pricing options

8/9/2019 Pricing Options

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Option Pricing

Before venturing into the world of trading options, investors should have a goodunderstanding of the factors that determine the value of an option. These include the

current stock price, the intrinsic value, time to expiration or the time value, volatility,interest rates and cash dividends paid.

There are several options pricing models that use these parameters to determine thefair market value of the option. Of these, the Black-Scholes model is the most widelyused. In many ways, options are just like any other investment in that you need tounderstand what determines their price in order to use them to take advantage ofmoves the market.

Main Drivers of an Option's Price

Let's start with the primary drivers of the price of an option: current stock price, intrinsicvalue, time to expiration or time value, and volatility. The current stock price is fairlyobvious. The movement of the price of the stock up or down has a direct - although notequal - effect on the price of the option. As the price of a stock rises, the more likely theprice of a call option will rise and the price of a put option will fall. If the stock price goesdown, then the reverse will most likely happen to the price of the calls and puts.

Intrinsic Value

Intrinsic value is the value that any given option would have if it were exercised today.Basically, the intrinsic value is the amount by which the strike price of an option is in the

money. It is the portion of an option's price that is not lost due to the passage of time.The following equations can be used to calculate the intrinsic value of a call orputoption :

Call Option Intrinsic Value = Underlying Stock'sCurrent Price Call Strike Price

Put Option Intrinsic Value = Put Strike Price

Underlying Stock's Current Price

The intrinsic value of an option reflects the effective financial advantage that wouldresult from the immediate exercise of that option. Basically, it is an option's minimumvalue. Options trading at the money orout of the money have no intrinsic value.
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Time Value

The time value of options is the amount by which the price of any option exceeds theintrinsic value. It is directly related to how much time an option has until it expires as

well as the volatility of the stock. The formula for calculating the time value of an optionis:

Time Value = Option Price Intrinsic Value

The more time an option has until it expires, the greater the chance it will end up in themoney. The time component of an option decays exponentially. The actual derivation of

the time value of an option is a fairly complex equation. As a general rule, an option willlose one-third of its value during the first half of its life and two-thirds during the secondhalf of its life. This is an important concept for securities investors because the closeryou get to expiration, the more of a move in the underlying security is needed to impactthe price of the option. Time value is often referred to as extrinsic value.

Time value is basically the risk premiumthat the option seller requires to provide theoption buyer the right to buy/sell the stock up to the date the option expires. It is like aninsurance premium of the option; the higher the risk, the higher the cost to buy theoption.

An option's time value is also highly dependent on the volatility in that the marketexpects the stock will display up to expiration. For stocks where the market does notexpect the stock to move much, the option's time value will be relatively low. Theopposite is true for more volatile stocks or those with a high beta, due primarily to theuncertainty of the price of the stock before the option expires. In the table below, youcan see the GE example . It shows the trading price of GE, several strike prices and theintrinsic and time values for the call and put options.
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General Electric is considered a stock with low volatility with a beta of 0.49 for thisexample.

Amazon.com Inc. (AMZN) is a much more volatile stock with a beta of 3.47 (see Figure2). Compare the GE 35 call option with nine months to expiration with the AMZN 40 calloption with nine months to expiration. GE has only $0.20 to move up before it is at themoney, while AMZN has $1.30 to move up before it is at the money. The time value ofthese options is $3.70 for GE and $7.50 for AMZN, indicating a significant premium onthe AMZN option due to the volatile nature of the AMZN stock.


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This makes - an option seller of GE will not expect to get a substantial premium

because the buyers do not expect the price of the stock to move significantly. On the

other hand, the seller of an AMZN option can expect to receive a higher premium due to

the volatile nature of the AMZN stock. Basically, when the market believes a stock will

be very volatile, the time value of the option rises. On the other hand, when the market

believes a stock will be less volatile, the time value of the option falls. It is this

expectation by the market of a stock's future volatility that is key to the price of options.

The effect of volatility is mostly subjective and it is difficult to quantify. Fortunately, there

are several calculators that can be used to help estimate volatility. To make this even

more interesting, there are also several types of volatility - with implied andhistorical

being the most noted. When investors look at the volatility in the past, it is called either

historical volatility or statistical volatility. Historical Volatility helps you determine the

possible magnitude of future moves of the underlying stock. Statistically, two-thirds of all

occurrences of a stock price will happen within plus or minus one standard deviation of

the stocks' move over a set time period. Historical volatility looks back in time to show

how volatile the market has been. This helps options investors to determine which

exercise price is most appropriate to choose for the particular strategy they have in

mind.

Implied volatility is what is implied by the current market prices and is used with the

theoretical models. It helps to set the current price of an existing option and assists

option players to assess the potential of an option trade. Implied volatility measures

what option traders expect future volatility will be. As such, implied volatility is an

indicator of the current sentiment of the market. This sentiment will be reflected in the

price of the options helping options traders to assess the future volatility of the option

and the stock based on current option prices.

The Bottom Line

A stock investor who is interested in using options to capture a potential move in a stock

must understand how options are priced. Besides the underlying price of the stock, the

key determinates of the price of an option are its intrinsic value - the amount by which

the strike price of an option is in-the-money - and its time value. Time value is related tohow much time an option has until it expires and the option's volatility. Volatility is of

particular interest to a stock trader wishing to use options to gain an added advantage.

Historical volatility provides the investor a relative perspective of how volatility impacts

options prices, while current option pricing provides the implied volatility that the market

currently expects in the future. Knowing the current and expected volatility that is in the
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price of an option is essential for any investor that wants to take advantage of the

movement of a stock's price.

BlackScholes model of Option Pricing

The BlackScholes model is a mathematical description of financial markets and derivativeinvestment instruments. The model develops partial differential equations whose solution, the

BlackScholes formula, is widely used in the pricing of European-style options.

The model was first articulated by Fischer Black and Myron Scholes in their 1973 paper, "ThePricing of Options and Corporate Liabilities." The foundation for their research relied on workdeveloped by scholars such as Jack L. Treynor, Paul Samuelson, A. James Boness, Sheen T.

Kassouf, and Edward O. Thorp. The fundamental insight of BlackScholes is that the option is

implicitly priced if the stock is traded. Robert C. Merton was the first to publish a paperexpanding the mathematical understanding of the options pricing model and coined the term

BlackScholes options pricing model.

Model assumptions

The BlackScholes model of the market for a particular equity makes the following explicitassumptions:

It is possible to borrow and lend cash at a known constant risk-free interestrate. This restriction has been removed in later extensions of the model.

The price follows a Geometric Brownian motion with constant drift andvolatility. This often implies the validity of the efficient-market hypothesis.

There are no transaction costs or taxes. Returns from the security follow a Log-normal distribution. The stock does not pay a dividend (see below for extensions to handle

dividend payments). All securities are perfectly divisible (i.e. it is possible to buy any fraction of a

share). There are no restrictions on short selling. There is no arbitrage opportunity Options use the European exercise terms, which dictate that options may

only be exercised on the day of expiration.

From these conditions in the market for an equity (and for an option on the equity), the authorsshow that "it is possible to create a hedged position, consisting of a long position in the stock and

a short position in [calls on the same stock], whose value will not depend on the price of the

stock." Several of these assumptions of the original model have been removed in subsequentextensions of the model. Modern versions account for changing interest rates (Merton, 1976),

transaction costs and taxes (Ingerson, 1976), and dividend payout (Merton, 1973).
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