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 = 120.00 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 120.00.

3

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

4

1. The Bounds of Call Pricing: Intrinsic Value

X

Attain

able

rang

e

Intrinsic Value

5

1) Upper Constraint on Long Call Premium

• Price of underlying asset

• If you can buy the asset cheaper than the option to buy it, simply buy the asset

X

Upper option Premium Limit

S’ or S1 Price of Underlying asset

Premium

6

2) Lower Limit on the value of Long Call Option = Intrinsic Value

Numeric ExampleExercise price or X = $1.5Case 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 S - X =1.6 - 1.5 = 0.1- this option is “in the money”

with Premium = 0.08 for instance, you buy the call, exercise and sell the FOREX. Net Profit = 0.02

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 S - 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.

7

Illustration of the lower limit on the value of Long Call Option

X

S1

In the moneyOut of the money

If the current S< X, then it is a “out of the money” option

If the current S>X, then it is a “in the money” option

S: current market price of the underlying asset

S1: future market price of the underlying asset

S and S1 will be close to each other if nothing happens

8

Note that the most popular strike price is equal to the most widely expected future spot rate, or Forward Rate

Recall

S1 e = (360 day) F

9

2. The Actual Call Pricing lies between the upper and the lower bounds: The difference between the lower limit and the actual value is Time Value

X S1

Actually observed premiums

ime value

An out-of-money option (S-P <0) might have some positive time value to the potential buyer: S1 -P might be positive at the expiration.

10

3. Intuitive Explanation for the value/Premium of Call Option1) Option Premium

• Intrinsic Value

Is S (as a predictor of S1) higher than X?

• Time Value

Will S1 go further above S?

= profits in case the option is exercised (=S1– X) times its probability

- is closely related to the two components:

11

2) What determines the Time Value?

• Time left - The more time left until expiration, the more chance for changes in assets

prices

• Volatility- The more volatile, the better (why not worse?)

• Strike Price versus Market Price of the Underlying Assets - The larger the difference, the less likely it is to be exercised - Is this option deep/or slightly in- /or out-of-the-money (reality check)

• Interest Rate- You have to pay the option price up front and to take payoff later at expiration

date

12

4. Mathematical Model for Option Premium: Black & Scholes Model

d

SE r t

t

FHG

IKJlne j

2

2

C SN dE

eN d trt ( ) ( )

2) Find N(d) and N(d-t) in the table

Calculate

) Calcualte

13

Numerical example: Solving for Call Premium

• For now let’s take the example of Stock Option- Later this can be easily altered for FX Option Pricing

• S = 53 (Current Stock Price of’ ABM’ Co)• X or E = 50 (Strike or Exercise Price) • t = 6 months (Time Left until Expiration)• r = 8% (domestic interest rates)• = 0.2 (Standard Deviation/volatility of S)

14

Step 1: Solve for d

d

SE r t

t

d

FHG

IKJ

FHG

IKJ

ln

ln ..

.

. .

e j

e j

2

2

2

5350 08

22

05

2 05= 0.7656

15

Step 2: Finding Area

From the given Statistical Table

-Find N(d)

N (7656) = 0.7793

-Find N(d – t)

N (0.6242) = 0.7324

16

Step 3: Solve for Premium or C

C SN dE

eN d t

Ce

rt

( ) ( )

. .. .

53 779350

73240 08 0 5

C = 41.30 - 35.18 = 6.12

17

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

http://www.cfo.com/tool/1,,,00.html?tool=/calc/BSCurrency/input.jsp

18

FX Option Pricing

• Black-Scholes model can be easily adapted into FX/ Currency Option Pricing

• Interest rates of Domestic and Foreign Countires should be used

• Use the formula or www.cfo.com ‘s currency option calcualtor

: what is the premium for 360 days 1.30 Strike Option of USD Call/Canadian Dollar Put?

19

5. What is the merit of Black Scholes 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/dS1

• 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.

20

Hedging against Changes in Market Situations

• Change in Prices

• d C/ d S = Delta

• d (d C/dS) / dS = Gamma

• Change in Volatility

• d C/ d Sigma = Vega

• d C/ d t = Theta

21

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 linedC/dS.

• Develop hedge ratio that results in no change when underlying asset price changes

22

Numerical Example of Delta Hedging

• Suppose that you are a Canadian importer, and have US $ 1million payable; each call option covers US $1,000; The call premium is $3 with its delta ratio of 0.5. How many options do you have to buy for hedging?

• Risk: In case of a 1% increase in US dollars against Canadian dollars, your loss is US $ 10,000. This FX risk has to be covered.

• As the delta is 0.5, an increase in the price of underlying asset by 1% will raise the value/price of one unit of call option by 0.5% or US $5 (= $1000 x 0.005).

• Coverage: If you buy 2000 units of call options, your profits from the raised price(premium) of the options will be US $5 x 2000 =US $10,000, which covers the initial risk.

• The total cost of hedging is US $3 x 2000 uints = US $6,000.• In fact, the delta value =0.5 is the inverse of the required hedging ratio

=1/0.5 =2. For the cover of US $1 million, you have to buy the options of the nominal claims of US $2 millions. Thus, you buy $2 m/ $1000 = 2000 units for full coverage.