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 put-options 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

20,000

25,000

30,000

35,000

40,000

20 25 30 35 40

Stock Price ($)

Value of Holding ($)

No Hedging

Hedging

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

-3000

-2000

-1000

0

1000

2000

3000

4000

5000

6000

7000

8000

15 20 25 30

Stock price ($)

Val

ue

of

ho

ldin

gs

($)

Buy stock

Buy options

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)