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Session 9bDecision Models -- Prof. Juran1Decision Models -- Prof. Juran2OverviewFinance Simulation ModelsSecurities PricingBlack-ScholesElectricity OptionMiscellaneousMonte Carlo vs. Latin HypercubeReview of BinomialDecision Models -- Prof. Juran2Decision Models -- Prof. Juran3Finance ExampleA 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. Juran3Decision Models -- Prof. Juran4Finance ExampleThe Black-Scholes formula calculates the price of a European options based on the following inputs: today's stock pricethe duration of the option (in years)the option's exercise pricethe risk-free rate of interest (per year)the annual volatility (standard deviation) in stock priceDecision Models -- Prof. Juran4Decision Models -- Prof. Juran5

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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:

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Example: Diageo (DEO)Decision Models -- Prof. Juran7Decision Models -- Prof. Juran8Assume 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 formulaCrystal Ball modelDecision Models -- Prof. Juran8Decision Models -- Prof. Juran9

Data file: deo-data.xlsx

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Remove unnecessary columns and calculate monthly returns:

Decision Models -- Prof. Juran10Decision Models -- Prof. Juran11Black-Scholes assumes that the future price is the following random function of the current price:

Decision Models -- Prof. Juran11Decision Models -- Prof. Juran12Another way to look at it:

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Decision Models -- Prof. Juran15Now 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. Juran15Decision Models -- Prof. Juran16Notes on the formulas:B11:(random future DEO price)

B12:(payoff from the option)

B15:(present value of the payoff)

Decision Models -- Prof. Juran16Decision Models -- Prof. Juran17A green cell: B10 is now a standard normal random variableA blue cell: B15 is now the present value of the random cash flow from the option

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Decision Models -- Prof. Juran18Decision Models -- Prof. Juran19OK, 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. Juran19Decision Models -- Prof. Juran20Black-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. Juran20Decision Models -- Prof. Juran21Example: Asian OptionGeorge Brickfields 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. Juran21Decision Models -- Prof. Juran22Lisa thinks that an Asian option would work nicely for Georges 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. Juran22Decision Models -- Prof. Juran23In this case, Lisa wants to offer George a one year Asian option with a target price of $0.059.

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. Juran23Decision Models -- Prof. Juran24What 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. Juran24Decision 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):Decision Models -- Prof. Juran25Decision Models -- Prof. Juran26

Decision Models -- Prof. Juran26Decision Models -- Prof. Juran27Returns are approximately normal.Well use the sample mean and sample standard deviation from the data (0.001768 and 0.073462, respectively).Decision Models -- Prof. Juran27Decision Models -- Prof. Juran28

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95% confident that the true fair price is between $2,916.87 and $2,937.45. Could narrow the interval by running a longer simulation.Decision Models -- Prof. Juran32Decision Models -- Prof. Juran33Monte Carlo vs. Latin Hypercube

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Example: Standard normal distribution(mean = 0, standard deviation = 1)Divided into 8 equal-probability rangesDecision Models -- Prof. Juran34Decision Models -- Prof. Juran35

*Probability that the first random independent observation falls into any one range is 0.125.Decision Models -- Prof. Juran35Decision Models -- Prof. Juran36

*Probability that the first two observations fall into any one range is 0.1252 = 0.01563.*Decision Models -- Prof. Juran36Decision Models -- Prof. Juran37

*Probability that the first three observations fall into any one range is 0.1253 = 0.001953.A small (but not zero!) chance of an unrepresentative sample.**Decision Models -- Prof. Juran37Decision 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. Juran38Decision Models -- Prof. Juran39Monte Carlo vs. Latin Hypercube

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Decision Models -- Prof. Juran45Binomial Random Variable

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Decision Models -- Prof. Juran48Decision Models -- Prof. Juran49SummaryFinance Simulation ModelsSecurities PricingBlack-ScholesElectricity OptionMiscellaneousMonte Carlo vs. Latin HypercubeReview of BinomialDecision Models -- Prof. Juran49