OFRM III Topic 2-1 - Lecture notes 2 PDF

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TOPIC 2 BASIC OPTION VALUATION Standard Notation Option Boundaries Factors Determining Option Value Put-Call Parity Elementary Pricing of a Call

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STANDARD NOTATION • • • •

S0 X T r

~ stock price ~ exercise price ~ time to expiration ~ risk-free rate for maturity T with continuous compounding • Usually use most liquid government short-term debt instrument • In Australia it is common to use Bank Bill Swap Reference Rate or shortterm govt bond futures. • Technically, you should use a BBSRR with a duration close to that of the option contract. However, for the examples in this topic, lets assume: • BBSRR = 5%

• • • •

D ST Ca Ce

~ present value of dividends during option’s life ~ stock price at expiration ~ American Call ~ European Call

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COMPOUNDING FREQUENCY • The compounding frequency used for an interest rate is the unit measurement of time • The more frequent you compound, closer you are replicating real-time.

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CONTINUOUS COMPOUNDING • At the limit as we compound more and more frequently we obtain continuously compounded interest rates • $100 grows to $100ert when invested at a continuously compounded rate R for time T • $100 received at time T discounts to $100e-rt at time zero when the continuously compounded discount rate is R • Discrete compounding involves (1+r)t • Discrete discounting involves (1+r)-t

COMPOUNDING/DISCOUNTING WITH DERIVATIVES

• In practice continuous compounding/discounting is always used for derivative calculations. • However, in this course you should be able to switch between the two whenever necessary • Example, • PV(X) = (X)(1+r)-t • PV(X) = (X)e-rt

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CALL OPTION BOUNDARIES • Absolute Minimum Value of a Call = $0 • Ce(S0,T,X)  Max{0, S0-PV(X)} • Ca(S0,T,X)  Max{0, S0-X}

Call Price

Intrinsic / Exercise/ Parity Value: S0  X S0 > X X

Stock Price (S0)

S0-X

if S0 > X

0

if S0  X

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CALL OPTION BOUNDARIES • Example of Arbitrage • Consider 2 portfolios, A & B • Payoff for A • European Call • Bond worth X(1+r)-T • Payoff for B • Share

ST < X 0

ST  X ST– X

X

X

ST < X ST

ST  X ST

• As Portfolio A  Portfolio B then Ce + B  S • Ce  S0-PV(X) or Ce  S - B

CALL OPTION BOUNDARIES

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• ANZ Jun Call Contract ($15.74, (86/365), $16.0) = $0.52 • Is this option in or out of the money? • Instead of $15.74, let’s pretend S 0 =$17.0 • Intrinsic Value = $1 • Else Arbitrage Profit exists!

• Why is the market price for Call = $0.52 and not $0? • Time Value / Speculative Value ~ reflects uncertainty of stock price at expiration • The longer till expiration, the higher the time value • Time Value = C(S0,T,X) - Max(0,S0-X) = $0.52 for ANZ

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CALL OPTION BOUNDARIES • Absolute Maximum Value of a Call = $S0 • C(S0,T,X)  S0 • If C = S0 then must have infinite maturity

S

Call Price

Stock Price (S0)

European

Call Price

S

Stock Price (S0)

American

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CALL OPTION BOUNDARIES • What is the value of an option at expiration? • C(ST,T,X) = Max(0, ST-X)

Max(0,ST-X)

Call Price

X

Stock Price at expiration (ST)

• Is this appropriate for both European and American Options? • Yes

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FACTORS DETERMINING CALL OPTION VALUE • The Effect of Time to Expiration • Two American calls differing only by time to expiration, T1 and T2 where T1 < T2. • Ca2(S0,T2,X)  Ca1(S0,T1,X)

• At T1 Ca1 = Max(0, S1 – X) whereas this will be the minimum for Ca2 • Due to Time value of the call option. • Time value maximized when at-the-money • Minimized when deep in-/out-of- the money. • Time value decays closer to expiration

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FACTORS DETERMINING CALL OPTION VALUE Call Option Value Relative to Time Call Price

Max{(0,S0-PV(X)}

Max(0,S0-X) Stock Price (S0)

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FACTORS DETERMINING CALL OPTION VALUE Example: ANZ Last Sale Price $15.74 Series

Ex Price

Bid

Ask

Last Sale

Vol 000’s

Open Int

Implied Buyer

Volatility Seller

Delta

Annual % Return

Jun Oct

16.00 16.00

.47 .68

.57 .80

.47 .59

216 3

30 1040

16.21 15.78

19.35 18.68

.51 .55

14.02 11.67

Ceteris paribus, a call option is worth less when expiration is closer.

FACTORS DETERMINING CALL OPTION VALUE

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• The Effect of Exercise Price

• Two European calls differing only by strikes of X1 and X2 (where X1 < X2). • Which is greater, Ce1(S0,T,X1) or Ce2(S0,T,X2)? • By how much? • The difference in exercise prices will be greater than or equal to the difference in call prices.

• Consider 2 portfolios, A (money spread) & B •

ST < X1

Payoff for A

X1  ST < X2

X1 < X2  ST

•

Buy Ce1(S0,T,X1)

0

ST – X 1

ST– X1

•

Short Ce2(S0,T,X2)

0

0

-(ST– X2) =X2-X1>0

ST < X1

Payoff for B •

Buy Bond (X2 – X1)

(1+r)-T

• PV(X2-X1)  C1 –C2 • C2  C1 –B

X2 – X1> 0

X1  ST < X2 X2 – X1> 0

X1 < X2  ST X2 – X1> 0

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FACTORS DETERMINING CALL OPTION VALUE Example: ANZ Last Sale Price $15.74 Series

Ex Price

Bid

Ask

Last Sale

Vol 000’s

Open Int

Implied Buyer

Volatility Seller

Delta

Annual % Return

Jun Jun

15.50 16.00

.72 .47

.84 .57

.73 .47

3 216

6 30

15.51 16.21

19.49 19.35

.65 .51

14.42 14.02

Ceteris paribus, a call option is worth more the lower the exercise price.

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FACTORS DETERMINING CALL OPTION VALUE • Limits on the Value Difference in Premiums • We know that • (X2 - X1)(1+r)-T  Ce(S0,T,X1) - Ce(S0,T,X2) • Therefore: • X2 - X1 Ce(S0,T,X1) - Ce(S0,T,X2) • X2 - X1 Ca(S0,T,X1) - Ca(S0,T,X2) • The benefit from buying a call with a lower exercise price will not be greater than the difference in exercise prices. • Example: ANZ $16.00 call costs $0.52. Therefore, the maximum I would pay for an ANZ $15.50 call would be $0.52 + $0.50 = $1.02 • The mid-price of the Option is actually $0.78. • As $1.02>$0.78 the value difference statement is true.

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FACTORS DETERMINING CALL OPTION VALUE • Effect of Dividend Payments • With options pricing, we are NOT interested in the impact of a dividend announcement, but rather the actual payment of the dividend. • We state that once a stock goes Ex-dividend, the new stock value equals S0 – Div • Prior to dividend being paid, S0’ = S0 – PV(Div) • Therefore: • S0 – PV(Div)  Ce(S0,T,X)  Max{0, S0 – PV(X) – PV(Div)} • Example, assume dividend of $0.2 in two days • $15.74 - PV($0.2)> • ANZ Call($15.74, 0.2356, $16.00) > • Max(0, $15.74 - $16.00(1.05)-0.2356 - $0.2(1.05-2/365))

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FACTORS DETERMINING CALL OPTION VALUE • The option to exercise early • Are American calls worth more than European? • Yes - if it is worth to exercise ( dividends) • Ca(S0,T,X)  Ce(S0,T,X) • Example, call may only be in the money when (S0 – X) and not when {[S0 – PV(Div)] - X}. Better to exercise before dividend date.

• Would you ever exercise an American early on a stock that does not pay dividends? • Exercise now you receive (S0 – X). • However, value of call is intrinsic value PLUS time value!

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FACTORS DETERMINING CALL OPTION VALUE

• Effect of Interest Rates • Calls save you money • Rather than buying stock, invest extra cash in term deposit. • Higher r, the higher the value for a call.

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FACTORS DETERMINING CALL OPTION VALUE • Effect of Stock Volatility • The higher , the higher the value of a call. • Why? • No downside risk. • Maximum loss on a call? • Maximum loss on a stock?

• Higher  increases chances of an out-of-money option to become in-the-money.

PUTS

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PUT OPTION BOUNDARIES • Absolute Minimum Value of a Put = $0 • Pe(S0,T,X)  Max{0, PV(X) - S0} • Pa(S0,T,X)  Max{0, X - S0}

Put Price

Intrinsic / Exercise/ Parity Value: SX X

0

if S0  X

X – S0

if S0 < X

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PUT OPTION BOUNDARIES • Example of Arbitrage • Consider 2 portfolios, A & B • Payoff for A • Short European Put • Bond worth X(1+r)-T

ST < X - (X- ST) X

ST  X 0 X

• Payoff for B • Share

ST < X ST

ST  X ST

• As Portfolio A  Portfolio B then PV(X) - Pe  S • Pe  PV(X) - S

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PUT OPTION BOUNDARIES • ANZ Jun Put Contract($15.74, (86/365), $16.0)  $0.60 • What is its intrinsic value? • $0.26. This would have to be the minimum value for Pa • What’s the minimum value for a Pe? • {$16.00(1.05)

–0.2356

- $15.74} ~ (PV(X)-S) = $0.077

• Why is Pa = $0.60 and not $0.26? • Time Value / Speculative Value ~ reflects uncertainty of stock price at expiration • The longer till expiration, the higher the time value • Time Value = Pa(S0,T,X)

- Max(0,X-S0) = $0.60 - $0.26 = $0.34 for ANZ

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PUT OPTION BOUNDARIES • Absolute Maximum Value of a Put = $X • Pe(S0,T,X)  PV(X) • Pa(S0,T,X)  X

X for a Pa

Put Price

PV(X) for a Pe Stock Price (S0)

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PUT OPTION BOUNDARIES • What is the value of an option at expiration? • P(ST,T,X) = Max(0, X- ST) Max(0,X -ST) Put Price

X Stock Price at expiration (ST)

• Is this appropriate for both European and American Options? • Yes

FACTORS DETERMINING PUT OPTION VALUE

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• The Effect of Time to Expiration • Two American puts differing only by time to expiration, T1 and T2 where T1 < T2. • Pa2(S0,T2,X)  Pa1(S0,T1,X) • At T1 Pa1 = Max(0, X - S1) whereas this will be the minimum for Pa2 • Due to Time value of the put option. • Time value maximized when at-the-money • Minimized when deep in-/out-of the money. • Time value decays closer to expiration

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FACTORS DETERMINING PUT OPTION VALUE American Put Option Value Relative to Time Put Price As you reach the termination date, the put price curve approaches the intrinsic value.

PV(X)

Max(0,X – S0) Stock Price (S0)

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FACTORS DETERMINING PUT OPTION VALUE Example: ANZ Last Sale Price $15.74 Series

Ex Price

Bid

Ask

Last Sale

Vol 000’s

Open Int

Implied Buyer

Volatility Seller

Delta

Annual % Return

Jun Oct

16.00 16.00

.55 .66

.65 .78

.57 .76

105 26

2 11

16.74 17.03

19.92 20.01

-0.52 -0.49

9.30 7.33

Ceteris paribus, an American put option is worth less when expiration is closer.

FACTORS DETERMINING PUT OPTION VALUE

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• The Effect of Exercise Price • Two European puts differing only by strikes of X1 and X2 (where X1 < X2). • Which is greater, Pe1(S0,T,X1) or Pe2(S0,T,X2)? • Difference in put price will not be greater than the difference in the exercise prices: • Consider 2 portfolios, A & B • • •

• •

Payoff for A Write Pe1(S0,T,X1) Take Pe2(S0,T,X2)

ST < X1 -X1 + ST X2 -ST

X1  ST < X2 0 X2 -ST

SUM

X2 – X1>0

X2-ST>0

Payoff for B Buy Bond (X2 – X1) (1+r)-T

ST < X1 X2 – X1> 0

X1  ST < X2 X2 – X1> 0

Therefore, portfolio B  portfolio A

X1 < X2  ST 0 0

X1 < X2  ST X2 – X1> 0

FACTORS DETERMINING PUT OPTION VALUE

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Example: ANZ Last Sale Price $15.74 Series

Ex Price

Bid

Ask

Last Sale

Vol 000’s

Open Int

Implied Buyer

Volatility Seller

Delta

Annual % Return

Aug Aug

16.00 15.50

.55 .32

.65 .40

.57 .35

105 105

2 1

16.74 16.87

19.92 19.55

-0.52 -0.37

9.30 9.70

Ceteris paribus, a put option is worth less the lower the exercise price.

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FACTORS DETERMINING PUT OPTION VALUE • Limits on the Difference in Premiums • We know that • (X2 - X1)(1+r)-T  Pe(S0,T,X2) - Pe(S0,T,X1)

• Therefore: • X2 - X1 Pe(S0,T,X2) - Pe(S0,T,X1) • X2 - X1 Pa(S0,T,X2) - Pa(S0,T,X1) • The cost from buying a put with a higher exercise price will not be greater than the difference in exercise prices. • Example: ANZ $16.00 put costs $0.60. ANZ $15.50 put costs $0.36. Therefore, • (16 – 15.5) > (0.60 – 0.36)

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FACTORS DETERMINING PUT OPTION VALUE

• Effect of Dividend Payments • Stock Value now equals S0 – PV(Div) • Therefore: • PV(X)  Pe(S0,T,X)  Max{0, PV(X) – [S0 – PV(Div)]}

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FACTORS DETERMINING PUT OPTION VALUE

• The option to exercise early • Are American puts worth more than European? • Pa(S0,T,X)  Pe(S0,T,X) • There is always a sufficiently low stock price that will make it optimal to exercise an American put early. • Dividends on the stock reduce the likelihood of early exercise.

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FACTORS DETERMINING PUT OPTION VALUE

• Effect of Interest Rates • Puts delay the sale of a stock • Rather than selling stock now, you do so in the future at a cost of r. • Higher r, the lower the value for a put.

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FACTORS DETERMINING PUT OPTION VALUE

• Effect of Stock Volatility • The higher , the higher the value of a put. • Why? • No downside risk. • Maximum loss on a put?

• Higher  increases chances of an out-of-money option to become in-the-money.

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PUT-CALL PARITY Put-Call Parity Call

Put

Stock

Risk-Free Bond

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PUT-CALL PARITY • Consider 2 portfolios, A & B • Payoff for A • Take European Put • Buy Share

• Payoff for B

ST  X X – ST ST

ST  X

• Take European Call • Buy Bond worth X(1+r)-T

• S0 + Pe(S0,T,X) = Ce(S0,T,X) + X(1+r)-T

0 X

ST > X 0 ST

ST > X ST - X X

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PUT-CALL PARITY • Other common expressions: • Pe(S0,T,X) = Ce(S0,T,X) + X(1+r)-T - S0 • S0 + Pe(S0,T,X) - X(1+r)-T = Ce(S0,T,X)

• What about American Options? • Ca(Ś0, T, X) + X + PV(Div) ≥ • S0 + Pa(Ś0, T, X) ≥ • Ca(Ś0, T, X) + X(1+r)-T

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PUT-CALL PARITY • Example 1

• ANZ Put($15.74, (86/365), $16.0)  $0.60 • ANZ Call($15.74, (86/365), $16.0)  $0.52 • 30 Day Bank Swap Rate = 5% • Why is Put > Call price? • Does Put-Call Parity hold? • Call + Bond[$X]  Put + Share  Call + PV(Bond[$X]) • 0.52 + 16  0.60 + 15.74  0.52 + 15.82 • 16.52  16.34  16.34 • Even if the above does not hold: • incorrect risk-free rate! • no account for dividend payments!

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PUT-CALL PARITY • Example 2 • How would you show an at-the-money call option must be worth more than an at-the-money put option on the same stock with similar maturities (no dividends)? • C = P + S - PV(X) • As the options are at-the-money, S=X • C=P+r

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ELEMENTARY PRICING OF A CALL • The example below tries to look at a simple way we can price options. To help us out, let’s make some assumptions: • 1. A share is currently priced at $10 • 2. To make things simple, we assume that in one year the share will be worth either $11 or $13 • 3. The risk free rate of interest (rf) is 12% per annum. • 4. You can buy a call option that also expires in a year, with an exercise price of $12 What is the Price of the Call?

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ELEMENTARY PRICING OF A CALL • To Solve this problem, lets try to replicate the payoff of a call by holding a portfolio of bonds and stocks…. • 1. Buy a risk free asset worth the present value of the expected minimum share value in one year {$11} : 11/(1+rf) = $9.82 • 2. Now, our minimum payoff in one year’s time is $11 • 3. If, however, the share in one year is worth $13, the payoff of one call option is only $1 { $13 - $12 } • 4. Therefore, buying 2 call options {payoff = $2} plus buying a risk free asset with a payoff of $11, will yield the same result as owning the share • Share Value rf asset value • $11 $11 $13 $11 • S0 = 2 * C0 + $11 / ( 1.12) • C0 = $0.09

Value of 2 Calls Total $0 $11 $2 $13...