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BLACK SCHOLES MODELSaumya Goel 208Rohit Solanki 218INTRODUCTIONDiscovered in 1973By Fischer Black and Myron ScholesMathematical model of financial marketEquation: Gives theoretical estimate of price of an option over timeUses Delta HedgingThe Black Scholes Model is one of the most important concepts in modern financial theory. The Black Scholes Model is considered the standard model for valuing options. A model of price variation over time of financial instruments such as stocks that can, among other things, be used to determine the price of a European call option. The model assumes that the price of heavily traded assets follow a geometric Brownian motion with constant drift and volatility. When applied to a stock option, the model incorporates the constant price variation of the stock, the time value of money, the option's strike price and the time to the option's expiry.Fortunately one does not have to know calculus to use the Black Scholes model.2ASSUMPTIONS.

These assumptions are combined with the principle that options pricing should provide no immediate gain to either seller or buyer

3GREEKSGreeksRelationship with Option Price (V)Delta (Spot Price S) UA Call Premium Put PremiumVega (Volatility )Vega of call & put are identical and positiveTheta (Time to Expiration T)Call and put lose value as expiration approachesRho (Risk Free Rate r)Positive relation with callNegative relation with putGamma Gamma of put and call are equalThe sensitivities (denoted by Greek letters, so also called as Greeks) of option price to a change in underlying asset parameters that determine the option prices. These are also called as risk sensitivities, risk measures or hedge parameters.The underlying asset parameter include its price, time period, interest rate and the volatility. There are five measures of sensitivities: Delta, Gamma, Theta, Rho and Vega.

4EQUATION

Where, C = Call option price, P = Put option price S = Spot price of underlying asset X = Strike price of the option r = risk-free interest rate T t = Time to expiration expressed in years = Volatility of returns on the underlying asset N(d) = Cumulative standard normal distribution e = Exponential function (2.7183)

EXAMPLEThe stock price is $42. The strike price for a European call and put option on the stock is $40. Both options expire in 6 months. The risk free interest is 6% per annum and the volatility is 25% per annum. What are the call and put prices?

EXAMPLES = 42, X = 40, r = 6%, =25%, T=0.5

= 0.5341 = 0.3573

=4.7144

=1.5322

LIMITATIONSAssumes that the risk-free rate and the stocks volatility are constant.Assumes that stock prices are continuous and that large changes (such as those seen after a merger announcement) dont occur.Assumes a stock pays no dividends until after expiration.Analysts can only estimate a stocks volatility instead of directly observing it, as they can for the other inputs.Tends to overvalue deep out-of-the-money calls and undervalue deep in-the-money calls.Tends to misprice options that involve high-dividend stocks.To deal with these limitations, a Black-Scholes variant known as ARCH, Autoregressive Conditional Heteroscedasticity, was developed. This variant replaces constant volatility with stochastic (random) volatility. A number of different models have been developed all incorporating ever more complex models of volatility. However, despite these known limitations, the classic Black-Scholes model is still the most popular with options traders today due to its simplicity.

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