Option Pricing Dr. J.D. Han

2 *Currency Option in Practice USD call/JP Yen put “Face values in dollars = $10,000,000 Option call/put = USD call or JPY put Option Expiry = 90 days Strike or Exercise Price(FOREX rate) = Exercise = European” The buyer of this option has a right to buy USD $10 million by delivering JP Y 1,200 millions (USD call); He has a right to sell his JP Y 1,200 million for USD $1 (JPY put). This option will be exercise only when the actual price of a US dollar in terms of Yen goes above

3 What is the value/premium of an option to buy one unit of a foreign currency at specified/strike exchange rate?

You may recall: When an option is being written or sold, its premium is paid by the buyer to the writer: For the long call option, the premium is indicated by In the graph: 4

This premium is not arbitrary drawn, but is based on the calculation of “Option Pricing” or the “Pricing of an Option”. Let’s think about the i)Outstanding = secondary market; ii)European; iii) Call option: What will be the price that you are willing to pay from the secondary option market, which was ? 5

6 1.The Bounds of Call Pricing: X Attainable range Lower Bound S Upper Bound

7 1) Upper Constraint on Long Call Premium Premium < Price of underlying asset: “leverage is larger than one.” If you can buy the asset cheaper than the option to buy it, simply buy the asset X Upper Premium Limit S Premium

8 2) Lower Limit on the value = Intrinsic Value = Gross Profits Numeric Example Exercise price or X = $1.5 Case 1: If current Spot FOREX Rate turns out to be equal to $1.6: If this call is American, you can exercise immediately and get the gross profit by St St -X = = 0.1- this option is “in the money” Case 2: If current Spot FOREX Rate turns out to be equal to $1.4: If this call is American, you get the gross loss by St St - X = - 0.1, and this option is out of the money. Thus you do not exercise the option now. You have paid the premium and that is all the loss you are going to have.

9 Illustration of the lower limit on the value of Long Call Option X In the money Out of the money If the current S t < X, then it is a “out of the money” option If the current S t =X, then it is a “at the money” option If the current S t >X, then it is a “in the money” option S t : current market price of the underlying asset S t+1 : future market price of the underlying asset S t+1 will be probably distributed around S t S

**Intrinsic Value is not the same as the Net Pay-off The difference is the Premium, which is sunk cost. The second time buyer at the option market does not respect what the first option buyer has already paid as the premium, and is willing to pay only what the option is worth to him. *Having said that, the first buyer would not pay a lot more or less than the second buyer would pay. 10

Another component of the Market Value of an Option People are buying an option by looking at the current market S t+1,, not S t+1. However, the real possible profits are coming from the difference of S t+1 – X. This means that, from S t, they have to guess what S t+1 will be and whether they will make a profit. The time value = Expected Gains between t and t+1. 11

S t+1 will be distributed around S t with a normal distribution. Mean value of S t+1 is S t : E(S t+1 - X) = S t – X If today the option is such that current spot rate is higher than the exercise price (S t >X), you can expect that, on average, in the future spot rate will be higher than the exercise price(E(S t+1 )> X). 12

The expected gains between t and t+1: If the market is random and the possible loss and gains are symmetrical, the expected gains should be ( ). For the option market where the possible gains and loss are not symmetrical, the expected gains should be always ( ) for the buyer of the call option. 13

14 2. The Actual Call Pricing lies between the upper and the lower bounds: Intrinsic Value + Time Value X Actually observed premiums We call this portion of value/price/premium of option to be “  ime value” An out-of-money option (S t- -X<0) have some positive time value to the potential buyer: there is a chance that S t may change into S t+1 so that S t+1 - X might be positive at the expiration. S

15 3. Intuitive Explanation 1) Option Premium Intrinsic Value (St – X) today Is S t (as a predictor of S t+1 ) higher than X? Time Value Will S t+1 go further above S t ? Expected Value/Price of Option = profits in case the option is exercised (=S t+1 – X in the future) times its probability is closely related to the two components:

16 2) What determines the Time Value? Time left - The more time left until expiration, the more chance for S going up. Volatility - The more volatile, the better chance for S going up. Strike Price versus Market Price of the Underlying Assets - Is this option deep/or slightly in- /or out-of-the-money (reality check) - At-the-money option has the largest time value. Interest Rate - You have to pay the option price up front and to take payoff later at expiration date

17 4. Mathematical Model for Option Premium: 1) Foundation: “Black & Scholes General Option Pricing Model” Note that S t = Eo S = E in our case.

18 2) FX Option Pricing Black-Scholes model is adapted into FX/ Currency Option Pricing by M. Garman and S. Kohlhagen

19 2

20 **Do we have to memorize the formula? No. The option pricing model by B-S is computer- programmed into a spread sheet type of calculator. It is available free of charge from CBOE, and elsewhere eg) FX Option Pricing can be found in SCurrency/input.jsp SCurrency/input.jsp

21 Use the formula or currency option calculatorhttp:// or a more updated one by J.D. Hana more updated one by J.D. Han Example: what is the premium for 360 days 1.30 Strike Call Option of US dollar/Canadian dollar for the assumed volatility of  ?

22 5. What is the merit of Black Scholes or Garman-Kohlhagen model ? It gives the Delta ratio: The Delta ratio is the slope of the tangent line of the Actual Option Value or Price: As underlying asset’s price increases by one unit, call option premium will increase by the slope of the line= dC/dE The Delta ratio helps us figure out the Hedging Ratio: How much options do you have to buy for hedging? In fact, the delta value is the inverse of the required hedging ratio.

23 6. Delta Hedging Theory As underlying asset’s price increases by one unit, the price/value of call premium will increase by the slope of the line dC/dE. Develop hedge ratio that results in no change when underlying asset price changes

What does it mean? When S goes up by 1% Change in total price of an option =change in intrinsic value (1%) + change in time value(less than 1%) Therefore, the total price goes up less than 1%. Δ of an option = % changes in total price from the previous period / 1% change in S. 24

25 *Numerical Example of Delta Hedging Suppose that you are a Canadian importer, and will need US $ 1million one year later to pay to your suppliers; You wish to hedge yourself from the FOREX risk by using the FOREX options in Philadelphia Exchanges. Initial FOREX Risk from Business: In case of a 1% rise in E or the value of US dollars against Canadian dollars, your loss is US $ 10,000. This FX risk has to be covered. Draw the Payoff line. If you hedge with the forward or futures, you have to buy US $1 million. When S changes, the loss from initial FOREX risk is offset by the gains from Hedging.

However, Option Hedging is slightly different in its coverage. First, you will have to hedge by buying a call option of U.S. currency (= by buying a put option of Canadian currency). * If you are using the PHLX(NASDAQ) in the U.S., you should buy a put option of Canadian currency (to be paid in U.S. dollars): Long Put option of Cd $. The next crucial question is how much of external value or coverage of FOREX option of U.S. dollars do you have to buy? -Suppose that its delta ratio of the Canadian currency is 0.50 (graph). - On your original business front as an importer, An 1% change in S will raise the FOREX loss by 1% of U.S. 1 million. However, on the hedging front, a 1% increase in S raises the value of a call option only by 0.5% of U.S. 1 million: - In order to make the (potential) loss = gains, you need the option coverage of U.S. $2 millions. - The so-called ‘coverage ratio’ = 1/ delta In PHLX, one currency call option of Canadian dollars covers Canadian $100,000 or roughly U.S. $80,000 right now at the current S. Coverage: For full coverage of U.S. $ 2 millions, you will have to buy 25 units of Canadian dollar Put options. 26

27 7. More Hedging against Changes in Market Situations Change in FX Rates d C/ d E = Delta d (d C/d E) / dE = Gamma Change in Volatility d C/ d Sigma = Vega d C/ d t = Theta