The BLACK-SCHOLES OPTION PRICING MODEL

In order to understand the model itself, we divide it into two parts. The first part, SN(d1), derives the expected benefit from acquiring a stock outright. This is found by multiplying stock price [S] by the change in the call premium with respect to a change in the underlying stock price [N(d1)]. The second part of the model, Ke(-rt)N(d2), gives the present value of paying the exercise price on the expiration day. The fair market value of the call option is then calculated by taking the difference between these two parts.
 

Assumptions of the Black and Scholes Model:

1) The stock pays no dividends during the option's life:  Most companies pay dividends to their share holders, so this might seem a serious limitation to the model considering the observation that higher dividend yields elicit lower call premiums. A common way of adjusting the model for this situation is to subtract the discounted value of a future dividend from the stock price.

2) European exercise terms are used:  European exercise terms dictate that the option can only be exercised on the expiration date. American exercise term allow the option to be exercised at any time during the life of the option, making american options more valuable due to their greater flexibility. This limitation is not a major concern because very few calls are ever exercised before the last few days of their life. This is true because when you exercise a call early, you forfeit the remaining time value on the call and collect the intrinsic value. Towards the end of the life of a call, the remaining time value is very small, but the intrinsic value is the same.
 

3) Markets are efficient:  This assumption suggests that people cannot consistently predict the direction of the market or an individual stock. The market operates continuously with share prices following a continuous Itô process. To understand what a continuous Itô process is, you must first know that a Markov process is "one where the observation in time period t depends only on the preceding observation." An Itô process is simply a Markov process in continuous time. If you were to draw a continuous process you would do so without picking the pen up from the piece of paper.
 

4) No commissions are charged:  Usually market participants do have to pay a commission to buy or sell options. Even floor traders pay some kind of fee, but it is usually very small. The fees that Individual investor's pay is more substantial and can often distort the output of the model.
 

5) Interest rates remain constant and known:  The Black and Scholes model uses the risk-free rate to represent this constant and known rate. In reality there is no such thing as the risk-free rate, but the discount rate on U.S. Government Treasury Bills with 30 days left until maturity is usually used to represent it. During periods of rapidly changing interest rates, these 30 day rates are often subject to change, thereby violating one of the assumptions of the model.
 

6) Returns are lognormally distributed:  This assumption suggests, returns on the underlying stock are normally distributed, which is reasonable for most assets that offer options.

The Black and Scholes Model:

Delta is a measure of the sensitivity the calculated option value has to small changes in the share price.  Option prices and the share price of the underlying asset (stock)_ do not change at exactly the same rate.  Delta is the measurement of how much one changes as a percentage of the other.  When an option trader has to sell, for example, 1000 calls, she immediately protects herself by buying the stock in sufficient quantity to cover her position in naked calls (an unlimited loss potential).  Option traders enter all of their trades into a sophisticated, had-held computer designed specifically for the purpose of keeping track of their portfolios.  Traders ideally want "a delta-zero hedge."

Investopedia says this:  Delta Hedging
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An options strategy that aims to reduce (hedge) the risk associated with price movements in the underlying asset by offsetting long and short positions. For example, a long call position may be delta hedged by shorting the underlying stock. This strategy is based on the change in premium (price of option) caused by a change in the price of the underlying security. The change in premium for each basis-point change in price of the underlying is the delta and the relationship between the two movements is the hedge ratio.

For example, the price of a call option with a hedge ratio of 40 will rise 40% (of the stock-price move) if the price of the underlying stock decreases. Typically, options with high hedge ratios are usually more profitable to buy rather than write since the greater the percentage movement - relative to the underlying's price and the corresponding little
time-value erosion - the greater the leverage. The opposite is true for options with a low hedge ratio.


Related Links:
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Article:  Does Delta Neutral Trading Work?
http://www.investopedia.com/articles/trading/02/040202.asp

Article:  Getting to Know the "Greeks"
http://www.investopedia.com/articles/optioninvestor/02/120602.asp


Related Terms:
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Delta
http://www.investopedia.com/terms/d/delta.asp

Delta Neutral
http://www.investopedia.com/terms/d/deltaneutral.asp

Hedge
http://www.investopedia.com/terms/h/hedge.asp


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Gamma:

Gamma is a measure of the calculated delta's sensitivity to small changes in share price.
 

Theta:

Theta measures the calcualted option value's sensitivity to small changes in time till maturity.
 

Vega:

Vega measures the calculated option value's sensitivity to small changes in volatility.
 

Rho:

Graphs of the Black and Scholes Model:

This following graphs show the relationship between a call's premium and the underlying stock's price.

The first graph identifies the Intrinsic Value, Speculative Value, Maximum Value, and the Actual premium for a call.
 

The following 5 graphs show the impact of deminishing time remaining on a call with:
S = $48
E = $50
r = 6%
sigma = 40%

Graph # 1, t = 3 months
Graph # 2, t = 2 months
Graph # 3, t = 1 month
Graph # 4, t = .5 months
Graph # 5, t = .25 months
 

Graph #1

Graph #2

Graph #3

Graph #4

Graph #5

Graphs # 6 - 9, show the effects of a changing Sigma on the relationship between Call premium and Security Price

S = $48
E = $50
r = 6%
sigma = 40%

Graph # 6, sigma = 80%
Graph # 7, sigma = 40%
Graph # 8, sigma = 20%
Graph # 9, sigma = 10%
 

Graph #6

Graph #7

Graph #8

Graph #9

After the Black and Scholes Model:

Since 1973, the original Black and Scholes Option Pricing Model has been the subject of much attention. Many financial scholars have expanded upon the original work. In 1973, Robert Merton relaxed the assumption of no dividends. In 1976, Jonathan Ingerson went one step further and relaxed the the assumption of no taxes or transaction costs. In 1976, Merton responded by removing the restriction of constant interest rates. The results of all of this attention, that originated in the autumn of 1969, are alarmingly accurate valuation models for stock options.