Plan for Today. Properties of Options. Put-Call Parity. Early Exercise. Binomial Trees. Technical Analysis II. RSM 6310, Class 3, Kevin Wang

(1)

Plan for Today

RSM 6310, Class 3, Kevin Wang

————————————

B Properties of Options B Put-Call Parity

B Early Exercise B Binomial Trees

B Technical Analysis II

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1

(2)

Properties of Stock Options

—————————————————————————————

Notations

Notation

Options, Futures, and Other Derivatives, 8th Edition,

c: European call option price

p: European put option price

S₀: Stock price today K: Strike price

T: Life of option

: Volatility of stock price

C: American call option price

P: American put option price

S_T: Stock price at option maturity

D: PV of dividends paid during life of option r Risk-free rate for

maturity T with cont.

comp.

2

(3)

Properties of Stock Options

—————————————————————————————

Notations

Notation

T: Life of option

comp.

2

(4)

Properties of Stock Options

—————————————————————————————

Notations

Notation

T: Life of option

comp.

2

(5)

Properties of Stock Options

—————————————————————————————

Notations

Notation

T: Life of option

comp.

2

(6)

Properties of Stock Options

—————————————————————————————

Notations

Notation

T: Life of option

comp.

2

(7)

Time Value Pattern

• Option price/premium is a sum of intrinsic value and time value.

(8)

Time Value Pattern

• Intrinsic value:

− For a call option, the intrinsic value is max(S₀ − K, 0).

− For a put option, the intrinsic value is max(K − S0, 0).

(9)

Time Value Pattern

• Intrinsic value:

− For a call option, the intrinsic value is max(S₀ − K, 0).

− For a put option, the intrinsic value is max(K − S0, 0).

• Time value:

In general, the time value of a given option decreases over time when there is no change in the stock price.

3

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Example 1: Facebook 2019 Jan 18 call options

S

₀

= $150 on November 2, 2018

Strike Option price Intrinsic value Time value

135 18.90 15 3.90

140 15.11 10 5.11

145 11.65 5 6.65

150 8.75 0 8.75

155 6.25 0 6.25

160 4.42 0 4.42

165 2.95 0 2.95

4

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Example 1: Facebook 2019 Jan 18 call options

S

₀

= $150 on November 2, 2018

135 18.90 15 3.90

140 15.11 10 5.11

145 11.65 5 6.65

150 8.75 0 8.75

155 6.25 0 6.25

160 4.42 0 4.42

165 2.95 0 2.95

4

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Example 1: Facebook 2019 Jan 18 call options

S

₀

= $150 on November 2, 2018

135 18.90 15 3.90

140 15.11 10 5.11

145 11.65 5 6.65

150 8.75 0 8.75

155 6.25 0 6.25

160 4.42 0 4.42

165 2.95 0 2.95

4

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Example 2: Tesla post-earnings vertical spread

S

₀

= 250 on August 8, 2014

Tesla ^- S₀ = 250 on August 8, 2014

Symbol Open qty Avg price Date/time

TSLA Sep 20 2014 230 Call TSLA Sep 20 2014 250 Call

9 1 24.51 08/08/2014 2:52:06PM

9 (1) 12.31 08/08/2014 2:52:07PM

StrikeOptionprice Intrinsicvalue TimeValue

230 24.51 20 4.51

250 12.31 0 12.31

Difference 12.2012.20 Ͳ7.80Ͳ 7.80

9 1 24.51 08/08/2014 2:52:06PM

9 (1) 12.31 08/08/2014 2:52:07PM

230 24.51 20 4.51

250 12.31 0 12.31

Difference 12.2012.20 Ͳ 7.80Ͳ7.80

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Example 2: Tesla post-earnings vertical spread

S

₀

= 250 on August 8, 2014

9 1 24.51 08/08/2014 2:52:06PM

9 (1) 12.31 08/08/2014 2:52:07PM

230 24.51 20 4.51

250 12.31 0 12.31

Difference 12.2012.20 Ͳ7.80Ͳ 7.80

9 1 24.51 08/08/2014 2:52:06PM

9 (1) 12.31 08/08/2014 2:52:07PM

230 24.51 20 4.51

250 12.31 0 12.31

Difference 12.2012.20 Ͳ 7.80Ͳ7.80

2014 Nov 22 Call Options on Google

S0 = 599 on September 24, 2014

Strike Option price Intrinsic value Time Value

585 28.25 14 14.25

590 25.50 9 16.50

595 22.50 4 18.50

600 19.85 0 19.85

605 17.60 0 17.60

610 15.35 0 15.35

615 13.20 0 13.20

S0 = 250 on August 8, 2014

Strike Option price Intrinsic value Time Value

230 24.51 20 4.51

250 12.31 0 12.31

Difference 12.20 ‐ 7.80

⁵

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Effect of variables on option pricing

Effect of Variables on Option Pricing

25

c p C P

S

₀

K T 

r D

+ + –

? + ? + +

+ + + + + – + – – – – +

– + – +

6

(16)

Effect of variables on option pricing

Effect of Variables on Option Pricing

25

c p C P

S

₀

K T 

r D

+ + –

? + ? + +

+ + + + + – + – – – – +

– + – +

6

(17)

Effect of variables on option pricing

Effect of Variables on Option Pricing

25

c p C P

S

₀

K T 

r D

+ + –

? + ? + +

+ + + + + – + – – – – +

– + – +

6

(18)

Effect of variables on option pricing

Effect of Variables on Option Pricing

25

c p C P

S

₀

K T 

r D

+ + –

? + ? + +

+ + + + + – + – – – – +

– + – +

6

(19)

Dividend

S

₀ ^u@ @

@

@ @

@

@ @ @

H H A

A @

@ @

Div

@ @ @

@

@ @

@

@ @

S

_T

u

u u u

0 1 2

u

A A A K

ex-dividend

(20)

Effect of variables on option pricing

Effect of Variables on Option Pricing

25

c p C P

S

₀

K T 

r D

+ + –

? + ? + +

+ + + + + – + – – – – +

– + – +

6

(21)

Effect of variables on option pricing

Effect of Variables on Option Pricing

25

c p C P

S

₀

K T 

r D

+ + –

? + ? + +

+ + + + + – + – – – – +

– + – +

6

(22)

Effect of variables on option pricing

Effect of Variables on Option Pricing

25

c p C P

S

₀

K T 

r D

+ + –

? + ? + +

+ + + + + – + – – – – +

– + – +

6

(23)

Put-Call Parity

• We focus on the case of no dividends.

− Portfolio LHS: European call + zero-coupon bond paying K at T

− Portfolio RHS: European put + the stock

• Both are worth max(ST, K) at the options’ maturity.

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Put-Call Parity

• We focus on the case of no dividends.

− Portfolio LHS: European call + zero-coupon bond paying K at T

− Portfolio RHS: European put + the stock

• Both are worth max(ST, K) at the options’ maturity.

• They must therefore be worth the same today. So c + Ke^−rT = p + S0

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Values at expiration

S_T ≥ K S_T < K Portfolio LHS: Call option ST − K 0

Zero-coupon bond K K

Total ST K

(26)

Values at expiration

S_T ≥ K S_T < K Portfolio LHS: Call option ST − K 0

Zero-coupon bond K K

Total ST K

Portfolio RHS: Put option 0 K − ST

Share of Stock ST ST

Total ST K

8

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Applications of the parity

• In applications, one can manipulate the equation:

c + PV(K) = p + S₀.

− The parity is useful for understanding relations among various option positions.

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Applications of the parity

• In applications, one can manipulate the equation:

c + PV(K) = p + S₀.

− The parity is useful for understanding relations among various option positions.

• Create synthetic (long or short) stock positions

• Compare option trading strategies - Equivalences exist among various spreads and combinations.

9

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Equivalence

• A synthetic long stock position can be created:

S0 = c − p + PV(K).

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Equivalence

• A synthetic long stock position can be created:

S0 = c − p + PV(K).

• Equivalent relations exist among various positions.

− Selling a put:

−p = S0 − c − PV(K)

− A covered call is equivalent to a naked short put.

10

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Writing an out-of-the-money AAPL put

(For the CNBC video clip)

- 6

ST

Profit

u

490

u

K

515

u

S0 520

25

6

?

11

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Bull call vertical vs. bull put vertical

c1 + PV(K1) = p1 + S0

c2 + PV(K2) = p2 + S0 (K1 < K2)

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Bull call vertical vs. bull put vertical

c1 + PV(K1) = p1 + S0

c2 + PV(K2) = p2 + S0 (K1 < K2)

⇒ c1 − c2 = (p1 − p2) + PV(K2 − K1)

Note again we are looking at payoffs at the end, not the premium or the cost at the beginning.

12

(34)

Bull Call Vertical vs. Bull Put Vertical

(Ignoring the initial costs)

6

ST

Payoff

u

K₁

u

K₂

13

(35)

Equivalence between call and put verticals

Bull vertical Bear vertical

Call spread C₁ − C₂ C₂ − C₁

(debit) (credit)

Put spread P₁ − P₂ P₂ − P₁

(credit) (debit)

K1 < K2. The strike of Ci or Pi is Ki.

14

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Equivalence between call and put verticals

Call spread C₁ − C₂ C₂ − C₁

(debit) (credit)

Put spread P₁ − P₂ P₂ − P₁

(credit) (debit)

K1 < K2. The strike of Ci or Pi is Ki.

14

(37)

Back to Example 2 - Tesla

S = 263 on Sept. 18, 2014

Orders

Orders PositionsPositions Executions BalancesBalances ActivityActivity InboxInbox

 

TSLA Sep 20 2014 230 Call TSLA Sep 20 2014 250 Call AAPL Sep 20 2014 92.5 Call AAPL Sep 20 2014 100 Call YHOO Oct 18 2014 40 Call YHOO Oct 18 2014 43 Call

Symbol Strategy Fill qty Fill price Time placed Currency Sec type Total value Description

Vertical call 1 34.78 09/18/2014 2:20:19PM USD Option 3,478.00 TESLA MOTORS INC Vertical call 1 14.87 09/18/2014 2:20:19PM USD Option 1,487.00 TESLA MOTORS INC Vertical call 2 9.37 09/18/2014 1:39:00PM USD Option 1,874.00 APPLE INC

Vertical call 2 1.94 09/18/2014 1:39:00PM USD Option 388.00 APPLE INC Vertical call 5 4.28 09/12/2014 2:05:42PM USD Option 2,140.00 YAHOO INC Vertical call 5 2.75 09/12/2014 2:05:42PM USD Option 1,375.00 YAHOO INC Account: 26662730 - Margin power

WATCH LIST CHART OPTIONS ACCOUNT RESEARCH QUOTE BUY/SELL

Snap As of 2:23:07PM

Account # 26662730 - Marg… ON OFF Logout

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18/09/2014 https://iqessential.questrade.com/

• The trade was closed out before the expiration.

( )

(38)

Back to Example 2 - Tesla

S = 263 on Sept. 18, 2014

Orders

Orders PositionsPositions Executions BalancesBalances ActivityActivity InboxInbox

 

TSLA Sep 20 2014 230 Call TSLA Sep 20 2014 250 Call AAPL Sep 20 2014 92.5 Call AAPL Sep 20 2014 100 Call YHOO Oct 18 2014 40 Call YHOO Oct 18 2014 43 Call

Symbol Strategy Fill qty Fill price Time placed Currency Sec type Total value Description

Vertical call 1 34.78 09/18/2014 2:20:19PM USD Option 3,478.00 TESLA MOTORS INC Vertical call 1 14.87 09/18/2014 2:20:19PM USD Option 1,487.00 TESLA MOTORS INC Vertical call 2 9.37 09/18/2014 1:39:00PM USD Option 1,874.00 APPLE INC

Vertical call 2 1.94 09/18/2014 1:39:00PM USD Option 388.00 APPLE INC Vertical call 5 4.28 09/12/2014 2:05:42PM USD Option 2,140.00 YAHOO INC Vertical call 5 2.75 09/12/2014 2:05:42PM USD Option 1,375.00 YAHOO INC Account: 26662730 - Margin power

WATCH LIST CHART OPTIONS ACCOUNT RESEARCH QUOTE BUY/SELL

Snap As of 2:23:07PM

Account # 26662730 - Marg… ON OFF Logout

Page 1 of 1 IQ Essential

18/09/2014 https://iqessential.questrade.com/

• The trade was closed out before the expiration.

• The Tesla trade above, a debit bull call vertical, is equivalent to the credit put vertical trade:

− Long the 230 put and short the 250 put.

15

( )

(39)

Equivalence between Condor and Iron Condor

Condor C₁ − C₂ −C₃ + C₄

(debit) (credit)

Iron Condor P₁ − P₂ −C₃ + C₄

(credit) (credit)

K1 < K2 < K3 < K4. The strike of Ci or Pi is Ki.

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Equivalence between Condor and Iron Condor

Condor C₁ − C₂ −C₃ + C₄

(debit) (credit)

Iron Condor P₁ − P₂ −C₃ + C₄

(credit) (credit)

K1 < K2 < K3 < K4. The strike of Ci or Pi is Ki.

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Bounds for option prices

Example: Consider an in-the-money European call on a stock that does not pay any dividend over the life of the option.

• The intrinsic value of the call is S0 − K.

• The time value of the call is c − (S0 − K).

• Is the time value of the call always positive?

What if the call is American?

17

(42)

c + Ke

^−rT

= p + S

₀

(Put-Call Parity)

⇒ c = p + S

₀

− Ke

^−rT

c − (S

₀

− K) = p + K 1 − e

^−rT

(43)

c + Ke

^−rT

= p + S

₀

(Put-Call Parity)

⇒ c = p + S

₀

− Ke

^−rT

c − (S

₀

− K) = p + K 1 − e

^−rT

≥ K 1 − e

^−rT

> 0.

The time value has a positive lower bound.

(44)

Early Exercise

• Usually there is some chance that an American option will be exercised early.

• An exception is an American call on a non-dividend paying stock.

• Two questions:

− The call should never be exercised early. Why?

− However, that does not apply to a put. Why not?

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Example

• For an American call option:

S0 = 100; T = 0.25; K = 60; D = 0

• The call is quite deep ITM. Should you exercise now?

• What should you do in each of the two cases below.

− You are bullish and predict that the stock price will increase for the next 3 months.

− You are bearish and do not feel that the stock price will run well for the next 3 months.

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Reasons for not exercising a call early (in case of no dividends)

• No dividend income is sacrificed.

• Payment of the strike price is delayed.

• Holding the call provides insurance against stock price falling below strike price.

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Reasons for not exercising a call early (in case of no dividends)

• No dividend income is sacrificed.

• Payment of the strike price is delayed.

• Holding the call provides insurance against stock price falling below strike price.

• Time value of the call is positive - if you want to get rid of the call, you should sell it, rather than exercise it.

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Same for American puts?

• A numerical illustration:

S₀ = 60; T = 0.25; r = 10% K = 100; D = 0

(49)

Same for American puts?

• A numerical illustration:

S₀ = 60; T = 0.25; r = 10% K = 100; D = 0

• Early exercise of an American put is possibly optimal even if there is no dividend.

• As the stock price falls, early exercise of an American put becomes more attractive. When the stock price is low enough, early exercise becomes optimal. Why?

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Binomial Trees

A numerical example in a one-step tree:

w

1 PP

PP PP

PP PPq

Stock price = $20

Stock price = $22

Stock price = $18

A stock price is currently $20.

In 3 months it will be either $22 or $18.

22

(51)

Option pricing in a one-step tree

How to calculate the price of a call option?

w

1 PP

PP PP

PP PPq

Stock price = $20

Stock price = $22 Option price = $1

Stock price = $18 Option price = $0

A 3-month call option on the stock has a strike price of 21.

23

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Setting up a riskless portfolio

A portfolio that is long ∆ shares and short 1 call option:

w

1 PP

PP PP

PPP P q

22∆ − 1

18∆

(53)

Setting up a riskless portfolio

A portfolio that is long ∆ shares and short 1 call option:

w

1 PP

PP PP

PPP P q

22∆ − 1

18∆

The portfolio is a riskless bond when

22∆ − 1 = 18∆ ⇒ ∆ = 0.25.

24

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Valuing the riskless bond

• The synthetic portfolio, long 0.25 shares and short 1 call option, is simply a riskless bond.

• The value of the portfolio in 3 months is:

B = 22 × 0.25 − 1.00 = $4.50.

(B is equivalent to the face value of a long bond.)

• The value of the portfolio today is (r = 0.12):

Be^−rT = 4.50e^{−0.12×0.25} = $4.367.

25

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Valuing the option

• The synthetic bond, long 0.25 shares and short 1 call option, is worth

$4.367.

• The value of ∆ shares of the stock is:

$5.00 (= 0.25 × $20).

(56)

Valuing the option

• The synthetic bond, long 0.25 shares and short 1 call option, is worth

$4.367.

• The value of ∆ shares of the stock is:

$5.00 (= 0.25 × $20).

• The call is a mix of the stock and the bond.

The call’s value = 5.00 − 4.367 = $0.633.

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Given the tree of one stock, can we apply this riskless

portfolio approach to price another stock?

(58)

Given the tree of one stock, can we apply this riskless portfolio approach to price another stock?

r

* HH

HH HH

HH H

HH H j

Stock A Call

22

18

1

0

(59)

Given the tree of one stock, can we apply this riskless portfolio approach to price another stock?

r

* HH

HH HH

HH H

HH H j

Stock A Call

22

18

1

0

r1 PPP

PP PPq

r1 PP

PP PPPq

Stock B

50 45 50 45

(60)

Generalization

How to price a derivative in a one-step tree?

w

1 PP

PP PP

PP PPq

Stock price = S0

Option price = f

Stock price = Su

Option price = fu

Stock price = Sd

Option price = fd

A derivative has payoffs that are dependent on the stock price.

27

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Setting up a riskless portfolio

A portfolio that is long ∆ shares and short 1 derivative:

w

1 PP

PP PP

PP PPq

S_u∆ − f_u

Sd∆ − fd

(62)

Setting up a riskless portfolio

A portfolio that is long ∆ shares and short 1 derivative:

w

1 PP

PP PP

PP PPq

S_u∆ − f_u

Sd∆ − fd

The portfolio is riskless when

S_u∆ − f_u = S_d∆ − f_d ⇒ ∆ = fu − fd

Su − Sd

.

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Valuing the derivative

• The face value of the synthetic bond at time T:

B = S_u∆ − f_u = S_df_u − S_uf_d Su − Sd

.

• The value of the riskless bond today is Be^−rT = S0∆ − f.

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Valuing the derivative

• The face value of the synthetic bond at time T:

B = S_u∆ − f_u = S_df_u − S_uf_d Su − Sd

.

• The value of the riskless bond today is Be^−rT = S0∆ − f.

• The derivative is a mix of the stock and the bond.

f = S0∆ − Be^−rT.

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An alternative approach

Substituting for ∆ from

∆ = f_u − f_d Su − Sd

,

(66)

An alternative approach

Substituting for ∆ from

∆ = f_u − f_d Su − Sd

, we obtain

f = [pf_u + (1 − p)f_d]e^−rT where, let Su = S₀u, S_d = S₀d,

p = S0e^rT − Sd

Su − Sd

= e^rT − d u − d .

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Treat p as a probability

What if we interpret p and 1 − p as the up and down probabilities?

w

1 PP

PP PP

PP PPq

p

1 − p S₀

f

S₀u f_u

S0d f_d

Then the value of a derivative is its expected payoff discounted at the risk-free rate. This is known as risk-neutral valuation.

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Risk-neutral valuation

• What is a risk-neutral world? Risk-neutral pricing?

(69)

Risk-neutral valuation

• Binomial trees give us a general result:

− For pricing an option, we can proceed as if we are in a risk-neutral world. We discount the expected payoff from the option by the risk-free rate.

(70)

Risk-neutral valuation

• Binomial trees give us a general result:

− For pricing an option, we can proceed as if we are in a risk-neutral world. We discount the expected payoff from the option by the risk-free rate.

• On the other hand, for the underlying asset (stock), the risk-neutral probability p should ensure that the expected stock price at time T is equal to S0e^rT.

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Back to the numerical example

w

1 PP

PP PP

PP PPq

p

1 − p S₀ = 20

f =?

S₀u = 22 f_u = 1

S0d = 18 f_d = 0

(72)

Back to the numerical example

w

1 PP

PP PP

PP PPq

p

1 − p S₀ = 20

f =?

S₀u = 22 f_u = 1

S0d = 18 f_d = 0

Let the expected stock return be equal to the risk-free rate:

20e^0.12×0.25 = 22p + 18(1 − p) ⇒ p = 0.6523.

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(73)

Alternatively, we can directly use the formula for p:

p = e^rT − d

u − d = e^0.12×0.25 − 0.9

1.1 − 0.9 = 0.6523.

(74)

Alternatively, we can directly use the formula for p:

p = e^rT − d

u − d = e^0.12×0.25 − 0.9

1.1 − 0.9 = 0.6523.

Now applying the risk-neutral valuation approach:

Since p = 0.6523 and 1 − p = 0.3477, we have f = [pf_u + (1 − p)f_d]e^−rT

= (0.6523 × 1 + 0.3477 × 0)e^{−0.12×0.25}

= 0.633.

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Irrelevance of the stock’s expected return

• When we are valuing an option in terms of the price of the underlying asset, the probabilities of up & down movements in the real world are irrelevant.

• This is an example of a general result for option pricing that the expected return on the underlying asset in the real world is irrelevant. Why?

(76)

Irrelevance of the stock’s expected return

• When we are valuing an option in terms of the price of the underlying asset, the probabilities of up & down movements in the real world are irrelevant.

• This is an example of a general result for option pricing that the expected return on the underlying asset in the real world is irrelevant. Why?

• In general, how could it be alright to assume that we are in a risk-neutral world when pricing derivatives?

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Two-step trees

A 20

w1 PP

PP PP

PP PPq

B 22

w1 PP

PP PP

PP PPq

C 18

w

1 PP

PP PP

PP PPq

D 24.2

w

E 19.8

w

F 16.2

w

• Each time step is 3 months and r = 12%.

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(78)

Example 1: Valuing a call (K = 21)

A 20

w

1.2822

1 PP

PP PP

PP PPq

B 22

w

2.0256

1 PP

PP PP

PP PPq

C 18

w

0.0

1 PP

PP PP

PP PPq

D 24.2

w

3.2

E 19.8

w

0.0

F 16.2

w

0.0

(79)

A 20

w

1.2822

1 PP

PP PP

PP PPq

B 22

w

2.0256

1 PP

PP PP

PP PPq

C 18

w

0.0

1 PP

PP PP

PP PPq

D 24.2

w

3.2

E 19.8

w

0.0

F 16.2

w

• Value at node B 0.0

= e^{−0.12×0.25}(0.6523 × 3.2 + 0.3477 × 0) = 2.0256

(80)

A 20

w

1.2822

1 PP

PP PP

PP PPq

B 22

w

2.0256

1 PP

PP PP

PP PPq

C 18

w

0.0

1 PP

PP PP

PP PPq

D 24.2

w

3.2

E 19.8

w

0.0

F 16.2

w

• Value at node B 0.0

= e^{−0.12×0.25}(0.6523 × 3.2 + 0.3477 × 0) = 2.0256

• Value at node A

= e^{−0.12×0.25}(0.6523 × 2.0256 + 0.3477 × 0) = 1.2822

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Example 2: Valuing a European put (K = 21)

A 20

w

1.0592

1 PP

PP PP

PP PPq

B 22

w

0.4049

1 PP

PP PP

PP PPq

C 18

w

2.3794

1 PP

PP PP

PP PPq

D 24.2

w

0.0

E 19.8

w

1.2

F 16.2

w

4.8

• K = 21, each time step = 3 months

• r = 12%, u = 1.1, d = 0.9, p = 0.6523

38

(82)

What if this is an American put?

A 20

w

1.2687

1 PP

PP PP

PP PPq

B 22

w

0.4049

1 PP

PP PP

PP PPq

C 18

w

3.0000

1 PP

PP PP

PP PPP

PP PP

PPPq

D 24.2

w

0.0

E 19.8

w

1.2

F 16.2

w

4.8

• Value at node C increases from 2.3794 to 3.

• Value at node A increases from 1.0592 to 1.2687.

39

(83)

Example 3 on two-step trees

• The current stock price is $80.

Over each of the next two three-month periods, the stock price is expected either to rise by 10 percent or fall by 10 percent.

The risk-free rate, expressed with continuous compounding, is 12 percent per annum.

• Price a 6-month European put option with an exercise price of

$80 using the risk-neutral valuation method.

• If instead it were an American option, what would its price be?

40