Risk Management Using Derivatives Securities 1
Definition of Derivatives A derivative is a financial instrument whose value is derived from the price of a more basic asset called the underlying asset. Examples of underlying assets: shares, commodities, currencies, credits, stock market indices, weather temperatures, results of sport matches or elections, etc. Examples of derivatives are: Options put and call options, forwards, futures, and swaps 2
Derivative Markets Derivative markets are markets for contractual instruments whose performance is determined by how another instrument or asset performs Cash market or spot market maximum delivery two working days Forward markets related to forward and/or futures contract 3
The Role of Derivative Markets 1. risk management hedging Because derivative prices are related to the prices of the underlying spot market goods, they can be used to reduce or increase the risk of investing in the spot items 2. Price discovery futures and forward markets are an important means of obtaining information about investors expectations of future prices 3. Operational advantages Lower transaction cost, have greater liquidity than spot markets (futures & options), allow investors to sell short more easily 4. Market efficiency 5. Speculation 4
Types of Derivative Securities: Options Contract Forward Contract Futures Contract Swap 5
Derivatives Securities: OPTIONS CONTRACT 6
Definition of Options Arrangement or agreement between the seller and the buyer in which the buyer has the right to buy (call option) or sell (put option) an underlying assets at some time in the future at a price stipulated at present. 7
Option Terminology Buy - Long Sell - Short Call Put Key Elements Exercise or Strike Price Premium or Price Maturity or Expiration 8
Market and Exercise Price Relationships In the Money - exercise of the option would be profitable Call: market price>exercise price Put: exercise price>market price Out of the Money - exercise of the option would not be profitable Call: market price<exercise price Put: exercise price<market price At the Money - exercise price and asset price are equal 9
American vs European Options American - the option can be exercised at any time before expiration or maturity European - the option can only be exercised on the expiration or maturity date 10
Different Types of Options Stock Options Index Options Futures Options Foreign Currency Options Interest Rate Options 11
Payoffs and Profits on Options at Expiration - Calls Notation Stock Price = ST Exercise Price = X Payoff to Call Holder (ST - X) if ST >X 0 if ST < X Profit to Call Holder Payoff - Purchase Price 12
Payoffs and Profits on Options at Expiration - Calls Payoff to Call Writer - (ST - X) if ST >X 0 if ST < X Profit to Call Writer Payoff + Premium 13
Figure Payoff and Profit to Call at Expiration 14
Figure Payoff and Profit to Call Writers at Expiration 15
Payoffs and Profits at Expiration - Puts Payoffs to Put Holder 0 if ST > X (X - ST) if ST < X Profit to Put Holder Payoff - Premium 16
Payoffs and Profits at Expiration - Puts Payoffs to Put Writer 0 if ST > X -(X - ST) if ST < X Profits to Put Writer Payoff + Premium 17
Figure Payoff and Profit to Put Option at Expiration 18
Optionlike Securities Callable Bonds Convertible Securities Warrants Collateralized Loans Levered Equity and Risky Debt 19
Option Values Intrinsic value - profit that could be made if the option was immediately exercised Call: stock price - exercise price Put: exercise price - stock price Time value - the difference between the option price and the intrinsic value 20
Factors Influencing Option Values: Calls Factor Effect on value Stock price increases Exercise price decreases Volatility of stock price increases Time to expiration increases Interest rate increases Dividend Rate decreases 21
Black-Scholes Option Valuation C o = S o e -dt N(d 1 ) - Xe -rt N(d 2 ) d 1 = [ln(s o /X) + (r d + s 2 /2)T] / (s T 1/2 ) d 2 = d 1 - (s T 1/2 ) where C o = Current call option value. S o = Current stock price N(d) = probability that a random draw from a normal dist. will be less than d. 22
Black-Scholes Option Valuation X = Exercise price. d = Annual dividend yield of underlying stock e = 2.71828, the base of the natural log r = Risk-free interest rate (annualizes continuously compounded with the same maturity as the option. T = time to maturity of the option in years. ln = Natural log function s = Standard deviation of annualized cont. compounded rate of return on the stock 23
Figure 15-3 A Standard Normal Curve 24
Call Option Example S o = 100 X = 95 r =.10 T =.25 (quarter) s =.50 d = 0 d 1 = [ln(100/95)+(.10-0+(.5 2 /2))]/(.5.25 1/2 ) =.43 d 2 =.43 - ((.5)(.25) 1/2 =.18 25
Probabilities from Normal Dist. N (.43) =.6664 Table 17.2 d N(d).42.6628.43.6664 Interpolation.44.6700 26
Probabilities from Normal Dist. N (.18) =.5714 Table 17.2 d N(d).16.5636.18.5714.20.5793 27
Call Option Value C o = S o e -dt N(d 1 ) - Xe -rt N(d 2 ) C o = 100 X.6664-95 e -.10 X.25 X.5714 C o = 13.70 Implied Volatility Using Black-Scholes and the actual price of the option, solve for volatility. Is the implied volatility consistent with the stock? 28
Put Option Value: Black-Scholes P=Xe -rt [1-N(d 2 )] - S 0 e -dt [1-N(d 1 )] Using the sample data P = $95e (-.10X.25) (1-.5714) - $100 (1-.6664) P = $6.35 29
Put Option Valuation: Using Put-Call Parity P = C + PV (X) - S o = C + Xe -rt - S o Using the example data C = 13.70 X = 95 S = 100 r =.10 T =.25 P = 13.70 + 95 e -.10 X.25-100 P = 6.35 30
Using the Black-Scholes Formula Hedging: Hedge ratio or delta The number of stocks required to hedge against the price risk of holding one option Option Elasticity Call = N (d 1 ) Put = N (d 1 ) - 1 Percentage change in the option s value given a 1% change in the value of the underlying stock 31