Black-Scholes Option Pricing Model
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.
Black-Scholes Option Pricing Model Essay Example
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 ? = volatility of share price
N() = area under the normal curve d1 = [ ln(Ps/X) + r T ] / ? vT d2 = d1 – ? vT Call Option Pricing Example XYZ is trading for $75. Historically, the volatility is 20% (? ). 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 of the Call Premium: Inputs |Inputs (^10%) |C (call price) |Change | |Ps |$82. 5 |$14. 3 |^ | |X |$77. 0 |$ 4. 0 |v | |T |6. 6 months |$ 8. 3 |^ | |r |4. 4% |$ 8. 1 |^ | |? |22% |$ 8. 3 |^ |
Spot Price: The option price depends on the market price of the underlying asset. If the price of the underlying asset increases, the premium of a call option will rise and the premium of a put option will fall. If the price of the underlying asset drops, the premium of a put option will rise and the premium of a call option will fall. Exercise Price: This is the price level at which the option holder has the right to buy or sell the underlying asset. The price of an option is naturally related to the exercise price.
When exercise price increases, spot price also increases and moving for the option to go in-the-money, call options become less valuable. Time to Expiration: The time (in years) until the option expires and the holder is no longer entitled to exercise the option. The longer day the option have, the more exercise opportunity and the more profit option’s holders can enhance. Both call and put options become more valuable with the increase of the duration until the expiration, because the larger the interval of time the greater chances has the holder to exercise the option in a favorable way.
Risk free Interest Rate: The risk free interest rate for the period until the option expires. The interest rate affects the option price according to the type of the option, through the means of the time value of the money. The increase of the interest rate determines the decrease of the present value of the exercise price and, as a consequence, the value of the call option increases and the value of the put option decreases. In practice, if the interest rate increases, the price of the call option increases because the underlying asset’s price is lower, the difference being invested at the interest rate without risk.
For the put option, the increase of the interest rate determines the decrease of the option’s price because the purchase of the derived asset postpones the cashing of the asset’s price until the option is exercised, and the increase of the interest rate represents an opportunity loss (Craiova and Ispas). Volatility: Volatility, which is an unpredictable factor that is very difficult to be measured. Volatility expresses the expectations to fluctuations in the price of the underlying asset.
The volatility has an influence on the value of an option because it is one of the factors that determine the probability to what extent the option will end in the money, and thus the size of payoff at expiry; The higher the volatility, the higher the value of the option price. The option price will therefore rise if the volatility of the underlying asset’s market price increases. The option pricing model also used to determine another useful number besides the option price. The variable N(d1) is called the hedge ratio, or the delta, for the call option.
The hedge ratio tells how much the option price will change when the underlying stock price changes by some small amount. For example, our option for XYZ had a hedge ratio of about . 76. This means that if the price of XYZ increased (or decreased) by one dollar per share, the price of the call option would increase (or decrease) by about $0. 76. This information is very useful for option traders who are trying to combine stocks and options into portfolios that will have offsetting movements.
The number of call contracts or stocks held can be adjusted using the hedge ratio to produce protected portfolios. The hedge ratio will change whenever the stock price changes and also as the time to maturity decreases. Thus, the information provided by the hedge ratio is good for only small stock price changes, and for only a short period of time. REFERENCES Black, F. and Scholes, M. (1973): The Pricing of Options and Corporate Liabilities, The Journal of Political Economy, 81(3), pp. 637-54. Ray, S. (2012): A Close Look into Black-Scholes Option Pricing Model, Journal of Science, 2(4), pp. 172-78.
Jay, S (2001): The Greeks are coming, Financial Training Company (Midlands): ACCA Study School Lecturer and member of Paper 3. 7 marking team. Hull, J. C. (2012): Options, Futures, and Other Derivatives, 8th edition, United States of America: Prentice Hall McDonald, L. R. (2013): Derivatives Markets, 3rd edition, United States of America: Prentice Hall Simion, D. and Ispas, R: Aspects Regarding the Influence of Volatility on the Option’s Price, (unpublished) thesis, Faculty of Economics and Business Administration, University of Craiova. Numa Option Calculator http://www. numa. com/cgi-bin/numa/calc_op. pl