Options VI

1
Options VI
Options on indices, foreign currency, and
futures Pricing using variations of
Black-Scholes and Binomial
2
European Options on Stockswith continuous
dividends

• The probability distribution of the stock price
  at maturity T is the same in the following cases
• 1. The stock starts at price S0 and gives
  dividends continuously with yield q
• 2. The price of the stock starts at S0eq T and
  does not pay dividends

3
European Options on Stockswith continuous
dividends (cont)

• European options can be priced if we set the
  stock price in t0 at
• S0eq T
• and pretending that they dont pay dividends.

4
Synthesizing a stock that does not pay dividends

• Consider the case of a stock that in tt,
• 0 lt t lt T
• pays a dividend of Dt proportional to the
  value of the stock
• Dt q St
• The cost in t 0 of getting the stock with no
  dividends, e.g. el cost of buying an asset that
  pays ST in T is
• S0 (1-q)

5
Synthesizing a stock that does not pay dividends

• - Buy 1-q in t0
• - In tt we get (1-q) Dt (1-q) q St
• - Immediately after the dividend payment in tt
  the price of the stock decreases to (1-q) St
• - Invest dividends, buy q units of the stock
• (1-q) Dt / (1-q) St (1-q) q St / (1-q)
  St q
• - Then, in tT we have one unit of the stock.
• - Note that we dont need to know the value of
  St

6
Synthesizing a stock that does not pay dividends

• - Consider a stock that pays dividends in 0 lt t1
  lt t2 lt T with proportions q1 and q2 .
• - In t0 start with (1- q1)(1- q2) units
• - In t1 we get (1- q1)(1- q2) q1 St1 and
  buy
• (1- q1)(1- q2) q1 St1 / St1 (1- q1) (1- q2)
  q1
• - In t2 we get (1- q1)(1- q2) (1- q2) q1 q2
  St2 and buy
• (1- q1)(1- q2) (1- q2) q1 q2 St2 / St2 (1-
  q2) q2
• - In tT we have
• (1- q1)(1- q2) (1- q2) q1 q2 1

7
Synthesizing a stock that does not pay dividends

• Generalizing the previous case, consider a stock
  that pays dividends in 0 lt t1 ltlt tn lt T with
  proportions q1 ,, qn .
• In order to get one unit of the stock without
  dividends at tT we have to buy
• (1-q1) (1-q2) . (1-qn) units of the stock in
  t0.
• Alternatively we can use q for the (continuously
  compounded ) dividend yield and we get
• (1-q1) (1-q2) . (1-qn) exp(-qT )
• Where q is the average continuously compounded
  dividend yield
• q - (1/T) ln(1-q1)ln (1-q2) . ln(1-qn)
• (1/T) q1 q2 . qn for big n

8
Synthesizing a stock that does not pay dividends
Numerical Example

• T 0.5 (half year)
• Consider a stock that pays dividends
• t1 2/12 y t2 4/12
• q1 0.02, q2 0.04
• (1-q1) (1-q2) 0.98 0.96 exp(-qT )
• q - (1/T) ln(1-q1)ln (1-q2) 0.12205
• (1/T) q1 q2 . qn 0.12
• Two 2 and 4 payments in 6 months are
  equivalent to an annual dividend yield of 12 .

9
Binomial Model Stock with yield q
S0u u
p
S0 e -qT
S0d d
(1 p )
fe-rTpfu(1-p)fd
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Binomial Model Stock with yield q

• Risk neutral pricing of a stock that grows at a
  rate of r-q instead of a rate r
• The probability of an up, p, satisfies
• p S0u (1-p) S0d S0e (r-q)T
• p ( e (r-q)T d ) / ( u d)

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Binomial Model Comparison

• Without dividends
• ST /S0 u o ST /S0 d
• pS0u(1-p)S0dS0e rT
• p (e rT d ) / (u-d)
• With dividends (ST doesnt include dividends)
• ST /S0 u o ST /S0 d
• pS0u(1-p)S0dS0e (r-q)T
• p (e (r-q)T d ) / (u-d)

12
Binomial Model Comparison

• Without dividends
• ST /S0 u o ST /S0 d
• p (e rT d ) / (u-d)
• With dividends
• ST /S0 u o ST /S0 d
• p (e (r-q)T d ) / (u-d)
• Hence
• u u e-qT, d d e-qT ? p p

13
Binomial Model Comparison

• Without dividends
• ln(ST /S0 ) ln(u) o ln(ST /S0 ) ln(d)
• With dividends
• ln(ST /S0 ) ln(u) o ln(ST /S0 ) ln(d )
• Suppose that
• u u e-qT, d d e-qT
• ln(u) -qT ln(u), ln(d) -qT ln(d)
• s2 Var(ln(ST /S0 )) is independent of q.

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Black-Scholes Formula Call

• Without dividends
• C S N( d1 ) - Ke-rT N(d1 - sT1/2 )
• d1 log(S/K e-rT )/s T1/2 ½s T1/2
• With dividend yield q
• C e-qTS N( d1 ) - Ke-rT N(d1 - sT1/2 )
• d1 log(S/K e-(r-q)T )/s T1/2 ½s T1/2

15
Using the previous formulas
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Options on Indices

• Typical contracts on indices 100index
• Most common
• SP 100 (American) OEX
• SP 500 (European) SPX
• Contracts are settled in cash

17
Pricing European Options on Indices

• We use the formulas for an option on a stock
  that pays dividends continuously
• S0 index current level
• q expected average yield during the option
  contract

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Options on Currency

• Foreign interest rate rf (e.g. EURO).
• Purchasing an unit of foreign currency costs
  (e.g. EURO) S0 dollars.
• The return of investing in foreign currency is rf
  units of that currency.
• Hence, investing in foreign bonds gives a
  dividend yield at a rate rf
• rf is a dividend yield, because it is in terms
  of the units of the asset (foreign currency)

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Pricing European Options on Currency

• A foreign currency is an asset that pays a
  dividend yield equal to rf
• We use the formula for an option that pays
  dividends with yield equal to q
• S0 current exchange rate
• q r
• Assume that the exchange rate behaves as a stock
  Random Walk
• s is the standard deviation of the exchange rate
  log difference.

20
Formulas for an European Option on Currency

21
Mechanics of a Call Futures Options

• When a call futures option is exercised, the
  holder gets
• A long position in futures
• An amount of cash equal to the difference between
  the current future price and the strike price of
  the option.

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Mechanics of a Put Futures Option

• When the holder exercises a put futures option he
  gets
• A short position in the future
• An amount of cash that equals the difference
  between the strike price and the current future
  price.

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Payoffs of a Future Option

• Assume that the position on the future is closed
  immediately
• Call Payoff F0-K
• Put Payoff K-F0
• where F0 is the current future price (at the
  moment of exercising the option)

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Payoffs of a Future Option (cont)

• Remember the relationship between the price of
  the underlying S and the price of the future on
  the underlying asset F
• F0 S0 erT
• FT ST
• Assume that the expiration date is the same for
  the option and for the future.
• In this case the pay-off of the future option is
  the same as the pay-off of the option on the
  underlying asset.
• Finally, if they are European options, the value
  is the same.

25
Put-Call Parity for Options on Futures

• - Consider 2 portfolios
• 1. European Call Ke-rT cash
• 2. European Put long futures F0e-rT cash
• Since they have the same pay-off in T, they
  should have the same value
• CKe-rT PF0 e-rT
• - Remember that F0 S0 erT

26
Future Options Pricing and Hedging using the
futures market

• Well analyze pricing and hedging of a future
  option trading in the futures market instead of
  the underlying asset market.
• The reason to do this is that in a lot of cases
  the futures market is more liquid than the market
  of the underlying asset.
• This will affect the formulas that we use for
  pricing since they will be in terms of the price
  and volatility of the future.
• For example, remember that the prices of the
  futures converge to the price of the underlying
  from above and that subscribing a futures
  contract has zero value.

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Example with a Binomial tree

• One month Call Future Option with strike price
  29.

Price Future 33 Option 4
Future 30 Price of the Option?
Price Future 28 Option 0
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Neutral Hedge using the future

• Consider long D futures short 1 call
  option
• The portfolio is risk free (neutral hedge) if
• 3D 4 -2D o D 0.8

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Pricing the Portfolio( 6 annual interest rate,
maturity one month )

• The portfolio includes
• long 0.8 futures short 1 call option
• The value (cash flow) of the portfolio in 1 month
  is (sold bond)
• -1.6
• Then the value of the portfolio today is -1.6
  e 0.06/12 -1.592

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Pricing the option

• A portfolio with
• long 0.8 futures short 1 option
• has a cash flow -1.592
• The current value of the future is zero.
• The value of the option has to be 1.592

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Generalizing the Binomial example

• In this case u and d are the gross changes in the
  value of the future.
• Option has maturity T

F0u u
F0
F0d d
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Generalization (cont)

• Consider the portfolio long D futures short
  1 derivative
• The portfolio is risk free if

(F0u - F0 ) D u
(F0d - F0) D d
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Generalization (cont)

• The portfolio has a risk free cash flow in T
  equal to
• F0u D F0D u
• The value of the portfolio today is
• Then
• F0u D F0D ue-rT

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Generalization (cont)

• Substituting D we get
• p u (1 p )d erT
• where

35
Binomial Model Futures Options

• The effects of trading in the futures market are
• We need to alter the binomial model because the
  future price is zero.
• The formulas of the risk neutral probabilities
  are different (because they are in terms of the
  futures u and d)

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Binomial Model Futures Options
Denote (u, d) the up and down for the future and
(u,d) for the underlying asset, and analogously p
and p. Since F0 S0 erT FT ST We
have u FT / F0 u e-rT d FT / F0 d
e-rT Thus p (1- d) /( u - d ) (erT d )
/ (u-d)
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Pricing European Options on Futures

• Use the formula for a stock option that pays
  dividend yield
• S0 Current future price (F0)
• q interest rate (r )
• Con q r the expected price of the future is
  constant (using the risk neutral probabilities
  derived previously).

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Future Option as a Stock with qr

• A futures contract does not require initial
  investment.
• In a neutral risk world its expected value should
  be zero.
• Then the futures appreciation rate should be
  zero.
• Hence futures contracts can be thought as a stock
  that pays continuous dividend yields at a rate r

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Blacks formula

• The formulas for European options on futures are
  known as Blacks formula

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Blacks formula alternative explanation
Black-Scholes formula C S0 N( d1 ) - Ke-rT
N(d1 - sT1/2 ) d1 log(S0 /K e-rT )/s T1/2
½s T1/2 Price of the future F0 S0 erT C
e-rT(S0erT)N( d1 ) - Ke-rT N(d1 - sT1/2 )
e-rTF0 N( d1 ) - K N(d1 - sT1/2 ) d1
log(e-rT (S0erT)/K e-rT )/s T1/2 ½s T1/2
log( F0 / K )/s T1/2 ½s T1/2
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Price of options on futures vs. price of options
on spots

• When futures prices are higher than spots prices
  (normal market), an American Call on futures has
  a higher value than a similar call on the spot.
  An American Put on futures has a lower value than
  a similar put on the spot. Why?
• When the futures prices are lower than the spots
  prices (inverted market) the contrary occurs.

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Summary

• Indices on stocks, currency and futures can be
  thought as stocks that pay a continuous dividend
  with yield q
• Indices, q average expected yield during the
  life of the option
• Currency, q r (foreign bond yield with
  maturity equal to the life of the option)
• Futures, q r (domestic bond yield with
  maturity equal to the life of the option)