Fourth Edition1

Chapter 16

Option Valuation

Fourth Edition2

Outline

• Valuation– Intrinsic and time values– Factors determining option price– Black-Scholes Model

• How valuation helps trading (optional)– Hedge ratio (Delta) and option elasticity– Other variables

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1. VALUATION

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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

• However, option price is always higher than or equal to its intrinsic value

• Time value - the difference between the option price and the intrinsic value

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Time Value of Options: Call

Option value

XStock Price

Value of Call Intrinsic Value

Time value

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Factors Influencing Option Values: CallsIf this variable increases Value of a call optionStock price increasesExercise price decreasesVolatility of stock price increasesTime to expiration increasesInterest rate increasesDividend Rate decreases• Interest affects the PV(x), your obligation to pay in the future.

Higher interest, the less you need to pay in today’s value, the higher the value of call

• Div is a drag on stock price, call holder want stock price to be higher

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Factors Influencing Option Values: Puts

If this variable increases Value of a Put optionStock price decreasesExercise price increasesVolatility of stock price increasesTime to expiration increasesInterest rate decreasesDividend Rate Increases• Interest affects the PV(x), your sell price in the future. Higher

interest, the less you get paid in today’s value, the lower the value of put

• Div is a drag on stock price, put holder want stock price be low

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Black-Scholes Option Valuation

Co = SoN(d1) - Xe-rTN(d2)

d1 = [ln(So/X) + (r – + 2/2)T] / (T1/2)

d2 = d1 - (T1/2)

whereCo = Current call option value.

So = Current stock price

N(d) = probability that a random draw from a normal dist. will be less than 1.

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Black-Scholes Option Valuation

X = Exercise price. = Annual dividend yield of underlying stocke = 2.71828, the base of the nat. 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 functionStandard deviation of annualized cont.

compounded rate of return on the stock

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Call Option Example

So = 100 X = 95

r = .10 T = .25 (quarter)

= .50 = 0

d1 = [ln(100/95)+(.10-0+(5 2/2))]/(5.251/2)

= .43

d2 = .43 - ((5.251/2)

= .18

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Probabilities from Normal Dist.

N (.43) = .6664

Table 17.2

d N(d)

.42 .6628

.43 .6664

.44 .6700

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Probabilities from Normal Dist.

N (.18) = .5714

Table 17.2

d N(d)

.16 .5636

.18 .5714

.20 .5793

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Call Option Value

Co = Soe-TN(d1) - Xe-rTN(d2)

Co = 100 X .6664 - 95 e- .10 X .25 X .5714

Co = 13.70

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Put Option Value: Black-Scholes

P=Xe-rT [1-N(d2)] – S0 [1-N(d1)]

Using the sample data

P = $95e(-.10X.25)(1-.5714) - $100 (1-.6664)

P = $6.35

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2.HOW VALUATION HELPS TRADING

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Hedge ratio

• Hedge ratio: The change in the price of an option for a $1 increase in stock price. Hedge ratio is also called delta

• If we graph option value as a function of stock price, hedge ratio is the slope

• For call, 0<delta<1, for put -1<delta<0

• In Black-Schole model, hedge ratio for call is N(d1), for put is N(d1)-1

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How to use hedge ratio in trading

• Hedge ratio (delta) help to understand your potential gain and loss for options positions

• Leverage– Option elasticity: (%change of option price)/(%

change of stock price)

– Option elasticity=(delta/option price)/(1/stock price)

– Elasticity measures your leverage (with options) vs. investing in stocks

• My own measurement: delta/option price– Measures % change of option value for $1 change

of stock price

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Important measurements in trading

• Delta: the change in an option price for one dollar increase in stock price

• Gamma: the change of Delta for one $ increase in stock price

• Theta: the change in an option price given a one-day change in time. Always negative, Good for option sellers.

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Important measurements in trading

• Rho: the change in an option price for one % change in risk free rate ( not a big concern in trading. 1% rate is huge change, compared with $1 change of underlying stock price)

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Important measurements in trading• Vega: sensitivity to volatility. The

change in an option price for 1%change in implied volatility– Vega declines overtime– Example:

• June 2010 S&P index Put, exercise price: 800• Index now: 1015; option Price/premium: $33

Vega: 2.3;implied volatility 35%• If implied volatility increase by 10% from 35%

to 45%. (CBOE Volatility Index soars as Wall St slumps)

• Put price: 2.3*10+33=$56

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Important measurements in trading• Calculate option price change

Stock AAPLOption 2012 Jan $200 PutNow(Time 0) 5/7/10AAPL 235.86$ Option 34.55$ implied volatility0(%) 46Delta -0.2732Vega 1.0232

Next trading day(Time 1) 5/10/2010Stock price 200stock price changeOption price change due to stock price

Implied volatility1 60volatility changeOption Price change due to increased volatility

Total Option Price change -$ Option Price 1

Gain per put contract wirte 0

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Important measurements in trading

Variables Relationship with Call Option Value

Relationship with put Option Value

Sensitivity variables

Importance in Trading

Exercise Price - + Stock Price + - Delta, Gamma Very Time to Maturity

+ + Theta Very

Volatility + + Vega Very Risk Free Rate + - Rho Dividend Yield - +