Option Valuation with Binomial Lattices corrected version Prepared by Lara Greden, Teaching Assistant ESD.71

Transcription

1 Option Valuation with Binomial Lattices corrected version Prepared by Lara Greden, Teaching Assistant ESD.71 Note: corrections highlighted in bold in the text. To value options using the binomial lattice with risk-neutral pricing method, two things happen: 1. the risk-neutral probability (q) is used to calculate an expected value, AND 2. the risk-free rate (r f ) is used to discount a future expected value to the present. In the homework problem (which did not use risk-neutral pricing), the regular probability p was used to calculate the expected value of future cash flows for a node, and an assumed discount rate was used to discount that future expected value to the present value for that node. During recitation, I made the error of saying that q replaced both the probability and the discount rate. Rather, q replaces p (for expected value calculation), and r f replaces a risk-adjusted r (for discounted present value calculation). (Note that r f is also involved in the calculation of q.) Refer to the derivation of the risk-neutral pricing approach using a replicating portfolio to understand how one arrives at a pricing method that uses both q and r f. (see Lecture notes Nov. 16 th Arbitrage-Enforced Pricing of Options slide 28-end & Recitation notes Nov. 18 th ). The corrections, as compared to the exercise done in class, are in Step 4 option valuation. Corrected numbers for the entire exercise are provided within this document. The (corrected) spreadsheet is also available on the course website. Objectives: 1. Demonstrate the binomial lattice, risk-neutral approach to calculating option value. 2. Review the dynamic programming process that determines the 'strategy' for exercising the option and, thus, the value of the option. 1

2 Example: valuation of a copper mine inclusive of the option to close the mine in any time period. Methodology: Binomial lattice method with risk-neutral pricing Dynamic programming is used to solve the 'option value lattice.' o The essence of DP is that you do not have to consider all possible ways of arriving at a possible end-state. o Rather, we recognize that in each time period, we will be in some particular state, and there exists an optimal strategy if management finds themselves in that particular state. Review major assumptions: 1. Assume that the evolution of the price of copper can be modeled with a binomial lattice. o We are assuming that the evolution process is stationary (constant volatility), that each state leads to only 2 others over a period, and that a later state is a multiple of the earlier state. o Furthermore, we assume that price evolution is path independent and completely random (e.g. no mean reversion). 2. Assume that we can use arbitrage enforced pricing, or the risk-neutral approach, to value the option. o Here, we are assuming that a 'replicating portfolio' exists that provides the exact same returns as the project with option. o In this case, the replicating portfolio would consist of 'units of the revenues from the mine' and risk-free loans. o We assume that the 'units of revenues from the mine' can be freely traded in a market. 2

3 o Notice how the financial option assumptions are stretched when applied to a real option... Model Inputs: current price of copper, standard deviation of copper price, periodic rate of increase in copper price, period length, amount of copper produced, variable unit cost, fixed cost, and the risk-free rate of return. Input Data Price Data Start Price 1250 $/ton standard dev σ 0.15 /year periodic increase,ν 0.04 /year period length, T 1 year Revenue Model Amount Produced 5,000 tons/year Variable unit cost 0 $/ton Fixed Cost 6,000,000 $/year Risk-free rate of return, r 0.05 /year Calculated lattice parameters: u, d, q, and p. Calculations u 1.16 =exp(σ T) d 0.86 =1/u q =(1+r-d)/(u-d) p = (ν/σ T) 3

4 Step 1 (Price Outcome Lattice): Construct a lattice to model the movement of the prices over time. uuus o uus o S o us o ds o uds o uuds o o ddus o Solution dds o ddds o t t Step 2 (Probability Lattice - optional for now): Construct the probability lattice for the outcomes; however, note that you do not use this lattice to calculate the option value. (p u ) 3 (p u ) 2 1 p u (1-p u ) p u (1- p u ) (p u ) 2 (1- p u ) 2 p u (1- p u ) 2 Solution (1- p u ) 2 (1- p u ) 3 4

5 Step 3 (Revenue Lattice): Construct the 'outcome' or 'state of the system' lattice. Here, it is the revenue that would be received from operating in each state. o According to formula: Revenue = tons (price - variable cost) - (fixed cost) o We will find that the negative revenue in the down state of the first time period does NOT necessarily mean we should close the mine. Solution 0 1,261,464 2,436,618 3,801, , ,000 1,261,464-1,369, ,575-2,014,824 t t 5

6 Step 4 (Option Lattice): Calculate the value of the project with the option using the risk-neutral probability method and dynamic programming. o Here's how it works: We, the 'virtual managers' start at the possible states in the final time period, where we decide whether or not to keep the mine open (depending on the revenues that would be realized). C uuu = Max[value open, 0] C Solution for t=3 nodes 3,801,951 open 1,261,464 open 0 closed 0 closed t t 6

7 o Then, we move back in time one period. In each state of the second-to-last time period, the 'risk-neutral adjusted' expected value of the revenue from the mine in the future period (discounted by the risk-free rate) is added to the current revenue. Notice both q and r f are used. C uu = Max [(revenue i + (1/(1+r f ))(qc uuu + (1-q)C uud )), 0] Value alive (i.e. closed) Value dead (i.e. closed) q C uuu C 1-q C uud Solution 2,817,126 4,705,566 5,158,949 3,801, ,005,213 1,261, open open open open closed open open closed closed closed t t 7

8 o Do for all nodes in that time period, than move back to the nodes in the previous time period and repeat. o We continue to work backwards to 'today,' time zero. The results are twofold: a) the total value of the project with the option, and b) the optimal strategy for closing the mine if we find ourselves in any particular state. Step 5: Subtract the 'base case (no option) project value' from the 'value with option' to determine the value associated with the option to close. o Need to calculate the NPV of the project without the option (using an assumed discount rate) to complete this step. o In this example, we will use the expected value of revenue in each year. (We will not consider uncertainty.) NPV of project without uncertainty and without option Assumed discount rate 0.12 /year Time period Expected Revenues 0 500, ,000 1,030,400 NPV $1.6 M Base Case Project with option value of option alone $1.6 M $2.8 M =($ ) = $1.2 M 8

9 Summary To determine option value using a binomial lattice, use dynamic programming by working backwards, starting at the final time period. Determine whether to exercise option or not by looking at the (riskneutral) adjusted expected value of future cash flows discounted to the present using the risk-free rate. Do not need to separately consider all possible future cash flows radiating from each node because of the dynamic programming approach. Binomial lattice gives a record of the strategy to follow in any particular state as well as the value of the project with the option. What if a manager finds themselves in a state in between one of the modeled states? 9