23
Feb 18

## Call options and probabilistic intuition - dependence on time

As before, we change one variable - in this case it's time to expiration $T$, while holding others fixed, and see how the call price changes. We've assumed that at each $T$, the stock price distribution is triangular, symmetric and centered at $S(0)$. This time on top of that assumption we add the following.

Assumption 2. The standard deviation of the stock price distribution is proportional to $T$.

This standard deviation is a measure of riskiness and unpredictability of the stock price. The farther into the future, the more difficult it is to predict.

## Areas of influence of the initial stock price

Figure 1. Areas of influence for two times

As a result of our assumptions, we can visualize areas of influence for two times in Figure 1. Points $S(0),\ A_1,\ A_2$ are on the same straight line. Similarly, points $S(0),\ B_1,\ B_2$ are on the same straight line. This is because by Assumption 2, the triangles $\triangle S(0),A_1,B_1$ and $\triangle S(0),A_2,B_2$ are similar. Note that the triangle $\triangle A_2, B_2,C_2$ is flatter than $\triangle A_1, B_1,C_1$ because in both cases the area should be 1.

## Finding ranges of initial stock prices that can lead to ITM areas

Figure 2. Initial price ranges

In Figure 2 we find initial price ranges that may contribute to the ITM areas. For example, for time $T_1$ we find the highest initial price $S_1$ such that its area of influence doesn't intersect the ITM area $\{K. Only initial prices above $S_1$ can contribute to $\{K (such prices are shown by the green arrow). Similarly, only initial prices above $S_2$ can contribute to $\{K (such prices are shown by the blue arrow). Since the blue area is wider than the green area, the call expiring at $T_2$ must be more expensive than the one expiring at $T_1$.

## Fast time decay of OTM option

Figure 3. Call price as a function of time to expiration

If an OTM option continues to be OTM as expiration approaches, the likelihood of it becoming ITM quickly declines and the call price erodes to zero. This is illustrated in Figure 3 (from Mathematica), where time is measured in years, so the chart traces the call price as time to expiration declines from half a year to zero. The practical recommendation is to sell the option before it dies on you, if the odds of the stock moving higher are low.

19
Feb 18

## Call options and probabilistic intuition - dependence on strike

Awhile ago I gave a course in options following the excellent book by Hull. Frankly, at that time I didn't understand a fraction of what a trader needs to understand about options. I plan to make several posts to explain the intuition behind the Black-Scholes formula for options valuation. The geniality of this formula consists in the fact that it correctly reflects the likelihood of various events. More precisely, the more likely a certain event, the higher the call option price. The level of explanation will be midway between the theoretical view of Hull and practical view of option traders. We start with a series of definitions.

## Why do we want options?

A stock price moves up and down, and this movement is pretty erratic. You don’t want to buy the stock, because it’s an expensive investment and the price may fall right after you buy the stock. However, in case the price goes up, you want to gain from that movement, without the financial commitment required by the stock purchase. You are willing to pay a relatively small amount for the guarantee that if the stock goes up, you will be able to buy it at a fixed (presumably, lower) price. At the same time, you don’t want to buy the stock if it goes down. This is the idea behind the call option.

## Basic definitions

A call option gives its buyer the right, but not the obligation, to buy the stock at the price fixed in the option contract. This price is called a strike price and denoted $K$. The contract is valid until a certain date called an expiration date and denoted $T$. More generally, $T$ may denote any time before expiration and $S(T)$ will then denote the stock price at that time (it is random and depends on time, while the strike price is fixed for the life of the contract and is deterministic).

There are two parties in the transaction: the call buyer owns a long call and the call seller owns a short call. We discuss only long positions; for short positions most of the time the opposite is true.

At any point in time during the life of the option, the strike price $K$ may be higher than the stock price. In this case people say that it is out of the money (OTM). If $K=S$, the option is at the money (ATM). If $K, the option is in the money (ITM). The call buyer wants the call to be ITM at expiration.

If at expiration the call option is ITM, the long call holder buys the stock at price $K$ and sells at $S(T)$, profiting $S(T)-K-c$, where $c$ is the call price paid at the outset. If at expiration the call option is ATM or OTM, the holder doesn't gain from buying the stock and gives up the right to buy it, because buying is not an obligation. The option expires worthless and the call holder’s loss is $c$.

The market price $c$ of a call option depends on 5 variables: the strike price $K$, stock price $S$, time to expiration $T$ (measured in years), volatility of the stock price $\sigma$ and risk-free interest rate $r$ (annualized). Our purpose is to understand the option price dynamics when one of the arguments of the call option changes, while others are kept fixed.

The call price changes during the life of the option, depending on market conditions. The call holder can sell it at any time prior to expiration. The call price at that time depends on the event $\{S(T)>K\}$ that the option is in the money. In all our explanations we will use this event. The higher its likelihood, the higher the call price.

## So, how does the call price depend on the strike price?

Figure 1. In the money ranges for two strikes

Consider two strike prices $K_1. Since $\{K_2 (see Figure 1), the $K_1$ call should be more expensive than the $K_2$ call because the market values higher events with higher probabilities. As the strike increases, the call price falls. This is illustrated in Figure 2.

Figure 2. Call as a function of strike price

Figure 2 was produced in Mathematica using the Black-Scholes formula, which is valid for European options that can be exercised only at expiration. Our simple probabilistic argument is true also for American options that can be exercised any time prior to expiration. In Figure 2 we assume that the stock price is \$50 (the other assumptions are Expiration=0.5, Interest Rate=0.1, Volatility=0.5, Dividend=0.05). The red part of the curve corresponds to in the money strikes, and the out of the money strikes are shown by the blue part of the curve.

Note also that in Mathematica it's better to use the command FinancialDerivative than to program the Black-Scholes formula directly. The latter for some reason sometimes doesn't work.