Option Pricing

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Option Pricing

• Dr. J.D. Han

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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) 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.

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

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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

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• This premium is not arbitrary drawn, but is based
  on the calculation of Option Pricing or the
  Pricing of an Option.
• Lets 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 ?

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The Bounds of Call Pricing
Upper Bound
Lower Bound
Attainable range
S
X
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1) Upper Constraint on Long Call Premium

• Premium lt 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

Premium
Upper Premium Limit
X
S
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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 -X
  1.6 - 1.5 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 - 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.

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Illustration of the lower limit on the value of
Long Call Option
St current market price of the underlying
asset St1 future market price of the underlying
asset St1 will be probably distributed around St
Out of the money
In the money
X
S
If the current Stlt X, then it is a out of the
money option If the current StX, then it is a
at the money option If the current St gtX, then
it is a in the money option
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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.

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Another component of the Market Value of an Option

• People are buying an option by looking at the
  current market St , not St1.
• However, the real possible profits are coming
  from the difference of St1 X.
• This means that, from St , they have to guess
  what St1 will be and whether they will make a
  profit.
• The time value Expected Gains between t and
  t1.

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• St1 will be distributed around St with a
  normal distribution.
• Mean value of St1 is St
• E(St1 - X) St X
• If today the option is such that current spot
  rate is higher than the exercise price (St gtX),
  you can expect that, on average, in the future
  spot rate will be higher than the exercise
  price(E(St1 )gt X).

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The expected gains between t and t1

• 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.

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2. The Actual Call Pricing lies between the upper
and the lower bounds Intrinsic Value Time Value
Actually observed premiums
We call this portion of value/price/premium of
option to be Time value

• S

X
An out-of-money option (S t- -Xlt0) have some
positive time value to the potential buyer there
is a chance that St may change into St1 so
that St1 - X might be positive at the
expiration.
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3. Intuitive Explanation 1) Option Premium
Expected Value/Price of Option profits in case
the option is exercised (St1 X in the future)
times its probability is closely related to the
two components

• Intrinsic Value
• (St X) today
• Is St (as a predictor of St1) higher than X?

• Time Value
• Will St1
• go further above St?

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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

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4. Mathematical Model for Option Premium

• 1) Foundation Black Scholes General Option
  Pricing Model
• Note that
• St Eo
• S E in our case.

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2) FX Option Pricing

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

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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/BS
  Currency/input.jsp

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• Use the formula or
• http//www.cfo.com/ currency option calculator
• or a 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 s20?

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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 assets 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.

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6. Delta Hedging Theory

• As underlying assets price increases by one
  unit, the price/value of call premium will
  increase by the slope of the linedC/dE.
• Develop hedge ratio that results in no change
  when underlying asset price changes

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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.
• ? of an option changes in total price from
  the previous period / 1 change in S.

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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.

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• 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 large of size
  of contract 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.

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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