1.2 Valuing Options Contracts

In general, the price of a contract is determined by the difference between the strike price of the contract and the price of the underlying (the spot price), the time to expiration, and the assumed future volatility of the underlying price. The interest rate of return of US Treasuries also dictates the price of puts and calls through put-call parity, but for most of this manual we will ignore the impact of this condition. Be aware that changes in monetary policy that leads to notable changes in interest rates can impact options pricing.

The simplest price condition is at expiration, where the value of the contract is simply based on the spot price of the underlying $S$ and the strike price of the contract $K$ , where contracts that are in the money are

$S - K$ (calls)

$K - S$ (puts)

and contracts that are out of the money expire worthless. For example, consider a SPY 600c contract where the price of SPY is $607 at expiration. In this case, the owner of the call contract has the right to call away 100 shares of SPY for a total price of $60,000 when the current value of those shares is $60,700. That call contract is then worth $700, or the difference between spot and strike. Conversely, for a SPY 610p, the buyer of the contract has the right to sell 100 shares of SPY at $610 per share when the SPY is only at $607, making the value of that contract $300. In another example, a SPY 610c would be worth $0.00 at expiry when the spot is $607, because it would not be in the best interest of the call owner to pay $3.00 extra per share.

Knowing the price at expiry is easy to calculate, but not particularly helpful since contracts are generally written long before expiration. So how might someone price the contract if the value isn’t determined until expiration, which hasn’t happened yet?

This is where assumed future volatility, or implied volatility (IV) comes into play. All options pricing must be based on assumptions about the future path the price of the underlying takes. This path is defined by the IV, which represents the expected volatility range, quoted in 1 year increments. IV scales with the square root of time, so if you want to convert IV to a particular range of days, you multiply IV by sqrt(days) and divide by sqrt(252). For example, if an AAPL contract has an IV of 0.40, or 40%, the seller of that contract is assuming the stock price of AAPL will have a daily volatility of 2.52%, and a monthly volatility of 11.5%. Note that a year has 252 trading days and each month has roughly 21 trading days.

Once you have an assumed volatility range for a stock over some period of time, you can then come up with a pricing model by introducing a probability distribution that describes how a stock is likely to traverse that volatility range over time. Black, Merton, and Sholes derived the price of options using a log-normal distribution, for which Scholes and Merton won the Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel in 1997. The equation is given as

C(S,\tau) = N(d_+)S - N(d_-)Ke^{-r\tau}

d_+ = {1 \over \sigma \sqrt{\tau}}\bigg[\ln\bigg({S \over K}\bigg) + \bigg(r + {\sigma^2 \over 2}\bigg)\tau\bigg]

d_- = d_+ - \sigma \sqrt{\tau}

N(d_+) = {1 \over 2}\bigg[1 + erf\bigg({-d_+ \over \sqrt2}\bigg)\bigg]

Where $C$ is the price of a call option, $N()$ is the integral of a normal distribution, $S$ is the underlying spot price, $K$ is the option strike price, $r$ is the risk free interest rate, $τ$ is the time to expiry in years, and $σ$ is the implied volatility.

The value of puts can be found using the put-call parity assumption, which eliminates leverage arbitrage

P(S,\tau) = C(S,\tau) -S + Ke^{-r\tau}

where $P$ is the value of the put.

Figure 1.1 below shows a plot of option price as a function of spot price using the Black Scholes model for both a call and put. The controls below the figure allow you to see how changing the strike price, IV, and the days to expiry changes the price of the option as a function of the spot price. When the IV is low or the days to expiry trends to zero, we recover the simple valuation |S-K| described above. As the IV increases, the sharp point at the strike price begins to smear out, representing the probability that out of the money options could end up in the money at expiration.

Notice what happens when IV is zero. Regardless of the days to expiry, the price of the options always remains simply |S-K|. Thus, the price of an option is determined by the IV. You are left with a “Chicken or the Egg” scenario, where you must ask if the price of the option determines the IV, or the IV determines the price of the option? In practice, they are essentially the same thing.