How to Calculate Options Prices and Their Greeks. Ursone Pierino

distribution depicted in the two-dimensional chart 1.2. The best performance is at 50 in the Future, resulting in a P&L of around $45,000 and the worst case scenario is when the Future is at 40 or at 60, when the loss will mount up to around $55,000. However, when the maturity is 365 “days to expiry” and the market starts moving and consequently the volatility will, for instance, increase, the performance will overall be positive, there will be some profit at the 50 level in the Future. This is the smallest amount, but still a few thousand up: a profit of around $35,000 when the Future is at 60 and around $10,000 when the Future is at 40. These P&L numbers keep changing during the lifetime of the strategy. For instance, the $35,000 profit at 60 in the Future (at 365 days to expiry) will turn into a $55,000 loss at expiry when the market stays at that level – losses in time, called the time decay or theta. Also, when the trade has been set up, the P&L of the portfolio increases with higher levels in the Future, so there must be some sort of delta active (change of value of the portfolio in relation to the change of the Future). Next to that, the P&L distribution displays a convex line between 50 and 65 at 365 days to expiry, which means that the delta will change as well; changes in the delta are called the gamma.

Chart 1.3 Combination trade, long 20,000 40 puts and 60 calls, short 10,000 50 puts and 50 calls

      So the P&L distribution of this structure is heavily influenced by its Greeks: the delta, gamma, vega and theta – a very dynamic distribution. Thus, without an understanding of the Greeks this structure would not be understandable when looking at the P&L distribution from a more dimensional perspective.

      It is of utmost importance that one realises that changing market conditions can make an option portfolio with a profitable outlook change into a position with an almost certainly negative P&L; or the other way around, as shown in the example above. Therefore it is a prerequisite that when trading/investing in options one understands the Greeks.

      This book will, in the first chapters, discuss probability distribution, volatility and put call parity, then the main Greeks: delta, gamma, vega and theta. The first order Greeks, together with gamma (a second order Greek), are the most important and will thus be discussed at length. As the other Greeks are derived from these, they will be discussed only briefly, if at all. Once the regular Greeks are understood one can easily ponder the second and third order Greeks and understand how they work.

      In the introduction of a chapter on a Greek, the formula of this Greek will be shown as well. The intention is not to write about mathematics, its purpose is to show how parameters like underlying, volatility and time will influence that specific Greek. So a mathematical equation like the one for gamma,

, should not bring despair. In the chapter itself it will be fully explained.

      In the last chapter, trading strategies will be discussed, from simple strategies towards complex structures. The main importance, though, is that the trader must have a view about the market; without this it is hard to determine which strategy is appropriate to become a potential winner. An option strategy should be the result of careful consideration of the market circumstances. How well the option strategy performs is fully related to the trader making the right assessment on the market's direction or market circumstances. A potential winning option strategy could end up disastrously with an unanticipated adverse market move. Each strategy could be a winner, but at the same time a loser as well.

      The terms “in”, “at” or “out” of the money will be mentioned throughout the book. “At the money” refers to an option which strike is situated at precisely the level of the underlying. When not meant to be precisely at the money, the term “around the at the money” will be applied. “Out of the money” options are calls with higher strikes and puts with lower strikes compared to the at the money strike; “in the money” options are calls with lower strikes and puts with higher strikes compared to the at the money strike.

      The options in the book are treated as European options, hence there will be no early exercise possible (exercising the right entailed by the option before maturity date), as opposed to American options. Obviously, American option prices might differ from European (in relation to dividends and the interest rate level), however discussing this falls beyond the scope of the book. When applying European style there will be no effect on option pricing with regards to dividend.

      For the asset/underlying, a Future has been chosen; it already has a future dividend pay out and the interest rate component incorporated in its value.

      For reasons of simplicity and also for making a better representation of the effect of the Greeks, 10,000 units has been applied as the basis volume for at the money options where each option represents the right to buy (i.e. a call) or sell (i.e. a put) one Future. The 10,000 basis volume could represent a fairly large private investor or a fairly small trader in the real world. For out of the money options, larger quantities will be applied, depending on the face value of the portfolio/position.

      Chapter 2

      The Normal Probability Distribution

The “Bell” curve or Gaussian distribution, called the normal standard distribution, displays how data/observations will be distributed in a specific range with a certain probability. Think of the height of a population; let's assume a group of people where 95 % of all the persons are between 1.10 m and 1.90 m, implying a mean of 1.50 m

. Looking at Chart 2.1, one can see that 95 % of the observations are within 2 standard deviations on either side of the mean (on the chart at 0.00), totalling 4 standard deviations. So 0.80 m (the difference between 1.90 and 1.10) represents 4 standard deviations, resulting in a standard deviation of 0.20 m.

Chart 2.1 Normal probability distribution

      With a mean of 1.50 m and a standard deviation of 0.20 m one could say that there is a likelihood of 68 % for the people to have a height between 1.30 and 1.70 m, a high likelihood of 95 % for people to have a height between 1.10 and 1.90 m and almost certainty, around 99.7 %, for people to have a height between 0.90 and 2.10 m. Or to say it differently; hardly any person is taller than 2.10 m or smaller than 0.90 m.

      STANDARD DEVIATION IN A FINANCIAL MARKET

      The same could be applied to the daily returns of a Future in a financial market. According to its volatility it will have a certain standard deviation. When, for instance, a Future which is trading at 50 with a daily standard deviation of 1 %, one could say that with 256 trading days in a year (365 days minus the weekends and some holidays), in 68 % of these days, being 174 days, the Future will move during each trading day between 0 and 50 cents up or down. Twenty-seven per cent (95 % minus 68 %) of the days (69 days) it will shift between 50 cents and a dollar up or down. There will be around 13 days where the Future will move more than 1 dollar during the day.

      In the financial markets where a Future trades at 50 (hence a mean of 50), a standard deviation of

(or simply

) will be applied, where σ stands for volatility and

stands for the square root of time to maturity (expressed in years).

      THE IMPACT OF VOLATILITY AND TIME ON THE STANDARD DEVIATION

      Volatility is the measure of the variation of a financial asset over a certain time period. An asset with high volatility displays sharp directional moves and large intraday moves; one can think of times when exchanges experience turbulent moments, when for instance geopolitical issues arise and investors seem to be panicking a bit. With low volatility one could think of the infamous summer lull; at some stage markets hardly move for days, volumes are very low and people are not investing during their summer holidays.

      For T, the square root of