Option Pricing Model The Black-Scholes-Merton Model Chapter 5 Option Pricing Model The Black-Scholes-Merton Model

The Black-Scholes model for calculating the premium of an option was introduced in 1973 in a paper entitled, "The Pricing of Options and Corporate Liabilities" published in the Journal of Political Economy. The formula, developed by three economists – Fischer Black, Myron Scholes and Robert Merton – is perhaps the world's most well-known options pricing model.

The knowledge required to develop the Black- Scholes-Merton model came from mathematics and physics. Black–Scholes–Merton model is a mathematical model of a financial market containing certain derivative investment instruments. From the model, the Black–Scholes formula, gives a theoretical estimate of the price of European- style options.

The model makes certain assumptions, including: The options are European and can only be exercised at expiration. Stock prices are random and log normally distributed The risk-free rate and volatility of the underlying are known and constant. There are no taxes, transaction costs, or dividends. Random variableis a variable whose value is subject to variations due to chance Log normally distribution A statistical distribution of random variables which have a normally distributed logarithm.

The Black–Scholes–Merton call option pricing formula gives the call price in terms of the Stock price Exercise price Time to expiration Risk-free interest rate Standard deviation.

Stock Price The higher the stock price, the more a given call option is worth A call option holder benefits from a rise in the stock price

Exercise Price The lower the striking price for a given stock, the more the option should be worth Because a call option lets you buy at a predetermined striking price

Time to Expiration The longer the time until expiration, the more the option is worth The option premium increases for more distant expirations for puts and calls

Risk Free Interest Rate The higher the risk-free interest rate, the higher the option premium, everything else being equal A higher “discount rate” means that the call premium must rise for the put/call parity equation to hold

Volatility or Standard deviation The greater the price volatility, the more the option is worth

The Black–Scholes–Merton Formula C = S0N(d1) – Xeˉͬ ͭ N(d2) Where d1 = ln(S0/ X) +( rc + σ²/2) T σ√T d2 = d1 - σ√T N(d1), N(d2) = cumulative normal probabilities σ = annualized volatility (standard deviation) of stock return rc = risk-free interest rate T = time until option expiration S0 = current stock price C = theoretical call premium

The model is essentially divided into two parts: The first part, SN(d1), multiplies the price by the change in the call premium in relation to a change in the underlying price. This part of the formula shows the expected benefit of purchasing the underlying outright.

The model is essentially divided into two parts: The second part, Xeˉͬ ͭ N(d2), provides the current value of paying the exercise price upon expiration (remember, the Black- Scholes model applies to European options that are exercisable only on expiration day). The value of the option is calculated by taking the difference between the two parts, as shown in the equation.

Using the normal distribution A standard normal random variable is called z statistic. One can take any normally distributed random variable, convert it to a standard normal or z statistic, and use a table to determine the probability that an observed value of the random variable will be less than or equal to the value of interest.

The entry in the 1.5raw\0.07 column is 0.9418. The table 5.1 gives the cumulative probabilities of the standard normal distribution. Suppose that we want to know the probability of observing a value of z less than or equal to 1.57. We go to the table and look down the first column for 1.5, then move over to the right under the column labeled 0.07 that is the1.5 and 0.07 add up to the z value, 1.57. The entry in the 1.5raw\0.07 column is 0.9418. cumulative probabilities الاحتمالات التراكمية

Example; Let us use the Black-scholes-Merton model to price the DCRB June 125 call. The inputs are; 𝑆 0 =125.94, 𝑋=125, 𝑇=0.0959, 𝑟 =0.0446 𝑎𝑛𝑑 𝜎=0.83. 1. computed d1; 𝑑 1 = 𝐼𝑛 125.94 125 +(0.0446+ 0.83) 2 2 0.0959 0.83 0.0959 =0.1743

2. computed d2; 𝒅 𝟐 =𝟎.𝟏𝟕𝟒𝟑 −𝟎.𝟖𝟑 𝟎.𝟎𝟗𝟓𝟗 =−𝟎.𝟎𝟖𝟐𝟕. 3. look up N(d1) N(0.17) = 0.5675. 4.look up N(d2) N(-0.08) = 1-N(0.08)= 1- 0.5319= 0.4681.

5. plug into formula for C C= 125.94(0.5675) - 125 𝒆 −𝟎.𝟎𝟒𝟒𝟔 𝟎.𝟎𝟗𝟓𝟗 𝟎.𝟒𝟔𝟖𝟏 =𝟏𝟑.𝟐𝟏.

Known Discrete Dividends The black-scholes-Merton model can price Euoropean options on stocks with known discrete dividends by substracting the present value of the dividends from the stock price.

Suppose that stocks pays a dividend of Dt at some time during the option‘s life. If we make a small adjustment to the stock price, the Black-Scholes-Merton model will remain applicable to the pricing of the option.

Calculating the Black-Scholes-Merton price when there are known discrete dividends 𝑆 0 =125.94, 𝑋=125, 𝑇=0.0959, 𝑟 =0.0446 𝑎𝑛𝑑 𝜎=0.83. One dividend of 2, ex-dividend in 21 days (May 14 to June 4). 1.Determine the amount of thee dividend; Dt =$2. 2.Determine the time to the ex-dividend date; 21 days, so t= 21/365 = 0.0575.

3. Substract the present value of the dividend from the stock price to obtain 𝑆′ 0 $125.94 – 2 𝑒 −0.0446(0.0575) =$123.9451. 4. Compute d1 𝑑 1 = 𝐼𝑛 123.9415 125 +(0.0446+ 0.83) 2 2 0.0959 0.83 0.0959 = 0.1122

5.Compute d2 𝒅 𝟐 =𝟎.𝟏𝟏𝟐𝟐 −𝟎.𝟖𝟑 𝟎.𝟎𝟗𝟓𝟗 =−𝟎.𝟏𝟒𝟒𝟗. 6. Look up N(d1) = N(0.11) = 0.5438. 7. Look up N(d2) =N(-0.14)= 1-N(0.14) = 1- 0.5557 = 0.4443. 8.C= 123.9451(0.5438)-125 𝒆 −𝟎.𝟎𝟒𝟒𝟔 𝟎.𝟎𝟗𝟓𝟗 𝟎.𝟒𝟒𝟒𝟑 =𝟏𝟐.𝟏𝟎.

Put option pricing models The Black-Schole-Merton European put option pricing model; 𝑃= 𝑋𝑒 −𝑟𝑇 1−𝑁 𝑑2 − 𝑆 0 (1−𝑁(𝑑1) Where d1 and d2 the same as in the call option pricing model. In the previous example of the DCRB June 125 call, the values of N(d1) and N(d2) were 0.5692 and 0.4670 respectively. P=125 𝑒 − 0.0446 0.0959 1−0.4670 −125.94 1−0.5692 =12.08.

Problems Using the Black-Scholes Model Does not work well with options that are deep-in-the-money or substantially out- of-the-money Produces biased values for very low or very high volatility stocks Increases as the time until expiration increases May yield unreasonable values when an option has only a few days of life remaining