Question

11. 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-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.

Assumption of black sholes model

• Lognormal distribution: The Black-Scholes-Merton model assumes that stock prices follow a lognormal distribution based on the principle that asset prices cannot take a negative value; they are bounded by zero.
• No dividends: The BSM model assumes that the stocks do not pay any dividends or returns.
• Expiration date: The model assumes that the options can only be exercised on its expiration or maturity date. Hence, it does not accurately price American options. It is extensively used in the European options market.
• Random walk: The stock market is a highly volatile one, and hence, a state of random walk is assumed as the market direction can never truly be predicted.
• Frictionless market: No transaction costs, including commission and brokerage, is assumed in the BSM model.
• Risk-free interest rate: The interest rates are assumed to be constant, hence making the underlying asset a risk-free one.
• Normal distribution: Stock returns are normally distributed. It implies that the volatility of the market is constant over time.
• No arbitrage: There is no arbitrage. It avoids the opportunity of making a riskless profit.

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).

Expression for option delta

Formula for the calculation of a call option's delta. The delta of an option measures the amplitude of the change of its price in function of the change of the price of its underlying.

Formula

δ=N(d1)

where:d1=ln(SK)+(r+σ22)tσt√

Legend

K	Option strike price
N	Standard normal cumulative distribution function
r	Risk-free interest rate
σ	Volatility of the underlying
S	Price of the underlying
t	Time to option's expiry Gamma : Gamma in the Black-Scholes Model The use of the Greeks was popularized in the Black Scholes Model. ... Gamma and the other Greek metrics help show how sensitive the value of derivatives is to changes in the value of the underlying asset. Gamma, as noted above, is itself a derivative of one of the other Greeks – delta.