Financial Mathematics 2
The plan for Tuesday October 5, 2010
• Practical matters
• Forwards: Hull Sec. 1.6-8
• Options: Hull Sec. 1.5, 1.8
• The rest of Hull Ch. 1 is self-reading.
(We’ll get back to ”futures”.)
• Valuing forward contracts by (no-)arbitrage
arguments: CT1 Unit 12
Practical matters
The admin’ does not want us to move
Workshops around ”willy-nilly”. Those of
you with time-table conflicts contact Louise
Feaviour (room 8.19b). Until further notice
we stick to the orginal plan.
Hand-out: Course Work #1. Due at lectures
on Thursday October 14.
Who would want to use/trade in
forward contracts?
• Hedgers. Hull’s p. 10 example: A US company will pay
£10 million for imports from Britain in 3 months and
decides to hedge using a long position in a forward
contract.
• Speculators. Hull’s example p. 12 (For ”futures” read
”forward”.) But clearer in a minute w/ options.
• Arbitrageurs: people who attempt to make risk-free
profits by exploiting relative mis-pricing between
assets/products/contracts. More on these shortly.
Options
Call-option: The right, but not the obligation,
to buy the underlying for the (strike- or
exercise-)price K at the future
(expiry-)date T.
Put-option: Right, not obligation, to sell.
Pay-off-diagrams: Hockey-sticks.
Unlike forward contacts, call- and putoptions cost money up front. Clearly, they
have to. (Why?)
Why Study Options?
Used by
• Hedgers (put ~ portfolio insurance)
• Speculators
Embedded in many other financial contratcs
(pensions, mortgages, …)
We will not study how options are priced, i.e.
why they cost, what they cost.
Hedging w/ Put-Options
An investor owns 1,000 Microsoft shares
currently worth $28 per share.
A two-month put-option with a strike price
of $27.50 costs $1.
The investor decides to hedge by buying
1,000 put options (“10 contracts”)
Portfolio Value in Two Months
with and without Hedging
40,000
Value of
Holding ($)
35,000
No Hedging
30,000
Hedging
25,000
Stock Price ($)
20,000
20
25
30
35
40
Speculating with Call-Options
An investor with $2,000 to invest feels that
Amazon.com’s stock price will increase
over the next 2 months.
The current stock price is $20 and the price
of a 2-month call option with a strike of
22.5 is $1
He can put his $2,000 into
• 100 shares of Amazon.com stock
• 2,000 strike-22.5, expiry-2M call-options
Profit or loss from speculating on the Amazon.com stock price
8000
7000
Value of holdings ($)
6000
5000
4000
3000
2000
Buy stock
1000
Buy options
0
-1000
15
20
25
-2000
-3000
Stock price ($)
30
Valuation of Forward Contracts
How are spot and forward prices related?
A simple yet powerful principle: Absence of
arbitrage. Or: There is no such thing as a
free lunch. CT1 Unit 12, Sec 1
Base-case:
Fwd(t,T) = exp(r*(T-t))*Spot(t)
Extensions of Forward Valuation
CT1 Unit 12
• Sec. 2.3: Fixed intermediate cash-flows on
the underlying (~ fwd on coupon bond)
• Sec. 2.4: Dividend yield (~ currency
underlying; ~commodities w/ storage
costs)
• Sec. 2.6: Value between initiation (t) and
expiry (T) (motivates introduction of
futures contracts)