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Option Pricing

Option Pricing. Junya Namai. Agenda. Current Option Price for Netflix Binomial Model for Stock Binomial Options P ricing for Call Option Binomial Options P ricing for Put Option Binomial Options P ricing for Call Option – Multi period Black-Scholes Model Quiz Questions.

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Option Pricing

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  1. Option Pricing JunyaNamai

  2. Agenda • Current Option Price for Netflix • Binomial Model for Stock • Binomial Options Pricing for Call Option • Binomial Options Pricing for Put Option • Binomial Options Pricing for Call Option – Multi period • Black-Scholes Model • Quiz • Questions

  3. Current Option Price for Netflix • http://finance.yahoo.com/q/op?s=NFLX&m=2013-05

  4. Binomial Model for Stock t0 t1 $80 P(u) = 0.6 up P(d) = 0.4 down $55 r = 0.08 = = $64.81

  5. Binomial options pricing for Call Option K = $70 t0 t1 Max(0, Price - K) P(u) = 0.6 $10 $80 P(d) = 0.4 up r = 0.08 down $55 $0 = = $5.556

  6. Binomial options pricing for Put Option K = $70 t0 t1 Max(0, K-Price) P(u) = 0.6 $0 $80 P(d) = 0.4 up r = 0.08 down $55 $15 = = $5.556

  7. Call Option - Multi Period t0 t1 t2 t3 t4 Max(0, Price-K) $20 $90 0.6 0.6 0.4 $10 $80 0.6 0.6 0.4 0.6 0.6 0.4 0.4 $70 $0 0.4 0.6 0.4 0.6 0.4 0.6 0.4 $0 K = $70 $60 0.6 P(u) = 0.6 0.4 r = 0.08 P(d) = 0.4 0.4 $0 $50

  8. Call Option - Multi Period Path call t4 1 4ups $90 $20 4 3ups + 1down $80 $10 6 $0 2ups + 2downs $70 4 $60 $0 3downs + 1up 1 $0 $50 4downs

  9. Call Option - Multi Period

  10. Binomial Distribution (Pascal's triangle)

  11. Black-Scholes Formula (5 parameters) • Stock Price • Exercise (Strike) Price • Time to Expiration • Volatility of Stock • Risk-Free Rate

  12. Black-Scholes Formula • Value of call option = • cumulative normal probability density function • = exercise price of option; PV(EX) is calculated by discounting at the risk-free interest rate rf • t = number of periods to exercise date • P = price of stock now • = standard deviation per period of (continuously compounded) rate of return on stock

  13. Black-Scholes Formula • P=430, EX=430, =0.4068, t=0.5(6 months), rf=.05 • = = 0.1956 • = 0.195 – 0.4068 = -0.0921 • = N(-0.0921) = 1-N(0.0921) = 0.4633 • Use Normsdist function in Excel • = 0.5775430 – (0.4633430/1.015) = 52.04 • $52.04

  14. Binomial vs Black-Scholes • Binomial • Flexible • Finitesteps • Discrete • Values American • Values complexities • Black-Scholes • Limited • Infinite • Continuous

  15. Quick Quiz 1 • If volatility of stock price becomes higher, does the option price go up or down? • Black-Scholes Calculator

  16. Lognormal Distribution

  17. Quick Quiz 2 • If interest rates becomes higher, does the option price go up or down?

  18. Question

  19. Reference • http://stattrek.com/probability-distributions/binomial.aspx • http://en.wikipedia.org/wiki/Binomial_distribution • http://www.tradingtoday.com/black-scholes?callorput=c&strike=70&stock=70&time=180&volatility=48&interest=8 • http://en.wikipedia.org/wiki/Binomial_options_pricing_model • http://www.optiontradingpedia.com/free_black_scholes_model.htm • http://www.optiontradingpedia.com/free_black_scholes_model.htm • http://easycalculation.com/statistics/binomial-distribution.php • http://www.hoadley.net/options/bs.htm

  20. Risk-Neutral Valuation (Backup) Expected return rf = 1.5% Expected return 1.5 = 33p - 25(1-p) 1.5 = 33p + 25p -25 p = 45.6%

  21. Risk-Neutral Valuation (Backup)

  22. Risk-Neutral Valuation (Backup)

  23. Up and Down Changes to STD • 1+upside change = u = • 1+downside change = d = 1/u • e = 2.718 • = standard deviation of stock returns • h = interval as fraction of a year • To find the standard deviation given u, we turn the formula around

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