__Introduction__

A market estimated to be worth up to $600 trillion in notional value [1], the derivatives market is simply a financial behemoth. With contracts deriving from across the asset spectrum, the derivative contract is a useful asset for any portfolio manager and can be used to either speculate on the upward or downward movement of an underlying asset’s price, or for hedging purposes. Such is the importance of the industry that it has been linked to the recent market rally in the S&P 500 via a deluge of bullish sentiment in call options on certain technology stocks – driven by Softbank and retail investors. As broad as the derivatives market is, this article aims to summarise one key type of derivative: the options contract by using examples, as well as put-call parity, and the ‘Greeks’ – key intrinsic mathematical derivatives related to the options contract and other characteristics of the underlying stock. This article focuses specifically on options contracts related to equities.

__Options contracts__

As best defined by Investopedia, an option contract takes two types: a call and a put [2].

A call gives the buyer of a contract the right, but not obligation, to buy the underlying asset at a specific price on or before a specific date. In the case of equities, this is the underlying stock. The aim of this particular trade is for the price of the underlying asset to increase.

For example, an investor can buy a call contract giving them the right – but not obligation – to buy 100 shares of Apple (AAPL) at a price of $125 per share.

A put gives the buyer of the contract the right, but not obligation, to sell the underlying asset (in this case the underlying stock) at a specific price on or before a specific date. If an investor buys a put option, they are speculating that the price of the underlying stock will fall.

Conversely to the previous example, an investor can buy a put contract giving them the right – but not obligation – to sell 100 shares of Apple (AAPL) at a price of $125 per share.

It is worth noting that the profit from trading both call and put options is affected by a premium which buyers pay to sellers. The premium itself is determined by many different factors.

The two types of contract offer effective insurance given the specified price and date. The price specified in this article is defined as the strike price, the relationship between which and the price of the underlying stock is called moneyness. Contracts which are in-the-money are those where the trader would stand to make a profit from the contract at a certain point in time. Conversely, contracts out-of-the money mean that the trader would lose money from the trade in question. For example:

A trader buys a call option for Apple (AAPL) stock with a strike price of $125 expiring on 11/09/2020. The price as of 10/09/2020 (the date of writing this article) was $117.32, meaning the contract is currently out-of-the-money as the strike price is above the current market price. Conversely, a put contract with a strike price of $125 is in-the-money. In this case, there would be an arbitrage opportunity as the trader effectively sells 100 shares of Apple at $125 per share and can then buy Apple shares at $117.32 per share. If the market price was higher than $125, then there would be a reversal of moneyless between the two trades: the aforementioned call would become in-the-money and the put contract would become out-of-the-money.

__The Greeks__

The “Greeks” are key first-, second- and third-order financial derivatives which affect options pricing. Arguably the most important Greek is Delta. This is the rate of change of the option price with respect to changes in the price of the underlying asset, given by:

where V represents the price of the option and S represents the price of the underlying asset.

The value of delta is always between zero and an absolute value of 1. For a long call (and short put) it is positive; conversely, it is negative for a long put (and short call). The closer delta is to one, the more correlated the option price is with the underlying stock price.

With European options, the notion of put-call parity states that the sum of the absolute deltas of a call and a put with the same strike price and expiry date is always one. Therefore, an investor can easily discern the delta of a particular contract by the following formulas assuming the above is correct:

Delta of call = Delta of put + 1

Delta of put = Delta of call – 1

Delta itself is also used to gauge the probability of the contract expiring in the money due to the Mathematical nature of derivative contracts. So a call contract with a delta of 0.25 has a 25% chance of expiring in the money; a put contract with a delta of -0.25 has a 25% chance of expiring in the money.

Vega is another first order derivative, showing the rate of change of the option price with respect to changes in implied volatility, which itself is defined as “the market’s forecast of a likely movement in a (underlying) security’s price” – effectively an estimation rather than indication of future volatility. Prices rise when implied volatility rises, leading to the following combinations:

Type of option trade | Sign of Vega |

Buy call | Positive |

Sell call | Negative |

Buy Put | Negative |

Sell Put | Positive |

Theta is the final first order derivative discussed in this article, which is the rate of change of the option price with respect to the time decay of the option contract. Assuming ceterus paribus, the value of an option will fall as it approaches its expiry, hence Theta tends to be negative. The rationale is fairly simple:

Take two options with similar characteristics. The option with the latest expiry date will tend to be worth more as it has more time to expire in-the-money. Hence the shorter the time to expiry, the less valuable the option contract.

__Why has the importance of options increased in recent months?__

It has recently been revealed that Softbank was behind the significant increase in options purchases in recent months, buying up call contracts worth up to $30bn in notional value [3]. This is in conjunction with an increase in retail trading of options which has led to a significant increase in the value of options traded on US stocks to $300bn a day.

The sellers of these call options – most often investment banks – hedge their exposure. If they sold these contracts “naked”, then unlimited losses would be enabled. However, this sudden increase in the amount of calls bought has led to a proportional increase in purchases of the underlying stocks (to act as a hedge), hence fuelling what many investors saw as a tech bubble.

With the recent effects of excessive options trading on equity markets, central banks and financial regulators may need to up their game in order to avoid potential market bubbles and increased volatility throughout underlying markets.