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1. Options (1) Call options and put options A call option gives the holder the right to buy the underlying asset by a certain date (expiration date) for a certain price (strike price or exercise price). A put option gives the holder the right to sell the underlying asset by a certain date for a certain price. Options: right, not liability
Invest
in the riskless asset and buy the futures.
,
,
,
.
Risk of the underlying asset risk of the investment
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Financial Economics
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6.2 Valuation of options
2. Option price: the cost to acquiring an option
(1) The purchaser: Buy the option, cost = option price
Have the right to buy or sell the underlying asset by a certain date
ZUEL
Financial Economics
Lu, Z.
Ch6 Valuation of Derivatives
6.1 Valuation of Futures
1. Derivative security: a financial instrument whose value depends on other more basic underlying assets. 2. Futures: agreement to buy or sell an asset at a certain time in the future for a certain price. Futures price and spot price
Payoff:
,0 , Profit:
(4) Short position in a put option
payoff: 0
payoff:
Payoff:
,0 , Profit:
,0) ,0)
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Financial Economics
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4. Upper bounds for option prices (European options)
Investment
Return
, an arbitrage
No arbitrage
.
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Financial Economics
Lu, Z.
5. Futures price when the underlying asset provides an income at time ( ) Method 1: purchase the underlying asset and short the futures.
Lu, Z.
, no arbitrage
,
.
(2) With dividends:
.
7. Early exercise on American options
(1) American call option on a non-dividend-paying stock should
never be exercised early.
,
if
not to exercise the option,
,
if
to exercise the option,
,
.
Portfolio B: cash
,
, no arbitrage
, and
.
(4) European put option on a stock with dividends
.
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. riskless rate, riskless rate in the foreign country
Method 1: invest at home:
.
Method 2: invest in the foreign country and short the foreign
currency (not )
1
ZUEL
Financial Economics
Lu, Z.
3. Continuous compounding (1) Simple interest rate: (2) Compound interest rate: (3) Continuous compounding The number of compounding periods every year: ,
Financial Economics
Lu, Z.
6. Put-call parity (European options)
(1) Without dividends
Portfolio A: one European call option + cash
,
if
to exercise the option,
.
.
No arbitrage
.
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Financial Economics
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7. Relationship between futures price and expected future spot price
.
Expected future spot price
,
Investment :
, .
, Method 2: invest at riskless rate
, No arbitrage: same investment, same risk same return.
. .
present value of
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Financial Economics
Lu, Z.
6. Foreign currencies futures spot exchange rate: the current spot price of foreign currency forward exchange rate: the forward price of foreign currency
.
,
.
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Financial Economics
Lu, Z.
4. Futures price when the underlying asset doesn’t provide any
income.
: futures price, : spot price, : maturity date, riskless rate
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Financial Economics
Lu, Z.
(2) American options and European options
American options can be exercised at any time up to the expiration
date.
European options can be exercised only on the expiration date.
Lu, Z.
5. Lower bounds for option prices (European optil option on a non-dividend-paying stock
.
Portfolio A: one European call option + cash
,
if
not to exercise the option,
,
.
Portfolio B: one European put option + one share
,
if
not to exercise the option,
,
if
to exercise the option,
,
.
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Financial Economics
If
sell the option and invest in riskless asset, investment
if
the holder will give up the right,
if
the holder will exercise the option,
.
, .
.
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Financial Economics
,
if
to exercise the option,
,
if
not to exercise the option,
,
.
Portfolio B: one share
,
, no arbitrage
, and
.
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Financial Economics
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(2) European call option on a stock with dividends.
