The binomial option pricing model pdf

The latter is a key to providing incentives for on an appropriate investment opportunity. Texaco also aspires to combine gone a transformation from an academic specialty the real options approach with corporate risk manage- to a business-oriented practice with strong analyti- ment. Finally, Texaco aims, where practical, to incor- cal and organizational foundations. Texaco found porate real options into the portfolio planning process that this superior version of real options had in order to provide the best overall mix of investments several appealing features.

First, it began with a and strategic positioning at the corporate level. For robust and broader strategic framing of the oppor- maximum benefit, Texaco believes in a more holistic tunity in question, something very important to and orchestrated deployment of real options.

While Texaco management. Second, it drew heavily on this is a substantial challenge, Texaco continues to familiar tools such as DCF, decision analysis, and move forward in this endeavor. In sum, while some firms have succeeded at integrated into performance evaluation and com- integrating real options deep into their capital pensation.

Similarly, while the financing decisions of the firm should carefully reflect instruments or contingent financing plans based on formal real options analyses the risk-return profile of its portfolio of assets-in-place as well as its growth options, such as decision trees. Given these twin, but some- taken by the many different corporate managers we times conflicting, goals, substantial effort must be spoke with are quite varied. Nevertheless, whether spent on putting together the right team, choosing by choice or by default, it is clear that there is a the most appropriate project s to experiment with, common path to the successful adoption of real and benchmarking the analysis.

Real options models gates must be passed early on to sustain momen- have immediate appeal to strategic planners since tum in the adoption process. The key steps in this the dynamic modeling of uncertainty and decisions process are: is broadly consistent with, and tends to reinforce, conducting one or more pilot projects that are their intuition.

In contrast, financial analysts are explicitly experimental; more apt to follow conventional methods em- getting buy-in from senior-level and rank-and- ployed by investment bankers and other analysts file managers based on the pilot projects; on Wall Street. Few investment bankers have to codifying real options through an expert working date embraced real options analysis, in large part group, specialist training, and customization; and because of the difficulty in applying real options to institutionalizing and integrating real options value a whole company rather than a single project.

One of the strongest arguments that can be At the completion of each stage, companies made for real options is that it provides a frame- choose whether or not to continue. Nevertheless, work that bridges the longstanding gap between for the vast majority of the managers interviewed, strategy and finance.

In fact, companies that have the expectation is that the firm will proceed through been quick to embrace real options models have all these stages, though at a pace that may differ appreciated that it presents not only a valuation considerably among companies.

In general, com- tool but a framework to incorporate knowledge panies seem to be pleased with their initial work from various parts of the organization into the with real options, and are eager to move on to investment decision-making process. As a result, in further stages, perhaps held back temporarily by exploring the use of real options in the organiza- lack of senior level buy-in, as we discuss below. They also have expertise in Pilot Projects understanding the technical risks faced by the company that are often at the heart of the analysis.

Very few companies want to, or should, pro- The marketing group has experience in understand- ceed with the adoption of real options in one leap. Scientists and engineers are the appro- need to be carefully designed not only to maximize priate experts to estimate success probabilities for Getting approval for a limited budget to perform a pilot project experiment has not seemed to pose a major problem in most firms, particularly if there is an advocate who has credibility with senior management.

All of these aspects of a project failure whether legitimate or not can mean the end must be integrated into a successful real options of the process.

Consequently, firms that have suc- analysis. Frequently, outside consultants with experi- Assuming that the pilot projects demonstrate ence in implementing real options models and the feasibility and desirability of a real options processes are brought in to facilitate and participate approach, the next critical step is senior-level buy- in the pilot project analysis.

One or more successful pilots can go a long way The selection of an appropriate pilot project is toward demonstrating the value of real options. In most firms, one can cesses without management support for such readily identify projects for which the use of real changes. Projects that involve high volatility, decision-making process is the buy-in of key senior- large irreversible investments, and significant flex- level executives.

In companies with projects e. First, and perhaps most obvious, senior trading where real option ideas provide a natural fit, executives are often too busy handling issues requir- support for the adoption of real options analysis seems ing immediate attention to be willing to spend time to come much more easily and quickly.

The valuations generated by the alterna- of Long-Term Capital Management. This and about how to create flexibility in the future. Most firms are equally able to identify projects for which traditional DCF investment needs to be made immediately there are few or insignificant analysis provides relatively accurate valuations.

These include projects where there downstream decisions , or because the level of uncertainty is relatively low. While it is true that real sentations by participants in the pilot stage, circu- options will likely deliver a higher value than would lation of other internal white papers, and even a conventional DCF analysis that ignores options intranet sites dealing specifically with real options.

Such projects were often accepted tend to experience very quick acceptance of these based on overly optimistic, qualitative assessments techniques, presumably because of the ability of of strategic value such as the creation of future these individuals to understand probability distri- growth options, or flexibility. Real options provides butions and their penchant for logic-based pro- a way to inject discipline into this decision process cesses.

One particularly effective technique is to by quantifying these benefits, providing a tougher combine a pilot project success story with a senior standard. Given these challenges, how then is senior- level buy-in achieved? Our interviews suggest a few Codifying Real Options lessons. First, rather than beginning with the techni- cal superiority of the analytic approach, the argu- The combination of successful pilot projects ments in favor of real options must be couched in and widespread buy-in sets the stage for broader simple, compelling terms.

For the senior manage- use and refinement of the real options approach. Second, to the extent possible, direct evidence effort required to use real options. The key elements of in the form of the pilot projects or examples from such a plan include: outside the firm. Some firms creating a working group of experts to coordi- have employed outside consultants to help commu- nate the real options process; and nicate such a story to senior management, but the training specialists in key business areas.

As with senior process to work with, and not against, existing management, rank-and-file employees are invari- processes. At the same time, the process should ably busy with their existing roles and obligations, also be scaleable. Middle- and lower-level employees may look analyzed in a very systematic and detailed manner, at real options as one more burden, especially if not only to decide whether or not to proceed with senior management has not made an appropriate the investment, but also how best to design the commitment to incremental resources.

Furthermore, facility for future flexibility needs. In many firms, these employees may have witnessed a variety of this is the primary application of real options. Again, senior management be a major concern. In contrast, an in-house venture capital group These issues can be compounded by inertia— that has five employees and is evaluating 20 poten- the tendency to rely on known techniques—and by tial deals each quarter has to be able to conduct a real a lack of knowledge about real options that in turn options analysis in an accelerated time frame.

For will exaggerate its complexity. A handful of compa- the decision-making process in relation to the size nies we spoke with had already exposed hundreds of of the investment is critical.

As real options begins their employees to the real options methodology, and to be adopted by different groups in an organization, had provided more detailed training to dozens of key templates and guidelines should be developed for personnel who would be conducting the analyses. These templates and guidelines should be coordinated as much as pos- Once real options has been codified, the firm sible with other management processes.

This work- have begun to explore this stage in the adoption of ing group, consisting of several experts, tends to be real options, and they are arriving there in somewhat housed in the treasury department or in an in-house different ways. Some firms were already using consulting group. As analysts throughout the orga- decision analysis and related techniques in the nization begin to apply real options techniques, this s, and transformed these processes into real group of experts can oversee the necessary quality options analysis techniques in the s.

These firms control. Other firms were using option valu- developed software package, users may not fully ation techniques to price contractual options, and understand what the inputs represent, or how to the evaluation of physical assets to incorporate their properly estimate them.

Specifically, since inputs option features became a natural extension. Inte- such as volatility, mean reversion, jumps, conve- grating the risk management of the physical and nience yields, and correlations do not appear in contractual sides of their business followed naturally standard DCF applications, there is justifiable cause as well. To our knowledge, only one or two firms for concern that lack of experience could lead to have made major external pronouncements to key misspecification of these inputs.

Most of the firms in our in the training of a broad cross-section of employees. But, given that the to be well trained in real options, and to be able to overwhelming majority have only recently become recognize when an evaluation calls for real options interested in real options, it is not surprising that full- and when it does not.

Ac- however, for individuals from operations, market- cording to many of our interviewees, however, this ing, and strategy to understand the benefits of the appears to be only a matter of time, the inevitable real options framework, and to develop a mindset result of the diffusion of real option techniques that aims to create and preserve real options when throughout the organization. Volatility estimates, or the probabilities of the stochastic variables reaching For instance, some options are so deep-in-the-money, i.

The degree of correlation between uncertain variables is another important variable that must be carefully estimated. For instance, while revenues and expenses associated with a new product may both be highly uncertain, if they are closely correlated, the margin on selling the product may be much less uncertain.

Unfortunately, it is impossible to obtain defini- tive proof that this process indeed leads to the right Some observers may feel that real options decision in an individual case, since one can remains primarily an academic pursuit. Others feel observe only whether the outcome is good or bad, that its fortunes are tied to the condition of the new and the outcome is tied as much to the uncertain economy, and that its prospects have dimmed with world as to the decision.

However, over long the plunge in Nasdaq prices. Our discussions with periods of time, companies that make better deci- corporate practitioners paint a different picture. It sions should prosper relative to those whose appears that a quiet, modest evolution is occurring decision processes are deficient.

Furthermore, if in important parts of many companies. These firms firms are able to create options and then monetize have built on the academic foundations, drawn their value by divesting assets or businesses—or by upon research and consulting resources, and are selling equity positions in specific businesses in actively adopting the real options approach.

Will this a discount to their value will be more quickly process accelerate or stall, and how will the practice recognized. While the benefits of real options will deter- Based on our interviews, the evidence seems mine its long-term acceptance, other factors can quite clear.

First, real options will serve not simply affect the rate of adoption dramatically. Two phe- as an analytical tool, but as a general way of thinking nomena indicate that this rate will increase. The first and, in a more rigorous and intensive way, as an is a network effect; as more companies speak the organizational process.

Second, and related, there language and use similar valuation techniques for will likely be increasing convergence among the evaluating projects internally, other companies will various real options approaches, particularly the follow their example. While real real options will likely depend on the answer to options is a difficult tool to apply accurately in one very important question: Does real options valuing an entire company, it is likely that these help managers make better investment decisions— techniques will gradually take hold on Wall Street.

Based on our interviews, there and language by which value will be measured and is an overwhelming—in fact, seemingly unani- communicated in the future. Companies that issued stock on an internally generated dot-com were able to quickly realize the value of their Internet growth options. Smith School of Business atriantis rhsmith. Analysis, a wholly owned subsidiary of PricewaterhouseCoopers adam.

How has it been applied analytically? How have the applications changed as you gain 2. How and when did your firm learn about it? What was it about real options that interested 5. What kind of follow-up process do you have your firm? How was interest in real options communicated 6. How does real options interact with other within your firm?

Who in your firm has expressed the most interest in real options? No other outcomes are possible over the next month for this stock's price. All rights reserved. To order copies, send an to No part of this publication may be reproduced, stored in a retrieval system, used in a spreadsheet, or transmitted in any form or by any means electronic, mechanical, photocopying, recording, or otherwise without the permission of the Darden School Foundation.

Consider what happens when we make the following investments in the stock and the call option. Assuming no arbitrage opportunities, an investor who makes this investment should earn exactly the risk-free rate of return.

The process used to price the option in this example is exactly the same procedure or concept used to price all options, whether with the simple binomial option model or the more complicated Black-Scholes model. The assumption is that we find and form a risk-free hedge and then price the option off of that risk-free hedge. The key assumption is that the risk less hedge will be priced in such a way that it earns exactly the risk-free rate of return, which is where arbitrageurs come in to play.

It is the activity of these individuals, looking for opportunities to invest in a risk less asset and earn more than the risk-free rate of return that insures that options are priced according to the no-arbitrage conditions. In general, there are two approaches to using the binomial model, the Risk-less Hedge Approach and the Risk-Neutral Approach. Either approach will yield the same answer, but the underlying approach differs.

This note examines each approach. Riskless Hedge Approach Estimating the risk less hedge In the example, the hedge ratio, or the number of units of the stock held per call option was given. The next step is to demonstrate how we would find the appropriate hedge ratio for a stock. Assume the following payoff structure for a stock over the next month. Assuming no arbitrage opportunities, the investment,. More formally, this ratio is an estimate of the rate of change in the value of the option relative to the change in the stock price.

In a more formal framework, this is the Delta of the call option. Effectively our total position is Delta neutral. You can see that, by dividing the month up into 2 periods, we wind up with 3 possible outcomes at the end of the month.

The approach to solving this problem is really no more difficult than the original one. The strategy for solving this multi-period problem is to break it up into a number of simple two-period models. We work backwards! Once again, the first step is to assume that we buy H units of the stock with 2 weeks to go and write one call option. The number of units of the stock we need to buy is the risk-free hedge. Thus, as before, we determine the payoff structure.

We now move back to the first 2-week period. We first find the value of the H that leads to a riskless payoff. The basic argument in the risk neutral approach is that since the valuation of options is based on arbitrage and is therefore independent of risk preferences; one should be able to value options assuming any set of risk preferences and get the same answer.

As such, the easiest model is the risk neutral model. In the risk neutral approach, given a stock price process tree we try to estimate these probabilities for a risk neutral individual and then use these risk neutral probabilities to price a call option. For example, we will use the same price process as the original risk less hedge example. It is not easy but it can be shown that while the approaches appear to be different, they are the same.

As such, either approach can be used. Estimating the Binomial Stock price processes. One of the difficulties encountered in implementing the binomial model is the need to specify the stock price process in a binomial tree. While it is not transparent, when we use the Black-Scholes model we are assuming a very explicit functional form for the stock price.

If we are willing to make the same assumptions when we are using the binomial model we can construct a binomial model of the price. Thus, given a volatility estimate we can construct the price process for that security. Once the price process for the underlying security is determined it is possible to use the binomial model to price options on that security. Summary Given the emphasis on the Black-Scholes model it may seem strange but most sophisticated option pricing models use some form of the Binomial Option Pricing model and not the Black-Scholes model.

The approach is to estimate the price process of the underlying asset over the maturity of the option and then overlay the option payoffs given the values of the underlying asset. Solution of 8 with initial conditions 1 is the equations 4 , in which the parameters to the equations 7. These solutions are well-known Black-Scholes formulas calculate the price of European Call and Put when multiplied by. For the numerical solution of equation 8 we can use the correspondingfinite difference method.

In the simplest case, the first and second partial derivatives are approximated by the following finite differences: 9 Substituting 9 into 8 , we obtain the following calculation formula to pass the time to : 10 Note that formula 10 refers to the so-called explicit finite difference schemes, [1] in which the values of a subsequent layer calculated directly by the values of the previous layer.

In the so-called implicit schemes for finding the values of the function at the next layer has to solve a system of linear equations. The advantage of the explicit scheme is reduced compared with the implicit number of calculations. The disadvantage is that such a scheme may be unstable, as occurs, for example, by using the binomial method for options with barriers.

For implementation 10 necessary to choose two parameters: time step and space step. In the binomial method is chosen only time step. Just select the number of steps from 0 to expiration, and the time step is: 11 Space step is chosen so that the transition from the previous step to the subsequent used not three values offunction but two. The method itself is called binomial because of this circumstance. However, it can easily be interpreted in terms of the theory of probability. Indeed, from 13 it follows that the price of an option at a subsequent point in time is the mathematical expectation of option prices in the two neighboring grid points, down one step and up one step.

The transition probabilities of the nodes up and down are appropriate coefficients in Note also that according to 13 in the binomial method one uses not all the nodes in time and space of rectangular grid, but the so-called triangular tree, at the foundation of which lies the point in time for which the option price is calculated and at each time step function values are considered to be only half of the spatial units.

Obviously, finite difference method can be used for the European option. But in this case there is an explicit analytical formula Black-Scholes to make it impractical.