Session 9b

Decision Models -- Prof. Juran2 Overview Finance Simulation Models Securities Pricing –Black-Scholes –Electricity Option Miscellaneous –Monte Carlo vs. Latin Hypercube –Review of Binomial

Decision Models -- Prof. Juran3 Finance Example A European call option on a stock earns the owner an amount equal to the price at expiration minus the exercise price, if the price of the stock on which the call is written exceeds the exercise price. Otherwise, the call pays nothing. A European put option earns the owner an amount equal to the exercise price minus the price at expiration, if the price at expiration is less than the exercise price. Otherwise the put pays nothing.

Decision Models -- Prof. Juran4 Finance Example The Black-Scholes formula calculates the price of a European options based on the following inputs: –today's stock price –the duration of the option (in years) –the option's exercise price –the risk-free rate of interest (per year) –the annual volatility (standard deviation) in stock price

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6 Notice the use of “if” statements in cells E10:E11 and B13, so that the same model can be used for both puts and calls. A Black-Scholes calculator:

Decision Models -- Prof. Juran7 Example: Diageo (DEO)

Decision Models -- Prof. Juran8 Assume today is the first trading day of October and that DEO is selling for $57.98 per share. What is a fair price for a six-month call option with a strike price of $60.00? Assume the risk-free rate is 10%. Two approaches: Black-Scholes formula Crystal Ball model

Decision Models -- Prof. Juran9 Data file: deo-data.xlsx

Decision Models -- Prof. Juran10 Remove unnecessary columns and calculate monthly returns:

Decision Models -- Prof. Juran11 Black-Scholes assumes that the future price is the following random function of the current price:

Decision Models -- Prof. Juran12 Another way to look at it:

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Decision Models -- Prof. Juran15 Now we create a new sheet that uses estimated parameters from the data to calculate the future price of DEO and the resulting cash flow from the option. The present value of the expected payout is $0.17, but Black-Scholes says $4.14. Why?

Decision Models -- Prof. Juran16 Notes on the formulas: B11: (random future DEO price) B12: (payoff from the option) B15: (present value of the payoff)

Decision Models -- Prof. Juran17 A green cell: B10 is now a standard normal random variable A blue cell: B15 is now the present value of the random cash flow from the option

Decision Models -- Prof. Juran18 Simulation Results

Decision Models -- Prof. Juran19 OK, so simulation can do the same thing as Black-Scholes. Maybe Black-Scholes is easier and/or quicker than running a simulation. So, why do we need the simulation at all?

Decision Models -- Prof. Juran20 Black-Scholes is an analytical result; if specific assumptions hold true, then we can calculate the expected value of the payout on an option. Analytical solutions do not exist in general for all types of financial instruments. In the absence of analytical results, Monte Carlo simulation offers an alternative approach. Analytical solutions may exist for expected value, but not for other important parameters.

Decision Models -- Prof. Juran21 Example: Asian Option George Brickfield’s business is highly exposed to volatility in the cost of electricity. He has asked his investment banker, Lisa Siegel, to propose an option whereby he can hedge himself against changes in the cost of a kilowatt hour of electricity over the next twelve months.

Decision Models -- Prof. Juran22 Lisa thinks that an Asian option would work nicely for George’s situation. An Asian option is based on the average price of a kilowatt hour (or other underlying commodity) over a specified time period.

Decision Models -- Prof. Juran23 In this case, Lisa wants to offer George a one year Asian option with a target price of $ If the average price per kilowatt hour over the next twelve months is greater than this target price, then Lisa will pay George the difference. If the average price per kilowatt hour over the next twelve months is less than this target price, then George loses the price he paid for the option (but he is happy, because he ends up buying relatively cheap electricity).

Decision Models -- Prof. Juran24 What is a fair price for Lisa to charge for 1 million kwh worth of these options? Use the historical data provided and Monte Carlo simulation to arrive at a fair price.

Decision Models -- Prof. Juran25 Analysis of historical data: Our model will be based not on the actual prices, but on monthly percent changes in price (a.k.a. returns):

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Decision Models -- Prof. Juran27 Returns are approximately normal. We’ll use the sample mean and sample standard deviation from the data ( and , respectively).

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Decision Models -- Prof. Juran32 95% confident that the true fair price is between $2, and $2, Could narrow the interval by running a longer simulation.

Decision Models -- Prof. Juran33 Monte Carlo vs. Latin Hypercube

Decision Models -- Prof. Juran34 Example: Standard normal distribution (mean = 0, standard deviation = 1) Divided into 8 equal-probability ranges

Decision Models -- Prof. Juran35 * Probability that the first random independent observation falls into any one range is

Decision Models -- Prof. Juran36 * Probability that the first two observations fall into any one range is = *

Decision Models -- Prof. Juran37 * Probability that the first three observations fall into any one range is = A small (but not zero!) chance of an unrepresentative sample. * *

Decision Models -- Prof. Juran38 * Latin Hypercube ensures that each range gets one observation before any range gets a second observation (but with more than 8 ranges). *******

Decision Models -- Prof. Juran39 Monte Carlo vs. Latin Hypercube

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Decision Models -- Prof. Juran45 Binomial Random Variable

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Decision Models -- Prof. Juran49 Summary Finance Simulation Models Securities Pricing –Black-Scholes –Electricity Option Miscellaneous –Monte Carlo vs. Latin Hypercube –Review of Binomial