Practical Introduction:

In the early 1970’s, Fischer Black, Myron Scholes and Robert Merton wrote their seminal paper outlining the equations and assumptions that have become the backbone of option pricing. While the equations require a decent math background (Scholes and Merton did win the Nobel Prize for their work), the underlying concepts are somewhat easier to digest.

For the sake of completeness, we have printed the equations below. A brief explanation of the terms will follow.

Equations

C(S,T) = SN(d1) – Ke-rt N(d2) where C(S,T) is simply the function for a Call option

P(S,T) = Ke-rt N(d2) – SN(-d1) where P(S,T) is simply the function for a Put option

where:

d1 = (ln(S/K) + (r + σ2/2)T)/σ√T

d2 = (ln(S/K) + (r – σ2/2)T)/σ√T = d1 – σ√T

(ln is simply the natural logarithm operator)

where:

N is the normal distribution function
σ is the implied volatility level
S = Stock price
K = Strike Price
r = the interest rate
T = time to expiration

So What Does it All Mean?

Without getting too deep into the full mathematics, we can make some inferences regarding what the model actually tells us regarding option pricing. What we will see is that the model is simply a mathematical representation of things we already know. Let’s take the terms one by one and explain in common English what they represent to the average trader.

N:

N is the normal distribution function. The vast majority of the financial world has come to terms with admitting that the only predictable thing about stock prices is that they are randomly distributed. This random distribution is better known as the Standard Normal distribution, also commonly referred to as the “bell curve”. It is no coincidence that this is the distribution pattern chosen by Black, Scholes, and Merton.

What is interesting about the formulas for calls

[C(S,T)] and puts [P(S,T)] is that they attempt to price the options from the relative distances between the stock price S and the strike price K. As such, depending on that distance between the stock price and the strike price (S – K or K – S), and depending on the implied volatility level, the model gives us a value for an option at any distance away from the strike.

For those of you who are more mathematically advanced, it may be evident to you that the put formula is simply the call formula rearranged via the use of the put-call parity relationship.

σ:

σ (sigma) is the implied volatility level associated with the options. This one is fairly self explanatory, as most traders are aware that implied volatility levels dictate how much time value an option may have. The effects of σ are normally distributed peaking in at-the-money options. For any given period of time, the further away we get from at the money, the lesser the effect of σ on the options.

You may note that the σ term is identical in both the formula for the call and for the put. From this, one may infer that a change in implied volatility levels affects both call and put options of the same strike equally. This will become more evident once we consider the equations for the option greeks, especially that of the greek vega.

S:

S refers to the stock price. For any given percentage value, higher stock prices imply a range or move of a larger magnitude than lower stock prices. Therefore, will all else being equal, the present stock price is a determining factor in the pricing of the options.

K:

K refers to the strike price. It is important due to the fact that option prices follow a normal distribution with the at the money level (where S = K) being the mean. Since the difference between the stock and the strike price [(S – K) or (K – S)] is essential to option pricing, K must be included in the calculations. Depending on whether K or S is the greater number, calls and puts will derive their respective values. The model makes it prohibitive for options to trade below zero. However, the model does allow for options to trade below real (intrinsic) value.

r:

r refers to the interest rate used in the model. In pure Black Scholes, this rate refers to the risk free rate of return. However, in practical application this rate refers to whatever rate a trader pays or receives, depending on whether he is borrowing or lending money. It is also important because options are actually priced off the future value of the current stock price at option expiration. Therefore, higher interest rates imply higher future prices in the stock for any given period of time.

A closer look at the interest rate component shows that higher interest rates make call options trade at higher premiums but they make puts trade at a discount. Depending on the interest rate level and depending on the remaining time (extrinsic) value in a put option, it is possible under certain conditions for put options to trade below real value.

T:

T is the time to expiration. As just stated, options are priced off the future value of the current stock price. Therefore, the more time remaining until expiration, the larger the interest effect will be. Furthermore, the greater the time until expiration, the greater the expected stock movement for a given implied volatility level. Said another way, the more time we give ourselves, the more opportunity there is for a stock to move.

These are the basic terms used in the Black Scholes model. That said, there are some assumptions made in the model that do not necessarily translate into realistic option pricing. We will not go into details on the assumptions at this point, but the key ones are as follows:

 

A Brief Word on Dividends

The traditional Black-Scholes model does not take dividends into account. Since the model was published in 1973, various iterations of the model have arisen including some which attempt to take dividends into account. Initial models were fairly crude and simply included a discount variable d which was subtracted from the interest component r in order to account for dividends. However, this model had limitations, as it merely modeled a steady dividend stream as opposed to a more realistic discrete quarterly sum.

In order to model American-settled options, the binomial model was used. It led to decent modeling techniques as long as you had the time to go through seemingly endless iterations. Even with today’s computer power, one needs a fast computer and more importantly, a talented set of programmers to obtain a fairly fast and seamless binomial model. Many software developers shy away from using binomial models for fear of delivering a program that is too slow to be of any practical use.

Eventually, models using approximations such as the ever popular Bjerksund-Stensland model came about. The model uses some good approximations to model dividend payments. As with any other model, this one has its limitations, and depending on the programmers, the limitations may be quite visible. This is especially true as one gets close to the ex-dividend date on a stock that pays a fairly large dividend.

Of course, there is no perfect model. Each is subject to its own set of limitations and each is only as good as its initial assumptions. It is up to the trader to understand that his model has limitations and that one cannot expect a model to always spit out the correct value. Nevertheless, models allow us to measure and make predictions based on our assumptions. Most traders are content to realize that regardless of what price the market gives us for a particular option, that is the right price. The market is never wrong. However, a good options model affords us the luxury of being able to predict what we cannot currently see – future option prices. At this point, it is important to focus our attention on the “sensitivities” within the Black Scholes model, or what we commonly refer to as the greeks.

The Option Greeks

There is more to a model than the ability to spit out a value. A good model has some semblance of predictive power. For example, what happens to our option value if the stock price changes? What if implied volatility levels change? What if I have a position with more than one option? What happens when positions get so complex that we have options at different strikes and expiration dates? What if the stock moves? What if it does not?

To answer these and many other questions, we will take a quantitative look at the option greeks – delta, gamma, theta, vega, and rho. We will begin with the basic definition for each greek. However, as the reader will see, the greeks are also derived from the basic Black Scholes equations. We will show and explain those derivations as well.

Delta = (Change in the option price)/(Change in the underlying)

Delta is usually expressed per one dollar move in the underlying (stock, future, etc…). It is important to note that rarely do options change value at the same speed as the underlying changes value. All else being equal, if the underlying changes in value by a certain amount, the option will normally change in value by some lesser amount. That change in the option in relation to the change in its underlying security is what we refer to as the delta.

Calls are said to have a positive delta, since all else being equal, when the underlying rises, call options tend to rise, thus exhibiting a positive relationship with the underlying.

Puts are said to have a negative delta since, all else being equal, when the underlying rises, put options tend to lose value, thus exhibiting a negative relationship with the underlying.

The delta value may range anywhere from 0 to 100 and may be positive or negative. Options that are very far out of the money tend to move very slowly with respect to the stock, hence they are said to have a low delta. In the money options tend to move very close with the stock price and hence are said to have a high delta. At the money options tend to have a 50 delta – right in the middle of the range where we would expect them to be.

Quantitatively, delta is the first derivative of the Black Scholes equations with respect to stock. Since we are taking the partial derivative with respect to the stock price, any terms without a stock term (S) are considered constants and the derivative of a constant is simply zero. Hence, we arrive at:

call delta = δC/δS = N(d1)
and
put delta = δP/δS = N(d1) – 1

Note that N(d1) is a positive value, so N(d1) – 1 must be a negative value. Furthermore, when you add up the absolute value of the call delta and the put delta, that number adds up to one. This is consistent with traditional explanations of the greeks. Furthermore, note that delta is often defined as the probability that an option will end up in the money. The fact that delta may be thought of as a probability explains why it is a function of N(d1) and hence of the normal distribution.

Gamma (Change in the delta)/(Change in the underlying)

Gamma may be thought of as the “delta of the delta”. It is usually expressed per one dollar move in the underlying (stock, future, etc…). A call and a put of the same strike and expiration month will have the same gamma. In fact, any long option, regardless of whether it is a call or a put, will exhibit a positive gamma number. By the same logic, any short option, regardless of whether it is a call or a put, will exhibit a negative gamma number.

For any given expiration month, gamma will be at its highest level at the money. Therefore, all else being equal, as an option moves closer to the money from either direction, its gamma will increase. All else being equal, as an option moves further away from the money in either direction, its gamma will decrease.

Furthermore, the greater the amount of time a given option has until expiration, the less its gamma tends to be.

Quantitatively, gamma is the second derivative of the Black Scholes equations with respect to stock. In essence, it is the delta of the delta. Since we are taking the partial derivative with respect to the stock price, any terms without a stock term (S) are considered constants and the derivative of a constant is simply zero. Hence, we arrive at:

call gamma = δ2C/δS = δN(d1)/δS
and
put gamma = δ2P/δS = δ(N(d1)-1)/δS

Note that since the derivative of a constant is zero, it may be inferred that we are taking the second partial difference with respect to N(d1) in both cases. Therefore, one can see that the gamma of both a call and a put of the same strike must be identical.

If we actually went through all of the algebra, we would arrive at the following calculation for gamma:

gamma = (N'(d1))/Sσ√T

Note a few things about the formula above. First, the T term is in the denominator, meaning that the more time there is to expiration the smaller the overall gamma level. Second, the σ term is also in the denominator, meaning that higher implied volatility levels also lead to lower gamma levels. Finally, the same may be said for greater stock prices since the S term is also in the denominator. These findings are all consistent with what we know from a basic education on the greeks.

Theta (Change in the option)/(Change in Time)
(Change in Time is usually one day)

Options decrease in value as their time to expiration approaches. Theta is a measure of that daily decay. All else being equal, a call and a put of the same strike and expiration month will have the same theta.

Long options will exhibit a negative theta value, while short options exhibit a positive theta value. This is because as we move closer and closer to expiration, we have less time for our option to move into a more valuable position. After all, if we had two options that accomplished the same thing, we would prefer to own the one that gave us a little more time. Naturally, with each passing day implying less time in our option, it makes sense that our options would lose some value. This daily loss is what we quantify as theta.

Quantitatively, theta is the first derivative of the Black Scholes equations with respect to time (remember that the call and put functions are functions of both stock price S and time T). Since we are taking the partial derivative with respect to time, any terms without a time term (T) are considered constants and the derivative of a constant is simply zero.

call theta = δC/δT and put theta = δP/δT

The actual formulas are extremely involved, but we will note them in order to be consistent:

Call theta = (– SN'(d1) σ – rKe-rt N(d2))/(2√T)

Put theta = (– SN'(d1) σ + rKe-rt N(d2))/(2√T)

Note that call and put theta are almost identical. The slight difference in their calculation stems from the interest component in options. Note also that as time to expiration (T) decreases, theta values increase. Furthermore, due to the fact that the time relationship is not a linear one but rather a quadratic, as we get closer to expiration, theta increases in value at an increasing rate. This is also consistent with a basic education on the greeks.

Vega (Change in the option price)/(One percentage point change in implied volatility)

In other words, vega is an option’s sensitivity to changes in implied volatility. The higher the implied volatility level, the more the underlying is expected to move. Therefore, the more expensive the options will become.

For any given expiration month, vega will be at its highest level at the money. Therefore, all else being equal, as an option moves closer to the money from either direction, its vega will increase. All else equal, as an option moves further away from the money in either direction, its vega will decrease.

Furthermore, the greater the amount of time remaining until expiration in a given option, the greater the vega will be. This makes sense since implied volatility is a way to measure expected movement over time. Given a certain propensity for a stock to move, the more time that stock is alloted, the greater the range we can expect the stock to cover.

Quantitatively, vega is the first derivative of the Black Scholes equations with respect to implied volatility. Since we are taking the partial derivative with respect to σ, any terms without a σ term are considered constants and the derivative of a constant is simply zero. We may then realize that:

vega = SN'(d1)√T

Note that vega is the same for both call and put options. Note further that there are some interesting characteristics about the equation for vega. First, there is no denominator. Therefore, an increase in the stock price (S) leads to an increase in vega. Second, an increase in the time to expiration (T) also leads to an increase in vega. This makes perfect sense if we consider that implied volatility represents a way to measure expected range as a percentage of the stock price. Thus, for a given percentage number, the greater the stock price, the greater the expected range. By the same logic, given more time to move randomly, a stock may tend toward a greater expected range as well.

What is equally interesting is that which is not visible. Looking at the equation for vega, we can see that there is no σ term present! That is because when we take the partial derivative with respect to σ, the actual σ term cancels out and is dropped from the model. This means that while vega is the change in the option value with respect to a change in the implied volatility level, it is important to keep in mind that a change in the implied volatility level does NOT translate into a change in vega either way.

Rho (Change in the option price)/(100 basis point change in interest rates)

Rho is usually the least important of the option greeks. Given that interest rates do not usually change in hundreds or thousands of basis points at once, we can usually ignore the effects of Rho. However, they may have a noticeable impact for option traders who have a large exposure to LEAPS options.

Calls have a positive rho value while puts have a negative rho value. We can state that quantitatively as the formulas for Rho are as follows:

Call Rho = KTe-rtN(d2) Put Rho = –KTe-rtN(-d2)

Note that rho is directly related to both the option strike price and the time to expiration. Traders who have traded LEAPS options in times of volatile interest rate changes may note relatively dramatic changes to the values of their overall positions. This is because the greater the time to expiration (T), the greater the rho value. This is easily seen in the above equations.

What Was the Purpose of All of This?

We have received many inquiries regarding the Black Scholes model, so we thought we would take some time to explain the terms in as easy a language as we could. As you can see, some concepts simply do not have easy language attached to them. However, the point was to give the reader some familiarity with the underlying equations that govern option pricing and an understanding of why we can sound so confident when making statements regarding the behavior of option positions.

We have broken down the main equations into their key components. Having done that, we see that there is one recurring theme in both the general equations and in the greeks – the term N – the normal distribution function. Stock prices are independent, identically distributed events, or stated differently, they follow a random walk. It follows that we named our firm after the preferred method of stock behavior. Trading is challenging enough with a proper model. We felt no need to make things harder on ourselves.

We hope that this month’s article has been useful in helping you understand how options are priced and why they are priced the way they are. Our intent was not to give you a lesson on partial differential equations, but rather to give you a rudimentary understanding of what it means when someone refers to Black Scholes, and to show you that the math behind the model can be explained for the most part in simple terminology.

Thank you for reading this article and working to be a better trader. For more information on option greeks and how to profit utilizing them read our text Option Greeks for Profit.