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# 1 Topic 11 Options on Stock Indi

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```									                                                                                           11.1                                                                                               11.2
European Options on Stocks Paying
Continuous Dividends
• The following two cases lead to identical stock
Topic 11                                                                                price distribution at time T
– The stock starts at price S and pays a continuous
Options on Stock Indices, Currencies                                                                       dividend yield at rate q
– The stock starts at price Se–q T and pays a continuous
dividend yield at rate q
Chapter 12
• A Simple Rule
We can value European options by reducing the
stock price to Se–q T and then treat it as though
there is no dividend

Financial Engineering, Lingnan College, 1999 Undergraduate Class, Spring 2002, by Prof. NIU Hong   Financial Engineering, Lingnan College, 1999 Undergraduate Class, Spring 2002, by Prof. NIU Hong

11.3                                                                                               11.4
European Options on Stocks Paying                                                             Extension of Bounds and Call-Put Parity
Continuous Dividends,cont’d                                                                               in Chapter 8
Lower Bound for calls:                c ≥ S − PV ( D) − Xe − rT
− qT                      − rT
c = Se             N ( d1 ) − Xe             N (d 2 )
−qT
c ≥ Se− qT − Xe −rT
ln(Se            / Xe− rT )       1
where              d1 =                                    + σ 2T
σ T                    2                              Lower Bound for puts                    p ≥ Xe − rT − Se− qT
d 2 = d1 − σ T                                                                                                  p ≥ Xe − rT − ( S − PV ( D ))
Put-Call Parity                c + Xe− rT = p + S − PV (D)
p = Xe− rT N ( −d 2 ) − Se− qT N ( −d1 )
c + Xe− rT = p + Se−qT
Financial Engineering, Lingnan College, 1999 Undergraduate Class, Spring 2002, by Prof. NIU Hong   Financial Engineering, Lingnan College, 1999 Undergraduate Class, Spring 2002, by Prof. NIU Hong

11.5                                                                                               11.6
Extension of Bounds and Call-Put Parity                                                            Extension of Bounds and Call-Put Parity
in Chapter 8, proving                                                                          in Chapter 8, proving, cont’d
• Portfolio A: one European call plus cash Xe −rT
• At the time of purchasing the option, its value is
–   cash grows to X at time T at risk-free rate                                                   c, hence
–   if S T >X, exercise option, value of A is S T
c + Xe −rT ≥ Se− qT
–   if S T <X, option is worth nothing, value of A is X
–   At T, portfolio A is worth max (S T, X)                                                     • Using the simple rule, we also obtain
−qT
• Portfolio B: e                    shares
– with reinvestment, B is worth X at time T                                                                               p ≥ Xe − rT − Se − qT
• Value of A is at least worth as much as B
c + Xe− rT = p + Se−qT
Financial Engineering, Lingnan College, 1999 Undergraduate Class, Spring 2002, by Prof. NIU Hong   Financial Engineering, Lingnan College, 1999 Undergraduate Class, Spring 2002, by Prof. NIU Hong

1
11.7                                                                                               11.8
The Binomial Model With
The Binomial Model With
A Continuous Dividend Yield, continued
A Continuous Dividend Yield                                                                  • In a risk-neutral world the stock price grows
S 0u                                          at r-q rather than at r when there is a
p             ƒu                                           dividend yield at rate q
S0
• The probability, p, of an up movement must
ƒ
(1 –               S 0d                                          therefore satisfy
p)        ƒd                                                     pS 0u+(1 – p)S 0d=S0e (r-q)T
so that
e(r−q ) T − d
f=e-rT[pf u+(1– p)fd ]                                                           p=
u− d
Financial Engineering, Lingnan College, 1999 Undergraduate Class, Spring 2002, by Prof. NIU Hong   Financial Engineering, Lingnan College, 1999 Undergraduate Class, Spring 2002, by Prof. NIU Hong

11.9                                                                                               11.10
Hong Kong Stock Options (American)                                                                  Hong Kong Stock Options                                                  Cont’d

Stock         No. of shares/contract                                                               • Information about HK Stock options
CITIC Pacific      1000                                                                                 – Time to expiration: 1, 2, 3, 6, 9 months
Cheung Kong        1000                                                                                 – Trading time: 10 to 12:30am; 2:30 to 3:55pm
HSBC                 400
– Tick: \$ 0.1
HK Telecom           400
Swrie Pacific A      500                                                                                – Expiration dates: the last stock trading day of the
month. The last option trading day is the day
China L&P            500
before the expiration date
Henderson Land 1000
Hopewell           1000                                                                                 – Strike prices: at least 5 strike prices are quoted;
Hutshison          1000                                                                                   At least two in the money and two out of money.
SHK Prop.          1000
Financial Engineering, Lingnan College, 1999 Undergraduate Class, Spring 2002, by Prof. NIU Hong   Financial Engineering, Lingnan College, 1999 Undergraduate Class, Spring 2002, by Prof. NIU Hong

11.11                                                                                              11.12
Index Options                                                                                                     Example
• Some Index Options
– S&P 100 (American)                                                                           • A S&P call on an index with a strike price
– S&P 500 (European)                                                                             of 560
– HSI (European)                                                                                 What is the payoff for one contract
• Multiplier is the value of one index point                                                          exercised when index is 580 if index is S&P
– S&P 100, S&P 500 is \$100                                                                       100 or 500,
– HSI is HK\$50
– Payoff of 120 point gain in                                                                    and if the index is HSI?
one S&P contract = \$100                                120
• Contracts are settled in cash

Financial Engineering, Lingnan College, 1999 Undergraduate Class, Spring 2002, by Prof. NIU Hong   Financial Engineering, Lingnan College, 1999 Undergraduate Class, Spring 2002, by Prof. NIU Hong

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11.13                                                                                               11.14
Valuing European Index Options
LEAPS                                                         S = current index level;     σ= index volatility
q = average dividend yield expected during the option life
• LEAPS(Long-term Equity AnticiPation
c = Se − qT N ( d1 ) − Xe − rT N ( d 2 )
Securities )
– long term options on index last up to 3 years                                                  p = Xe − rT N ( − d 2 ) − Se −qT N ( − d 1 )
– December expiration dates                                                                                                  ln( S / X ) + (r − q + σ 2 / 2 )T
– multiplier is \$10                                                                              where             d1 =
σ T
ln( S / X ) + (r − q − σ 2 / 2)T
d2 =
σ T
Financial Engineering, Lingnan College, 1999 Undergraduate Class, Spring 2002, by Prof. NIU Hong    Financial Engineering, Lingnan College, 1999 Undergraduate Class, Spring 2002, by Prof. NIU Hong

11.15                                                                                               11.16

Currency Options                                                                       The Foreign Interest Rate
• Currency options trade on the Philadelphia                                                      • We denote the foreign interest rate by rf
Exchange (PHLX)                                                                                 • When a U.S. company buys one unit of the
• There also exists an active over-the-counter                                                      foreign currency it has an investment of S 0
(OTC) market                                                                                      dollars
• Currency options are used by corporations to
• The return from investing at the foreign rate
buy insurance when they have an FX exposure
is rf S 0 dollars
• This shows that the foreign currency
provides a “dividend yield” at rate rf

Financial Engineering, Lingnan College, 1999 Undergraduate Class, Spring 2002, by Prof. NIU Hong    Financial Engineering, Lingnan College, 1999 Undergraduate Class, Spring 2002, by Prof. NIU Hong

11.17                                                                                               11.18
Valuing European Currency Options
Valuing European Currency Options
Set S = current exchange rate, Foreign interest rate rf
• A foreign currency is an asset that                                                                    c = Se
− rf T
N ( d 1 ) − Xe − rT N ( d 2 )
provides a continuous “dividend yield”
equal to rf                                                                                                                     ln( S / X ) + ( r − r              + σ 2 / 2 )T
f
• We can use the formula for an option on                                                                where          d1 =
σ T
a stock paying a continuous dividend
yield :                                                                                                                         ln( S / X ) + ( r − r              − σ 2 / 2 )T
f
d2 =
Set S 0 = current exchange rate                                                                                                                 σ T
Set q = rƒ                                                                                        and
− rf T
p = Xe − rT N ( − d 2 ) − Se                       N (− d 1 )
Financial Engineering, Lingnan College, 1999 Undergraduate Class, Spring 2002, by Prof. NIU Hong    Financial Engineering, Lingnan College, 1999 Undergraduate Class, Spring 2002, by Prof. NIU Hong

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