**Chapter 15Option Valuation** • Put-Call Parity • The Black-Scholes-Merton Option Pricing Model • Varying the Option Price Input Values • Measuring the Impact of Input Changes on Option Prices • Implied Standard Deviations • Hedging a Stock Portfolio with Stock Index Options • Implied Volatility Skews • Summary & Conclusions

**Put-Call Parity** C = Call option price P = Put option price S = Current stock price K = Option strike price r = Risk-free interest rate T = Time remaining until option expiration Buy the stock, buy a put, and write a call; the sum of which equals the strike price discounted at the risk-free rate

**More Put-Call Parity** If the stock pays a dividend before option expiration:

**More Put-Call Parity** If the stock pays a dividend before option expiration:

**Black-Scholes-Merton Option Pricing Model** Value of a stock option is a function of 6 input factors: 1. Current price of underlying stock. 2. Dividend yield of the underlying stock. 3. Strike price specified in the option contract. 4. Risk-free interest rate over the life of the contract. 5. Time remaining until the option contract expires. 6. Price volatility of the underlying stock. The price of a call option equals:

**B-S-M Option Pricing Model (cont’d)** Where the inputs are: S = Current stock price y = Stock dividend yield K = Option strike price r = Risk-free interest rate T = Time remaining until option expiration = Sigma, representing stock price volatility The price of a put option equals:

**B-S-M Option Pricing Model (cont’d)** Where d1 and d2 equal:

**B-S-M Option Pricing Model (cont’d)** Remembering put-call parity, the value of a put, given the value of a call equals: Also, remember at expiration:

**B-S-M Option Pricing Model Example** Assume S = $50, K = $45, T = 6 months, r = 10%, and = 28%, calculate the value of a call and a put option. From a standard normal probability table, look up N(d1) = 0.812 and N(d2) = 0.754

**Varying the Option Input Values**

**Varying Option Input Values (cont’d)** • Stock price: • Call: as stock price increases call option price increases • Put: as stock price increases put option price decreases • Strike price: • Call: as strike price increases call option price decreases • Put: as strike price increases put option price increases

**Varying Option Input Values (cont’d)** • Time until expiration: • Call & Put: as time to expiration increases call and put option price increase • Volatility: • Call & Put: as volatility increases call & put value increase

**Varying Option Input Values (cont’d)** • Risk-free rate: • Call: as the risk-free rate increases call option price increases • Put: as the risk-free rate increases put option price decreases • Dividend yield: • Call: as the dividend yield increases call option price decreases • Put: as the dividend yield increases put option price increases

**Measuring the Impact of Changes - Delta** • Delta measures the impact of a change in the stock price on the value of the option. • A $1 change in stock price causes the option to change by delta dollars. • Delta is positive for calls and negative for puts

**Measuring the Impact of Changes - Eta** • Eta measures the percentage impact of a change in the stock price on the value of the option. • A 1% change in stock price causes the option to change by eta percent. • Eta is positive for calls and negative for puts.

**Measuring the Impact of Changes - Vega** • Vega measures the impact of a change in stock price volatility on the value of the option. • A 1% change in sigma causes the option price to change by vega percent. • Vega is positive for calls and puts • Where:

**Measuring the Impact of Changes - Example** • From the previous example: P = $50, K = $45, T = 6 months, r = 10%, = 28%, N(d1) = 0.812, N(d2) = 0.754, C = $8.32, and P= $1.13. • Call Delta = 0.812, so for every $1.00 the stock price increases, the call option increases by $0.81 • Call Eta = 0.812 x (50 / 8.32) = 0.812, so for a 1% increase in stock price, the call option increases by 4.88% • Call Vega = $50 x 0.26998 x (.5)1/2 = 9.545, for a 1% increase in sigma, the call option will increase by 9.55 cents.

**Other Measures of Sensitivity** • Gamma: measure of delta sensitivity to a stock price change. • Theta: measure of option price sensitivity to a change in time to expiration. • Rho: measure of option price sensitivity to a change in the interest rate.

**Implied Standard Deviation** • Implied Standard Deviation (ISD) • Implied Volatility (IV) • Use current option price to compute an estimate of the stock’s standard deviation.

**Implied Volatility Skews** • Volatility smiles • Relationship between implied volatility's and strike prices • Steep negative slope between ISD’s and strike prices for both calls & puts • Stochastic volatility • Black-Scholes-Merton option pricing model assumes constant volatility • Use at-the-money options

**Hedging a Stock Portfolio with Options** • To calculate the number of index option contracts need to hedge an equity portfolio: • To maintain hedge, must rebalance portfolio

**Problem 15-6** A call option is currently selling for $10. It has a strike price of $80 and 3 months to maturity. What is the price of a put option with a $80 strike and 3 months to maturity? The current stock price is $85, and the risk-free rate is 6%. Solution: Using put-call parity:

**Problem 15-7** What is the value of a call option if the underlying stock price is $100, the strike price is $70, the underlying stock volatility is 30%, and the risk-free rate is 5%. Assume the option has 30 days to expiration. Solution: S = $100 sigma = .30 K = $70 T = 30 days r = .05 solving for C = $30.29

**Problem 15-19** Suppose you have a stock market portfolio with a beta of 1.4 that is currently worth $150 million. You wish to hedge against a decline using index option. Describe how you might do this with puts and calls. Suppose you decide to use SPX calls. Calculate the number of contracts needed if the contract you pick has a delta of .50, and S&P 500 index is at 1200. Solution: You can either buy puts or sell calls. In either case, gains or losses on your stock portfolio will be offset by gains or losses on your options contracts. [cont’d next slide]

**Problem 15-19 (cont’d)** Solution:

**Problem 15-20** Using an options calculator, calculate the price and the following “greeks” for a call and a put option with 1 year to expiration: delta, gamma, rho, eta, vega, and theta. The stock price is $80, the strike price is $75, the volatility is 40%, the dividend yield is 3%, and the risk-free rate is 5%. Solution: S = $80 sigma = 0.40 K = $75 T = 365 days r = .05 y = 3% [cont’d next slide]

**Problem 15-20 (cont’d)** Solution: Call Put Value $15.21 $8.92 Delta 0.640 -0.330 Gamma 0.011 0.011 Rho 0.360 -0.353 Eta 3.366 -2.963 Vega 0.285 0.285 Theta 0.016 0.015