Hi, in the previous lecture, we calculated NPV of a project by considering value of real options using the decision tree approach. In that approach, optimal decisions were made in each state. If the outcome was good, we expanded the business. If the outcome was bad, we abandoned the project. Also, in calculating the NPV, we had to assign probabilities to all possible states of the outcome. In previous examples, we had two possible states, good and bad, and assume that they have equal probabilities of occurring, 50% each. Alternatively, we can use a well established option pricing model named Black-Scholes to evaluate real options. Black-Scholes option pricing model is probably the most popular option pricing model, which is widely used to price options on various underlying assets, including stocks. With the Black-Scholes option pricing model, we can easily calculate an option's value by entering values of five determinants of an option's price in the formula. In this approach, we'll calculate real options value by using the option pricing model. Before we look at the formula, we'll first think about the five determinants of an options price, which are inputs in the Black-Scholes option pricing formula. In this lecture, we'll focus on the call option, which gives the holder the right to buy the underlying asset. First, the current price of the underlying asset should affect a call option's value. Let's take an example of a call option on stocks. When you hold a call option, whose exercise price is $30, and which matures in three months, you feel much happier when the current stock price is $40 than when the price is $20. The reason is because as the current price of underlying asset is higher, the chance that you can gain profits by exercising the option on the expiration date becomes greater. Hence, the current price of the underlying asset should positively affect the call option price. The exercise price of the option also determines the option's value. The lower the exercise price, the higher the call option's value will be. A lower exercise price not only increases the chance that the option will be exercised, but also increases the potential payout from the option. The third determinant is standard deviation or variance of the underlying asset's value. An option value is always positively affected by the volatility of the underlying asset's value. Since an option holder exercises her option only if it is profitable to do so, she's always protected from the downside risk, but it's always exposed to the upside potential. High variance means a higher chance that the underlying asset's price will be either very high or very low in the future. This is an ideal situation for an option holder because she can benefit from a very high ending stock price, while she is not concerned about the very low ending stock price at all. If the ending stock price is lower than the exercise price, she will just choose not to exercise the option. The fourth factor is time to expiration date. The time to expiration date is also positively related to to the option's value. And the reason is very similar to to the one we had in number three. We know that option holders always appreciate volatility in underlying asset value. As you have more time until expiration date, you can expect that the price movement of the underlying asset will be even more volatile. The final input in the Black-Scholes option pricing model is the interest rate on the risk-free asset. This rate is used to calculate the present value of the exercised price in the formula, but we are not going to talk too much about the details here. Just note that the risk-free rate is positively related to the call options price. So, to summarize, the five inputs of the option pricing model are S, K, the standard deviation, σ, time to maturity, T, and the risk-free rate. One amazing thing about the Black-Scholes model is that if we know those five numbers, we can get the fair price of any option very easily. The Black-Scholes model uses the following formula to price call options. You'll see formula variables such as S, K, T, or σ in the formula. I know this formula is overwhelming, but don't let this formula frighten you. We just want to know how to use this model in Excel, and we really don't have to understand the math behind the formula. To understand what N(d1) and N(d2) mean, however, we need a little bit of explanation. And it's a value of the cumulative standard normal distribution, or simply CDF of the standard normal distribution. In Excel, once we calculate for values of d1 and d2, we can get the values of N(d1) and N(d2) by using the function named NORM.S.DIST. Inside the function, we first enter the value, like d1 or d2, for example. Then after hitting a comma, just type true to indicate that we want to get the cumulative distribution function here. Okay, so, why don't we try to find the call option price by applying the Black-Schole's model in Excel? The stock price is $28 now, and the exercise price of the option is $30. That means the option is not profitable at this time. The option holder should hope that the stock price will be higher then $30 on the expiration date. The expiration date is one year from now. And we also know that the risk-free rate is 4%, and the standard deviation of the underlined asset's value is 20%. When we build a model in Excel, we would like to start by making a table for the five inputs at the top. Once you make a good template spreadsheet, by letting sales below reference the numbers in the table, you'll be able to price any call option, using the Black-Scholes model from now on, by simply changing the input in the table at the top. Calculating d1 is the hardest part. Note that I used ln function to take the natural log of S over K, as in the d1 formula, and also used SQRT function to express the square root of T in the denominator. And we'll use this number shortly when we get the value of N(d1) in row 11. Then, we can calculate d2 as well by following the formula. The next step is to find values of the cumulative standard normal distribution of d1 and d2. Inside the NORM.S.DIST function, we reference cell C8, which has the value of d1. Then we type true to indicate that we want the CDF value, not the PDF value. So here is the value of N(d1). Using the same way, we can find the value of N(d2), too. And now, the last step is to find the call option price by applying the Black-Scholes option pricing formula. In the formula, the exercise price, K, is multiplied by the continuous compounding discount factor. And that's why the Excel formula includes the EXP function. So finally, we got the fair price of this call option using the Black-Scholes model. According to the model, the fair call option price is $1.87. So what does this number tell us? Right now, the option seems to be worthless at a glance, because the exercise price is $30, while the current stock price is only $28. How could we benefit from having the right to purchase the stock at $30 when the market price is lower than 30? Nevertheless, this option has the value of about $2.00, because we still have one year until the expiration date. And the stock price is volatile, as the standard deviation of 20% suggests. We can expect that there should be a fair chance that the ending stock price is above $30, so that the option can make money at the end.
