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 of the price of the underlying. This path is defined by the IV, which represents the expected volatility range, quoted in one 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 options are given a moderate implied volatility and time to expiration, and so option near spot and those out of the money are worth more than they would be at expiration, denoted by the lines |S-K|. As the value of the underlying spot shifts, so too do options prices at each strike. For example, a call and a put bought at the $90 strike when spot is also$ 90 are both worth about $5. When spot moves to $110, the put is now worth $0.50 and the call is worth $20.

valuespot

Figure 1.2 shows what happens to options prices when the expected or implied volatility shifts from low to high. When IV is vanishingly small, spot is not expected to move at all between the present and the expiration time of the contract, and so the contracts are simply worth their distance from spot. However, as IV begins to rise, contracts that are currently out of the money increase in value, as the probability that they will go in the money over time increases. Note that this effect can even lift the price of very far out of the money options if IV increases enough. The $75 put is $25 out of the money, but in this scenario goes from a few cents to a few dollars simply from a change in implied volatility. This 100x return creates a huge risk for people who write these contracts, as they stand to lose 100x more than they made from the sale of the contract! We will return to this volatility risk in various ways further along.

valueiv

Figure 1.3 shows options pricing as time to expiration goes from low to high for a moderate value of IV. The impact on options price is very similar to the effect of increasing IV. The primary difference is subtle in these animations, but increasing time to expiration has a square root impact on options price, while the impact of IV is much closer to linear. An important outcome of this feature is that options with more time to expiration “decay” in value much more slowly, giving traders more time to realize an anticipated move without losing too much of the premium they spent on the contract.

valuetime

Although we have discussed the role of spot, IV, and time to expiration on options pricing, options pricing is really primarily determined by IV. Consider the case where IV is zero. It does not matter how many days to expiration, the contracts will always be worth |S-K|, and if that IV is realized, the spot will not move through the life of the contract. 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.