Davis
Edwards on Trading 



Fair Value and Hedging 
Financial Risk Management 
Model Risk Management 




Introduction to Options
This web page introduces some basic concepts associated with one type of financial contract called an option. Options are a type of derivative that gives one trader the right to buy or sell an asset at a fixed price at some point in the future.
Portions of this discussion were adapted from the book, “Energy Trading and Investing, 2nd Edition” by Davis W. Edwards, published by McGrawHill 2016.
Understanding
option payoff diagrams
Understanding
Early Exercise: European and American Options
Closed
form option pricing formulas
An option is a contract (or part of a contract) between two traders. In their simplest form, options gives one of the traders (the option buyer) the right, but not the obligation, to buy or sell property at some future date (the expiration date) at a fixed price (the strike price). If the option buyer decides to buy or sell the underlying property, the trader is said to be exercising their option. The property that may be bought or sold as a result of the option is called the underlying asset, or more commonly, the underlying. The right to buy is called a call option. The right to sell is called a put option. Options have an upfront cost, called a premium, which is paid when the buyer purchases the option. For the buyer, options have limited downside risk. Buyers will either lose their premium or they will make a profit. The seller earns the premium and carries all of the risk. The seller of the option contract also called the option writer. Although options sound like a good deal to the buyer, there is the issue of the premium. The premium can be very large and modern option pricing theory has spent a lot of time figuring out how much premium is needed to compensate the option seller for the risk of selling the option.
In addition to being a traded product, options are important because they can represent many types of investment decisions. For example, a combination oil well and refinery might be able to produce gasoline at $4.00/gallon. Owning this installation would give the owner the option to “buy” gasoline at $4.00/gallon for immediate resale. By comparing the cost of building and operating the installation to the economic value of the option, it is possible to determine whether this would be a good investment. Using the mathematics developed in this section to value physical assets is real option modeling.
Financial option contracts are an allornothing investment. It is possible to buy a million dollars’ worth of options and lose everything. This is like buying insurance. Most often, the purchaser will pay a premium and have the contract expire worthless. Occasionally, the contract will pay off big when an unusual event occurs. Even though the size of the downside is small (losing the premium) compared to the potential upside (a huge profit), the odds of making a profit are stacked against the buyer. Similar to buying insurance, buying options is generally unprofitable. It typically only makes sense as part of a broader strategy. The option writer is taking on a risk from the buyer and needs to be compensated for taking that risk.
From a transaction standpoint, option trading requires both a buyer and a seller. The seller (the option writer) takes on the possibility of a big loss in exchange for money up front. The buyer pays a premium to the writer for that service. If the option pays off, the writer will need to find the cash to pay the buyer. With options, money is not magically created; it is simply transferred between the two parties. The option buyer is long the option (because the buyer benefits when the option increases in value) or being long volatility (since a higher probability of rare events means an opportunity for a big profit). The option writer is short the option (seller benefits if the price of the option decreases in value) or being short volatility (since a higher probability of rare events creates more risk to the seller).
Although this page introduces the mathematics necessary to price options, there are wellknown mathematical formulas available to price all common options (and most of the uncommon ones). An example of an option pricing library can be found on the “Closed form option pricing formulas” part of this web page.
The net amount of money transferred between the buyer and seller at the expiration is the payoff of the option. Every option has a strike price  a fixed price at which trading can occur in the future. For example, a call option involves the right to buy the underlying asset at the strike price. The owner of a call option benefits when the price of the underlying asset rises above the strike price. This allows the owner to buy at the underlying asset at a lower price than is otherwise available. The owner can also immediate profit by reselling the asset at the current price after buying it at the (lower) strike price.
A put option works similarly. A put option gives the owner of the option the right to sell the underlying asset at a fixed price. If the market price of the underlying is greater than the fixed price, a put option is worthless. No one will willingly sell at a lower price than necessary. However, if the fixed price is higher than the market price, the put buyer makes a profit by selling at a higher price.
Mathematically, these payoffs can be represented as formulas (See Figure 1 – Option Payoff Formulas).
Call Option 
Put Option 
Call Payoff = Max(0, Asset Price – Strike Price) 
Put Payoff = Max(0, Strike Price – Asset Price) 
From the buyer’s perspective, call options give the right to buy an asset at the strike price at some point in the future. 
From the seller’s perspective, put options give the right to sell an asset at the strike price at some point in the future. 
Figure 1  Option Payoff Formulas
A graphical depiction of these formulas is called a payoff diagram. These diagrams will often illustrate payoffs for both the buyer and a seller. Since the payoff is a transfer of money from one trader to another, the seller will have the negative of the buyer’s payoff. These graphs will look like a dogleg or a hockey stick. On one side of the strike price, the payoff will be zero. On the other side, it will linearly increase. Additionally, sometimes these diagrams will include the premium that was paid from the buyer to the seller. This will shift the payoff at expiration vertically on the graph. See Figure 2 – Call Options Payoff and Figure 3 – Put Option Payoff for examples of these diagrams.
Figure 2  Call Option Payoff
Figure 3  Put Option Payoff
It is possible to combine the payoffs from multiple option transactions. This is the basis of many types of option trades. The most important combination of payoffs is the combination of put and call options to form forwards. The ability to combine calls and puts to form a forward forces a link between these two products. For example, by simultaneously buying a call and selling a put, a trader can replicate the payoff from owning a forward (see Figure 4 – Combining Option Payoffs)
Figure 4  Combining Option Payoffs
The most important implication of this is that if a trader knows the price of a call option and a forward contract, then it is possible to determine the price of a put option. In fact, knowing the prices of any two of the three instruments makes it possible to calculate the price of the third.
Options are often described in terms of their intrinsic and extrinsic value. The intrinsic value is the money that could be obtained by exercising the option immediately—forgoing the chance at more money. The extrinsic value of an option is the value of the option that comes from holding on to it longer.
For example, consider a $1 lottery ticket that pays the winner $1 million and gives the owner an ability to flip a coin for more money. If the ticket holder wins the coin toss after winning the first million, he gets another $2 million. However, if he loses, he gives up the initial $1 million. After winning the first million, the lottery ticket is no longer riskless. The owner has the option of taking a million dollars and walking away. This is now the intrinsic value of the option. The owner of the ticket no longer has a dollar at stake—he has a million dollars at stake! The higher the intrinsic value of the option the more risk is involved.
Continuing this example, a savvy trader might realize that a 50/50 chance of making $3 million compared to losing $1 million is a profitable investment. If losing a million dollars wasn’t a hardship (maybe a multibillionaire was playing the game), it would make good economic sense to take the coin toss every time it was offered. The extrinsic value of the option is the value of not exercising the option. Mathematically, the coin flip is really a 50/50 chance of making $3 million or going home with no money. There is also an option to walk away with $1 million. The coin flip represents a 50 percent chance of winning $3 million, giving the lottery ticket an expected value of $1.5 million. Since exercising the option immediately would give $1 million risk free, the value of taking the extra coin flip (the extrinsic value) is the value of the entire lottery ticket ($1.5 million) less the intrinsic value ($1 million), or approximately $500,000 dollars.
Not wanting to risk losing a million dollars, and being an extrasavvy trader, the owner of the winning lottery ticket might decide to sell the lottery ticket instead of exercising his option to take the riskfree million dollars. For example, the owner of the lottery ticket might decide to offer it for sale for $1.4 million. After all, a 50/50 chance at $3 million for $1.4 million is still a profitable investment. In this way, the owner of the option could get most of the benefit from the coin flip without actually exposing himself or herself to the risk of losing the million dollars.
From an economic perspective, the initial cost of the lottery ticket determines whether playing this lottery is a good or bad investment. For example, a free ticket would be a great investment. However, as the price of the ticket rises, this would become a progressively worse investment. The potential for a big payoff at low risk does not make a good investment by itself. If the upfront costs are too high, both lottery tickets and options become riskfree ways to lose money!
With many options, it is possible to exercise the option only at maturity. This type of option is a European option. With other options, American options, it is possible to exercise the option early. This will lock in some of the profits (the intrinsic value) but give up any possibility of any additional profits (the extrinsic value). If there is a possibility that it will be more profitable to exercise the option than sell it, an American option will have more value than a corresponding European option. Early exercise typically occurs only when an option is in the money. If an option is out of the money, there is usually no reason to exercise early  it would be better to sell the option (in the case of a put option, to sell the option and the underlying asset).
Early exercise worthwhile in some circumstances. First, particularly when interest rates are high, money today is worth more than money in the future. Second, some assets will pay intermediate cash flows (like a stock dividend) or have value outside of their price (for example, voting rights). These payments will only benefit the owner of the asset, not owners of options on the asset. As a result, the decision of whether to exercise early is primarily a question of interest rates and carrying costs.
Carrying costs, or cost of carry, is a term that means intermediate cash flows (or other value) that is a result of holding an asset. For example, dividends on stocks are a positive cost of carry (owning the asset gives the owner a cash flow). Since an option has some value (the extrinsic value) that would be given up by exercising the option, exercising an option prior to maturity is a trade off between the option's extrinsic value (the remaining optionality) and the combined benefits of holding cash now (time value of money) and any benefit of holding the asset (carrying costs).
The early exercise feature of American equity put options may have value when:
· The cost of carry on the asset is low  preferably zero or negative.
· Interest rates are high
· The option is in the money
The early exercise feature of American equity call options may have value when:
· The cost of carry on the asset is positive
· Interest rates are low or negative
· The option is in the money
With commodities, things are a slightly different. There is typically no cost of carry since the underlying is a forward or a futures contract. It does not cost any money to enter an atthemoney commodity forward, swap, or futures contract. Additionally, these contracts do not have any intermediate cash flows. As a result, the primary benefit of early exercise is to get cash immediately (exercising an inthemoney option) rather than cash in the future. In high interest rate environments, the money received immediately may exceed the extrinsic value of the contract. Mathematically, this difference occurs because the strike price is not being present valued for immediate execution (it is specified in the contract as a fixed number) but the payoff of a European option is discounted. Because of that, early exercise is uncommon for most commodity options. Typically, it only occurs when interest rates are high. Generally, interest rates have to be higher than 15%20% for American commodity options to differ substantially in value from European options with the same terms.
The early exercise feature of American commodity options has value when:
· Interest rates are high
· Volatility is low (lowers the extrinsic value making it more likely that timevalueofmoney exceeds the extrinsic value)
· The option is in the money
Option pricing is a bit nonintuitive.
Modern option theory involves replicating the payoff of an option by holding some amount of the underlying security. Adopting this approach fundamentally shifted how the financial market approached options. Previously, valuation focused on trying to forecast the future. From the point of view of an option seller, shifting away from price forecasting is a big deal. Option sellers don’t have the ability to pick and choose what they trade. They have to be willing to trade whatever product the buyer is looking to purchase. Consistently predicting the future is a nearly impossible task if you have to predict the price of every asset at every time horizon. There are two steps to most option pricing models. The first is predicting the statistical distribution of likely movements in the underlying price. The second step is determining how to replicate the option under those probability conditions.
It is often easier to predict a distribution of likely future prices than it is to predict a single price. For example, an probabilistic description of prices might be that 60% of the time, tomorrow’s prices will be within 20 cents of today’s price, 20% of the time the price will move between 20 and 70 cents, and that 20% of the time, the price will move more the 70 cents. This type of estimate is much more likely to be accurate than a directional estimates like, ‘the price of oil will rise tomorrow’.
Most models start with the assumption that today’s price for a contract is the best consensus estimate of tomorrow’s price for the same contract. A riskfree profit argument is the most common justification for this assumption. For example, if the price of a natural gas future was $10 today and a trader knew that it would be $20 the next day, the trader could buy the contract today and resell it the next. At that point, the contract could be sold for a riskfree profit. This assumption that today’s price is the best estimate of tomorrow’s price leads to the conclusion that the returns (percent changes to the current price) are distributed around today’s price.
A second common assumption is that volatility of returns will stay constant over the life of the option. This is mostly because this is mathematically convenient while being accurate for most products. For example, that the magnitude of returns will be the same a week from now as it is tomorrow. Combining these two assumptions makes it relatively easy to construct a model of likely future underlying prices. The simplest model of future prices is a binomial tree (see Figure 5  A Binomial Tree). A binomial tree assumes that the underlying prices can rise or fall a certain percent every day. That percent stays constant over the life of the option. Prices start at today’s price, and slowly diverge from it over time.
Figure 5  A Binomial Tree
In a binomial tree model, the price of the underlying changes over time. In the first period, the example starts at $100, and will go to either $110 or $85 in the next period. Although downside moves may be bigger than upside moves, there is a higher probability that prices will rise. As a result, the average expected price in periods 2 and 3 is still $100.
When described in mathematical literature, mathematical models are described by variables, see Figure 6  A Binomial Tree (with more general notation). For example, the probability that the price rises might be denoted by p and the probability that prices fall is denoted as (1 – p). The amount that prices raise might be abbreviated u, and the amount that prices fall might be abbreviated v. The price of the underlying is Sx with the subscript indicating it’s a price at a specific period. Also, the first period will usually be period 0, the second period 1, and so on.
Figure 6  A Binomial Tree (with more general notation)
A model of interest rates can be calculated using the same binomial framework that was used for prices. With constant interest rates, this graph isn’t terribly interesting, but will be used later on to figure out correct value for the option premium (see Figure 6 – Example of a Binomial Tree for Interest Rates). In this example, with a 5% interest rate between periods, a dollar at time_{t} is always worth 1.05 at Time_{t+1}.
Figure 7  Example of a Binomial Tree for Interest Rates
The key insight to option pricing is that it is possible to combine underlying and interest rates levels (bond prices) to replicate an option payout if there is a way to compare their payoffs under likely future conditions. It is not necessary to predict actual future prices. It is sufficient to predict a likely distribution of possibilities like in a binomial tree model. Then, the underlying prices and riskfree return can be compared to the payoff of a call option in the same circumstances. The combination of underlying and interest rate payoffs replicate the payoff of the option (see Figure 8  Replication of a Call Option). Using the values from the earlier examples:
Figure 8  Replication of a Call Option
The temptation is to say the value of the option is $6, because it has a 60% chance of paying $10. However, and this is the key to option pricing, the option seller is taking on more risk than the option buyer. If the prediction of prices is wrong, the option seller has more money at stake. As a result, a $6 price benefits the buyer more than the seller. The chance for the option seller to take a huge loss doesn’t really show up on two period binomial models.
Predicting future prices is both subjective and inaccurate. Option sellers need to have some assurance that they are being fairly compensated for the risk that they are taking on. This is done by calculating the cost of eliminating the risk of writing the option contract. The risk can be removed by replicating the option with other instruments. Most commonly, these instruments are the underlying security and a riskfree investment like a government bond.
The general structure of the trade would be to purchase some quantity of underlying and riskfree investment at the starting period and use those profits to offset losses from the option (See Figure 9  An Arbitrage Relationship). The underlying and riskfree investment positions will be liquidated at the expiration of the option. There are two unknowns (the quantity of the underlying, and the quantity of the risk free investment) and two equations (the payoff of the option in positive and negative cases). The goal is to buy a certain quantity of the underlying and risk free investment at time zero that will duplicate the option payoff at time 1.
Figure 9  An Arbitrage Relationship
To solve this equation it will be necessary to eliminate one of the variables from the equation. This can be done by adding the two equations. The easiest variable to eliminate will be B (the units of riskfree investment) since its payoff is the same in every ending scenario. Eliminating B can be done by subtracting the first equation from the second (See Figure 10  Solve For Underlying).
Figure 10  Solve for Underlying
Plugging this value for A into either equation will allows us to finish solving the problem by calculating a value for B (See Figure 11  Solving for Risk Free Investment)
Figure 11  Solving for RiskFree Investment
The value of the option is the cost of entering the trades at the initial period. This can be done by calculating the price of purchasing the right amount of the underlying and riskfree investment at the starting point. For example, if one unit of underlying cost $100, it would cost $40 to purchase .4 units of underlying. Combining the two numbers ($40 – $32.88) gives a $7.22 cost as the fair value of the option (See Figure 12  Fair Value of Option).
Figure 12  Fair Value of Option
Some commonly used terms that describe options:
· American Option. The term American applied to an option means an option that can be exercised at any time.
· Asian Option. An Asian option is an average price option. At expiration, the value of an Asian option depends on an average of the underlying’s price over some period of time. Typically, Asian options cannot be exercised early. For example, an Asian option might pay the difference between a strike price and the average cost of peak power over a month.
· At the Money. An option where the strike price of the option equals the current price of the underlying. Exercising an atthemoney option will not result in zero profit – it will be an exchange of two equal quantities.
· Derivative. In its most general form, a derivative is any financial contract whose value is derived from the price of something else. Options are a common type of derivative.
· European Option. The term European applied to an option means that the option can only be exercised at the expiration. A European option cannot be exercised early.
· Extrinsic Value. The time value of the option. This can be found by subtracting the intrinsic value of the option from the current price of the option.
· In the Money. An option with a positive intrinsic value. An option that would be profitable to exercise immediately.
· Intrinsic Value. The amount of money that the option buyer would receive if the option was immediately exercised.
· Out of the Money. An option with zero intrinsic value. Alternately, an option that would lose money if exercised immediately.
· Premium. The money paid by the purchaser of an option to the seller of an option.
· Strike Price. The exercise price of an option contract. Alternately, the price where the call buyer can purchase the underlying asset or the put buyer can sell the underlying asset.
· Synthetic Option. The process of duplicating the payoff of an option by using a combination of other options and forwards.
· Underlying. An asset that is used to determine the price of a derivative. It is possible for the underlying to be a derivative of some type (like a forward or a future), and for options to have more than one underlying (spread options).
I have written Pythonbased option pricing library. It contains additional documentation around closed form solutions (formulas that can be evaluated in a finite number of steps) for a variety of common financial instruments. This library includes European Options, American Options, European Options, and Spread Options as well as Implied Volatility Calculators. The code is written as a Jupyter notebook which combines documentation with code. If you have not used Python or Jupyter – these are phenomenal tools for modeling and data science. Learning these tools would be an excellent investment of time for anyone interested in financial modeling. You can view the library as HTML (Closed Form Option Pricing) or download a Jupyter notebook from my GitHub site (https://github.com/dedwards25/Python_Option_Pricing).