More Option Greeks

More weird symbols you find in equations

Yeah, we weren’t done yet. Our last article on Greek letters in finance covered the most foundational ones, but is by no means an exhaustive list of the metrics used in option pricing, hence today’s article. We’re going to cover 2 more option Greeks that frequently operate in addition to the previous ones. Let’s give a warm welcome to our new symbols:

Vega V

It goes without saying that markets can become volatile and therefore impact their derivatives’ values. That’s where vega comes in, representing the rate of change between an option’s price and the implied volatility (an estimate of how volatile an asset is going to be) of the underlying asset. So if a stock’s implied volatility is expected to change by 10% and then cause a 10 cent shift in the option price, the vega is then 0.10. 

The concept of implied volatility is crucial not just when measuring a stock’s vega, but in all of finance. Although it’s a strictly theoretical guess, implied volatility can give you a sense of how the market as a whole feels about a given asset. Because these sentiments ultimately determine prices, traders often find it more useful to act on the market’s implied volatility as opposed to coming up with their own forecasts. 

As for how this fits into options, a higher implied volatility increases the option’s value whereas a lower implied volatility decreases it. A volatile asset is more likely to experience extreme values and therefore create more opportunities for the option to be in the money (the option possesses intrinsic value and is profitable), but these instances are less likely for more stable assets.

On that note, time is also a major factor in determining an option’s vega. Options with longer times until expiration will have higher vega values, as more time means more potential for the underlying asset’s price to change in the option holder’s favour. This is why vega is at its max for options with longer expiration times that are at the money (where the strike price is at or close to the underlying asset’s price): an option’s value can become either worthless or profitable depending on how the underlying asset’s price moves, and this sensitivity will be exacerbated over a long period of time.

Source

As shown in the graph above, the option’s vega increases across the board with the time until expiration, regardless of the stock price – the 30 day option has a higher vega than the 15 day option, which has a higher vega than the 5 day one (assume a $25 strike price).

Rho Ρ

One of the most important macroeconomic indicators is the interest rate, as it subsequently affects pretty much everything else: consumer spending, manufacturing output, and of course, financial markets. Rho is how much the option’s value changes following a 1% change in interest rates, specifically the country’s risk-free rate (in the US, that’s usually denoted with the Treasury bill’s rate).

Changes in the interest rate affects call and put options differently. Purchasing stock call options (the right, but not the obligation, to buy an asset at a certain price) is almost always cheaper than purchasing the actual shares. This means that if an investor chose to spend, say, $10 out of $100 on stock call options (as opposed to the entire $100 on the shares), they can invest the remaining $90 in an interest-bearing asset such as a bond or savings account. And when interest rates rise, so does the income from those assets. Meanwhile, it’s still possible for the call option to yield large profits with the right strike price. Hence, higher interest rates allow investors to still potentially profit with options while still keeping most of their money safe, thereby increasing the demand and price for call options.

Conversely, put options’ (the right, but not the obligation to sell an asset at a certain price) prices tend to move inversely with interest rates. Remember, put options are basically substitutes for short selling the underlying assets, and short selling stocks immediately puts money in your pocket (you borrow shares, immediately sell them on the open market and hope to buy them back at a lower price. See our article on the matter for more details), money that can be invested in interest-bearing assets. Buying put options, on the other hand, doesn’t give you that investable money. In a high-interest rate environment, you could actually make more money short selling stocks the old fashioned way instead of with put options. As such, the demand and price for put options decrease as interest rates rise. This is why call and put options have positive and negative rho values, respectively.

Because options with longer expirations have more time for interest rates to fluctuate, thus increasing their exposure to interest rate changes, rho increases as the time to expiration increases. 

Source

This graph shows the relationship between a call and put option’s rho value and the underlying stock price (i.e. how in or out of the money those options are). Why is the rho value larger for in the money options? Because they have intrinsic value, which can easily change with interest rates as explained earlier. 

But out of the money options have no intrinsic value and are instead driven by their time value (how much of an option’s premium, that is, the cost of purchasing them, is attributable to the time remaining until expiration). When options are fully out of the money, the investor can only hope for the underlying asset’s price to move in their favor, and the chances of this happening increase with a longer timespan until expiration. The time value isn’t at all affected by the risk-free interest rate, and so rho decreases the more out of the money the option becomes.

Econ IRL

The importance of households’ inflation expectations isn’t really disputed. With this particular metric influencing consumer spending, wage rates, and even the realized inflation rate, economists pay attention to it especially during recessions. The problem, however, is that there are several components to the households’ inflation expectations, which may raise debates as to which one is the most important. This week’s paper seeks to answer precisely that question.

Using French household survey data, the researchers compared the impact of the 2 main ways households form inflation expectations: the intensive margin (the precise quantitative inflation rate) vs the extensive margin (i.e. whether households expect prices to rise or remain stable). They find that among the extensive margin responses, heightened inflation expectations increased the consumption of durable goods (which is what you’d expect from consumers who think prices are going to rise in the near future). 

Numeric differences in the intensive margin, on the other hand, didn’t impact consumption patterns anywhere near as much. 

‘Till next time, 

SoBasically