# Option (finance)

An option referred to in the economic right, a certain thing at a later date at an agreed price to buy or sell. Therefore it is also called conditional future. It is expressly a right and not a duty, ie the option holder who has bought the option at a specified price ( the premium ) from the option writer ( seller) of, decides unilaterally whether to exercise the option against the writer or the option to expire can. A great importance have options in the area of financial transactions.

- 5.1 In the money
- 5.2 Out of the money
- 5.3 On the Money

- 6.1 Delta
- 6.2 gamma
- 6.3 Theta
- 6.4 Vega
- 6.5 Rho
- 6.6 lever
- 6.7 Omega

- 7.1 factors
- 7.2 Asymmetric profit and loss
- 7.3 Calculation of the option price 7.3.1 Distribution -free no-arbitrage relations 7.3.1.1 put-call parity

- 7.3.2.1 Consideration of Interest

- 8.1 dividends

## Survey

The standard options ( also English plain vanilla options called ) are call options (calls) or sell options ( puts). The buyer has the right - but not the obligation - to exercise certain points in time, a certain amount of reference value at a predetermined exercise price ( strike or English ) to buy or sell. The seller of the option (also writer, writer, artist) receives the purchase price of the option. He is in the case of exercise required to the underlying asset at the predetermined price to sell or to buy.

One differentiates between options, as with all contracts, two types of exercise: payment and delivery (English physical delivery) and cash. Is payment and delivery has been agreed, delivers a Party (for a put the owner in a call, the writer ) the underlying asset, the other Party shall pay the exercise price as the purchase price. A cash payment, the writer will pay the difference in value that results from the exercise price and the market price of the underlying asset at the exercise date, the option holder. The reverse case, in which the owner pays to the writer, in the normal case can not occur because the owner option is not exercised in this case. The economic benefit to the owner is equal in both cases, with the exception of transaction, storage and delivery costs.

### Option types

In addition to the standard options still exist exotic options whose payoff profile depends not only on the difference between the price and the exercise price.

### Exercise types

Depending on the exercise dates, a distinction

- European option: the option can be exercised only on the maturity date;
- American option: the option can be exercised on any trading day before the due date;
- Bermuda option: the option can be exercised at a predetermined multiple times.

## Black- Scholes model

In 1973 the American scientist Fischer Black and Myron Scholes published almost simultaneously with Robert C. Merton in two independent articles methods for the exact determination of the "true " value of an option. Scholes and Merton received the 1997 Prize of the Swedish Riksbank for Economic Sciences in Memory of Alfred Nobel, often referred to as the Nobel Prize, "for a new method to determine the value of derivatives ", the Black- Scholes model. Black could not accept the award, he died in August 1995.

## Trade

An option can be used as an individual contract between the Optionee and the encoder option ( option writer ) to be completed. It is, as such, open to choice. Such concluded directly between two parties options is commonly called OTC options.

The largest part of world trade in options, however, consists standardized contracts that are traded on futures exchanges such as the EUREX in Europe or the CBOT in the USA. The standardization is to increase the liquidity of the options.

Ultimately, options still designed as securities are ( Warrant).

## Underlying

In the financial markets can be traded the following underlyings:

- Shares
- Indices
- Currencies
- Bonds
- Exchange - traded fund
- Raw materials
- Food
- Electrical energy
- Weather
- Other options

For closed-loop options trading, it is essential that the underlying securities are traded on liquid markets in order to determine at all times the value of the option. In principle, however, it is also possible that the underlying can be chosen arbitrarily, as long as it is possible to determine the necessary variables described in section sensitivities and ratios. These derivatives are, however, offered only by authorized dealers such as investment banks or brokers over the counter in OTC trading.

## Terms

### In the money

In the money (English in the money ) denotes an option, in which the market price of the underlying asset is higher than the exercise price (call option ) or the market price of the underlying asset is lower than the exercise price ( put option ).

### Out of the money

- Out of the money (English out of the money ) is an option that has no intrinsic value
- A call option is out of the money when the market price of the underlying asset is less than the exercise price
- A put option is out of the money when the market price of the underlying asset is greater than the exercise price

### On the money

An option is in the money (English at the money ) if the market price of the Underlying is equal to or nearly equal to the exercise price.

If the exercise price and compare with the spot price, it is called at- the-money spot. If the strike price with the same maturity forward rate is compared, then one speaks of at- the-money -forward.

See also: moneyness.

## Sensitivities and Key Figures - the so-called " Greeks "

### Delta

A measure of sensitivity, indicating what impact the price of the Underlying on the value of the option. The Delta is mathematically the first derivative of the option price according to the price of the underlying. So does a delta of 0.5, that a change of the underlying asset at € 1 ( in linear approximation ) causes a change in the option price of 50 cents. Delta is particularly important in connection with the so-called delta hedging.

### Gamma

The gamma of an option indicates how strongly their delta ( in linear approximation ) changes when the price of the underlying changes by one unit and all other variables do not change. Mathematically, the gamma is the second derivative of the option price according to the price of the underlying. For the holder of the option (ie both long call and long put for ) is always that gamma ≥ 0. The index is also used in hedging strategies in the form of gamma hedging into account.

### Theta

The theta of an option indicates how strongly the theoretical value of an option changes when the remaining term shortened by one day, and all other variables remain constant. For the holder of the option Theta is usually negative, a shorter period remaining thus always means a lower theoretical value.

### Vega

The Vega (sometimes called lambda or kappa, since Vega is no letter of the Greek alphabet ) of an option indicates how much the value of the option changes when the volatility of the underlying asset changes by one percentage point and all other variables remain constant.

### Rho

The Rho of an option indicates how much the value of the option changes when changes of the risk-free rate on the market by one percentage point. For call options Rho is positive, negative for put options.

### Lever

The lever is calculated by dividing the current price of the underlying by the current price of the option. The option relates to a multiple or fraction of the Underlying, this factor must be taken into account in the calculation. This is referred to by the reference ratio ( Ratio).

### Omega

Is obtained by multiplying the delta with the current lever a new lever size, the " effective lever " is found in the rate tables usually under the name Omega or. An option with a current leverage of 10 and a delta of 50 % has therefore "only" Omega of 5, that the glow increases by about 5% when the base increases by 1 %. However, here is again to be noted that both the Delta and the Omega, and most other indicators are constantly changing. Nevertheless, the Omega offers a relatively good picture of the opportunities the appropriate option.

## Assessment

### Factors

The price of an option depends firstly on its equipment features from here

- The current price of the underlying asset,
- The exercise price,
- The remaining term until the exercise date,

On the other hand of the underlying model for the future development of the underlying and other market parameters. Under the Black-Scholes model, the other variables

- The volatility of the underlying,
- The risk-free, short-term interest rate in the market,
- Expected dividend payments over the life- time.

The current price of the underlying asset and the exercise price determine the intrinsic value of the option. The intrinsic value is the difference between the exercise price and the price of the underlying. In the case of a call option in relation to an underlying with a current value of € 100 and an exercise price of 90 €, the intrinsic value is 10 €. In the case of a put option is the intrinsic value in the case described 0

In particular, the volatility has a major impact on the value of the option. The greater the fluctuations of the price, the higher the probability that the value of the Underlying changed significantly and thus increases the intrinsic value of the option or decreases. The general rule is that a higher volatility has a positive impact on the value of the option. In extreme limiting cases but it can behave exactly the opposite.

The remaining term affects the value of the option similar to the volatility. The more time is available until the exercise date, the higher the likelihood that changes the intrinsic value of the option. Part of the value of the option consists of this value. It is theoretically possible to calculate the value by comparing two options that are distinguished only by their duration and are otherwise identical. However, this requires the unrealistic case of a nearly perfect capital market.

The increase of the risk-free rate has a positive effect on the value of call options (call option ) and a negative effect on the value of put options ( a put option ), because according to established valuation methods, the probability of a price or value increase of Basisguts to the risk-free rate is coupled. The reason is that the money does not have to be invested in an underlying asset, thanks to the call, can be interest-bearing. The higher the interest of an alternative investment, the more attractive is the purchase of a call. With rising interest rates, this increases the amount exceeding the NAV value of the option, the time value. When you put the situation is reversed: the higher the interest rate, the lower the value of the put, because you theoretically have the underlying of the option would have to take the right to sell able to claim.

Dividend payments in the case of stock options have a negative impact on the value of a call option compared to the same share at Dividendenlosigkeit, as will be omitted dividends during the holding periods, which can theoretically be collected by exercising the option. Conversely, they have compared to the same dividend-paying stock has a positive impact on the value of a put option, because during the holding periods and no dividends can be collected, which states the option holder exercised immediately. In the case of options on currencies or commodities, the underlying interest rate of the currency or the " convenience yield " is used instead of dividends.

### Asymmetrical profit and loss

In the event of an adverse to him development in the price of the underlying asset of the owner of the option will not exercise his right to leave the option to expire. He loses more than the option price - thus realized a total loss - but has the possibility of unlimited profit with purchase options. This means that the potential losses to the seller in purchase options are unlimited. However, you might this loss as " loss of profit " ( covered short call ) look, unless the seller of the call option is not to buy the property of the respective Underlying (must so to fulfill and then deliver - uncovered sale of a call option ( uncovered short call ), where uncovering means that the position consists of only one instrument).

The following diagrams illustrate the asymmetric payoff structure. The options shown are identical in all factors. It is important for the understanding that the buyer of an option receives a long position and the seller of an option enters into a short position. In all four cases, the value of the option and the exercise price 10 100

In the previous diagram can be seen that the buyer (long) of the call has a maximum loss of 10, however, has unlimited profit opportunities. In contrast, the seller ( short) has a maximum gain of 10 with unlimited losses.

In the case of a put, the buyer (long) also has a maximum loss of 10 An error is often the transfer of the unlimited profit potential of the option to purchase the put option. The Basisgut but may possibly take the market value of zero. Characterized the maximum win is limited to the case of a rate of zero. Just like the call the seller ( short) has a maximum gain of 10, now with only limited losses if the price of the underlying null accepts. The difference between call and put is in how changed the payout is the underlying, and in the limitation of the maximum income / loss on sale of options.

### Calculation of the option price

In option pricing theory, there are basically two approaches to determine the fair option price:

- With the help of estimates without making assumptions about possible future share prices and their probabilities ( distribution -free no-arbitrage relations, See: Option Pricing Theory )
- Due to possible stock prices and risk-neutral probabilities. These include the binomial model and the Black- Scholes model

In principle, it is possible to model the stochastic processes which determine the price of the underlying asset in different ways. You can map these processes analytically in continuous time with differential equations and analytic discrete-time with Binomialbäumen. An analytical solution is not possible by future simulations.

The best known analytically continuous-time model is the model of Black and Scholes. The best known analytically discrete-time model is the Cox-Ross- Rubinstein model. A common simulation method is the Monte Carlo simulation.

#### Distribution -free no-arbitrage relations

A call option can not be worth more than the base value. Assume that the underlying security is traded today to 80 € and someone offers on this underlying an option that costs 90 €. No one would want to buy this option because the underlying is even more cheaply, which is obviously worth more than the option. For example, since a share include as underlying any obligations, these can be bought and disposed of. If necessary, they will be brought out again. This is an eternal option with exercise price € 80; a more valuable option is not feasible, so that the (call ) option can never be worth more than the underlying.

A put option can not be worth more than the present value of the exercise price. No one would for the right to be allowed to sell some 80 € to spend more than 80 €. Financial Mathematics correctly they must be € 80 discounted to present cash value.

This value limits are the starting point for determining the value of a European option, the put-call parity.

##### Put-call parity

The put -call parity is a relationship between the price of a European call and the price of a European put, if both have the same strike price and the same due date:

In which

- P: Price of European put option
- : Share Performance
- C: Price of European call option
- K: strike price of the call and put option
- R: risk-free interest rate
- T: Number of years
- D: Discounted dividend payments during the term of the options

If the put-call parity is violated, risk-free arbitrage profits would be possible.

Using the put-call parity allows the equivalence between simple option strategies and option positions show.

- Covered Call corresponds Put short, demonstrated in this example relation: i.e. share long and short call ( Covered Call ) is equal to a short put plus a cash amount.
- Opposite position (reverse hedge) of covered call corresponds to long put
- Protective Put corresponds to long call
- Counter-position to the protective put is the call short

#### Black- Scholes

The Black-Scholes formula for the value of European call and put options on underlying assets excluding dividend payments are

In which

In this formula S is the current price of the underlying asset, X is the exercise price, r the risk-free interest rate, T is the lifetime of the option in years, σ the volatility of S and the cumulative probability that a variable with a standard normal distribution is less than x.

If the underlying asset does not pay dividends, the price of an American call option is equal to the price of a European call option. The formula for c hence gives the value of an American call option with the same ratios under the assumption that the underlying asset pays no dividends. There is no analytical solution for the value of an American put option.

##### Considering interest

The gain or loss of options can be determined in consideration of interest as:

Being linear, as here, the money market interest rate will be used.

### Dilution protection

The assessment methods is implicitly assumed that the option right not lose by corporate actions of the corporation in value ( dilute ) can. This is guaranteed by the so-called anti-dilution protection with option trading.

## Optimal exercise

American options can be exerted at multiple time points. The exercise behavior is influenced by the factors interest on the strike price, a flexibility effect and the dividend. To differentiate is by calls and puts.

A positive effect means that is to be exercised, a negative effect that it is rewarding to be seen.

When interest on the strike price of the call is to effect negative, on the other hand, puts positive. The flexibility effect acts both negatively on calls as well as to put. The dividend event has a positive effect on calls, but a negative on puts.

### Dividends

- If no dividend paid, the exercise of a call option at the end of the term is always optimal.
- In the Waiting dividend payment by the last date for puts is still optimal.

## Criticism of the standard valuation methods

Usually, the evaluation methods are based on the assumption that changes in the value normally distributed ( " bell curve ") and are independent. According to Benoît Mandelbrot all are building upon models and valuation formulas (for example, the above Black-Scholes ) wrong. His investigations revealed that the rate changes exponentially distributed and interdependent, leading to much more violent price swings than provide the standard models.