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FINM2002 Options on Stock Indices and Currencies

创建日期：2024-05-05 17:24:37

所属分类：金融Finance

标签： FINM2002

Options on Stock Indices and Currencies

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FINM2002/7041 (Applied) Derivatives

Lecture 7
Options on Stock Indices and
Currencies;
Options on Futures Contracts
Hull et al: Chapter 15
Review of Previous Lecture
• Last week we examined the evolution of
stock prices, and found the expected price
of a stock based on this evolution.
• We also used the Black-Scholes Model to
price European put and call options on
both dividend and non-dividend paying
stocks.
2
Lecture Overview
• In today’s lecture we will discuss options on
stock indices and currencies. We will focus on
how to treat dividend yields when examining
options. More specifically, with reference to
stock index options and FX options, we will
address necessary alterations to:
– Lower bounds for options;
– Put-call parity;
– Binomial trees;
– Black-Scholes Model for dividend yielding stocks;
and,
– Portfolio insurance.
3
1. Options and Dividend Yields
• How do we treat a stock which pays a
known dividend yield?
– Take two identical stocks. Stock 1 pays no
dividends, and stock 2 pays a dividend yield
at rate q p.a.
– Both stocks will give an identical overall
return. The return from stock 1 will be in the
form of capital gains, and the return from
stock 2 will be in the form of capital gains and
dividends.
4
1. Options and Dividend Yields
• The payment of a dividend will cause stock 2’s price to
drop by an amount equal to the value of the dividend.
• So payment of a dividend yield at rate q causes the
growth rate to be less than that it would otherwise be, by
an amount of q.
• Thus if the price of a stock, with a dividend yield of q,
grows from S0 today to ST at time T:
– In the absence of dividends it would grow from S0 today to STeqT
at time T; or,
– In the absence of dividends it would grow from S0e-qT today to ST
at time T.
5
1. Options and Dividend Yields
• Therefore, we get the same probability
distribution for the stock price at time T in
each of the following cases:
– The stock price starts at price S0 and pays a
dividend yield at rate q; and,
– The stock price starts at price S0e-qT and pays
no dividend yield.
6
1. Options and Dividend Yields
• The rule which stems from the above
example is:
– We can value European options lasting for
time T on a stock paying a known dividend
yield at rate q by reducing the stock price to
S0e-qT , and then valuing the option as though
the stock pays no dividend.
7
1. Options and Dividend Yields
• The above rule can be applied to any
option on an underlying asset which pays
a known dividend yield.
• In this lecture we focus on two particular
options:
– Options on Stock Indices; and,
– Options on Currencies.
8
1. Options and Dividend Yields
• Stock indices paying a known dividend yield:
– Exchange traded stock indices are available in most
exchanges around the world;
– In Australia, options on the ASX200 are traded on the
Australian Securities Exchange; and,
– In this lecture, we assume that dividends on stock
indices are paid in terms of a dividend yield. They
can also be paid in terms of a dollar value.
9
1. Options and Dividend Yields
• Options on currencies:
– A foreign currency is analogous to a stock paying a
known dividend yield.
– The owner of the foreign currency option receives a
yield equal to the risk-free rate of interest in the
foreign currency.
– The options are traded on both OTC markets and on
an exchange, although they are mainly traded in OTC
markets.
– The Philadelphia Stock Exchange is the major
exchange for FX options.
– FX options can be either European or American.
10
2. Lower Bounds for Option Prices
• The payment of a dividend by the underlying asset,
lowers the price of that asset.
• The reduction in price is good for the holder of a put
option as it increases their payoff, while it is bad news
for the holder of a call option as it reduces their payoff.
• Thus, the lower bounds for both call and put options
are affected by the payment of dividends.
11
2. Lower Bounds for Option Prices
• The effect on lower bounds for European calls:
– In lecture 4 we saw that the lower bound for a European
call on a non-dividend paying stock was:
– Substituting S0e-qT for S0, the lower bound for a European
call option on a stock which pays a dividend yield is:
0
rTc S Xe−≥ −
0
qT rTc S e Xe− −≥ −
12
2. Lower Bounds for Option Prices
• We can prove the above result directly by considering
the following two portfolios:
– Portfolio A: one European call option plus an amount of
cash equal to Xe-rT
– Portfolio B: e-qT shares with dividends being reinvested in
additional shares.
– In portfolio A, if the cash is invested at the risk-free interest
rate, it will grow to X at time T. If ST > X, the call option is
exercised at time T, and portfolio A is worth ST.
13
2. Lower Bounds for Option Prices
– If ST < X, the call option expires worthless, and the portfolio is
worth X.
– Hence, at time T portfolio A is worth:
Max(ST,X)
– Due to the reinvestment of dividends, portfolio B becomes one
share at time T. Thus, it is worth ST at this time.
– Therefore, portfolio A must be worth at least as much as
portfolio B at time T.
– Hence:
0
0
rT qT
qT rT
c Xe S e
or
c S e Xe
− −
− −
+ ≥
≥ −
14
2. Lower Bounds for Option Prices
• The effect on lower bounds for European puts:
– In lecture 4 we saw that the lower bound for a European put
on a non-dividend paying stock was:
– Substituting S0e-qt for S0, the lower bound for a European put
option on a stock which pays a dividend yield is:
0
rTp Xe S−≥ −
0
rT qTp Xe S e− −≥ −
15
2. Lower Bounds for Option Prices
• As with the call option, this can be proved directly by
forming the following two portfolios:
– Portfolio C: one European put option plus e-qt shares with
dividends on the shares being reinvested in additional
shares
– Portfolio D: an amount of cash equal to Xe-rT
16
3. Put-Call Parity
• In order to incorporate the effect of dividend yields on
put-call parity:
– In lecture 4 we saw that put-call parity for a European call
and put option on a non-dividend paying stock was:
– Replacing S0 with S0e-qT yields the following put-call parity
relationship on a stock paying a dividend yield at rate q:
0
rTc Xe p S−+ = +
0
rT qTc Xe p S e− −+ = +
17
3. Put-Call Parity
• We can prove the above result directly by considering
the following two portfolios:
– Portfolio A: one European call option plus an amount of cash
equal to Xe-rT
– Portfolio C: one European put option plus e-qT shares with
dividends on the shares being reinvested in additional shares.
– Both portfolios must be worth max(ST,X) at time T. They must
therefore be worth the same today, and the put-call parity
result above holds.

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