Options, Forwards and Futures SSE MBA – Fin I, A. Simonov

Outline - Options Option features –trading –put and call options –payoffs Option strategies –protective put –spread –covered call –straddle –collar Option Valuation

Derivative Security whose payoff depends on the value of another asset Examples: Options Futures Warrants and convertibles Insurance policies

For other investors, the same derivative securities provide an insurance function For some investors, derivative securities offer the cheapest way to capitalize on information that the security will rise or fall in value Derivative Trading Speculation Insurance and Hedging

Options Call Option: gives the holder the right to purchase the underlying asset at a specified price on or before a specified date Put Option: gives the holder the right to sell the underlying asset at a specified price on or before a specified date The option is a right, not an obligation. What can you do with the option? –Sell it back at market price –Exercise it –Do nothing

Elements of Option Contracts Strike or Exercise Price: specified price Expiration Date: specified date Option Premium: the price of the option Option Writer: the seller of the contract (receives the option premium) Exercise Style American Options: can be exercised anytime before or on expiration date European Options: can be exercised on expiration date only

Underlying Asset Option can be written on value of: Stock Index –S&P500, NYSE Index, Nikkei 225, FTSE 100… Bond Prices and Yields Foreign Currency Futures

Options Trading Over-the-Counter (OTC) Especially exotic options Organized Exchanges Chicago Board Options Exchange (CBOE) Chicago Mercantile Exchange (CME) American Stock Exchange (AMEX) Standardized contracts: Stock option: a call option contract allows owner to buy 100 units of stock

History 1780 Calls first traded in USA 1934 SEC regulates option markets 1973 CBOE established Central marketplace Option Clearing Corporation Standardized contracts introduced 1974 Volume exceeded AMEX stock trading

European Call Option Today You buy a 6-month IBM call with strike price X = $90 You pay C = $10 for the option. In 6 months IBM stock is worth S T If S T <$90, no point in exercising the call Payoff = 0, Profit = -$10 If S T >$90, you exercise the option, pay $90 for the stock, can resell it at S T Payoff = S T – $90, Profit = S T – $90 – $10

Payoff in 6 Months IBM price S T Cost of option Payoff = Value at expiration Profit Payoff

Option Moneyness STST At-the-money Out-of-the-money Payoff In-the-money

European Put Option Today You buy a 6-month IBM put with strike price X = $90 You pay P = $5 for the put In 6 months IBM stock is worth S T If S T > $90, no point in exercising the put Payoff = 0, Profit = -$5 If S T < $90, you exercise the put, buy the stock at S T, resell it for $90 Payoff = $90 - S T,, Profit = $90 - S T -$5

Put Payoff Diagram Payoff STST Payoff At-the-money Out-of-the-moneyIn-the-money 90

Option Strategies Options can be combined to obtain an unlimited variety of payoff patterns Most common ones –Protective put –Spread –Covered Call –Straddle –Collar (problem set)

Protective Put Suppose that you buy IBM and a put option with exercise price X = $90 At maturity If S T < $90, you can sell the stock for $90 If S T > $90, the put is worthless and the portfolio is worth S T The put offers insurance against a decline in the stock price Cost of insurance = put premium

Spread Combination of two or more calls (or puts) on the same stock with different exercise prices or expiration dates Money Spread: buy a call (put) and write a call (put) with different exercise prices Time Spread: buy a call (put) and write a call (put) with different expiration dates Money Spread (bullish)

Buy a Call VTVT STST 2030 Payoff Profit Long (buy) January call with $20 exercise price

Write a Call VTVT STST 2030 Payoff Profit Short (write) January call with $25 exercise price

Money Spread VTVT STST 2030 Payoff Profit VTVT STST 2030 Payoff Profit Long (buy) January call with $20 exercise price Short (write) January call with $25 exercise price

Money Spread VTVT STST 2030 Payoff Profit VTVT STST 2030 Payoff Profit Long (buy) January call with $20 exercise price Short (write) January call with $25 exercise price

Money Spread VTVT STST 2030 Payoff Profit VTVT STST 2030 Payoff Profit VTVT STST 2030 Payoff Profit Long (buy) January call with $20 exercise price Short (write) January call with $25 exercise price “Bullish spread”

Covered Call Purchase of a stock position with a simultaneous sale of a call in that stock The call is covered since the call writer has already the stock to be sold in case the option is exercised Very popular among institutional investors –already have the stocks –cash on the call premium –lose some upside gain but maybe planning to sell at strike price anyway

Covered Call STST 2030 Profit Short (write) January call with $25 exercise price Payoff VTVT

Covered Call VTVT STST 2030 Payoff Profit VTVT STST 2030 Profit Long (buy) stock at current price of $24 Short (write) January call with $25 exercise price Payoff

Covered Call VTVT STST 2030 Payoff Profit VTVT STST 2030 Profit Long (buy) stock at current price of $24 ⅞ Short (write) January call with $25 exercise price Payoff

Covered Call VTVT STST 2030 Payoff Profit VTVT STST 2030 Payoff Profit VTVT STST 2030 Profit Long (buy) stock at current price of $24 Short (write) January call with $25 exercise price Covered Call Payoff

Straddle A long (short) straddle is established by buying (selling) a call and a put Profit from up and down movements of the underlying They are bets on volatility The worse scenario is the case of no or little volatility

Straddle VTVT STST 2030 Profit Long (buy) Feb call with exercise price of $25 Payoff

Straddle VTVT STST 2030 Payoff Profit Long (buy) put with exercise price of $25

Straddle VTVT STST 2030 Payoff Profit VTVT STST 2030 Profit Long (buy) put with exercise price of $25 Long (buy) call with exercise price of $25 Payoff

Straddle VTVT STST 2030 Payoff Profit VTVT STST 2030 Payoff Profit VTVT STST 2030 Profit Long (buy) put with exercise price of $25 Long (buy) call with exercise price of $25 Straddle Payoff

Option Valuation Arbitrage Pricing Put-Call Parity Option value as a function of the stock price –volatility –expiration Binomial model Black and Scholes formula

Principle of Arbitrage Pricing The principle of arbitrage pricing: if two portfolios of assets have identical cash flows, then they must have the same price if they do not have the same price then there exists an arbitrage opportunity –the opportunity of making profit without taking any risk The principle of arbitrage pricing is used in pricing derivative securities

Arbitrage Opportunity Consider the following two portfolios cash flows: Short I and Long II:

Arbitrage Opportunity (cont) Riskless profit: arbitrage opportunity As a result the price of portfolio II will increase and the price of portfolio I will decrease until they are equal

Put-Call Parity Arbitrage pricing implies a relationship between the premiums of a call and a put with the same strike price and expiration date: P: put premium, C: call premium S: stock price X: strike price r f : risk free rate T: time to expiration date Present Value of X

Example American Express stock –Stock price S = $24.88 (January 3 rd 2003) –Put premium P = $1.06 –Call premium C = $0.88 –Strike price X = $25 –Risk free (borrowing) rate r f = 1% –T = 47 (February 19 th 2003)

Put-Call Parity VTVT STST 40 Long (buy) stock Payoff

Put-Call Parity VTVT STST 40 Long (buy) put with $25 exercise price Payoff

Put-Call Parity VTVT STST 40 Long (buy) put with $25 exercise price Payoff VTVT STST 40 Long (buy) stock Payoff

Put-Call Parity VTVT STST 40 Long (buy) put with $25 exercise price Payoff VTVT STST 40 Long (buy) stock Payoff VTVT STST 40 Put plus stock Payoff

Put-Call Parity VTVT STST 40 Long (buy) put with $25 exercise price Payoff VTVT STST 40 Long (buy) stock Payoff VTVT STST 40 Put plus stock Payoff P+S = $1.06+$24.88

Put-Call Parity VTVT STST 40 Long (buy) put with $25 exercise price Payoff VTVT STST 40 Long (buy) stock Payoff VTVT STST 40 Put plus stock Payoff VTVT STST 40 Long (buy) call with $25 exercise price Payoff P+S = $1.06+$24.88

Put-Call Parity VTVT STST 40 Long (buy) put with $25 exercise price Payoff VTVT STST 40 Long (buy) stock Payoff VTVT STST 40 Put plus stock Payoff VTVT STST 40 Long (buy) call with $25 exercise price Payoff VTVT STST 40 Invest amount equal to present value of exercise price Payoff P+S = $1.06+$24.88

Put-Call Parity VTVT STST 40 Long (buy) put with $25 exercise price Payoff VTVT STST 40 Long (buy) stock Payoff VTVT STST 40 Put plus stock Payoff VTVT STST 40 Long (buy) call with $25 exercise price Payoff VTVT STST 40 Invest amount equal to present value of exercise price Payoff P+S = $1.06+$24.88

Put-Call Parity VTVT STST 40 Long (buy) put with $25 exercise price Payoff VTVT STST 40 Long (buy) stock Payoff VTVT STST 40 Put plus stock Payoff VTVT STST 40 Long (buy) call with $25 exercise price Payoff VTVT STST 40 Invest amount equal to present value of exercise price Payoff VTVT STST 40 Call plus money in bank Payoff P+S = $1.06+$24.88

Put-Call Parity VTVT STST 40 Long (buy) put with $25 exercise price Payoff VTVT STST 40 Long (buy) stock Payoff VTVT STST 40 Put plus stock Payoff VTVT STST 40 Long (buy) call with $25 exercise price Payoff VTVT STST 40 Invest amount equal to present value of exercise price Payoff VTVT STST 40 Call plus money in bank Payoff P+S = $1.06+$24.88 =25.94 ≈ = $.88+$24.97 = C + PV(X)

Option Value Before Maturity Option Value = Intrinsic Value + Time Value Intrinsic Value = Value of immediate exercise Time Value = Option value less intrinsic value – attributed to the fact that it still has positive time to expiration – related to volatility, also called volatility value Date t < T

Out-of-the-money option Intrinsic value = 0 Because of stock volatility, there is still a chance that the option will deliver a positive payoff at maturity: Option value = Time value > 0 Option value (time value) will increase with –stock volatility –time to expiration

Deep-in-the-money call Consider a very high stock price: S t >> X Call is very likely to be exercised Payoff at maturity is very likely to be: S T – X Call price at date t is mainly intrinsic value: S t – PV(X) Call Value (intrinsic value) will –increase with stock price –increase with interest rate (lower PV) –decrease with exercise price

Call Value Prior to Expiration CtCt StSt S t – X S t – PV(X) CtCt At-the-money In-the-money Out-of-the-money

Value of a Call Option Value of Call Stock Price S t ⇧ Exercise Price X ⇩ Volatility ⇧ Time to Expiration ⇧ Interest rate ⇧

Option Pricing Models  Black-Scholes (1973) option pricing model  One of the most important models of modern finance  Based on arbitrage pricing  Extremely useful to practitioners  Binomial (or two-state) model  Provides the intuition of the Black-Scholes model  Technically much simpler

Two-State Approach IBM is trading at P s = $100, and price will either increase to S + = $200 or decrease to S - = $50 by year end. The interest rate is r = 8%. What is the premium of a 1 yr IBM $125 call? IBM Stock P S =$100 S + = $200 S - = $50 T-Bills P B = $ 1 1+r = 1.08 Step 1. Security payoffs Call Call Premium C 0 = ? C + = $75 C - = $0

N S :# of shares of IBM in the portfolio. N B :# of T-Bills Step 2. Construct portfolio (N S, N B ) that replicates call N S · P S + N B · P B N S S + + N B (1+r) = C + N S S - + N B (1+r) = C - Portfolio Payoff Portfolio Price TodayYear End Two-State Approach (cont)

N B = C - - S - N S 1+r = 0 – 50 · = (T-Bills) Two-State Approach (cont) C + - C - S + - S - = = 0.5 Shares (IBM) Choose: N S = N S is called the hedge ratio or the Option Delta spread of possible option prices / spread of possible stock prices

Option Value C 0 = N S · P S + N B · P B = 0.5 · $100 + (-23.15) · $1 = $26.85 Step 3. Pricing options using the replicating portfolio [arbitrage pricing] Two-State Approach (cont)

Transition to Black-Scholes The option price gets more and more precise if we add more and more scenarios if we add more and more periods with higher frequency In the limit, we obtain the famous formula for the price of a European call option

C o = S o N(d 1 ) - X e -rT N(d 2 ) d 1 = [ln(S o /X) + (r +  2 /2)T] / (  T 1/2 ) d 2 = d 1 - (  T 1/2 ) C o = European Call premium S o = Stock price X= Strike r = Risk-free rate T = Time to expiration  = Volatility of stock returns ln= Natural log function N(d) = Standard normal distribution function Black-Scholes Formula

Consider a 3-mo IBM 95 Call. IBM is selling at $100. The input for using the Black-Scholes formula is: S o = 100X = 95r = 0.10 T = 0.25 (yr)  = 0.50 We have N (d 1 ) = ; N (d 2 ) = C o = S o N(d 1 ) - X e -rT N(d 2 ) = 100 · · e -(0.10 · 0.25) · = $13.70 Black-Scholes: Example

Summary Option characteristics –payoff –trading Option strategies –protective put –spread –covered call –straddle –collar Option valuation –put-call parity –binomial model –Black-Scholes formula

Outline`- Futures and Forwards Risk management with futures –should companies engage in risk management? Forwards and Futures –contract features –differences –valuation Valuation with cost and convenience yields Commodity futures –Basis risk Exchange Rate futures –Interest Rate Parity

Hedging Forwards and Futures can be used for hedging –Commodity price risk for a buyer (milling company) for a seller (wheat grower) –Foreign exchange risk Sell Swiss Francs (buy $) for a US company Buy Swiss Francs (sell $) for a Swiss company –Stock market risk –Interest rate risk

Examples A mining company expects to produce 1000 ounces of gold 2 years from now if it invests in a new mine: –Avoid that the loan for financing the investment cannot be repaid because the gold price moved A bank expects repayment of a loan in 1 year, and wishes to use proceeds to redeem 2-year bond –Lock in current interest rate between 1 and 2 years from now in order to avoid shortfall if interest rates have changed A hotel chain buys hotels in Switzerland, financed with a loan in US-dollars: –Make sure that the company can repay the loan, even if Swiss franc proceeds diminished because of exchange rate movement

Does Risk Management Add Value? No if stockholders can manage the risks as effectively as the firm can Yes if any of the following reasons apply: asymmetric information trading cost asymmetric taxation bankruptcy cost

Cash (Spot) Transaction

Forward Transaction

Forward Contract A forward contract is a contract made today for future delivery of an asset at a pre-specified price –no money or assets change hands prior to maturity –forwards are traded in the over-the-counter market The buyer (long position) of a forward contract is obligated to: –deliver the asset at the maturity date –pay the agreed-upon price at the maturity date The seller (short position) of a forward contract is obligated to: –deliver the asset at the maturity date –accept the agreed-upon price at the maturity date

Terminology of a Forward Contract Maturity Date: the pre-specified delivery date Forward Price: the pre-specified delivery price Contract Size: the pre-specified amount of assets for delivery Spot Price: price of the underlying asset Open Interest: number of contract outstanding

F F F Payoff Structure at Maturity

Forward vs Futures With forwards, cash only changes hands on the delivery date With futures, gains and losses are settled daily rather than at the expiration of the contract Initial margin: initial deposit on a margin account to be made when entering the contract Daily profit and losses are netted against the initial margin Maintenance margin: level below which a margin call is made and an additional deposit is required to bring the margin account to the initial margin level In this lecture we will abstract from margin issues Effectively we will consider only forwards

Example – Data from CBOT Buy one silver future contract at F 0 = $7.4 /oz Size of one contract is 5,000 oz of silver Initial margin = $2,025 per contract Maintenance margin = $1,500 DayFuture Price Daily ProfitMargin AccountDeposits 0F 0 = $7.40$2,025 1F 1 = $ · 5000 = $5002, = $2,5250 2F 2 = $ · 5000 = -$1,2502, ,250 = $1,275 $750 3F 3 = $ · 5000 = $2502, = $2,2750

Forward Valuation Consider two ways of getting an asset, e.g. gold, in the future A: long a gold forward –F 0 : the forward price B: borrow money at the rate r and invest it directly in gold –S 0 : gold price Initial cost of the each strategy?

Payoffs at Maturity Assume that holding gold does not entails any costs or benefits Strategy A payoff at maturity S T - F 0 Strategy B payoff at maturity S T - S 0 (1+r) Since both strategies are riskless and cost the same (nothing), by no arbitrage the two payoffs must be equal S T - F 0 = S T - S 0 (1+r) F 0 = S 0 (1+r)

Forwards Valuation Revisited The forward price can be written: F 0 = S 0 (1+r) = S 0 + r S 0 The forward price is equal to the spot price plus the financing cost of holding the spot There might be other costs of holding the underlying There might be benefits too

Forward Valuation with Costs and Benefits Repeat earlier analysis: two strategies A: long in the forward –F 0 : the forward price B: borrow money buy the underlying –S 0 : spot price –r : interest rate –OC : other costs of holding the underlying –B : benefits of holding the underlying

Payoffs at Maturity Strategy A payoff at maturity S T - F 0 Strategy B payoff at maturity S T - S 0 (1+r) – OC + B The two payoffs must be equal, hence F 0 = S 0 (1+r) + OC – B F 0 = S 0 + TC – B where TC = rS 0 + OC is the total cost of holding the underlying

Costs and Benefits Apart from the financing cost of holding the underlying rS 0, the other costs and benefits depend on the type of future Convenience and Cost Yield: benefits and costs expressed as fraction of the spot price when the convenience yield outweighs the cost yield, forward prices fall below spot prices UnderlyingCostsBenefits Commodity Transportation Storage, Upkeep Stock and Stock IndexDividends BondCoupons Foreign CurrencyInterest

Hedging Commodity Risk It’s June and a major industrial corporation knows that it will need to buy 1,000 barrels of oil in December It’s concerned about the price it will have to pay in December In June The corporation buys December oil futures at F 0 = $19.45/barrel In December The Corporation is perfectly hedged against changes in the oil price since has fixed the price at which it can buy the oil Payoff: - F 0 = - $19.45

Basis Risk Now suppose that the corporation needs to buy oil in November but there is no November future contract In June The corporation buys December futures at F 0 = $19.45/barrel In November Unwind future position by selling the Dec futures: F T-1 – F 0 Buy oil: -S T-1 Payoff = -F 0 + (F T-1 - S T-1 ) = -$ (F T-1 - S T-1 ) The corporation is exposed to the risk that the future and the spot price could be not the same in November: Basis Risk Basis

Foreign Exchange Futures Companies face currency mismatching: –Assets and liabilities in different currencies –Expect receipts and payments in different currencies. –Use currency futures to hedge Example: In May 97 a US mining company signed a contract to buy a large tunnel digging machine from a UK company: –Total price £35 million –Delivery and payment of the machine at the end of December 1997 –Dec future price for Pound = $1.637, contract size £62,500 Buy 560 Dec 97 EUR future contracts Certain cash outlay in Dec 97: £35M / = $21,380,574

Example (cont) - Alternative Strategy A certain cash outlay in December can be obtained by using an alternative strategy: May borrow US dollars at the 7 month US interest rate exchange US dollars in British pounds at the May spot exchange rate invest British pounds at the 7 month U.K. interest rate December pay back the US loan Both strategies should give the same December cash outlay