection expires, it will affect the company's profitability. Therefore, the following issues need to be considered as well:

- How far do sales fall when the product come off patent

- Will there be rise in marketing expenses to compete with generic products etc. (Cahill, 2003)

### The Impact Delay Have On The Value Of The Opportunity And The Way To Measure It

The new drug development project for the company has been typically analyzed based upon the project's expected future cash flows and discount rate. The net present value (NPV) computed on this basis is a measure of the project's value and acceptability (Arnold, 2002). The NPV for our new project is the difference between the underlying asset value (S) and the strike price for the project (K), which can be expressed as:

NPV = S - K = 350 - 500 = - ￡150 million (Arnold, 2002).

On the basis of this computation we can see that our project has a negative NPV. However, since the expected cash flows and discount rate may change over time, therefore so does the NPV (Ross, Westerfield & Jaffe, 2005). In such circumstances, projects with negative NPV could be able to turn to positive NPV in the future. In this sense, our company should defer this new drug development project. The decision to postpone this new project is equivalent to holding a call option, which provides the right, but not the obligation, to undertake the project sometimes in the future, when the company decides to do so (Ross, Westerfield & Jaffe, 2005). As mentioned before, the NPV for our new project is negative; hence this project is less valuable and attractive now. However, since our company has the exclusive rights to the new drug development for the next 20 years, which are the barriers to entry for this new product, therefore it is significantly valuable for our company to defer this new drug development project.

### The Option To Delay The Project

PV of cash flow

Project's NPV turns positive in this range

Initial investment in project

Project has negative NPV in this range PV of future cash flow

Source: Arnold, G. (2002) “Corporate Finance Management”.

The above payoff diagram is that of a call option (Arnold, 2002). The underlying asset is our new drug discovery project; the strike price is the initial investment required to take this project and the life of this call option is the period for which our company has rights to the project. Our company should delay to undertake this new project until the project's NPV turns positive.

### Implications Of The Delay

The key implication that emerges from delaying the project can be analyzed as follows. Although the new project has a negative NPV based upon expected cash flows currently, it might still a valuable project owing to the characteristics for the call option (Damodaran, 2006). In this regard, while a negative NPV should encourage our company to reject the project, it should not lead the company to conclude that its rights to the project are worthless because the company would gain by waiting and accepting the project in the future period in the hopes that the future expanded market will improve the expected cash flows and consequently the value of the project (Damodaran, 2006). However, the company has to weigh this off against the cost of deferring the project, which will be the cash flows that will be forsaken by not taking the project.

### Measurement Of The Delay Option

In general, the inputs needed to apply option pricing theory to value the delay option are quite similar to those required for other options (Vintila, 2007). These inputs are consist of the value of the underlying asset, the variance in that value, the time to expiration on that option, the initial strike price as well as the riskless discount rate (Vintila, 2007). However, the key difference to other option measurement is that the cost of delay which has also been named as the dividend yield is the critical input for valuing the option of delay (Vintila, 2007). More specifically, if the cash flows arising from the new project are evenly distributed over time, and the life of the patent is n years, then the cost of delay can be written as:

Annual cost of delay = 1 / n (Vintila, 2007).

As referred to our case, the new project rights are for 20 years, therefore the annual cost of delay is equal to one over twenty, which is 5% a year.

In addition, the value of underlying asset (S) equals to £350 million; the strike price (K) is £500 million; the variance in underlying asset's value (σ2) is equal to 0.05; the time to expiration (t) is 20 years; the dividend yield (y) equals to 1/20; and finally the riskless rate (r) is 7%.

Based on the above information, the value of the delay option can be expressed as:

Value of the delay option = S exp (-y) (t) N (d1) - K exp (-r) (t) N (d2)

where,

d1 = [ln (S/K+ (r - d + σ2/2)T]/ σ √T;

d2 = d1 - σ √T;

and N(d) represents the cumulated probability of normal distribution (Vintila, 2007).

These yield the following estimates for d and N(d):

d1 = 0.5433542; N(d1) = 0.70655 ;

d2 = -0.456615; N(d2) = 0.324

Plugging back into the delay option pricing model, we could get:

Value of the delay option = 350 exp (-0.05) (20) N (d1) - 500 exp (-0.07) (20) N (d2)

= 350×0.3679×0.70655 - 500×0.2466×0.324

= 90.9738 - 39.9487

= 51.025 million

In contrast, the NPV for this new project is negative ￡150 million; while the value of the delay option is positive ￡51.025 million , which is significantly above the NPV. In this case, the company will be better off waiting rather than developing the new drug immediately. However, it is important to note that the cost of delay will rise over time, and consequently make the development of new drug more likely.

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