1 / 8

Real Options

Real Options. Binomial Approximation Prof. Luiz Brandão brandao@iag.puc-rio.br 2009. Using Discrete Models. Continuous time models require relatively advanced math in order to manipulate the resulting differential equations for option value.

annice
Download Presentation

Real Options

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Real Options Binomial Approximation Prof. Luiz Brandão brandao@iag.puc-rio.br 2009

  2. Using Discrete Models • Continuous time models require relatively advanced math in order to manipulate the resulting differential equations for option value. • Nonetheless, we can model this processes more easily by dividing time in discrete periods, or time intervals. • As we adopt smaller time intervals, the values converge to the continuous time values. • This process is similar to what we usually do with the Discounted Cash Flow Methods, where continuous time cash flows are modeled as discrete monthly or annual lump sums. • In 1979, Cox, Ross e Rubinstein developed a discrete method that allows an approximation to the Geometric Brownian Motion.

  3. VP4 VP3 VP2 VP1 VP0 t0 t1 t2 t3 t4 The Binomial Model • The GBM process can be approximated by a binomial lattice • This allows us to use a simpler, discrete model • The parameters for the CRR model are:

  4. Example: • Using CRR, model the price evolution of an asset that follows a GBM process, assuming no dividends are paid out. • Value: $100 million • Volatility: 15% • WACC: 12%. • No dividends. • Parameters: u, d, e probability q:

  5. Example • We model initially only two periods. • Starting with the current value of the asset, we can determine possible future values and their probabilities • For this we use the CRR parameters we determined previously for the up and down movements. • Given that the process is a GBM, the distribution of values at any time t is lognormal 135,0 116,2 q 100 100 1-q 86,1 74,1

  6. Exemple • Once the distribuiton of the asset prices is modeled, we can determine back its present value. • To do this, we fold back the tree starting with the payoffs of the last period, weighing these values by their risk neutral probabilities. • These values are then discounted at the risk free rate. • Note that we obtain the same value we had initially, regardless of the discount rate we use 135,0 116,2 q 100 100 1-q 86,1 74,1

  7. 332,0 285,8 246,0 246,6 211,7 211,7 182,2 182,2 182,2 156,8 156,8 156,8 135,0 135,0 135,0 135,0 116,2 116,2 116,2 116,2 q 100 100 100 100 100 1-q 86,1 86,1 86,1 86,1 74,1 74,1 74,1 74,1 63,8 63,8 63,8 54,9 54,9 54,9 47,2 47,2 40,7 40,7 35,0 30,1 Exemple • We can extend this model to as many periods necessary to obtain greater precision 156,8 135,0 116,2 116,2 q 100 100 1-q 86,1 86,1 74,1 63,8

  8. Real Options Binomial Approximation Prof. Luiz Brandão brandao@iag.puc-rio.br 2009

More Related