Question

Black-Scholes.

a) Write down the Black-Scholes put option formula for a stock that has a continuous dividend yield. Be sure to elaborate on d1 and d2.

b) Derive an expression for option Delta

c) Derive Gamma

Answer 1

Ans ) Black sholes model ; Black-Scholes is a pricing model used to determine the fair price or theoretical value for a call or a put option based on six variables such as volatility, type of option, underlying stock price, time, strike price, and risk-free rate. The quantum of speculation is more in case of stock market derivatives, and hence proper pricing of options eliminates the opportunity for any arbitrage. There are two important models for option pricing – Binomial Model and Black-Scholes Model. The model is used to determine the price of a European call option, which simply means that the option can only be exercised on the expiration date.

Black sholes put options

An alternative form of valuation is to use the Black-Scholes formula for a put, which is:

P = Xe –r(T-t) [1-N(d2)] – S [1-N(d1)]

Where d1 and d2 are as given in the section deriving a call option.

Note that [1 – N(d2)] is the same as N(-d2) and [1 – N(d1)] is the same as N(-d1).

put option value is calculated as follows:

p = 31.6693(0.3446) – 35(0.2743) = 1.3127

Values for d1 and d2 must be computed first, and then a table for the standard normal distribution must be used to look up the N values. In actual practice, computer algorithms and handheld calculators use a polynomial equation that will give very accurate N value approximations.

Ans B ) options delta :

The Delta of an option is a calculated value that estimates the rate of change in the price of the option given a 1 point move in the underlying asset.

As the price of the underlying stock fluctuates, the prices of the options will also change but not by the same magnitude or even necessarily in the same direction. There are many factors that will affect the price that an option will change by e.g. Whether it is a call or put, the proximity of the strike to the underlying price, volatility, interest rates and time to expiry. This is why the delta is important; it takes much of the guess work out of the expected price movement of the option.

Ans C ) Gamma : Gamma. The option's gamma is a measure of the rate of change of its delta. The gamma of an option is expressed as a percentage and reflects the change in the delta in response to a one point movement of the underlying stock price. Options that are very deeply into or out of the money have gamma values close to 0.

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