Continuing our series on options and options trading, we focus on the behavioural patterns associated with options prices and how non-linear behaviour is an important consideration.

In the previous article we introduced the concept of options and how to trade them, along with some of the infrastructure involved.

With the core concepts introduced, we now move on to the basics of option pricing and the consequences of these models. As mentioned before, I will only discuss the pricing models themselves briefly, as that topic is well-covered in many other resources, in far more detail than is possible here.1

# Option Payoff Graphs

Before we discuss option pricing, it is worth discussing the concept of payoff. Simply stated, it is the realised profit earned from owning the option. This concept is a little more subtle than it first appears: when discussing profit do we include or ignore the amount paid to buy the option?

Going on personal experience, this term splits along the line of traders vs quants: traders include the cost of the option when discussing payoffs, but quants do not.

I imagine this is due to focus, traders think about trading profit, so it makes sense to include the cost. Quants try to build models and price instruments, so it is natural to ignore the price paid for the option and focus on the value of the option at expiration - how in the money is it?2

Being a quant, I will largely ignore the price paid for an option when discussing payoffs, unless explicitly noted otherwise.

The most basic distinction for options is whether it is a call or a put, i.e. it confers the right to buy or sell the stock. To get a feel for how options work, it is worth looking at some charts.

Suppose we have a call option for stock XYZ with a strike price of 100 USD: what does the payoff for the option look like as a function of the stock price $S$ of XYZ at expiration? Symbolically, it is $(S - 100)$ but bounded below at 0:

$$\text{payoff} = \text{max}(0, S - 100)$$

Why is this?

If the stock price is above 100 USD, say 105, then we exercise the option to buy XYZ for 100 USD and immediately sell those shares into the market for 105 USD, creating a 5 USD profit. Thus, the call option is worth 5 USD.

On the other hand, if XYZ is less than 100 USD, say 90, then we do not exercise the option as we can buy the stock cheaper than the strike price allows us. Thus, the option is worthless and has a payoff of 0.

The payoff chart for this particular option is shown in the chart below:

A put option is exactly the opposite: In the above scenario but with a put instead of a call, we have:

$$\text{payoff} = \text{max}(0, 100 - S)$$

If the stock is at 105 USD the option is valueless as we only have the right to sell at 100 USD. When the stock is at 90 USD, we can buy the shares for 90, exercise the put and sell them for 100, netting 10 USD profit. Thus, the put is worth 10 USD.

Payoff curves are often overkill for simple options once you have a grasp of the basics, but are still a very useful tool for option spreads: combinations of different option contracts with the same underlying. I will not discuss spreads too much in this series as that is a topic all in itself, but it is worth mentioning a few of the most common here as they are an excellent illustration of the use of payoff curves.

A straddle spread is the combination of a long (or short) call & put option with the same expiration and strike price. Straddle spreads tend to be used to trade volatility - the trader is betting on the size of the movement of the underlying rather than on the direction of the movement.

A call spread is the combination of a long and short call option at different strikes.3 If the long strike is lower than the short strike, it is a bullish spread since it profits from a rise in stock price. If the short strike is lower, it is a bearish spread. In either case, the maximum profit is capped at the difference between the strikes.

# Pricing Options

With the basics dealt with, we now start discussing the interesting parts of options: how to price them, and how their price depends on their inputs.

There are two main models for pricing options: the binomial model and the Black-Scholes model.

The binomial model uses a tree-like structure to model the price changes of the underlying over time, and has the advantage of making early-exercise features easy to implement. It is also computationally fast.

The Black-Scholes model is the workhorse of option pricing theory and results in the famous Black-Scholes partial differential equation for option prices:

$$\frac{dV}{dt} + \frac{1}{2} \sigma^2 S^2 \frac{d^2 V}{dS^2} + rS \frac{dV}{dS} - rV = 0$$

where:

• $V$ is the price of the option
• $S$ is the stock price
• $r$ is the risk-free interest rate
• $t$ is the time to expiration
• $\sigma$ is the expected volatility of the underlying over the lifetime of the option.

Notice the absence of the strike price, $K$ in the equation. Why is this?

When solving the above equation, $K$ is important in setting boundary conditions for the solution, but the fact it is not in the equation itself is a manifestation of the close connection between the price of a call and a put.

We have mentioned the idea of volatility a few times now without actually explaining it. As its name suggests, volatility is a measure of the size of the relative moves of the underlying price.

Quantitative finance models price changes in assets in terms of percentage changes, termed the returns of the asset. This is for a number of reasons:

• Percentage changes are often easier to understand and remember when it comes to interpreting the output of models
• It allows more natural comparisons, without needing to know the underlying price level as a reference point, and allows comparisons across asset classes
• Much of quantitative finance deals with time series, and a sequence of percentage changes tends to behave more independently than a series of price changes, making it more amenable to statistical methods.

A basic assumption of the Black-Scholes model is that returns of the underlying asset prices are distributed according to a lognormal distribution - the volatility of the asset is the standard deviation of this distribution.

Underlying assets that pay dividends can also be accounted for but we will keep things simple and assume no dividends are paid on the underlying.

All of the above inputs to the option pricing model are observable except for the volatility $\sigma$, since we do not know the value of this quantity until after the option expires.4

We ignore the philosophical consequences of this, and focus on the most important practical one: we can use volatility and the option price interchangeably. For a given set of observed values of $S$, $t$, $r$ and $K$, there is a one-to-one relationship between the option price and the volatility.

Used this way, the volatility is called the implied volatility (or implied vol or implied). Thus, implied vols are proxies for option prices, and are independent of the current stock price.

In financial markets, volatility levels tend to be more stable than price levels, so traders quote option prices in terms of implieds,5 plugging in the current stock value when executing the trade to get the dollar amount.

Before we move on to actual calculations, we have one more concept to discuss: units of time.

Markets in various asset classes are open from Monday to Friday. For equities, they open between the hours of 0930 to 1600. It often makes sense to measure time in terms of trading days rather than calendar days. Thus, years have 252 days in them (the approximate count of business days in a year).

As mentioned, volatility is a standard deviation and so scales by the square root of time.6 Thus, if we have a daily volatility of 1%, this becomes $\sqrt{252} = 15.8\%$ in annual terms. For simplicity, this factor is often rounded up to 16, meaning that an asset with an annualised volatility of 16% has a daily volatility of 1%.7

The convention is to express volatility in annualised terms.

# Calculating Option Prices

As mentioned in the previous post, there are many excellent books on option pricing, and I would not do the subject justice.8 Instead I want to focus on using R, and what we can learn from manipulating inputs and observing the effect on the price.

We will use QuantLib, an extensive open-source library containing a cornucopia of useful functions for quantitative finance. It is available in R through the RQuantLib package - this is what we use for all subsequent work.

Suppose we have an American (early-exercisable) call option on a stock XYZ at price level 100 USD with a month to expiry (20 trading days) with 24% annualised volatility. How do we calculate a price for this option?

interest_rate <- 0.01;

implied_vol <- 0.24
t           <- 20 / 252;
K           <- 100;

AmericanOption(type          = 'call'
,underlying    = 100
,strike        = K
,dividendYield = 0
,riskFreeRate  = interest_rate
,maturity      = t
,volatility    = implied_vol)

Concise summary of valuation for AmericanOption
value  delta  gamma   vega  theta    rho divRho
2.7563     NA     NA     NA     NA     NA     NA


According to the output, this option has a value of 2.76 USD.

The other quantities (delta, gamma, vega, theta, rho and divRho) are collectively known as the 'Greeks' and are the analytic derivatives of the price function with respect to the various parameters. They are termed such as they use Greek letters for their symbols, but also I suspect to avoid confusion from over-using the word 'derivative'.

For some reason, QuantLib does not calculate the Greeks using the American option routines, so let's check what we get for a European (non-early exercise) option:

EuropeanOption(type          = 'call'
,underlying    = 100
,strike        = K
,dividendYield = 0
,riskFreeRate  = interest_rate
,maturity      = t
,volatility    = implied_vol)

Concise summary of valuation for EuropeanOption
value    delta    gamma     vega    theta      rho   divRho
2.7563   0.5183   0.0585  11.3110 -17.3402   3.9531  -4.1752


The option price calculated is almost identical, and now we also have values for the Greeks. A quick finite difference calculation shows that the Greeks for the European options are equivalent to those for the equivalent American options, but I leave that as an exercise for the reader.

Now we look at the effect of underlying on the call price. We can check this by calculating this price for a sequence of prices and plotting the output. We plot this against the payoffs to get a sense of perspective for the price.

S_seq <- seq(50, 150, by = 0.1)

price_seq <- sapply(S_seq, function(iterS)
AmericanOption(type          = 'call'
,underlying    = iterS
,strike        = K
,dividendYield = 0
,riskFreeRate  = interest_rate
,maturity      = t
,volatility    = implied_vol)$value)  ## Option Intrinsic Value and Option Time Value Option prices can be split into two components, the intrinsic value (the value of the option were it to be immediately exercised) and the time value of the option which is the remainder. The time value is also referred to as the option premium. Options that are in the money always have positive intrinsic value. Options that are out of the money only have premium in them. If we look at a zoomed-in version of the previous plot, we can see how the premium behaves as the underlying changes. We will discuss this further when we talk about the Greeks, but visuals will work for now. Looking at the plot, we see the premium increases as the underlying approaches the strike price. At the strike price, the premium is at its maximum, and beyond that the intrinsic value becomes non-zero and takes an increasing proportion of the option value. ## Comparing Calls and Puts We mentioned earlier that there is a relationship between the price of a call and a put for a given set of parameters. This relationship, put/call parity, can be expressed in closed form for European options: $$C = P + S - K e^{-rt}$$ where: •$C$is the price of the call option, •$P$is the price of the put option, •$S$is the current price of the underlying, •$K$is the strike price, •$r$is the risk-free interest rate, •$t$is the time to expiration Let us investigate this by calculating calls and puts with strike price$K = 100$, for$S = 90, 100, 110$, starting with$S = 90$:  Put price for S = 90 Concise summary of valuation for EuropeanOption value delta gamma vega theta rho divRho 10.0941 -0.9333 0.0211 3.3041 -3.9811 -7.5796 6.7665 Call price for S = 90 Concise summary of valuation for EuropeanOption value delta gamma vega theta rho divRho 0.1747 0.0667 0.0211 3.3041 -4.9803 0.4695 -0.4835  A quick inspection of the numbers shows that gamma and vega are the same, and$\Delta_C - 1 = \Delta_P$. This will make sense once we have discussed the Greeks. The premium in both options are not the same, the put has$0.0941$of premium compared to$0.1747$in the call. Moving on to$S = 100$:  Put price for S = 100 Concise summary of valuation for EuropeanOption value delta gamma vega theta rho divRho 2.6758 -0.4817 0.0585 11.3110 -16.3410 -4.0959 3.8804 Call price for S = 100 Concise summary of valuation for EuropeanOption value delta gamma vega theta rho divRho 2.7563 0.5183 0.0585 11.3110 -17.3402 3.9531 -4.1752  The pattern for delta, gamma and vega we observed continues for the at-the-money options and the option premia are similar at values$2.6758$and$2.7563$respectively. Finally, we look at$S = 110$. Given the symmetry, it would not be too surprising to see a similar result for$S = 90$but with calls and puts switched:  Put price for S = 110 Concise summary of valuation for EuropeanOption value delta gamma vega theta rho divRho 0.2555 -0.0742 0.0187 4.3842 -6.4467 -0.6782 0.6576 Call price for S = 110 Concise summary of valuation for EuropeanOption value delta gamma vega theta rho divRho 10.3360 0.9258 0.0187 4.3842 -7.4459 7.3708 -8.2035  Now we see the premium in the put and call is$0.2555$and$0.3360$respectively, higher than the premium in the options when$S = 90$. This asymmetry arises as a consequence of the assumptions in Black-Scholes model. The asset is assumed to move according to a lognormal distribution. The volatility is the standard deviation of this distribution, but the mean is not zero, it is slightly positive due to the risk free rate. As a result there is a slight bias upwards in the price movements. Hence the higher premium on the upside prices for$S$.9 We discuss this further once we have an understanding of the Greeks. # The First and Second Derivatives - The 'Greeks' Every discussion of option pricing involves describing the 'Greeks' - derivatives of the option price with respect to different input quantities such as stock price and volatility. ### Delta:$\Delta = \frac{dV}{dS}$The delta,$\Delta$, of an option is the first derivative of the option price with respect to the underlying price. It can be interpreted in two ways: • it is the equivalent amount of shares the option corresponds to at this instant. Being long a 30-delta call option is equivalent to owning 30 shares of the underlying. • the absolute value of the delta is the probability of that option ending up being in the money Numerically, delta varies from -1 to 1, but as option contracts are for 100 shares, we generally multiply the delta by 100. Historically, this made it simpler for traders to know their exact exposure to the underlying stock in terms of shares. Also, the human brain finds it easier to think in terms of whole numbers than with decimals. Calls have a positive delta as they are a bullish instrument and so are like being long the underlying, whereas puts are bearish and have negative deltas. As mentioned, at the money options have strike prices and underlying prices that are equal or almost equal. In delta terms, at-the-money options have delta of$\pm 50$. The strike that is the closest will have delta closest to 50. At-the-money options are important as they are usually the most liquid: most trading activity for a given expiration occurs at the strikes closest to the current price of the underlying. We will discuss reasons for this in the next article. Conversely, options where the delta is close to 100 or -100 behave like the underlying and are often treated as such for risk management purposes. One counter-intuitive result of using options is that deltas tend to be simultaneously the most important Greek from a risk point of view while being the least interesting from a trading and portfolio management point of view. ### Gamma:$\Gamma = \frac{d^2V}{dS^2}$The Gamma,$\Gamma$, of an option is the second derivative of the price with respect to the underlying price - it describes the change in delta as the underlying changes. Similar to delta values, gamma values are multiplied by 100 when quoted, and represent the instantaneous change in delta when the underlying moves by 1 USD. Gamma is hugely important for options and holds a similar position to convexity in bond pricing. Its existence is one of the reasons why the behaviour of option prices can be counter-intuitive - the presence of a non-zero second derivative causes non-linear behaviour. To see the importance of gamma, suppose we have a straddle spread where the strike of the spread is at the money. Such a spread has almost no delta but a lot of gamma. This means that while the delta of the spread is zero right now, it is likely to change significantly as the underlying price moves. The gamma of an option is positive for both calls and puts. ### Vega:$\text{Vega} = \frac{dV}{d\sigma}$The vega of an option is the first derivative of the option price with respect to the implied volatility. It is quoted in units of dollar amounts and is scaled to represent the change in value of an option when the implied vol moves by 1 'vol click' i.e. when the vol moves from 24% to 25%. The RQuantLib functions calculate vega on the scale of changes of 1 unit of vol, 100 vol clicks, so this needs to be accounted for. The vega of an option is always positive. ### Theta:$\Theta = \frac{dV}{dt}$The theta,$\Theta$, of an option is the first derivative of the price with respect to time. For practical reasons it is usually expressed in terms of change in price per day, requiring a transformation of the output of the QuantLib routines as the default amount is the same unit of time for the maturity and volatility (usually annualised in years). Theta represents the passive change in option price if nothing else changes. It is always negative as option values decay as time passes. This is because the reduced lifetime of the options results in less opportunity for the underlying to move, and thus is worth less. ### Rho:$\rho = \frac{dV}{dr}$The rho,$\rho\$, of an option is the first derivative of the option price with respect to the interest rate.

This also needs the QuantLib output to be modified as it is more natural to think in terms of change in price per change in interest rate points (100 basis points).

Interest rate moves tend to be well signposted and most option trading tends to be for short timescales, so rho is not as important for most use cases as its effect is limited. It can be extremely important for very long-dated options though.

# Summary

In this article we introduced payoff graphs and looked at the charts for some option spreads. We also used the QuantLib library in R through the RQuantLib package to calculate option prices.

Finally, we introduced the Greeks and talked a little about why they are important.

In the next article, we will continue this discussion and show how calls and puts behave similarly. We will also talk a little about the consequences of the non-linearity in options.

1. Wilmott on Quantitative Finance is an excellent resource for this.

2. An option that is in-the-money is an option contract where the exercise value of the option is positive. If the exercise value is negative, the option is out-of-the-money. If the strike price and underlying price are almost the same, the option is at-the-money.

3. The options can also have different expirations, though this is generally termed a calendar spread.

4. Even then, it is probably more accurate to say that the 'true' value stays latent, and we instead observe a realization of it.

5. Some traders at the desks of the larger banks had a reputation for trying to get you to honour dollar prices on trades, even when based on stale stock prices. A common response was "why don't I just write you a cheque right now and save us all the time?"

6. It is probably no surprise to learn this is a large simplification: asset volatility does not scale smoothly across time. Intra-day volatility is often higher than that measured at longer time scales.

7. A common misinterpretation of this is that the 'average move' of the asset is then 1%, which is false. It is more like 0.80%. The correct interpretation is that you expect the movement to be 1% or less two-thirds of the time.

8. Not to mention that I would probably get some technical details wrong and look stupid...

9. At first glance, this may seem to only apply to calls, as the value of a put has a negative relationship to the underlying but that is not the case due to parity. This will be discussed more in the third article.

Cover image by marco_1186 via Flickr