Energy and public Policies

Transcription

1 Energy and public Policies Decision making under uncertainty Contents of class #1 Page 1 1. Decision Criteria a. Dominated decisions b. Maxmin Criterion c. Maximax Criterion d. Minimax Regret Criterion e. Expected value Criterion 2. Utility Theory basics a. Von Neumann-Morgenstern axioms b. Estimating individual utility functions c. Utility and attitude towards risk Contents of class #2 3. Decision trees a. Decomposing complex decision problems b. Incorporating risk aversion c. Bayes rule and decision trees 4. Not deciding Real options a. Delay option basic setup b. Pricing an option

2 3 Decision trees 3a. Decomposing complex decision problems Page 19 In decision analysis, decision trees and their closely related diagrams are used as a visual and analytical decision support tool, where the expected (or expected utility) values of competing alternatives are calculated. A decision tree consists of 3 types of nodes: 1. Decision nodes - represented by squares 2. Event nodes - represented by circles 3. End nodes - represented by triangles (not here) Problem : choose the best decision sequence [u ] given a set of conditional state-probabilities (s, p) and a reward function r =f (u, s) Example: A company wants to decide whether to invest in a new technology and has 3 alternatives: Alternative 1. Wait before investing to get more information on the market Alternative 2. Decide to invest immediately without further information on the market Alternative 3. Abandon the investment decision immediately without further information on the market Success would increase the company s asset position from 150 to 450 M (+300) and failure would decrease it from 150 to 50 M (-100). If the company waits to gather more information, waiting costs 30M.

3 Without further information, the company believes in having a 55% chance of success and a 45% chance of failure. Further information will point out to an investment success with 60% probability and to a failure with 40% probability. If information points out to success, then success becomes guaranteed with 85%. Page 20 If information points out to failure, then success becomes guaranteed with 10% probability only. Supposing the company is risk-neutral, what would be the best strategy to follow? We have two kinds of decisions: Wait Invest Don t wait Abandon and two kinds of events: Ο Pointed success Pointed failure Ο Success Failure

4 The decision process can be represented by the following tree of events and decisions: Page 21

5 Let us start by assigning asset values to the terminal branches (Success and Failure) For instance, the success after investing without waiting gives =450, the failure gives =50. The success after waiting and for a pointed success gives = 420, the failure gives =20. Page 22

6 Based on the terminal asset positions, we then compute the expected final asset positions for the 3 previous event bifurcations: = = 60 Page =270 And then we can evaluate the decision bifurcations by selecting the best decisions (stared below)

7 Moving to the last event bifurcation, one gets: =264 This is inferior to 270, so we decide not to wait before investing. Page 24

8 3b. Incorporating risk aversion Note that the optimal risk-neutral strategy determined before leads to a 45% chance of ending up with an asset position of 50M. Page 25 How should one incorporate risk aversion into the decision process? We start by assigning utility values to the extreme outcomes of the asset positions: u(450)=1 and u(20)=0 And then define a risk-averse utility function for the intermediate values u(420)=0.97 u(150)=0.48 u(120)=0.40 u(50)= fitted curve Utility Asset [M ]

9 Based on the terminal asset positions, we then compute the expected final asset positions for the 3 previous event bifurcations: = = Page = And then we can evaluate the decision bifurcations by selecting the best decisions (stared below) and continue to the root.

10 3c. Bayes rule and decision trees In the example above, there are two states (success S and failure F) which have different probabilities (0.55 and 0.45, respectively) P(S)=0.55 Page 27 P(F)=0.45 Let us call such probabilities, prior probabilities But there are other probabilities for the transitions in the tree those are dependent of the state, such as the probability of success S after a pointed success ps, which we write as P(S ps)=0.85. Such probabilities are called posterior probabilities. P(S ps)=0.85 P(S pf)=0.10 P(F ps)=0.15 P(F pf)=0.90 However, sometimes we don t have information on posterior probabilities but instead information on likelihoods of each state given some prior information (observation). For example, we might know that: 55 investments that were a success have been decided after waiting; from such 55 successes only 51 were pointed out as such.

11 The likelihoods can then be written as: P(pS S)=51/55 P(pF S)=4/55 Page 28 Suppose that one also has information on 45 failed investments and gets P(pS F)=9/45 P(pF F)=36/45 To complete the decision tree, we still need the posterior probabilities, which we can get with help from the Baye s rule: One gets ( )= P(pS S) P(S)=51/55(0.55)=0.51 ( )= P(pF S) P(S)= 4/55(0.55)=0.04 ( )= P(pS F) P(F)= 9/45(0.45)=0.09 ( )= P(pF F) P(F)=36/45(0.45)=0.36 And ( )= ( )+ ( )= =0.6 ( )= ( )+ ( )= =0.4 = ( ) ( ) = ( )

12 Which can be used to compute the posterior probabilities: P(S ps)=0.51/0.6=0.85 P(S pf)=0.04/0.4=0.10 Page 29 P(F ps)=0.09/0.6=0.15 P(F pf)=0.36/0.4=0.90

13 4 Not deciding Real options When there is flexibility to postpone decisions and uncertainty about the future, one might consider using real options Page 30 to value flexibility and to set the best timing to decide

14 4a. Delay option basic setup In finance, the binomial options pricing model (BOPM) provides a generalizable numerical method for the valuation of options. [Cox, J. C.; Ross, S. A.; Rubinstein, M. (1979). "Option pricing: A simplified approach". Journal of Financial Economics 7 (3): 229] Page 31 Essentially, the model uses a discrete-time (lattice based) model of the varying price over time of the underlying financial instrument. Binomial probability mass function:

15 4b. Pricing an option STEP 1: Create the binomial price tree The tree of prices is produced by working forward from valuation date to expiration. Page 32 At each step, it is assumed that the price will move up or down by a specific factor ( or ) per step of the tree. So, if is the current price, then in the next period the price will either be or. The up and down factors are calculated using the underlying volatility, the condition that the variance of the log of the price is, we have:, and the time duration of a step,, measured in years. From This property also allows that the value of the underlying asset at each node can be calculated directly via formula, and does not require that the tree be built first. The node-value will be: Where is the number of up ticks and is the number of down ticks.

16 STEP 2: Find Option value at each final node At each final node of the tree, i.e. at the expiration date of the option, the option value is simply its intrinsic, or exercise, value. Max {, 0} for a call option Page 33 Max {, 0} for a put option: Where is the strike price and is the spot price of the underlying asset at the period. STEP 3: Find Option value at earlier nodes Once the above step is complete, the option value is then found for each node, starting at the penultimate time step, and working back to the first node of the tree (the valuation date) where the calculated result is the value of the option. In overview: the binomial value is found at each node, using the risk neutrality assumption. The steps are as follows: (1) Under the risk neutrality assumption, today's fair price of a derivative is equal to the expected value of its future payoff discounted by the risk free rate. Therefore, expected value is calculated using the option values from the latter two nodes (Option up and Option down) weighted by their respective probabilities probability p of an up move in the underlying, and probability (1-p) of a down move. The expected value is then discounted at r, the risk free rate corresponding to the life of the option. Where is the option's value for the node at time, and

17 If one wants the prices binomial distribution to be representative of a stationary randomness we determine the probability p as = 1 + (1 )=1 If one wants prices tendency to evolve (geometric Brownian motion), we replace the unitary value by an exponential function of time. Page 34 (2) The binomial values represent the fair price of the derivative at a particular point in time (i.e. at each node), given the future evolution of the price from that point forward. It is the value of the option if it were to be held as opposed to exercised at that point. (3) Evaluate the possibility of early exercise at each node: if (1) the option can be exercised, and (2) the exercise value exceeds the Binomial Value, then (3) the value at the node is the exercise value.

18 Example: Page 35

19 Page 36

20 p 0,40 up prob. q 0,60 (1-p) r 0,10 risk free discount rate s 300 initial asset position u 1,5 up factor (down factor=1/u) i 375,00 investment Page 37 probabilities asset position 1,0000 0,4000 0,1600 0,0640 0,6000 0,4800 0,2880 0,3600 0,4320 0, ,0 1012, ,0 450,0 133,3 200,0 88,9 i option value 29,1 80,4 245,6 0,0 0,0 0,0 The red colored figure means that the figure results from exercising the option, i.e., making the investment. Other figures result from waiting for investing (if not zero) or abandoning the investment option (zeros)