Introduction to Options Mario Cerrato

Option Basics Definition A call or put option gives the holder of the option the right but not the obligation to buy (sell) an underlying security at fixed price on (European-style) or before (American-style) a specified date.

Terminology Holder Writer Strike or Exercise Price Premium Covered and Uncovered option positions At-, Out-, and In-the-Money options (ATM, OTM, ITM) Trading and Exercise of options

Option Types 1: Plain Vanilla European-style: CALLS PUTS

Gain when underlying security rises Maximum Loss = option premium Decision Rule: max[0, S T - K]

The pay out here is the mirror image of the long call pay out. The Holder determines the option value to the Writer. The pay out rule is: -1* max[0, S T - K]

Gain when underlying falls Again any loss limited to the option’s premium Decision Rule: max[0, K - S T ]

The pay out here is the mirror image of the long put pay out. The Holder determines the option value to the Writer. The pay out rule is: -1* max[0, K - S T ]

Exercise (strike) prices and ‘Moneyness’ Option users can select from several strike prices when adopting a strategy. The following diagram looks at this from the point of view of call options. Put profiles could be developed in a similar fashion.

ITM high premium ATM OTM low premium

Option Pricing One of the most widely used methods to price European options is the Black-Scholes-Merton method (BS). The BS is a mathematical formula for the theoretical value of an European call/put option. Black-Scholes and Merton showed, using a non-arbitrage argument, that investors are risk neutral, and therefore the appropriate interest rate to discount cash flows is the risk-free rate of interest. Based on the above argument, they derived a stochastic partial difference equation whose solution leads to the well known formulas for pricing call/put European options.

Option Pricing The Black and Scholes formulas for pricing European call/put options on non-dividend paying stock are:

Option Pricing There are quite few assumptions underlying the BS model: -Short selling allowed -No transaction costs -No dividend paid -No riskless arbitrage opportunities -Stock trading is continuous -Risk free rate of interest is constant for all maturities

The Greek Letters Greek letters measure a dimension of risk in an option position: Delta: Gamma: The Theta:

The Greek Letters Vega: Rho: Generally, traders can change the delta/gamma of an option by trading the underlying/options with different maturities. Vega, generally, is only closely monitored.

Option strategies Synthetic future Straddle Caps - Floors

Gain if price of underlying falls Gain if price of underlying rises BUT - two premiums payable

Using twice the required No. of Puts to fully hedge the portfolio a synthetic straddle is created. Put contracts Underlying security

Synthetic Futures: Short calls plus long puts Short ATM call Long ATM put Synthetic short future

Caps and Floors Underlying instrument Profit Loss A long position - a bought position Short OTM call Long OTM put collar

Hedging Covering the perceived risk of a market fall in the value of an underlying portfolio. If the assumption is made that the underlying portfolio is index-tracking - and that a suitable put option contract is available - buy the appropriate number of contracts to fully hedge the portfolio against a fall in value.

Gain when the underlying instrument falls in values Strike Maximum Loss

Pay out Profile synthetic The resulting pay out profile for this hedge (long underlying and long put) will result in the pay out profile of a call option - albeit a synthetic call option.

Long Equity Portfolio Long PUT option Synthetic CALL option Synthetic Call Option

Hedge Outcome The gain from the option market transaction offsets the loss in the cash market portfolio. The downside risk is protected but the upside potential is retained. BUT DOES THIS REALLY WORK IN PRACTICE?

An Example To mount a hedge in practice a number of values need to be used: The current value of the cash market portfolio. The relationship between the portfolio and the appropriate cash market index (  p ). The number of contracts to use to fully hedge the portfolio. The time frame over which the hedge is to be in place. The option premia for the selected strike price(s). An assumed scenario about the level to which the cash market index will fall.

FTSE 100 Index Options (European-Style Exercise) Unit of TradingContract valued at £10 per index point (eg value £65,000 at ) Delivery MonthsNearest eight of March, June, September, December plus such additional months that the nearest four calendar months are always available for trading. QuotationIndex points (e.g ) Minimum Price Movement (Tick Size & Value)0.5 (£5.00) Exercise DayExercise by 18:00 on the Last Trading Day only Last Trading Day Third Friday in delivery month. Settlement DaySettlement day is the first business day after the Last Trading Day A Typical Index Option Summary Contract Specification: LIFFE FTSE 100

Scenario Portfolio value: £6,000,000 Option price (premium): £110 Cash index: 6250  = 1 (index-tracking) Hedge Horizon: 35 days Contract specification: LIFFE FTSE 100 index Option For simplicity assume that the option contract matures in exactly 35 days. Extent of decline in the cash index: 5900.

No. of Put option contracts for a full hedge Calculating the number of contracts required to hedge a portfolio using Strike (exercise) Price:

The Hedge Arithmetic DateCash MarketOptions Market NowPortfolio ValueBuy PUT options (VP 0 ) HorizonPortfolio ValueSell (or exercise) (VP 1 )PUT options max[0, K - S T ] Result VP 0 – VP 1 Sell - Buy Net Result = Options Market + Cash Market

Cash Market Calculations Now Current portfolio valuation:£6,000,000 Horizon This can be estimated using:

Option Market Calculations Buy PUTS (to open) (No. of contracts)(Index point value)(Option Premium) (96)(£10)(110) = £105,600 Expiration assumed. (No. of contracts)(index point value)max[0, K - S T ] (96)(£10)( ) = £336,000

Hedge Outcome DateCash MarketOptions Market Now£6,000,000£105,600 (96)(£10)(110) Horizon£5,664,000£336,000 (96)(£10)( ) Result (£336,000) £230,400 Net Result = £230,400 - £336,000 = -£105,600

Analysis Hedge horizon - realistic? 1. Normally the hedge horizon will not match a contract maturity date exactly! If the puts were bought at high a volatility level and sold at low or normal volatility levels the results of the hedge will be affected. 2. Buying puts requires no margin but will be affected by bid- ask spreads. These are likely to widen when big market movements are expected. For example, under normal circumstances the bid-ask spread may be 8-10 (the market makers buy at 8, sell at 10) when market makers expect a move in the price of the underlying security the spread may increase to This will also impact on the hedge outcome. 3. Volatility is also of crucial importance

Market views on volatility