# Relationship between VIX and Vega

Assuming that all other factors (such as underlying price, strike price, etc.) remain unchanged, I want to see how a spike in VIX would affect the price of the average call option? Assume Vega is almost always positive.

• Indeed in the Black Scholes model Vega is always positive and increases with the (square root of) time to maturity. It might help if you can you be more specific as to your question. Jul 30 at 14:34
• VIX is a measure of the Implied Vol of the Call & Put options on the SPX500. If there is a spike in VIX, it means the IV has gone up. In my experience, VIX only really spikes if the SPX tanks, so that tells me it's the put options IV that drives the VIX. Logical deduction then tells me that the Call Option IV would also go up, because call option sellers would be wary of writing ATM Calls on the underlying that just took a dive. So in conclusion: spike in VIX = massive increase in Put options IV (= put options prices) = mild increase in Call options IV (= call option prices). Jul 30 at 14:44
• PS: try to get some time series of VIX against SPX500 Call prices: should confirm the above hypothesis with not too much effort. Jul 30 at 14:45
• @statwoman You can take this question to higher levels than the standard higher variance => higher option value. Think of the following points: (1) VIX measures the quadratic variation, not simply the variance. Thus, there's for example also a jump component. How does this impact option values? (2) VIX measures variation in the S&P, i.e., aggregate risk. How is a single-stock option impacted by changes in systematic vs. idiosyncratic volatility? Jul 30 at 16:54
• @JanStuller Look for example at this paper from Todorov and Tauchen. Equation (2.4) shows how the VIX can be decomposed into the expected integrated spot variance and a jump component. There are surely better papers on the details of the VIX construction. Think of it this way: when BS implied vol is used, then this IV always also captures jumps. The BS IV contains everything in one single number that other models can separate (sto vol, jumps, etc.) Having said this, your answer is surely the one OP is after :) Jul 30 at 18:38

Vega is the option's price sensitivity to the volatility (i.e. IV). In the graph below, vega is shown to be a strictly positive function in volatility, which means that at any point in the graph (i.e. for any value of IV, irrespective of whether the option is OTM, ATM or ITM), the option price:

• will increase in value if IV goes up (because Vega is positive)

• will decrease in value if IV goes down (because Vega is positive: i.e. higher IV is "good" for option price, lower IV is "bad for option price)

Although we notice that as the IV goes up, Vega has diminishing returns.

It's simple in the end: Vega is strictly positive in Vol => higher Vol therefore always means higher option price => higher VIX means higher Vol => higher Vol (IV) means higher option price (you can argue either because Vega is strictly positive in IV, or simply because Option Price is scaled IV anyway).

• all options have 1-year expiry, rates are set to 0.01 and spot is 100. The ITM call has strike 80, the ATM call has strike 100 and the OTM call has strike 120.
• Nice plots and explanation +1 Jul 30 at 20:42
• Thanks @AKdemy. And congrats on the new gold badge :) Jul 31 at 6:57

VIX almost always only spikes when SPX goes down as @Jan Stuller also mentions in a comment. Insofar the question is a bit counterfactual. I frequently use twin axis in the charts that follow. The position of the label corresponds to the axes the ticker belongs to.

These are essentially two question in real world scenarios.
1 ) VIX and Vega: VIX up, IVOL up, but Vega is unclear (frequently down)
2 ) VIX and call options price: VIX up, call option down

If you think of VIX and call options vol, these go more or less hand in hand. After all, a generic 1m ATM vol is not very different from VIX (in value, not in how it is computed). Below is a comparison, where the 1m ATM vol comes from a vol surface.

All else equal, higher vol means higher option price. You can find a few charts for vega and volga here. Volga, also called DvegaDvol or vega convexity, measures the rate of change to vega as volatility changes. This question asks about the value of a call option if vol increases (isolated). The question is flawed, as is the accepted answer but the other answers are all correct.

However, the decline in SPX is the most significant driver of option prices. Let's assume we bought the SPX 04/30/20 C3370 option. Simply plotting the price quoted by the primary exchange (Chicago) around the highest value of VIX in 2020 (2020-03-16) demonstrates this nicely.

Same against SPX

and for completeness sake, VIX against IVOL of the listed option.

This also holds for longer tenors. I priced with my in house tool a 1y ATM on 2020-02-24 which cost ~6.53%. On 2020-03-16 (highest VIX), it would have been worth ~0.77%, despite IVOL for the option increasing by ~5.189 percentage points (I am not including screenshots here, but the numbers are reliable).

Lastly, Vega plotted against VIX

That shape / direction depends a bit on where your option starts from in terms of moneyness. The link above (for vega and volga) shows that vega is highest for ATM options (volga lowest). Therefore, if you are owning a (deep) ITM call option with low vega, and VIX spikes (equivalent to SPX down), you will get closer to ATM (or even OTM). The closer you are to ATM, the higher it will become.

Below is a chart similar to @Jan Stuller that shows vega as a function of vol and also adds the call option price (this is in isolation, meaning all else equal). The relationship is clear here but not what usually will happen to Vega and prices.

• Btw; are your charts from Python? When you say "in-house tool" , you mean your in-house coded Python pricer? Jul 31 at 9:36
• These charts are made with Julia. Not very different from Python but it is easier in Julia to make simple interactive charts in my opinion. Also, Julia is fast, which can make a difference if you plot 3D Greeks with interactive sliders. These are all toy examples though. With in-house, I meant the pricing engines I have at work. They are mainly built in C++. It's a different world to get a full fledged tool (vol surface(s), IR curve(s), GUI etc). Jul 31 at 10:56
• Vola Dynamics nicely illustrates how complex things become in actual production environments. How to filter (noisy) data, treat American options, dividends, fit the surface, build interest rate curves, you name it. Once that is done, you need to figure out how to fetch that data in the correct way with your actual pricing engine. All of a sudden, even a simple vanilla OTC product turns out to be almost impossible to price properly without a lot of work and thought put into your tools if you think it through. Jul 31 at 11:02
• Thanks @AKdemy for the links and other info! Very interesting! PS: feel free to connect with me on LinkedIn :) Jul 31 at 14:06