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

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ec247 financial instruments and capital markets dr helen weeds 2013 14 spring term

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

Lecture 7: Futures and options

learning outcomes
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
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
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
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 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 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
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
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
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 $20,000
    • Difference is in the size of up-front investment required
      • futures allow the speculator to obtain leverage , i.e. to take out a large speculative position with a small stake
specification of a futures contract
Specification of a futures contract
  • Asset
    • e.g. a commodity, of specified grade or quality
  • Contract size
    • amount of the asset to be delivered
  • Delivery
    • delivery month
    • arrangements for delivery (e.g. location)
  • Price
    • how prices are quoted (currency and unit size, e.g. US$ 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
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
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
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 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
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
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
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
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
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
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 expiration1
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
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
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
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
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 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
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
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
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’