Option pricing model
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The document discusses option pricing models. It covers the Binomial model and the Black-Scholes model. The Black-Scholes model assumes the stock price follows a geometric Brownian motion and uses a partial differential equation to derive a closed-form solution for pricing European stock options. It requires parameters such as the stock price, exercise price, risk-free interest rate, time to maturity, and volatility. The model provides a theoretical fair value for options.
Option pricing model
- 1. . Prepared by: Vishal Jain MBA 3rd Sem. Option Pricing
- 2. The amount that an investor must pay for an option contract. This price is based upon factors such as the underlying security as well as the left until the options expiration date. The twopopular models are: 1. The Binomial Model 2. Black – Scholes Model
- 3. The model assumes, The price of asset can only go up or go down in fixed amounts in discrete time. There is no arbitrage between the option and the replicating portfolio composed of underlying asset and risk-less asset.
- 4. Current stock price = S Next Year values = uS or dS Bamount can be borrowed at ‘r’. Interest factor is (1+r) = R d< R < u(no risk free arbitrage possible) E is the exerciseprice
- 5. Depending on the change in stock value, option value will be Cu = Max (uS – E, 0) Cd = Max (dS – E, 0)
- 7. The Binomial Model converges to the Black- Scholes model as the number of time periods increases.
- 8. 1820s – Scottish scientist Robert Brown observed motion of suspended particles in water. Early 19th century – Albert Einstein used Brownian motion to explain movements of molecules, many research papers. 1900 – French scholar, Louis Bachelier wrote dissertation on option pricing and developed a model strikingly similar to BSM. 1951 – Japanese mathematician Kiyoshi Ito developed Ito’s Lemma that was used in option pricing.
- 9. Fischer Black and Myron Scholes worked in Finance Faculty at MIT Published paper in 1973. They were later joined by Robert Merton. Fischer left academia in 1983, died in 1995 at 57. 1997 – Scholes and Merton got Nobel Prize
- 10. Assumptions: The underlying stock pays no dividends. It is a European option. The stock price is continuous and is distributed lognormally. There are no transaction costs and taxes. No restrictions or penalty on short selling The risk free rate is known and is constant over the life of the option.
- 11. C0= S0N(d1) – E/ert N(d2) where, C0 = Present equilibrium value of call option S0 = Current stock price E = Exercise price e = Base of natural logarithm r = Continuously compounded risk free interest rate t = length of time in years to expiration N(*) = Cumulative probability distribution function of a standardized normal distribution
- 12. C = S N(d 1)– Ke-rtN(d2) where, C = Present equilibrium value of call option S = Current stock price K = Exercise price e = Base of natural logarithm r = Continuously compounded risk free interest rate t = length of time in years to expiration N(*) = Cumulative probability distribution function of a standardized normal distribution
- 13. ln(S0/E) + (r + ½σ2)t d1= σ √t ln(S0/E) + (r - ½σ2)t d2= σ √t where lnis the naturallogarithm
- 14. ln(S/K e- rt) d1= + 0.5 σ√t σ√t d2= d1- σ √t where lnis the natural logarithm
- 15. . Call Stock Put Risk-Free Bond Black-Scholes Call Option Pricing Model Put-Call Parity Black-Scholes Put Option Pricing Model
- 16. Find the Standard Deviation of the continuously compounded asset value change and the square root of the time leftto expiration Calculate ratio of the current asset value to the present value of the exercise price Consult the table giving %age relationship between the value of the Call Option and the stock price corresponding tothe value in steps 1and 2 Value of Put Option = Value of Call Option + PV of exercise price – StockPrice
- 17. To price European options on dividend paying options and American options on non-dividend paying stocks (Robert Merton, 1973 and Clifford Smith, 1976). American call options on dividend-paying stocks (Richard Roll, 1977; Robert Whaley, 1981; and Richard Geske and Richard Roll, 1984)