The BS model is based on the following assumptions
(1) It considers only those options which can be exercised at their maturity that is, European options.
(2) The market is efficient and there are no transaction costs and taxes. Options and shares arc infinitely divisible. Information is available to all investors with no costs.
(3) The risk-free rate or interest rate are known and constant during the period of option contract. Investors can borrow as well as lend at this rate.
(4) No dividend is paid on the shares.
(5) Share prices behave in a manner consistent with a random walk in continuous time.
(6) The prolixity distribution of financial returns on the share is normal.
(7) The variance standard deviation of the return is constant during the life of the option contract and is known to market participants/investors.
• The Black-Scholes equation is done in continuous time. This requires continuous compounding. The r is short-term annual interest rate compounded annually.
• N (d) is the cumulative normal distribution N(d) is called the delta of the option, which is.a measure of change in option price with respect to change in the price of the underlying asset.
• 0, a measure of volatility, is the annualized standard deviation of continuously compounded returns on the underlying stock. When daily sigma are given, they need to he converted into annualized sigma.
In Equation represents the option delta or hedge ratio (already explained). The ratio indicates number of shares required to be purchased for each option to maintain a fully hedged position. Further, the option holder is considered as a levered investor and, hence, is required to borrow an amount equal to the present value (PV) of exercise price at risk-free interest rate. The aspect of loan is represented on the right side of it indicates the PV of the exercise price times an adjustment factor of N (d). In simple terms, Equation 5.13 shows the following value of call option,
C= (Option delta * Share price) - Loan adjusted.