Welcome back, my name is Han Smit. I'm a professor of Corporate Finance at Erasmus School of Economics. In the previous webcast, you learned the basics of binomial option valuation. In this webcast, you will learn the multi-period binomial approach, which allows you to value your options with greater accuracy. This model has the advantage that it is adjusted to all kinds of real options in various contexts. By showing how the binomial model relates to the continuous time models, you'll understand the Black-Scholes valuation model and even beyond in this MOOC. The general multiplicative binomial option pricing approach was popularized by Cox, Ross and Rubinstein. It is based on the replication argument, except that stock price movements follow a more strict, multiplicative, binomial process. The underlying stock price follows a stationary multiplicative binomial process over successive periods, expressed as we see here. The stock price at the beginning of a given period, V, may increase by a multiplicative factor u, with a probability q to uV, or decrease with the complementary probability, 1- q, to dV at the end of the period. Thus u and d represent the continuously compounded or logarithmic rate of returns if the stock price moves up or down, with d is 1 divided by u. The call option is the risk neutral expectation of future call option prices and the risk-neutral probability is estimated as before. The above valuation procedure can easily be extended to multiple periods. If the time to expiration of the option, tau, is subdivided into n equal subintervals, each of length h is tau divided by n, and the same valuation process is repeated starting at the expiration and working backward recursively, to the general option pricing formula for n periods. And that would look as follows: The first part is the binomial distribution formula giving the probability that the stock will take j upward movements in n steps, each with risk-neutral probability p. The last part gives the value of the call option at expiration conditional on the stock following j ups each by u percent, and n - j downs, each with d percent within n periods. Where V is the underlying value of the option and E is the exercise price. The summation of all these possible, from j = 0 to n, options values at expiration, multiplied by the probability that each will occur, thus gives the expected terminal option value, which is then discounted at the risk-free rate over n periods. If we let m be then minimal number of upward moves j over n periods necessary for the call option to be exercised or finished in-the-money, and we break up the resulting term into two parts, then the binomial option pricing can be more conveniently rewritten as follows: We see here that this part is the complementary binomial distribution function, giving the probability of at least m ups out of n steps. One may initially object to this discrete period-by-period binomial valuation approach, since in reality stock prices may take more than just two possible values at the end of a period, while actual trading in the market takes place almost continuously and not only on a period-by-period basis. However, the length of a "period" can be chosen to be arbitrarily small by successive subdivisions. As the length of a trading period, h, is allowed to become increasingly smaller, approaching to zero for a given maturity, continuous trading is actually effectively approximated. In continuous-time limit, as the number of periods n approaches infinity, the multiplicative binomial process approximates the log-normal distribution or smooth diffusion Wiener process, as we see in continuous time formulas. By choosing the parameters u, d and b so that the mean and the variance of the continuously compounded rate of return of the discrete binomial process are consistent in the limit with their continuous time counterparts, the stock price will become log-normally distributed and the binomial distribution function will converge to the cumulative standard normal distribution function we know from continuous time models. And so the above binomial formula converges here to the continuous time Black-Scholes formula, as we can see here. Note that the structure of this model is similar to the levered position of N shares of the underlying asset financed by a loan. Consider for example, if tau is three months, so 0.25 years, and n is 12, a discrete multiplicative binomial process with u is 1.1 and weekly intervals of h is tau divided by n is 0.02 years, would be consistent in the limit with a log-normal diffusion process with an annual standard deviation with a sigma of 66%. In this MOOC, we can go actually beyond the Black-Scholes model. Now we understand the basic structure of Black-Scholes, we can consider other, more complex options. The first strategic question management must address in understanding the option value of a business alternative concerns the value characteristics: Does this business alternative, or underlying value, realize its value primarily from direct measurable cash inflows, or does it have an additional strategic option value? Commercial or cash flow projects that can be deferred would be classified as simple options. Application of option theory actually starts here. Other projects do not derive their value directly from cash inflows, but do have strategic value. For instance, a pilot project that might create a new market, R&D, or exploration investments in natural resources might derive their value from future multi-stage commercial opportunities. These projects are classified as compound options. For instance, an exploration license that allows the mining company to invest in exploration can be viewed as a compound option, or an option on an option. The investment in test and appraisal drillings in a mining program, while typically yielding a low return, actually creates an option to invest in subsequent production and development facilities. We can actually extend the continuous time Black-Scholes formula to a compound option, or a call on a call, for which we can again recognize the structure of a position in the underlying asset and a loan. With a compound option, the underlying asset itself is a leveraged position. This formula that you see over here looks complex, but if you look at it carefully you again recognize the structure of an option. The option, compound option, is valued as a position in the underlying asset, which is an option itself, partly financed with a leveraged position. To summarize: in this webcast you learned the foundation of binomial option valuation. A number of points are worth reviewing about the above call option valuation: One: in this webcast you learned to understand the multi-period binomial approach, which allows you to value options with greater accuracy. This model has the advantage that it is adjusted to all kinds of real options in various contexts. You can adjust the model to the business context. Two: the above binomial formula converges to the continuous-time Black-Scholes formula, as the length of a trading period, h, in the binomial model is allowed to become increasingly smaller, approaching to zero for a given maturity. And continuous trading is effectively approximated and the parameter settings of the multiplicative binomial process approaches the log-normal distribution, or smooth diffusion Wiener process, that we find in the continuous time models. Three: in the Black-Scholes model, you can recognize that the value of the financial option can be replicated with the dynamic strategy with adjustments made over time, depending on the change in the market price of the underlying asset. European option actually can be hedged almost perfectly, which allows pricing them with good accuracy, assuming a good estimation of volatility. Four: application of option theory actually just starts here. Many projects do not derive their value directly from cash inflows but generate new options. Real options theory allows to value these options through compound option valuation models. Five: real options refer to choices on whether and how to proceed with business investments. Real options analysis helps management decide on investments that might be delayed, expanded, abandoned or repositioned. A venture capitalist deciding to finance the next stage in a startup, a retail chain that has to decide how and where to expand, or a multinational company decides whether to abandon an unprofitable division, or to shift operations to a plant in another country, all involve real option decisions under uncertainty. I invite you to use real options theory to gain new insights when you deal with corporate investment strategy in an uncertain future. Now you've finished this week of real options valuation, you actually know the tenets of what we can do on real options valuation. I hope you can apply it. See you next week, bye.
