# Black-Scholes Option Pricing Model

A derivative is a financial instrument that has a value determined by the price of something else, such as options. The crucial idea behind the derivation was to hedge perfectly the option by buying and selling the underlying asset in just the right way and consequently "eliminate risk" (Ray, 2012). The derivative asset we will be most interested in is a European call option. A call option gives the holder of the option the right to buy the underlying asset by a certain date for a certain price, but a put option gives the holder the right to sell the underlying asset by a certain date for a certain price. The date in the contract is known as the expiration date or maturity date; the price in the contract is known as the exercise price or strike price. The market price of the underlying asset on the valuation date is spot price or stock price. Intrinsic value is the difference between the current stock market price and the exercise price or simply higher of zero. American options can be exercised at any time up to the expiration date. European options can be exercised only on the expiration date itself. (Hull, 2012). For example, consider a July European call option contract on XYZ with strike price $70. When the contract expires in July, if the price of XYZ stock is $75 the owner will exercise the option and realize a profit of $5. He will buy the stock for $70 from the seller of the option and immediately sell the stock for $75. On the other hand, if a share of XYZ is worth $67 the owner of the option will not exercise the option and it will expire worthless. In this case, the buyer would lose the purchase price of the option. One of the best-known and most widely used formulas in finance is the Black-Scholes option pricing model. It was originally developed in 1973 by two professors, Fischer Black and Myron Scholes. They designed the model to calculate the price of a European-style call option on non-dividend-paying stocks. Black-Scholes option pricing model assumes that the stock pays no dividends during the option’s life, European exercise terms are used, markets are efficient, no commissions are charged, interest rates remain constant and known and returns are log-normally distributed (Black and Scholes, 1973).

Black-Scholes gave the formula to estimate the value of a call option: C = Ps N(d1) - X e-rT N(d2)

where

C = price of the call option

Ps = spot price

X = exercise price

r = risk-free interest rate

T = current time until expiration

σ = volatility of share price

N() = area under the normal curve

d1 = [ ln(Ps/X) + r T ] / σ √T

d2 = d1 - σ √T

Call Option Pricing Example

XYZ is trading for $75. Historically, the volatility is 20% (σ). A call is available with an exercise of $70, an expiry of 6 months, and the risk free rate is 4%. ln(75/70) + (.04 + (.2)2/2)(6/12)

d1 = -------------------------------------------- = .70, N(d1) = .7580 .2 * (6/12)1/2

d2 = .70 - [ .2 * (6/12)1/2 ] = .56, N(d2) = .7123

C = $75 (.7580) - 70 e -.04(6/12) (.7123) = $7.98

Intrinsic Value = $5, Time Value = $2.98

Black-Scholes model’s inputs:

There are five main Black-Scholes Model inputs affect to options price. The inputs include the spot price, the exercise price, time to expiration, the interest rate, and the volatility of the stock. The intrinsic value of the stock is affected by the stock price increase or decrease. This determines if the stock option is valuable or not. For evaluate the influence of five factors in to the option’s price on stock, we calculated the option’s price using Numa Option Calculator on the Call Option Pricing Example to increase each input by 10% and discussed below: Determinants...

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