The Six Inputs to the Black-Scholes Option Pricing Model Report (Assessment)

Introduction

The Black-Scholes pricing model was established by Black and Scholes for estimating the option value. It is used by stock traders to determine the value of the options in the market. These options can either be calls or puts. This paper will explore the six inputs in the Black-Scholes pricing model, describe the important concepts of Delta, Gamma, and Vega and give an analysis of how the six inputs affect the call and put options.

The Six Inputs

The Black-Scholes option-pricing model was introduced in 1973 by Black and Scholes. This model values European call options on non-dividend-paying stocks. It gives an illustration of the pricing of an option. In the past option, prices were determined randomly but with the introduction of the Black-Scholes pricing model, option prices can be determined rationally. This model has been modified to fit options on bonds, dividend-paying stocks, indexes, commodities among other optionable instruments. The option pricing models used today have different means but realize the same result. Six inputs are considered in the Black-Scholes pricing model to generate a theoretical value: strike price, interest rate, dividends, expiration time, stock price, and volatility (Daigler, 1994).

In a free market, option price is determined by forces of demand and supply which affect the values of the six inputs. Five of these inputs are dynamic and one is constant (strike price). The five inputs determine the variation in the option price. They are independent of each other but can change harmoniously resulting in an increasing or net effect on the value of the option. Options traders have to understand the relationship that exists between these variable inputs such as interest, price, and time.

Delta

According to Shaw (2002), options traders use five figures which are represented by Greek letters: delta, gamma, theta, vega, rho, and delta. These figures are called option Greeks although some of them are not from the Greek alphabet for example Vega. The figures are one of the products of option pricing model and each symbolizes a precise sensitivity that influences the option value.

Delta can be defined as the rate of change of the value of an option in respect to changes in the stock price. It can also be defined as the estimate of the possibility of the expiration of an option or the derivative of the graph of an option value about the stock price. In simple terms, delta can be said to be the rate of change in the value of an option as a result of a change in the security price and is expressed as a percentage (Kolb, 2003). An option with 60 deltas implies that if the stock price changes, the value of the option will change by 60 percent. In many cases, calls have positive correlation with positive deltas. Their value decreases if the stock price decreases and vice versa. For example if a stock has a price of $50 and a call option of 60 deltas trading for at $5.00 call value any changes in the stock price affect the call value. If the stock price decreases by $1.00, the value decreases by 60 percent of the change in stock price, from $5.00 to $ 4.4.

Puts have a negative correlation to the changes in the stock price. If the price of stock increases, put value decreases and vice versa. Puts have negative deltas which can be expressed as -50-delta put: let’s assume that the stock price is $60 and the put value is $ 3.00. If stock price rises by $2, the put value will reduce from $3.00 to $2.00.

The value of a delta can be estimated by the use of option’s expiration. If the option has more time left before expiration, the delta gravitates to 50 which signifies the greatest level of uncertainty. Delta is also affected by the level of volatility for example if the stock price is $50 and volatility 10%, delta tends to be 1 whereas with the same level of volatility and a stock price of $40, delta tends to be 0 but if the level of volatility is increased, the value of delta changes. A high stock price with low volatility level results in a high delta and the opposite is true. This implies that the higher the price of stock and the lower the volatility level, the bigger the delta size and consequently the bigger the call value.

Gamma

The Greek letter Gamma shows the relationship between stock price, option value, and delta. It denotes the rate of change of delta relative to the rate of change of the price of the principal security. The following example shows the effect of the change in gamma to value of a delta

Let’s assume:

• the stock price is $50
• call delta is 50%
• gamma is 4%

If stock prices increase by $1, delta increases by the gamma amount from 50 to 54. As the stock price increases further to $52, delta increases to 58 (Williams & Hoffman, 2001). This can be represented as shown below

Stock price $50 —————-→ $51 ——————————→ $52

Call Delta 50 → 4 gamma —-→ 54 call Delta → 4 gamma ——→ 58 call Delta

The increase in the value of the delta affects the call value. At $ 50 stock price, call value is increased at a rate of 50 percent as represented by delta at that stock price. As stock value increases to $51, call value increases by 54 percent as represented by the call’s delta at $ 51 stock value and the trend continues.

Puts work in the same manner, but because they have a negative correlation, increases in stock price cause a decrease in the put delta equivalent to the value of the gamma. Traders who buy options acquire positive gamma which causes options to gain value at a faster rate than they lose it. The bigger the number of calls and puts bought by a trader, the higher the value of gamma acquired (Chriss, 1997).

Vega

Volatility is used by traders to determine the price of options. Option prices react to any changes in volatility no matter how small it is. A relatively small change in volatility causes a relatively large change in the price of an option, more so for the long-term options. The future values of volatility can be estimated which helps in estimating option values. Vega measures the rate of change in the price of an option relative to a unit change in stock volatility. This can be estimated by the use of the following formula (Kolb & Overdahl, 2007).

Vega = ύ = d (option price)/ d (volatility)

If the ύ = 0.05, an increase in volatility by 10% causes an increase of $ 0.5 in the option price.

Conclusion

The Black-Scholes pricing model is used today by traders to estimate the prices of options in the market. This is illustrated by the use of delta, gamma, and Vega. Changes in any of these figures affect stock prices, exploration, interest rates among others consequently affecting the value of the option. Calls have a positive correlation to these figures while puts have a negative correlation.

Reference List

1. Chriss, N. (1997). Black-Scholes and beyond: option pricing models. New York: McGraw-Hill Professional.
2. Daigler, R. T. (1994). Advanced options trading: the analysis and evaluation of trading strategies, hedging tactics, and pricing models. New York: McGraw-Hill Professional.
3. Kolb, R. W. (2003). Futures, options891, and swaps. New York: Wiley-Blackwell.
4. Kolb, R. W. & Overdahl, J. A. (2007). Futures, options, and swaps. New York: Wiley-Blackwell.
5. Shaw, S. (2002). Black-Scholes Option Valuation Factor Table at $1 of Both Exercise Price and Stock Option. Bloomington: Trafford Publishing.
6. Williams, M. S. & Hoffman, A. S. (2001). Fundamentals of the options market. New York: McGraw-Hill Professional.