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EC247 FINANCIAL INSTRUMENTS AND CAPITAL MARKETS Dr Helen Weeds 2013-14, Spring Term

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### EC247 FINANCIAL INSTRUMENTS AND CAPITAL MARKETSDr Helen Weeds2013-14, Spring Term

Lecture 7: Futures and options

LEARNING OUTCOMES

- What is a derivative?
- Futures and forwards
- Explain the nature of forwards and futures
- Use of futures in hedging and speculation
- Specification of a futures contract
- Convergence of futures price to spot price
- Margin accounts and marking to market
- Options
- Explain the nature of financial options
- Calls and puts; European and American exercise types
- Payoff from call and put options for holder and writer
- Option valuation: some simple relationships

What is a derivative?

- General definition of a ‘derivative’
- An asset whose performance is based on (derived from) the value of an underlying asset (the ‘underlying’)
- Derivative contracts
- give the right (and sometimes the obligation) to buy or sell a quantity of the underlying
- or to benefit in another way from a rise or fall in the value of the underlying
- The derivative contract is itself an asset, with its own value, and can be purchased or sold
- Either on an exchange or ‘over the counter’ (OTC)

Derivatives can be risky…

- Depending on how they are used, derivatives trading can generate enormous losses
- In 1994, Proctor & Gamble (!) lost $102m speculating on the movements of future interest rates
- In 1995, Barings (Britain’s oldest merchant bank) lost over £800m (and went bankrupt) as a result of trading in derivatives on the Nikkei Index (Japanese share index) by ‘rogue trader’ Nick Leeson
- In 1998, Long-Term Capital Management (LTCM) collapsed as a result of its options trading (its bailout was brokered by FRBNY)
- In 2008, SociétéGénérale lost €4.9bn due to unauthorised trading by JérômeKerviel, who took out large, unhedged positions in equity indices

FORWARDS AND FUTURES

- Forward contract
- An agreement between two parties to undertake an exchange at an agreed future date at a price agreed now
- Example
- Farmer grows a field of potatoes, to be harvested in 2 months’ time
- Crisps producer wants to know how much it will have to pay for potatoes, in order to set its prices and market the product
- Market price of potatoes varies over time
- Both parties can lock into a price that is agreed now, to reduce uncertainty and limit exposure to unforeseen price shocks
- The buyer at the future date is said to take a ‘long’ position
- The seller at the future date is said to takea ‘short’ position

Forwards and futures: differences

- Forwards are traded over the counter
- Private agreements, not regulated by an exchange
- Tailor-made, to suit the requirements of the parties
- amounts and delivery dates are flexible
- may be written for long-term maturities (e.g. 3 years)
- But risk of default by the other party (‘counterparty risk’)
- Futures: similar to forwards, but traded on an exchange
- The clearing house is the counterparty to the transaction: reduces risk of default
- Contract is standardised, and tends to cover shorter maturities only (e.g. up to 1 year)
- Contract is easier to trade
- Today: focus on exchange-traded derivatives [OTC: next week]

Forwards and futures: development

- Forwards contracts
- Holland, late 1500s: fish dealers bought and sold herring before it was caught
- Exchange-based trading
- England: Royal Exchange in London, 1571
- now the London Metal Exchange (LME)
- Japan: Dōjima Rice Exchange, 1710
- USA: Chicago Board of Trade (CBOT), 1848
- now part of the Chicago Mercantile Exchange (CME)
- Futures contracts
- 1865: CBOT began trading futures contracts (in grain)
- These were the first standardised derivatives contracts
- Today: most futures exchanges are entirely electronic

Uses of futures

- Hedging
- Using a futures contract to offset specific risks
- E.g. in April (before planting) a farmer sells a futures contract, committing him to supply a specific quantity of the crop in September (after harvest) at the agreed price
- Offsets the risk of a price fall between planting and harvest
- Similar considerations for a food processor that buys the futures contract
- Speculation
- Trading in futures contracts with the intention of profiting from price changes (rather than to hedge specific risks)
- E.g. buy a futures contract now, hoping that the price will go up
- Speculative trading increases liquidity, which benefits other traders

Hedging with futures

- Spot and forward quotes for the $/£ exchange rate(22 June 2012, $ per £)
- ImportCo, based in the US, knows it will have to pay £10m on 22 Sept 2012 for goods imported from a supplier in the UK
- It can hedge this risk by buying £10m on a 3-month forward contract at 1.5585 (‘offer’ price), costing it $15,585,000
- ExportCo, based in the US, is exporting goods to the UK; on 22 June 2012 it knows it will receive £30m in 3 months’ time
- It can hedge this risk by selling £30m on a 3-month forward contract at 1.5579 (the ‘bid’ price), gaining it $46,737,000

Speculating with futures

- Suppose a speculator thinks £ will strengthen relative to $ over the next 2 months
- Two possible strategies
- Purchase £250,000 now, in the spot market, at $1.5470
- £250,000 can be deposited in an interest-bearing account
- Take a long position in futures contracts maturing 2 months, at $1.5410
- this requires a (refundable) margin payment to be deposited up-front, say $20,000
- Difference is in the size of up-front investment required
- futures allow the speculator to obtain leverage , i.e. to take out a large speculative position with a small stake

Specification of a futures contract

- Asset
- e.g. a commodity, of specified grade or quality
- Contract size
- amount of the asset to be delivered
- Delivery
- delivery month
- arrangements for delivery (e.g. location)
- Price
- how prices are quoted (currency and unit size, e.g. US$ and cents)
- exchange usually imposes limits on daily price movements
- Position limits
- maximum number of contracts an individual can hold
- aims to prevent undue influence on the market (‘cornering’)

Convergence to the spot price

- As the delivery period approaches, the futures price converges to the spot price of the underlying asset
- Otherwise there would be an arbitrage opportunity
- E.g. if futures price > spot price
- sell (i.e. short) a futures contract
- buy the asset
- make delivery, and realise a profit

Margin accounts

- Default risk (counterparty risk)
- Each party to a futures contract (or the central exchange) faces a risk that the counterparty might back out of the deal, or be unable to pay
- Margin accounts are used to mitigate counterparty risk
- Initial margin: investor deposits a certain sum of money per contract in their margin account at the exchange/clearing house
- e.g. initial margin of $6,000 per contract for gold futures (for 100oz at a current futures price of $1,650/oz)
- Marking to market: at the end of each trading day, the margin account is adjusted to reflect the investors gain/loss
- e.g. if futures price falls from $1,650/oz to $1,641/oz, the investor has a loss of $9 x 100 = $900: margin account is reduced by $900
- Maintenance margin: if the balance in the investor’s margin account falls below this level (lower than the initial margin), it must be topped up to the initial level
- the investor faces a margin call
- the extra funds deposited are called a variation margin

OPTIONS

- What is an option?
- The holder has the right, but not the obligation, to buy/sell the underlying asset at a given price, on or before a specified date
- The option writer is obliged to carry out the trade if the holder wishes to do so
- The holder pays the writer a non-returnable premium for the option
- Hence if the option expires unexercised, the writer makes a profit

Call and put options

- Call option
- Gives the holder the right to buy the underlying asset, at a given price, at or before a specified date
- Put option
- Gives the holder the right to sellthe underlying asset, at a given price, at or before a specified date
- Features
- ‘Underlying’: could be a stock, index, commodity, currency, etc.
- ‘Strike price’
- ‘Expiration’ or ‘maturity’ date
- Each option comes in two exercise types
- ‘European’: may be exercised only at the expiration date itself
- ‘American’: may be exercised at any time before or at expiration
- American option value value of equivalent European option
- NB: an American option on a non-dividend paying stock is never exercised early, and has the same value as the European equivalent

OPTION PAYOFFS

- Consider payoffs from different options and to the two parties
- European call option
- Option holder’s payoff at expiration/maturity
- Holder’s total profit, taking account of option premium paid
- Option writer’s profit
- European put option
- Similar analysis

Example 1: European call option

- An investor buys a call option to purchase 100 shares with the following features
- strike price, E = £100
- current stock price, S = £98
- price of an option to buy 1 share, C = £5
- i.e. initial premium paid = £5 x 100 = £500
- At expiration, the stock price is £115
- £115 > £100: the option is exercised
- total gain = (£115 £100) x 100 = £1,500
- Taking account of the option premium paid initially
- net gain = £1,500 £500 = £1,000

What happens at expiration?

- The call option holder’s decision to exercise or not depends on the stock price at expiration,
- ‘In the money’: > E
- profitable to exercise the option
- payoff (ignoring option premium) = E
- ‘Out of the money’: < E
- not profitable to exercise the option
- payoff (ignoring option premium) = 0

Profit of call option holder

- To calculate the holder’s profit, we need to take account of
- initial premium paid for the option, C
- payoff at expiration,
- Profit

Writer of the call option

- The writer of the contract is obliged to trade if asked to do so
- Option is ‘zero sum’: what one side gains, the other loses
- Writer’s profit is the mirror image of the holder’s profit, around the horizontal axis
- Writer’s loss is potentially unlimited as the stock price goes up

Example 2: European put option

- An investor buys a put option to sell 100 shares with the following features
- strike price, E = £70
- current stock price, S = £65
- price of an option to buy 1 share, P = £7
- i.e. initial premium paid = £7 x 100 = £700
- At expiration, the stock price is £55
- £55 < £70: the option is exercised
- total gain = (£70 £55) x 100 = £1,500
- Taking account of the option premium paid initially
- net gain = £1,500 £700 = £800

What happens at expiration?

- The put option holder’s decision to exercise or not depends on the stock price at expiration,
- ‘Out of the money’: > E
- not profitable to exercise the option
- payoff (ignoring option premium) = 0
- ‘In the money’: < E
- profitable to exercise the option
- payoff (ignoring option premium) = E

Profit of put option holder

- To calculate the holder’s profit, we need to take account of
- initial premium paid for the option, P
- payoff at expiration,
- Profit

Writer of the put option

- The writer of the contract is obliged to trade if asked to do so
- As before, writer’s profit is the mirror image of the holder’s profit, around the horizontal axis
- Writer’s maximum possible loss is P – E

OPTION VALUATION

- What is the value of an option before it is exercised?
- Sophisticated answer: the Black-Scholes model [Robert Merton & Myron Scholes: Nobel prize 1997]
- Underlying asset value S follows a random walk
- Values an option over Sby constructing a (fully) hedged portfolio, which must then earn the risk-free interest rate
- Solution for a European call/put: the Black-Scholes formula
- ‘No arbitrage’ principle
- Two assets (or portfolios of assets) which have the same payoffs in all possible cases must sell at the same market price
- ‘No free lunch’
- Using this principle some simple results can be derived
- Upper and lower bounds on price of a European call option
- Put-call parity

Upper bound on call option price

For a European call option on non-dividend paying stock

- Upper bound: , the current price of the stock
- Call option gives the holder the right to buy one share of a stock at a given price
- The call option can never be worth more than the stock
- ‘No-arbitrage principle’: if then an arbitrageur could make a riskless profit by buying the stock at and selling the option at C
- Compare two portfolios
- A: one call option, C
- B: one share in the underlying stock, S
- Payoffs at maturity
- A: payoff = 0 if ; otherwise
- B: payoff = : this is greater than portfolio A

Lower bound on call option price

- Lower bound:
- Compare two portfolios
- A: one call option, C+ a bond providing payoff E at time T
- B: one share in the stock, S
- Payoffs at maturity
- A: payoff =
- bond matures to give E
- option’s payoff is
- B: payoff = : this is the same as or less than portfolio A
- Value of portfolio A must be weakly greater than portfolio B
- The (discounted) value of the bond today is
- Thus:
- Rearrange:

Six factors affect option prices

- Current stock price, S
- Call: +ve Put: ve
- Strike price, E
- Call: ve Put: +ve
- Time to expiration, T
- Call & put: +ve (assuming no dividend payments)
- Volatility (variance) of the stock price,
- Call & put: +ve (‘option’ curtails downside risk while keeping upside)
- Risk-free interest rate, r
- Call: +ve Put: ve (PV of future cash amount is less)
- Any dividends that are expected to be paid
- Call: ve Put: +ve (dividend payment lowers S)

Effect of volatility

- Greater variance increases dispersion of future stock prices
- Option cuts off one side of the distribution: (call) option holder does not exercise when
- When , wider distribution gives greater probabilities of higher payoffs: increases value of option

Put-call parity

- A relationship between the prices of European call and put options with the same strike price E and time to maturity T
- Consider two portfolios
- A: one call option, C + a bond providing payoff E at time T
- B: one put option, P + one share of the stock, S
- Payoffs at maturity
- A: payoff =
- bond matures to give E
- option’s payoff is
- B: payoff =
- option’s payoff is
- value of the stock is
- No arbitrage principle: portfolios have the same price, i.e.:

‘Put-call parity’

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