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VIII: Options 26: Options Pricing. Chapter 26: Options Pricing © Oltheten & Waspi 2012 Options Pricing Models  Binomial Model  Black Scholes Options.

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Presentation on theme: "VIII: Options 26: Options Pricing. Chapter 26: Options Pricing © Oltheten & Waspi 2012 Options Pricing Models  Binomial Model  Black Scholes Options."— Presentation transcript:

1 VIII: Options 26: Options Pricing

2 Chapter 26: Options Pricing © Oltheten & Waspi 2012 Options Pricing Models  Binomial Model  Black Scholes Options Pricing Model © Oltheten & Waspi 2012

3 Chapter 26: Options Pricing Binomial Model © Oltheten & Waspi 2012

4 Chapter 26: Options Pricing © Oltheten & Waspi 2012 Law of One Price  Two assets with the same risk and return characteristics will have the same price.  Arbitrage  Taking advantage of different prices in different markets of two assets of the same risk and return characteristics.

5 Chapter 26: Options Pricing Two Portfolios © Oltheten & Waspi 2012  Leveraged Stock  Shares of the underlying asset and debt  Option  Call option on the underlying asset Constructed to have the same risk and return characteristics Leveraged Stock Call

6 Chapter 26: Options Pricing Leveraged Stock © Oltheten & Waspi 2012 $50 Discovery Café $100 $25 P=½ R=3% Borrow $24.27 $25.73 Repay $25 $75 $0

7 Chapter 26: Options Pricing Option © Oltheten & Waspi 2012 C $75 Call Option $25 $0 P=½ $100 - $75 expire

8 Chapter 26: Options Pricing Binomial Model © Oltheten & Waspi 2012 $25.73 $75 $0 P=½ C $25 $0 P=½ Leveraged Stock $75 Call Option

9 Chapter 26: Options Pricing Multi-period Binomial Model $50 $59.46 $42.04 $70.71 $84.09 $100. $35.36 $29.73 $25 1 3 2 5 6 © Oltheten & Waspi 2012

10 Chapter 26: Options Pricing Black Scholes Option Pricing Model © Oltheten & Waspi 2012

11 Chapter 26: Options Pricing Black-Scholes Options Pricing Model © Oltheten & Waspi 2012

12 Chapter 26: Options Pricing Discovery Café © Oltheten & Waspi 2012

13 Chapter 26: Options Pricing © Oltheten & Waspi 2012 Example:  Discovery Café Call  P Exercise= $100  91 days to maturity  P stock = $90  σ=40%  r = 5% © Oltheten & Waspi 2012

14 Chapter 26: Options Pricing The first factor: d1 © Oltheten & Waspi 2012

15 Chapter 26: Options Pricing © Oltheten & Waspi 2012 Delta Hedge Ratio  N[d 1 ] = 0.35942  Delta Hedge Ratio = 36  If P Stock  $1 then P Call  $0.36  36 shares hedges 1 short call  Example: 36 shares *(+$1) = +$36 1 written call: -100 shares * (+$0.36) = -$36 Net effect = $0

16 Chapter 26: Options Pricing The second factor: d2 © Oltheten & Waspi 2012

17 Chapter 26: Options Pricing © Oltheten & Waspi 2012 In-the-Money  N[d2] = 0.28774  Probability (In-the-Money) = 28.8%  σ = 0.40 annual volatility  → σ√t = 0.40( √.25) = 0.20 over 91 days  20% of $90 = $18 volatility © Oltheten & Waspi 2012

18 Chapter 26: Options Pricing Discovery Café © Oltheten & Waspi 2012

19 Chapter 26: Options Pricing Black-Scholes Options Pricing Model © Oltheten & Waspi 2012

20 Chapter 26: Options Pricing Put Call Parity © Oltheten & Waspi 2012  Engineer two portfolios Put Portfolio Call Portfolio Same payout → same price

21 Chapter 26: Options Pricing © Oltheten & Waspi 2012 Put Call Parity  Put Portfolio  $100 Put Option  100 Shares DVC  Call Portfolio  $100 Call Option  NPV $10,000

22 Chapter 26: Options Pricing Put Portfolio © Oltheten & Waspi 2012 Put Exercised Put Expires

23 Chapter 26: Options Pricing Call Portfolio © Oltheten & Waspi 2012 Call Expires Call Exercised

24 Chapter 26: Options Pricing © Oltheten & Waspi 2012 Put Call Parity  Put Portfolio  $100 Put Option  100 Shares DVC  Call Portfolio  $100 Call Option  NPV $10,000

25 Chapter 26: Options Pricing © Oltheten & Waspi 2012 Put Call Parity  Put Portfolio  $100 Put Option  100 Shares DVC  Call Portfolio  $100 Call Option  NPV 100 shares P (Put) = p P (Shares) = $90 P (Call) = $3.93 NPV (Ex) = $100e - (0.05) (0.25)

26 Chapter 26: Options Pricing © Oltheten & Waspi 2012 Put Call Parity  P = $3.93 - $90 + $100 e -0.05*0.25  P =

27 Chapter 26: Options Pricing © Oltheten & Waspi 2012 Excel Spreadsheet  Built with Spreadsheet Exercise 26-1

28 Chapter 26: Options Pricing Implied Volatility © Oltheten & Waspi 2012 Black Scholes Options Pricing Model Volatility : σ Call Option Price Put Option Price Call Option Price Put Option Price Volatility : σ

29 Chapter 26: Options Pricing Volatility © Oltheten & Waspi 2012 Put Price Call Price

30 Chapter 26: Options Pricing Volatility Smile on GE Options © Oltheten & Waspi 2012

31 Chapter 26: Options Pricing © Oltheten & Waspi 2012 Exercise  26-5

32 Options IV

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