1. Links HOME Options Pricing Model-based pricing of options is a relatively new phenomenon. Until the early 1970's option premiums were determined by offer and demand. "Fair prices" were difficult to estimate and trade was scarce. Buyers thought they paid to much and sellers believed the premiums they received were not enough compensation for the risks they took. Black-Scholes model is mathematical with equilibrium prices determined through a model with five variables the strike price of the option the underlying market price the time until expiry of the option the price volatility of the underlying commodity the "no-risk" interest rate of the Black- Scholes model, in 1973, revolutionised the trade of options. In 1997, Merton and Scholes received the Nobel Prize in Economic Sciences for their work.

2. Links HOME The Cox-Ross-Rubinstein model was developed in the early 1980s. It is also called the binomial options pricing model, or alternatively, path- dependent options pricing model. The Cox-Ross-Rubinstein model The Cox-Ross-Rubinstein model is a relatively simple model for option pricing. Consider a board game, with one pin on the top and several pins below, as follows:     | $60 | 0 | 0 | 0 | 0 | 0 | Options Pricing

3. Links HOME If a ball is dropped at the top, there is a 50% chance that after hitting the first pin it will go to the left and a 50% chance that it will go to the right. Whatever it does, it will hit one of the two pins below, and again there will be a 50% chance that it will go to the left and a 50% chance that it will go to the right. This will be repeated when the ball hits the third pin and it ends up in one of the six slots at the bottom. When the ball enters into the slot at the left, the player wins US$ 60. If it ends up in any of the other slots, nothing would be won. The equilibrium price of a ball in this game can be calculated by multiplying the probability that a profitable outcome is reached with the profits one would have in a profitable case. In the game above, there is only one situation in which the outcome will be profitable, and the probability of this happening is one out of eight. What is the maximum price that a player will be willing to pay for the chance to buy one ball to play this game? Or in other words, what is the “equilibrium price” of a ball in this board game? US$10.00 US$ 7.50 US$ 5.00 A B C Question Question Options Pricing A (RIGHT) The probability for the ball to end up in this box is 0.5 * 0.5 = 0.125. The equilibrium price is therefore (0.125 * US$ 60) = US$ 7.50 (RIGHT) The probability for the ball to end up in this box is 0.5 * 0.5 = 0.125. The equilibrium price is therefore (0.125 * US$ 60) = US$ 7.50 B

4. Links HOME Exactly the same principle applies to options pricing. Consider the following "price tree": Day 1  100 Day 2  95  105 Day 3  90  100  110 Day 4  85  95  105  115 If this outcome is reached, the profit will be US$ 60. So the equilibrium price is 1/8 * US$ 60. In the long run, someone paying less than US$ 7.50 a ball to play this game would make a profit, someone paying more than US$ 7.50 would make a loss. Options Pricing

5. Links HOME On the first day, the price is US$ 100. There is a 50% chance that prices increase to 105, and a 50% chance that prices decrease to US$ 95. Whatever happens, the next day there is again a probability of the price moving US$ 5 up (50%) or down (50%), and the same again on day 3. On day four, the price is US$ 85, US$ 95, US$ 105 or US$ 115 - now, on day 1, it is simply not known which of these four prices will be reached on day 4. Suppose, when market prices become US$ 85, a gain of US$ 20 is made, the 3 other possible market prices having a zero result. What would be the proper price to pay for taking a chance on the possibility that prices move from US$ 100 to US$ 85? US$ 2.50 US$ 5.00 US$ 7.50 A B C Question Question Options Pricing (RIGHT) There is only one path to the price of US$ 85, there are three ways to the prices of US$ 95 and US$ 105 each, and the price of US$ 115 can be reached also only in one way. The relative chances of these prices being reached is: 1 to 3 to 3 to 1. The probability of a price of US$ 85 is therefore 1 out of 8. The price to be paid can be calculated by multiplying the probability that the outcome is worth something and the possible gain. So the price to be paid is 0.125 x US$ 20 = US$ 2.50 (RIGHT) There is only one path to the price of US$ 85, there are three ways to the prices of US$ 95 and US$ 105 each, and the price of US$ 115 can be reached also only in one way. The relative chances of these prices being reached is: 1 to 3 to 3 to 1. The probability of a price of US$ 85 is therefore 1 out of 8. The price to be paid can be calculated by multiplying the probability that the outcome is worth something and the possible gain. So the price to be paid is 0.125 x US$ 20 = US$ 2.50A (WRONG) The probability of a price of US$ 85 is 1 out of 8 and not 3 out of 8 (WRONG) The probability of a price of US$ 85 is 1 out of 8 and not 3 out of 8C (WRONG) The probability of a price of US$ 85 is 1 out of 8 and not 1 out of 4 (WRONG) The probability of a price of US$ 85 is 1 out of 8 and not 1 out of 4B

6. Links HOME Suppose that after 30 days, the price range is the following: With just these few data, the equilibrium prices of all call and put options can be calculated! Options Pricing

7. Links HOME Let us take the example of a put option with a strike price of 90, which is bought at the beginning of the period and can only be exercised on the 30th day. Such a put option has an intrinsic value of US$ 20 when, at the end of the period, the market price is US$ 70; and when at the end of the period, the market price is US$ 80, the put has an intrinsic value of US$ 10. Options Pricing

8. Links HOME Call option Call option The description so far is simplified. One factor to consider ccount is the interest rate - the value of US$ 100 after 30 days is lower than the value of US$ 100 on the first day. Also, for an American option, the option may be executed before the 30th day. The Cox-Ross- Rubinstein model can easily handle these complications. More importantly, the Cox-Ross-Rubinstein can handle "paths" with likelihoods that are not "normally distributed". Options Pricing

9. Links HOME The Black-Scholes model has much more difficulty with asymmetrical price behaviour. This difference is very important for commodity options, because commodity prices are "asymmetrically distributed“. Specifically, these prices tend to be skewed to the high side. There are normally bottom prices, below which market prices are not likely to fall. This is linked to government interventions and to production costs. In general, there is a large group of commodity users which need commodities whatever their price. This group remains in the market even if prices rise strongly, (at least in the short run) therefore the "brake" on price increases is relatively weak. The reasons for this are that Options Pricing

10. Links HOME Options Pricing What is the option’s equilibrium price on the first day, when the market price is US$ 100? US$ 2.00 US$ 1.00 US$ 1.80 None of the above A B C Question QuestionD (RIGHT) The equilibrium price of the option is the sum of the likelihood that the option will be worth something times the likelihood of a price change, so more than one outcome should be taken into account. In this case: equilibrium = (5% * US$ 20) + (10% * US$ 10) = US$ 2.00. (RIGHT) The equilibrium price of the option is the sum of the likelihood that the option will be worth something times the likelihood of a price change, so more than one outcome should be taken into account. In this case: equilibrium = (5% * US$ 20) + (10% * US$ 10) = US$ 2.00.A (WRONG) (WRONG)C B C D

11. Links HOME Consider the earlier scheme again, but now with different likelihoods of price increases and declines: Day 1  100 40% 60% Day 2  95  105 30% 70% 45% 55% Day 3  90  100  110 10% 90% 30% 70% 40% 60% Day 4  85  95  105  115 Again, simply by following the paths the value of options may be calculated. Suppose that in this case, the probability of a price rise is higher than of a price fall, and the percentages differ by step taken. Options Pricing

12. Links HOME Consider an at-the-money call option which can only be executed on the 30th day, the market price on the first day still being US$ 100. What is its equilibrium price? US$ 2.00 US$ 5.50 US$ 30.00 A B C Question  Question  What is the equilibrium price of an at-the-money put option, which can only be executed on the fourth day? US$ 0.72 US$ 1.14 US$ 2.84 A B C Question  Question  Options Pricing (WRONG) Since the price of the option is the sum of the likelihood that the option will be worth something times the likelihood of a price change. (WRONG) Since the price of the option is the sum of the likelihood that the option will be worth something times the likelihood of a price change.A (WRONG) Since the premium is not 30% of US$ 100, for the option’s value lies within the fact that if prices rise above US$ 100, a gain can be made. (WRONG) Since the premium is not 30% of US$ 100, for the option’s value lies within the fact that if prices rise above US$ 100, a gain can be made.C (RIGHT) The price of the option is the sum of the likelihood that the option will be worth something times the likelihood of a price change. In this case, the equilibrium price is (20% * US$ 10) + (10% * US$ 20) + (5% * US$ 15) = US$ 5.50. (RIGHT) The price of the option is the sum of the likelihood that the option will be worth something times the likelihood of a price change. In this case, the equilibrium price is (20% * US$ 10) + (10% * US$ 20) + (5% * US$ 15) = US$ 5.50.B (WRONG) (WRONG)A C (RIGHT) The option will have an intrinsic value when on the fourth day, the price is either US$ 85 or US$ 95. Simply by following the option paths through which these prices can be reached - there are three different paths - the equilibrium price can be calculated: (RIGHT) The option will have an intrinsic value when on the fourth day, the price is either US$ 85 or US$ 95. Simply by following the option paths through which these prices can be reached - there are three different paths - the equilibrium price can be calculated: 0.4 * 0.3 * 0.1 * US$ 15 =US$ 0.18 0.4 * 0.3 * 0.1 * US$ 15 =US$ 0.18 0.4 * 0.3 * 0.9 * US$ 5 = US$ 0.54 0.4 * 0.3 * 0.9 * US$ 5 = US$ 0.54 0.4 * 0.7 * 0.3 * US$ 5 = US$ 0.42 0.4 * 0.7 * 0.3 * US$ 5 = US$ 0.42 or, in total, US$ 1.14. B