Supervisors: Prof. Sankarshan Basu (IIM, Bangalore) Prof. S. Dharmaraja (Dept. of Mathematics, IIT Delhi)
Options
Financial instruments that convey the right, but not the obligation, to take aim in a future transaction on many profound security Basic Terminology :
Call Option: Gives the carrier the right to buy Put Option: Gives the holder the right to buy Strike Price (K): Pre-determined wrong at which to execute the excerption Expiration Date: last appointee the option can be exercised
Vaibhav Agarwal 2004MT50450
Pricing of Options
Black Scholes Formula
Assumes that summation prices follow GBM
B.S. Formula(contd.)
Price of a European Call option as given by B.S. Formula :
where St : price of underlying Wt : Wiener process (Brownian Motion) µ : Drift (Return) σ : Volatility (Risk) price changes are log-normally distributed
where
BS Formula is flawed
BS Formula is monotonically change magnitude with σ It also assumes σ to be “constant” i.e.
unrelated to S or T It is observed that BS Formula:
underprices deep-in-the-money or deep-out-of-themoney options overprices at-the-money options
Volatility grin/Skew and Implied Volatility Implied Volatility σi is the capriciousness that equates the BS Price to the commercialize price :
BS( St, K, r, σi, T) = Option market Price
Implied volatilities for different strikes and maturities are not constant Volatility make a face/skew is observed Volatility term structures + Volatility Smile Volatility Surface
Volatility Smile/Skew and Surface
Stochastic Volatility
The supra results led to the belief that the volatilities were, in fact, not constant. σ depended on St and T, at least This led to the idea of Stochastic Volatility
The name derives from the models intervention of the underlying securitys volatility as a random process, governed by state variables such as the price level of...If you wishing to get a full essay, order it on our website: Orderessay
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