### Table of Contents

## Option (finance)

Options are derivatives that allow a market participant to buy or sell an asset in the future against a fee, which is the option price. A “Call” means the intention to buy the asset, which would that if the price that asset is larger than at the time of acquisition, the buyer will make a profit because he can sell the asset at a higher price than he bought it. A “Put” is the opposite, meaning the intention to sell. In that case the buyer of the option benefits from a fall in price. Options have a strike price, which is the price which serves as the “basis” and a maturity at which the option can be excercised. In the case of a European option this can only be at the date of maturity, whereas in the case of an American option, the buyer of the option can excercise at any point between the purchase and the maturity. Options can be hold “long” or “short”, in the case of a “short”, which basically means a sale, the seller takes the opposite position of the buyer. For example in the case of a short call that would mean risking unlimited losses if the price of the underlying increases which is compensated by a fixed profit if the price of the underlying falls. Because of the unlimited risk involved in that position, banks tend to undertake a hedging strategy called “covered short call” which combines the short call with a long position in the underlying.

### Valuation

Because of the non-linear nature of options, valuation is not always simple. A common model to value options is the binomial model. In this case a “tree” of possible price developments is built, assuming a probability of 0.5 that the price goes up or down. The price is then determined by going from the price of the option at maturity (which is easy to determine) backwards through the tree to the present day. A more complex model is the Black-Scholes(and Merton-)-Model which uses standard distribution and the Black-Scholes-Equation to determien the options price. It is the model primarily used by large financial institutions and its prices are very close to the actual observed market prices.

### The greeks

The greeks are greek letters that measure certain sensitivities of options towards different factors. The most basic one is Delta which measures the sensitivity of the option price towards movements in the price of the underlying. A delta of 1 means that if the price of the underlying increases by 1, the price of the option does so, too. Vega is the sensitivity towards changes in the volatility of the underlying. Gamma is the derivative of delta, so the sensitivity towards changes in delta. Theta measures the sensitivity towards the passage of time, it is sometimes called “time decay”. Rho measures the sensitivity towards a change in the interest rate.

### American or European

“European” options can only be excercised at maturity, while “american” options can be excercised at any point in time till maturity is reached. This has implications on the valuation of those options, making an american option more valuable than a european option, ceteris paribus. However, in most situations it is not optimal to excerise an option before its maturity. Exeptions include situations like bankruptcy in the case of a put option or dividends in the case of a call option.

### Option Strategies

With the combination of options or options with underlyings one can theoretically create every pay-off profile possible and satisfy every kind of risk/return appetite. A simple examples would be the “straddle” in which a call and a put on the same underlying with the same strike price are bought. This results in a “V-shaped” payoff profile, which lets its buyer make a profit if there is a large movement in price of the underlying, regardless of the direction. A “Bull spread” involves buying a call with a lower strike price and selling a call with a higher strike price on the same underlying. This results in a fixed profit if the price increases but also limits losses if the price is significantly below both of the strike prices. It can also be modeled using puts.

### Counterparty Risk

An often ignored risk, when it comes to options is the counterparty risk. If one party goes bankrupt and cannot fulfill its obligations, the option becomes worthless. This risk is being tackled by making the parties require to maintain margin accounts and provide collateral.