CHAPTER 7 INVESTMENT III: OPTION PRICING AND ITS APPLICATIONS IN INVESTMENT VALUATION

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1 CHAPTER 7 INVESTMENT III: OPTION PRICING AND ITS APPLICATIONS IN INVESTMENT VALUATION Chapter content Upon completion of this chapter you will be able to: explain the principles of option pricing theory determine, using published data, the five principal drivers of option value (value of the underlying, exercise price, time to expiry, volatility and the risk free rate) discuss the underlying assumptions, structure, application and limitations of the Black Scholes model recognise real options embedded within a project, and classify them as one of the real option archetypes use the principles of the Black Scholes model to explain the value of real options to delay, expand, redeploy and withdraw in investment projects and calculate the value of the options. Preamble In general, the value of any asset is the present value of the expected cash flows on that asset. In this section, we will consider an exception to that rule when we will look at assets with two specific characteristics: They derive their value from the values of other assets. The cash flows on the assets are contingent on the occurrence of specific events. These assets are called options and the present value of the expected cash flows on these assets will understate their true value. We will describe the cash flow characteristics of options, consider the factors that determine their value and examine how best to value them.

2 Basics of Option Pricing An option provides the holder with the right to buy or sell a specified quantity of an underlying asset at a fixed price (called a strike price or an exercise price) at or before the expiration date of the option. Since it is a right and not an obligation, the holder can choose not to exercise the right and allow the option to expire. There are two types of options: call options and put options. An option : The right but not an obligation, to buy or sell a particular good at an exercise price, at or before a specified date. Call option :The right but not an obligation to buy a particular good at an exercise price. Put option: The right but not an obligation to sell a particular good at an exercise price. Exercise/strike price :The fixed price at which the good may be bought or sold. American option: An option that can be exercised on any day up until its expiry date. European option: An option that can only be exercised on the last day of the option. Premium : The cost of an option. Traded option: Standardised option contracts sold on a futures exchange (normally US options). Over the counter (OTC) option: Tailor made option usually sold by a bank (normally European options). Call and Put Options: Description and Payoff Diagrams A call option gives the buyer of the option the right to buy the underlying asset at a fixed price, called the strike or the exercise price, at any time prior to the expiration date of the option. The buyer pays a price for this right. If at

3 expiration, the value of the asset is less than the strike price, the option is not exercised and expires worthless. If, on the other hand, the value of the asset is greater than the strike price, the option is exercised - the buyer of the option buys the asset [stock] at the exercise price. And the difference between the asset value and the exercise price comprises the gross profit on the option investment. The net profit on the investment is the difference between the gross profit and the price paid for the call initially. A payoff diagram illustrates the cash payoff on an option at expiration. For a call, the net payoff is negative (and equal to the price paid for the call) if the value of the underlying asset is less than the strike price. If the price of the underlying asset exceeds the strike price, the gross payoff is the difference between the value of the underlying asset and the strike price and the net payoff is the difference between the gross payoff and the price of the call. This is illustrated in figure 7.1 below: If asset value<strike price, you lose is what you paid for the call Net Payoff on Call option Strike price Price of Underlying Asset A put option gives the buyer of the option the right to sell the underlying asset at a fixed price, again called the strike or exercise price, at any time prior to the expiration date of the option. The buyer pays a price for this right. If the price of the underlying asset is greater than the strike price, the option will not be exercised and will expire worthless. If on the other hand, the price of the underlying asset is less than the strike price, the owner of the put option will exercise the option and sell the stock a the strike price, claiming the

4 difference between the strike price and the market value of the asset as the gross profit. Again, netting out the initial cost paid for the put yields the net profit from the transaction. A put has a negative net payoff if the value of the underlying asset exceeds the strike price, and has a gross payoff equal to the difference between the strike price and the value of the underlying asset if the asset value is less than the strike price. This is summarized in figure 7.2 below. Net Payoff on put If asset value>strike price, you lose what you paid for the put. Price of Underlying Asset Strike Price Price of Underlying Asset Option value The key aspect to an option s value is that the buyer has a choice whether or not to use it. Thus the option can be used to avoid downside risk exposure without foregoing upside exposure. The value of an option is made up of two components. These are illustrated below for a call option: The intrinsic value The intrinsic value looks at the exercise price compared with the price of the underlying asset.

5 The value of the call option will increase as the share price increases. Conversely a lower exercise price would also give a higher option value. An option can never have a negative intrinsic value. If the option is out of the money, then the intrinsic value is zero. On the expiry date, the value of an option is equal to its intrinsic value. The Time Value Time to expiry. - As the period to expiry increases, the chance of a profit before the expiry date grows, increasing the option value. Volatility of the share price. - The holder of a call option does not suffer if the share price falls below the exercise price, i.e. there is a limit to the downside. - However the option holder gains if the share price increases above the exercise price, i.e. there is no limit to the upside. - Thus the greater the volatility the better, as this increases the probability of a valuable increase in share price. Risk free interest rate. - As stated above, the exercise price has to be paid in the future, therefore the higher the interest rates the lower the present value of the exercise price. This reduces the cost of exercising and thus adds value to the current call option value.

6 Exercise A pension fund manager is concerned that the value of the stock market will fall. Suggest an option strategy he could use to protect the fund value. [5] Suggested Solution Buying put options would allow the manager to limit the downside exposure. [draw diagrams to illustrate the protective put strategy] Drivers of Option Value The value of an option is determined by a number of variables relating to the underlying asset and financial markets. Current Value of the Underlying Asset: Options are assets that derive value from an underlying asset. Consequently, changes in the value of the underlying asset affect the value of the options on that asset. Since calls provide the right to buy the underlying asset at a fixed price, an increase in the value of the asset will increase the value of the calls. Puts, on the other hand, become less valuable as the value of the asset increase. Variance in Value of the Underlying Asset: The buyer of an option acquires the right to buy or sell the underlying asset at a fixed price. The higher the variance in the value of the underlying asset, the greater will the value of the option be1. This is true for both calls and puts. While it may seem counter-intuitive that an increase in a risk measure (variance) should increase value, options are different from other securities since buyers of options can never lose more than the price they pay for them; in fact, they have the potential to earn significant returns from large price movements. Dividends Paid on the Underlying Asset: The value of the underlying asset can be expected to decrease if dividend payments are made on the asset during the life of the option. Consequently, the value of a call on the asset is a decreasing function of the size of expected dividend payments, and the value of a put is an increasing function of expected dividend payments. There is a more intuitive way of thinking about

7 dividend payments, for call options. It is a cost of delaying exercise on in-the-money options. To see why, consider an option on a traded stock. Once a call option is in the money, i.e, the holder of the option will make a gross payoff by exercising the option, exercising the call option will provide the holder with the stock and entitle him or her to the dividends on the stock in subsequent periods. Failing to exercise the option will mean that these dividends are foregone. Strike Price of Option: A key characteristic used to describe an option is the strike price. In the case of calls, where the holder acquires the right to buy at a fixed price, the value of the call will decline as the strike price increases. In the case of puts, where the holder has the right to sell at a fixed price, the value will increase as the strike price increases. Time to Expiration on Option: Both calls and puts become more valuable as the time to expiration increases. This is because the longer time to expiration provides more time for the value of the underlying asset to move, increasing the value of both types of options. Additionally, in the case of a call, where the buyer has to pay a fixed price at expiration, the present value of this fixed price decreases as the life of the option increases, increasing the value of the call. Riskless Interest Rate Corresponding to Life of Option: Since the buyer of an option pays the price of the option up front, an opportunity cost is involved. This cost will depend upon the level of interest rates and the time to expiration on the option. The riskless interest rate also enters into the valuation of options when the present value of the exercise price is calculated, since the exercise price does not have to be paid (received) until expiration on calls (puts). Increases in the interest rate will increase the value of calls and reduce the value of puts. Nb. Note, though, that higher variance can reduce the value of the underlying asset. As a call option becomes more in the money, the more it resembles the underlying asset. For very deep in-the-money call options, higher variance can reduce the value of the option. Table 7.1 below summarizes the variables and their predicted effects on call and put prices.

8 Table 7.1: Summary of Variables Affecting Call and Put Prices Effect on Factor Call Value Put Value Increase in underlying Increases Decreases asset s value Increase in strike price Decreases Increases Increase in variance of underlying asset Increases Increases Increase in time to Increases Increases expiration Increase in interest Increases Decreases rates Increase in dividends Decreases Increases paid American versus European Options: Variables Relating To Early Exercise A primary distinction between American and European options is that American options can be exercised at any time prior to its expiration, while European options can be exercised only at expiration. The possibility of early exercise makes American options more valuable than otherwise similar European options; it also makes them more difficult to value. There is one compensating factor that enables the former to be valued using models designed for the latter. In most cases, the time premium associated with the remaining life of an option and transactions costs makes early exercise sub-optimal. In other words, the holders of in-the-money options will generally get much more by selling the option to someone else than by exercising the options. While early exercise is not optimal generally, there are at least two exceptions to this rule. One is a case where the underlying asset pays large dividends, thus reducing the value of the asset, and any call options on that asset. In this case, call options may be exercised just before an ex-dividend date if the time premium on the options is less than the expected decline in asset value as a consequence of the dividend payment. The other exception arises when an investor holds both the underlying asset and deep in-the-money puts on that asset at a time when interest rates are

9 high. In this case, the time premium on the put may be less than the potential gain from exercising the put early and earning interest on the exercise price. Option Pricing Models Option pricing theory has made vast strides since 1972, when Black and Scholes published their path-breaking paper providing a model for valuing dividendprotected European options. Black and Scholes used a replicating portfolio a portfolio composed of the underlying asset and the risk-free asset that had the same cash flows as the option being valued to come up with their final formulation. While their derivation is mathematically complicated, there is a simpler binomial model for valuing options that draws on the same logic. The Binomial Model The binomial option pricing model is based upon a simple formulation for the asset price process in which the asset, in any time period, can move to one of two possible prices. The general formulation of a stock price process that follows the binomial is shown in figure 7.3. Figure 7.3: General Formulation for Binomial Price Path Su 2 S Su Sud Sd Sd 2

10 In this figure, S is the current stock price; the price moves up to S u with probability p and down to S d with probability 1- p in any time period. Creating a Replicating Portfolio The objective in a replicating portfolio is to use a combination of risk-free borrowing/lending and the underlying asset to create a portfolio that has the same cash flows as the option being valued. The principles of arbitrage apply here and the value of the option must be equal to the value of the replicating portfolio. In the case of the general formulation above, where stock prices can either move up to S u or down to S d in any time period, the replicating portfolio for a call with strike price K will involve borrowing $B and acquiring of the underlying asset, where: = Number of units of the underlying asset bought = C u C d S u S d where, C u = Value of the call if the stock price is S u C d = Value of the call if the stock price is S d In a multi-period binomial process, the valuation has to proceed iteratively, i.e., starting with the last time period and moving backwards in time until the current point in time. The portfolios replicating the option are created at each step and valued, providing the values for the option in that time period. The final output from the binomial option pricing model is a statement of the value of the option in terms of the replicating portfolio, composed of shares (option delta) of the underlying asset and risk-free borrowing/lending. Value of the call = Current value of underlying asset Option Delta - Borrowing needed to replicate the option Illustration 7.1: Binomial Option Valuation Assume that the objective is to value a call with a strike price of 50, which is expected to expire in two time periods, on an underlying asset whose price currently is 50 and is expected to follow a binomial process:

11 Now assume that the interest rate is 11%. In addition, define = Number of shares in the replicating portfolio B = Dollars of borrowing in replicating portfolio The objective is to combine shares of stock and B dollars of borrowing to replicate the cash flows from the call with a strike price of 50. This can be done iteratively, starting with the last period and working back through the binomial tree. Step 1: Start with the end nodes and work backwards: Thus, if the stock price is $70 at t=1, borrowing $45 and buying one share of the stock will give the same cash flows as buying the call. The value of the call at t=1, if the stock price is $70, is: Value of Call = Value of Replicating Position = 70 - B = (70)(1)-45 =25 Considering the other leg of the binomial tree at t=1,

12 If the stock price is 35 at t=1, then the call is worth nothing. Step 2: Move backwards to the earlier time period and create a replicating portfolio that will provide the cash flows the option will provide. In other words, borrowing $22.5 and buying 5/7 of a share will provide the same cash flows as a call with a strike price of $50. The value of the call therefore has to be the same as the value of this position. Value of Call = Value of replicating position =(5/7)*(Current stock)- 22.5=(5/7)*(50)-(22.5)= The Determinants of Value The binomial model provides insight into the determinants of option value. The value of an option is not determined by the expected price of the asset but by its current price, which, of course, reflects expectations about the future. This is a direct consequence of arbitrage. If the option value deviates from the value of the replicating portfolio, investors can create an arbitrage position, i.e., one that requires no investment, involves no risk, and delivers positive returns. To illustrate, if the portfolio that replicates the call costs more than the call does in the market, an investor could buy the call, sell the replicating portfolio and guarantee the difference as a profit. The cash flows on the two positions will offset each other, leading to no cash flows in subsequent periods. The option value also increases as the time to expiration is extended, as the price movements (u and d) increase, and with increases in the interest rate.

13 While the binomial model provides an intuitive feel for the determinants of option value, it requires a large number of inputs, in terms of expected future prices at each node. As we make time periods shorter in the binomial model, we can make one of two assumptions about asset prices. We can assume that price changes become smaller as periods get shorter; this leads to price changes becoming infinitesimally small as time periods approach zero, leading to a continuous price process. Alternatively, we can assume that price changes stay large even as the period gets shorter; this leads to a jump price process, where prices can jump in any period. In this section, we consider the option pricing models that emerge with each of these assumptions. The Black-Scholes Model When the price process is continuous, i.e. price changes becomes smaller as time periods get shorter, the binomial model for pricing options converges on the Black- Scholes model. The model, named after its co-creators, Fischer Black and Myron Scholes, allows us to estimate the value of any option using a small number of inputs and has been shown to be remarkably robust in valuing many listed options. The Model While the derivation of the Black-Scholes model is far too complicated to present here, it is also based upon the idea of creating a portfolio of the underlying asset and the riskless asset with the same cashflows and hence the same cost as the option being valued. The value of a call option in the Black- Scholes model can be written as a function of the five variables: S = Current value of the underlying asset X = Strike price of the option t = Life to expiration of the option r = Riskless interest rate corresponding to the life of the option 2 = Variance in the ln (value) of the underlying asset The value of a call is then:

14 Where Value of call = S N (d 1 ) - X e -rt N(d 2 ) d 1 = [ ln(s/x) + (r + σ 2 /2) T ] / σ T 1/2 d 2 = d 1 - σ T 1/2 N(d) = equals the area under the normal curve up to d (see normal distribution tables to be provided in exam as attachment) e = , the exponential constant In = the natural log (log to be base e) Xe -rt = present value of the exercise price calculated by using the continuous discounting factors. Note that e -rt is the present value factor and reflects the fact that the exercise price on the call option does not have to be paid until expiration. N(d 1 ) and N(d 1 ) are probabilities, estimated by using a cumulative standardized normal distribution and the values of d1 and d2 obtained for an option. In approximate terms, the probabilities [N(d 1 ) and N(d 1 )] yield the likelihood that an option will generate positive cash flows for its owner at exercise, i.e., when S>X in the case of a call option and when X>S in the case of a put option. The portfolio that replicates the call option is created by buying N(d1) units of the underlying asset, and borrowing Xe -rt N(d2). The portfolio will have the same cash flows as the call option and thus the same value as the option. N(d1), which is the number of units of the underlying asset that are needed to create the replicating portfolio/delta hedge, is called the option delta. Some notes on estimating the inputs to the Black-Scholes model The Black-Scholes model requires inputs that are consistent on time measurement. There are two places where this affects estimates. 1. The first relates to the fact that the model works in continuous time, rather than discrete time. That is why we use the continuous time version of present value (exp -rt ) rather than the discrete version ((1+r) -t ). It also means that the inputs such as the riskless rate have to be

15 modified to make the continuous time inputs. For instance, if the oneyear treasury bond rate is 6.2%, the risk free rate that is used in the Black Scholes model should be Continuous Riskless rate = ln (1 + Discrete Riskless Rate) = ln (1.062) = or 6.015% 2. The second relates to the period over which the inputs are estimated. For instance, the rate above is an annual rate. The variance that is entered into the model also has to be an annualized variance. The variance, estimated from ln(asset prices), can be annualized easily because variances are linear in time if the serial correlation is zero. Thus, if monthly (weekly) prices are used to estimate variance, the variance is annualized by multiplying by twelve (fifty two). 3. Implied Volatility-The only input on which there can be significant disagreement among investors is the variance. While the variance is often estimated by looking at historical data, the values for options that emerge from using the historical variance can be different from the market prices. For any option, there is some variance at which the estimated value will be equal to the market price. This variance is called an implied variance Example 7.2 The current share price of B plc shares = $100 The exercise price = $95 The risk free rate of interest = 10% pa = 0.1 The standard deviation of return on the shares = 50% = 0.5 The time to expiry = 3 months = 0.25 Calculate the value of the above call option. Solution Step 1: Calculate d1 and d2. Given that d 1 = [ ln(s/x) + (r + σ 2 /2) T ] / σ T 1/2 d 1 = [ ln(100/95) + ( /2) 0.25 ] / [0.5 *0.25 1/2 ]=0.43

16 and that d 2 = d 1 - σ T 1/2 d 2 = *0.25 1/2 =0.18 Step 2: Use normal distribution tables to find the value of N(d 1 ) and N(d 2 ). N(d 1 ) = = N(d 2 ) = = Step 3: Plug these numbers into the Black-Scholes formula. Value of a call option = S N(d1) Xe -rt N(d2) = e ( ) = $13.70 Note This can be split between the intrinsic value of $5 (100 95) and the time value which is $8.70. Practice Exercise [other details as worked example above] Suppose that the risk free rate is 5% and the variance of the return on the share in the past has been estimated as 12%. Estimate the value of a six month call option at an exercise price of $1.48 (current share price = $1.64). Practice Exercise On March 6, 2016, Nhodo Systems was trading at $ Attempt to value a July 2016 call option with a strike price of $15, trading on the CBOT on the same day for $2.00. The following are the other parameters of the options: The annualized standard deviation in Nhodo Systems stock price over the previous year was 81.00%. This standard deviation is estimated using weekly stock prices over the year and the resulting number was annualized as follows: Weekly standard deviation = 1.556%

17 Annualized standard deviation =1.556%*52 = 81% The option expiration date is Friday, July 20, There are 103 days to expiration. The annualized Treasury bill rate corresponding to this option life is 4.63%. Is the call undervalued or overvalued and what are the implications? Model Limitations and Fixes The Black-Scholes model was designed to value options that can be exercised only at maturity and on underlying assets that do not pay dividends. In addition, options are valued based upon the assumption that option exercise does not affect the value of the underlying asset. In practice, assets do pay dividends, options sometimes get exercised early and exercising an option can affect the value of the underlying asset. Adjustments exist. While they are not perfect, adjustments provide partial corrections to the Black- Scholes model. Underlying assumptions The options are European calls. There are no transaction costs or taxes. The investor can borrow at the risk free rate. The risk free rate of interest and the share s volatility is constant over the life of the option. The future share price volatility can be estimated by observing past share price volatility. The share price follows a random walk and that the possible share prices are based on a normal distribution. No dividends are payable before the option expiry date (although it is possible to modify the model for US options). In practice these unrealistic assumptions can be relaxed and the basic model can be developed to reflect a more complex situation. 1. Dividends The payment of a dividend reduces the stock price; note that on the exdividend day, the stock price generally declines. Consequently, call options will become less valuable and put options more valuable as expected dividend

18 payments increase. There are two ways of dealing with dividends in the Black Scholes: Short-term Options: One approach to dealing with dividends is to estimate the present value of expected dividends that will be paid by the underlying asset during the option life and subtract it from the current value of the asset to use as S in the model. Modified Stock Price = Current Stock Price Present value of expected dividends during the life of the option Long Term Options: Since it becomes impractical to estimate the present value of dividends as the option life becomes longer, we would suggest an alternate approach. If the dividend yield (y = dividends/current value of the asset) on the underlying asset is expected to remain unchanged during the life of the option, the Black- Scholes model can be modified to take dividends into account. C= S e -yt N (d 1 ) - X e -rt N(d 2 ) Where d 1 = [ ln(s/x) + (r y + σ 2 /2) T ] / σ T 1/2 d 2 = d 1 - σ T 1/2 From an intuitive standpoint, the adjustments have two effects. First, the value of the asset is discounted back to the present at the dividend yield to take into account the expected drop in asset value resulting from dividend payments. Second, the interest rate is offset by the dividend yield to reflect the lower carrying cost from holding the asset (in the replicating portfolio). The net effect will be a reduction in the value of calls estimated using this model. Using the Black-Scholes model to value put options If you have calculated the value of a call option using Black-Scholes, then the value of a corresponding put option can be found using the put call parity formula. The put call parity equation is as follows:

19 Put call parity : P= C- S+ Xe rt Step 1: Value the corresponding call option using the Black-Scholes model. Step 2: Then calculate the value the put option using the put call parity equation. Example 7.3 Returning to the early exercise of B plc, where the current share price is $100, exercise price is $95, the risk free rate of interest is 10%, the standard deviation of shares return is 50% and the time to expiry is three months, calculate the value of a put option. Solution Step 1: We have already calculated the value of the call option at $ Step 2: Using put call parity equation: Put call parity: P= C- S+ Xe rt Value of a put = = $6.35 Application to US call options One of the limitations of the Black-Scholes formula is that it assumes that the shares will not pay dividends before the option expires. If this holds true then the model can also be used to value US call options. Example 7.3 Going back to our exercise about B plc, where we calculated that the value of the call option was $13.70 with three months to expiry and assuming that this is a US option and the holder wants to close the option today he has two basic choices:

20 1. Exercise today and benefit by the intrinsic value of $5.00 or 2. Sell the option back to the market and receive $13.70 (intrinsic value plus time value). Therefore if the US option holder exercises early he will forfeit the time value. Quite simply the option is worth more alive than dead. The right to exercise an American call early is irrelevant and the option will be exercised on the expiry date just like a European option. Therefore the values of European and US call options on shares not paying dividends are equal. Valuing US call options on shares that do pay dividends or US put options using the Black-Scholes model is reserved for higher modules. Application to shares where dividends are payable before the expiry date The Black- Scholes formula can be adapted to call options with dividends being paid before expiry by calculating a dividend adjusted share price : Simply deduct the present value of dividends to be paid (before the expiry of the option) from the current share price. S, becomes S PV (dividends) in the Black-Scholes formula. Example 7.4 You have been asked to value call options in D plc. The options are due to expire in five months time. However a dividend of 40 cents is due to be paid in three months time. The current share price is $10. The risk free rate is 10%. What value should be used for the dividend adjusted share price?

21 Solution Three months time t = (3/12) = 0.25 of a year, r = The present value of the dividends is equal to: 40e rt = 40e ( ) = say 39 cents. So S PV (dividends) = $ = $9.61, is used in the Black- Scholes model. Exercise A call option on V plc s shares has the following details: Exercise price $50 Current share price $40 Risk free rate 8% Time to expiry 3 months Dividend of $2 per share to be paid in three months time. Calculate a dividend adjusted share price for use in the Black-Scholes option pricing model. Application to foreign currency options the Grabbe variant The risk free interest rate is a necessary component of the Black- Scholes model to incorporate the time value of money. A complication that arises with currency options is that there are now two interest rates one for each currency. These can be incorporated by using the predicted forward rate in the formula as detailed below. The Black-Scholes formula can be adapted for forex as follows: Value of a currency call option = e -rt [F N(d 1 ) - XN(d 2 )] Value of a currency put option = e rt [X N(-d 2 )- FN(-d 1 )]

22 Where d 1 = [ln (F/X) + ½δ 2 T]/ δ T d 2 = d 1 s T. F = forward rate, calculated using interest rate parity (see below) X = exercise rate r = domestic interest rate (as usual) F and X need to be quoted as the price of the foreign currency (i.e. direct quotes) but indirect rates are used in the interest rate parity formula: Interest rate parity: F 0 = S 0 (1+i c ) (1+i b ) Example 7.5 A UK firm is looking to build a factory in Germany and will find out in 3 month s time whether its tender has been successful, in which case an immediate payment in euros will be necessary. The treasurer has decided to hedge using currency options. Sterling/euro spot rate (direct) 0.8 = 1 Sterling/euro spot rate (indirect) 1 = month LIBOR 3% p.a. 3 month LIBOR 4.5% p.a. Annual volatility of the Euro against Sterling 20% Required Calculate the value of a 3 month Euro/sterling call option with an exercise price of 0.80 = 1 (i.e. currently at the money ). Solution Step 1: Calculate the forward rate using interest rate parity (usually used with indirect exchange rates). Forward rate = spot (1 + foreign i)/(1 + domestic i) = 1.25 (1 + 3/12 3%)/(1 + 3/12 4.5%)

23 = Quoted as a direct rate this is 1/ = Step 2: Calculate d 1 and d 2 using direct exchange rates. Step 3: Use normal distribution tables to find the value of N(d 1 ) and N (d 2 ). N(d 1 ) = = N(d 2 ) = = Step 4: Plug these numbers into the Black-Scholes formula. Value of a call option = e -rt [F N(d 1 ) - XN(d 2 )] = e -( ) [ ] = = or pence per euro Exercise A US firm is suing a French rival over patent infringement and will find out in 9 month s time whether it has been successful, in which case an immediate receipt in euros will crystallise. The treasurer has decided to hedge using currency options. Dollar/euro spot rate (direct) $ = 1 Dollar/euro spot rate (indirect) $1 = month LIBOR 3% 3 month $ LIBOR 5% Annual volatility of the Euro against the dollar 25%

24 Calculate the value of a 9 month Euro/dollar put option with an exercise price of $ = 1 (i.e. currently at the money ). Delta hedges Delta hedging is an options strategy that aims to reduce (hedge) the risk associated with price movements in the underlying asset by offsetting long and short positions. For example, a long call position may be delta hedged by shorting the underlying stock. This strategy is based on the change in premium (price of option) caused by a change in the price of the underlying security. The change in premium for each basis-point change in price of the underlying is the delta and the relationship between the two movements is the hedge ratio. For example, the price of a call option with a hedge ratio of 40 will rise 40% (of the stock-price move) if the price of the underlying stock increases. Typically, options with high hedge ratios are usually more profitable to buy rather than write since the greater the percentage movement - relative to the underlying's price and the corresponding little time-value erosion - the greater the leverage. The opposite is true for options with a low hedge ratio. The figure N (d 1 ) is known as delta and can be used to construct what is known as a delta hedge. An investor who holds a number of shares and sells (an option writer) a number of call options in the proportion dictated by the delta/ the hedge ratio ensures a hedged portfolio. A hedged portfolio is one were the gains and losses cancel out against each other. Number of option calls to sell = Number of shares held/n(d 1 ). Similarly, if you have already written call options, then a delta hedge can be constructed by buying shares. Number of shares to hold = Number of call options sold N(d1).

25 Real Options in Investment Appraisal Flexibilty adds value to an investment.financial options are an example where this flexibility can be valued. Real options theory attempts to classify and value flexibilty in general by taking the ideas of financial options pricing and developing them. Conventional investment appraisal techniques typically undervalue flexibilty within projects with high uncertainty The Real Option Valuation combines a set of option pricing tools to quantify the embedded strategic value for a range of financial analysis and investment scenarios. Traditional discounted cash flow investment analysis will only accept an investment if the returns on the project exceed the hurdle cost of capital rate. This is a worthwhile exercise as input for valuing real options; however it ignores any strategic options that are commonly associated with many investment decisions. Real option valuation provides the ability to identify what options might exist in a business proposal and the tools to estimate the quantification of them. Options relating to project size Where the project's scope is uncertain, flexibility as to the size of the relevant facilities is valuable, and constitutes optionality. Option to expand: Here the project is built with capacity in excess of the expected level of output so that it can produce at higher rate if needed. Management then has the option (but not the obligation) to expand i.e. exercise the option should conditions turn out to be favourable. A project with the option to expand will cost more to establish, the excess being the option premium, but is worth more than the same without the possibility of expansion. This is equivalent to a call option. Option to contract : The project is engineered such that output can be contracted in future should conditions turn out to be unfavourable. Forgoing these future expenditures constitutes option exercise. This is the equivalent to a put option, and again, the excess upfront expenditure is the option premium. Option to expand or contract: Here the project is designed such that its operation can be dynamically turned on and off. Management may shut down part or all of the operation when conditions are unfavourable (a

26 put option), and may restart operations when conditions improve (a call option). A flexible manufacturing system (FMS) is a good example of this type of option. Option to switch/redeploy: It may be possible to switch the use of assets should market conditions change. Options relating to project life and timing Where there is uncertainty as to when, and how, business or other conditions will eventuate, flexibility as to the timing of the relevant project(s) is valuable, and constitutes optionality. Growth options are perhaps the most generic in this category these entail the option to exercise only those projects that appear to be profitable at the time of initiation. Option to delay/ defer: Here management has flexibility as to when to start a project. For example, in natural resource exploration a firm can delay mining a deposit until market conditions are favorable. The key here is to be able to delay investment without losing the opportunity,creating a call option on the future investment.this constitutes an American styled call option. Option to abandon: If a project has clearly identifiable stages such that investment can be staggered, then management have to decide whether to abandon or continue at the end of each stage.management may have the option to cease a project during its life, and, possibly, to realise its salvage value. Here, when the present value of the remaining cash flows falls below the liquidation value, the asset may be sold, and this act is effectively the exercising of a put option. Abandonment options are American styled. Sequencing options: This option is related to the initiation option above, although entails flexibility as to the timing of more than one inter-related projects: the analysis here is as to whether it is advantageous to implement these sequentially or in parallel. Here, observing the outcomes relating to the first project, the firm can resolve some of the uncertainty relating to the venture overall. Once resolved, management has the option to proceed or not with the development of the other projects. If taken in parallel, management would have already

27 spent the resources and the value of the option not to spend them is lost. The sequencing of projects is an important issue in corporate strategy. Options relating to project operation Management may have flexibility relating to the product produced and /or the process used in manufacture. This flexibility constitutes optionality. Output mix options: The option to produce different outputs from the same facility is known as an output mix option or product flexibility. These options are particularly valuable in industries where demand is volatile or where quantities demanded in total for a particular good are typically low, and management would wish to change to a different product quickly if required. Input mix options: An input mix option process flexibility allows management to use different inputs to produce the same output as appropriate. For example, a farmer will value the option to switch between various feed sources, preferring to use the cheapest acceptable alternative. An electric utility, for example, may have the option to switch between various fuel sources to produce electricity, and therefore a flexible plant, although more expensive may actually be more valuable. Operating scale options: Management may have the option to change the output rate per unit of time or to change the total length of production run time, for example in response to market conditions. Valuing real options Valuing real options is a complex process and currently a matter of some debate as to the most suitable methodology. Within this module we are expected to be able to apply the Black Scholes model to real options. Using the Black Scholes model to value real options The Black Scholes equation is well suited for simple real options, those with a single source of uncertainty and a single decision date. To use the model we need to identify the five key input variables as follows:

28 Exercise price/strike Price The capital investment required can be substituted for the exercise price or more precise corresponds to any (non-recoverable) investment outlays, typically the prospective costs of the project. In general, management would proceed (i.e. the option would be in the money) given that the present value of expected cash flows exceeds this amount; Share price The value of the underlying asset is usually taken to be the PV of the future cash flows from the project. This is usually based on management's "best guess" as to the gross value of the project's cash flows and resultant NPV. Time to expiry/option term This is straightforward if the project involves a single investment. It also may consider time during which management may decide to act, or not act, corresponds to the life of the option. As above, examples include the time to expiry of a patent, or of the mineral rights for a new mine. Volatility The volatility of the underlying asset (here the future operating cash flows) can be measured using typical industry sector risk. Risk free rate Many writers continue to use the risk free rate for real options. However, some argue that a higher rate be used to reflect the extra risks when replacing the share price with the PV of future cash flows. Dividends generated by the underlying asset: As part of a project, the dividend equates to any income which could be derived from real assets and paid to the owner. These reduce the appreciation of the asset.

29 Option style and option exercise Management's ability to respond to changes in value is modeled at each decision point as a series of options, as above these may comprise, i.e: the option to contract the project (an American styled put option); the option to abandon the project (also an American put); the option to expand or extend the project (both American styled call options); Switching options, composite options or rainbow options which may also apply to the project. Example 7.6 A US retailer is considering opening a new store in Zimbabwe with the following details: Estimated cost $ 12m Present value of net receipts $10m NPV $2m These figures would suggest that the investment should be rejected. However, if the first store is opened then the firm would gain the option to open a second store (an option to expand). Suppose this would have the following details: Timing (t) 5 years time Estimated cost (X/K) $20m Present value of net receipts(pa) $15m Volatility of cash flows (s) 28.3% Risk free rate (r) 6%. Assess the value of a call on the establishment of the second store. Solution The value of the call option on the second store is then calculated as normal using Black Scholes:

30 Step 1: Compute d 1 and d 2.[using formulae] d 1 =0.33 d 2 =-0.3 Step 2: Compute N(d 1 ) and N(d 2 ). N(d 1 ) = = N(d 2 ) = = Step 3: Use formula. Value of a call option = S N(d1) Xe -rt N(d2) = 15 x e = $3.8m Summary -(0.06 5) Conventional NPV of first store (2) Value of call option on second store 3.8 Strategic NPV 1.8 The project should thus be accepted. Exercise An online DVD and CD retailer is considering investing $2m on improving its customer information and online ordering systems. This is justified on the grounds that it will allow the business to extend the range of products offered. In particular the board is interested in selling gadgets and has estimated the following: Timing 1 years time Estimated cost $5m Present value of net receipt $4m Volatility of cash flows 40% Risk free rate 5% Advise the firm.

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