Vega/Implied Volatility

Hey guys, I'm a physics student and don't have any real background in finance, out of interest I've been reading Hull and I'm a little confused about Vega, I hope some of you guys can help me out here.
Hull doesn't specifically say that Vega is the derivative of the option price with respect to the implied volatility, he just says it's the derivative with respect to the volatility of the underlying. Now in other books Vega is defined as the rate of change of the value of the option with respect to the implied volatility. What I don't understand about this is that implied volatility is defined as the volatility that results from the option price with all the other variables held constant, so it's rather that the option price causes implied volatility to change, doesn't it? Or can you interpret implied volatility just as the future volatility that the market expects based on the option price? So if e.g actual volatility increases in the next week, can the increase in the value of the option as a result of that increased volatility be interpreted as vega?

WSO Elite Modeling Package

  • 6 courses to mastery: Excel, Financial Statement, LBO, M&A, Valuation and DCF
  • Elite instructors from top BB investment banks and private equity megafunds
  • Includes Company DB + Video Library Access (1 year)

Comments (32)

Aug 28, 2012 - 2:42pm

Vega is based off implied vol. in other words, it's the change in the price of the option for a given change in the implied vol when you keep everything else constant.

Aug 29, 2012 - 4:45am

I knew that, but what confuses me is that implied volatility is defined as the value for the volatility in a pricing model that is implied by the option price, so that changes in the option price cause implied volatility to change and not the other way around. If that definition is correct, the definition of Vega doesn't make any sense to me, because in that case its not implied volatility that causes the value of the option to change, but the option price causes imp. vol. to change.
Therefore I was wondering what exactly implied volatility means in the context of vega. Can someone please answer this question: "So if e.g actual volatility increases in the next week, can the increase in the value of the option as a result of that increased volatility be interpreted as vega?"

Aug 29, 2012 - 8:05am

"but what confuses me is that implied volatility is defined as the value for the volatility in a pricing model that is implied by the option price, so that changes in the option price cause implied volatility to change and not the other way around. If that definition is correct, the definition of Vega doesn't make any sense to me, because in that case its not implied volatility that causes the value of the option to change, but the option price causes imp. vol. to change."

Your answer lies within this statement. An option price is a function of spot, time to maturity, volatility, and your discount rate.BSM assumes volatility is held constant; however, in the real world we know that volaitlity is not constant. We assume that vega is constant within the formula - implied volatility. Each time you want to calculate the option price, you can choose to adjust your vega variable - or in this case, any other variable that is a function of the option price.

Hope this helps.

Learn More

300+ video lessons across 6 modeling courses taught by elite practitioners at the top investment banks and private equity funds -- Excel Modeling -- Financial Statement Modeling -- M&A Modeling -- LBO Modeling -- DCF and Valuation Modeling -- ALL INCLUDED + 2 Huge Bonuses.

Learn more
Aug 29, 2012 - 5:21am

Wait I think I get it now, let's use a hypothetical situation: Let's say that the market expects an event to occur that could cause massive volatility, so people will buy options as insurance, causing options to become more expensive. Now hypothetically if the price of the underlying would not have moved and the other variables in the option pricing model have not changed, you can say that this expectation of an increase in volatility caused to value of the option to increase, can't you? Would that be a correct interpretation of implied volatility and vega?

Aug 29, 2012 - 5:35am
Polymath89:
Wait I think I get it now, let's use a hypothetical situation: Let's say that the market expects an event to occur that could cause massive volatility, so people will buy options as insurance, causing options to become more expensive. Now hypothetically if the price of the underlying would not have moved and the other variables in the option pricing model have not changed, you can say that this expectation of an increase in volatility caused to value of the option to increase, can't you? Would that be a correct interpretation of implied volatility and vega?

Yes, think of it like this. Let's say you hold a 3 month option on some company. Tomorrow, the company comes out with some news that is going to make the stock double or cut in half in the next month (or at least people interpret it that way). The option prices are going to get way more expensive because the market is predicting more extreme outcomes. Even if the stock doesn't move, your option goes up in value.
Aug 29, 2012 - 5:52am
Polymath89:
Wait I think I get it now, let's use a hypothetical situation: Let's say that the market expects an event to occur that could cause massive volatility, so people will buy options as insurance, causing options to become more expensive. Now hypothetically if the price of the underlying would not have moved and the other variables in the option pricing model have not changed, you can say that this expectation of an increase in volatility caused to value of the option to increase, can't you? Would that be a correct interpretation of implied volatility and vega?

Yes. An increase in implied volatility will increase option price (I wouldn't necessarily say value). Vega is the measurement used to bridge the % shift in underlying volatility to the $ shift of the option.

DELETED_ACCOUNT
Aug 29, 2012 - 5:27am

Ok, you don't have enough info in that one sentence to determine if implied vol caused the option price to increase. A week has passed, if all other variables stayed constant, then implied vol has increased, because the option would have lost time value if implied vol stayed the same. If the other variables changed, then not necessarily.

Also, yes, you determine implied vol by calculating it via the market price.

Aug 29, 2012 - 8:09am

One last question though, it's expected volatility that causes option prices to change, not realized volatility, right?

This is correct in a way. The option price is primarily a function of the spot price (share price). If you think that the share price will move in either direction by an option because they have more leverage to the volatility of the spot price.

Aug 29, 2012 - 2:35pm

Vega is sensitivity to implied vol. How much you make on Gamma is effectively your realised vol.

Jack: They’re all former investment bankers who were laid off from that economic crisis that Nancy Pelosi caused. They have zero real world skills, but God they work hard. -30 Rock
Aug 31, 2012 - 5:42pm

was looking on bloomberg and saw this. does anyone know why overnight vol on friday always dips below overnight vol from mon-thur? wouldn't you expect friday overnight vol (which encompasses friday, saturday, and sunday) to be higher than mon-thur overnight vol?

Aug 31, 2012 - 7:08pm
whalesquid123:
was looking on bloomberg and saw this. does anyone know why overnight vol on friday always dips below overnight vol from mon-thur? wouldn't you expect friday overnight vol (which encompasses friday, saturday, and sunday) to be higher than mon-thur overnight vol?

this stems from one of my favourite interview questions...

an overnight (o/n) option on thursday expires in 1 calendar day (friday)
an o/n option on friday actually expires in 3 calendar days (monday).

markets are closed on saturday and sunday, so all things being equal an o/n option on thursday and friday should have the same premium because they have the same number of "real" days for spot to move around.

but if u plug a vol of 10% into the black-scholes formula for o/n option on thursday and plug the same vol into B-S on friday, you'll get a higher premium for the friday option because b-s counts 3 calendar days (b-s doesn't know anything about markets being closed on sat/sun)
so to make up for this, u must have a lower vol for the o/n on friday to bring the premium down to what it would be if it was thursday.

Sep 10, 2012 - 1:11am
FXTrader:
whalesquid123:
was looking on bloomberg and saw this. does anyone know why overnight vol on friday always dips below overnight vol from mon-thur? wouldn't you expect friday overnight vol (which encompasses friday, saturday, and sunday) to be higher than mon-thur overnight vol?

this stems from one of my favourite interview questions...

an overnight (o/n) option on thursday expires in 1 calendar day (friday)
an o/n option on friday actually expires in 3 calendar days (monday).

markets are closed on saturday and sunday, so all things being equal an o/n option on thursday and friday should have the same premium because they have the same number of "real" days for spot to move around.

but if u plug a vol of 10% into the black-scholes formula for o/n option on thursday and plug the same vol into B-S on friday, you'll get a higher premium for the friday option because b-s counts 3 calendar days (b-s doesn't know anything about markets being closed on sat/sun)
so to make up for this, u must have a lower vol for the o/n on friday to bring the premium down to what it would be if it was thursday.

but say it's friday today and you're quoting an overnight option that expires monday. instead of putting in monday's date and making the system think expiry is in 3 days, why not put in saturday's date and let the system price it as if it's a regular weekday? the dollar value of the premium should be the same, as should the vol, right? maybe i'm not understanding some convention of how options are quoted.

thanks

Sep 1, 2012 - 5:19am

Well, in FX we generally assume that the weekend days are NOT 0 variance. You assign some weighting to it as things CAN happen on the weekend, but it is often significantly less than a normal business day.

There are exceptions to this, take for example when the Greek Elections occurred on a Sunday; many people would say that Sunday was implying MUCH more than a normal business day.

Jack: They’re all former investment bankers who were laid off from that economic crisis that Nancy Pelosi caused. They have zero real world skills, but God they work hard. -30 Rock
Sep 2, 2012 - 1:07pm
Revsly:
Well, in FX we generally assume that the weekend days are NOT 0 variance. You assign some weighting to it as things CAN happen on the weekend, but it is often significantly less than a normal business day.

There are exceptions to this, take for example when the Greek Elections occurred on a Sunday; many people would say that Sunday was implying MUCH more than a normal business day.

Thanks, that's helpful to know. But if you did assume that weekends were 0 vol, would the math check out? Just wanna make sure I have it down right, conceptually

Sep 2, 2012 - 1:20pm
whalesquid123:
also, can someone explain the concept of volatility skew and how it relates to risk reversals? an intuitive explanation would be great. as well as floating vs. sticky skew?

i'm reading taleb's dynamic hedging and don't get it...

plot the vega of the risk reversal, and overlay your vol surface. Then see what happens when spot moves up / down

Sep 2, 2012 - 2:16pm

Yes, plot your vega profile over spot. Let's say you bought 25dlta downside on S&P and sold 25dlta topside to buy the RR. What happens to your vega as spot goes up/down? What do you anticipate would happen to volatility if spot goes up/down? To do this strategy to you anticipate paying or receiving a vol premium?

Jack: They’re all former investment bankers who were laid off from that economic crisis that Nancy Pelosi caused. They have zero real world skills, but God they work hard. -30 Rock
Sep 2, 2012 - 6:27pm
Revsly:
Yes, plot your vega profile over spot. Let's say you bought 25dlta downside on S&P and sold 25dlta topside to buy the RR. What happens to your vega as spot goes up/down? What do you anticipate would happen to volatility if spot goes up/down? To do this strategy to you anticipate paying or receiving a vol premium?

What does buying/selling 25d topside/downside mean? I only know the most basic stuff about options...

Sep 2, 2012 - 7:11pm

25 delta, as in the delta of the option will be 25% of notional. Topside meaning above current spot. So basically you're buying a Put and selling a Call in my original example.

Jack: They’re all former investment bankers who were laid off from that economic crisis that Nancy Pelosi caused. They have zero real world skills, but God they work hard. -30 Rock
Sep 3, 2012 - 12:30am

Ok so you are long a put at the lower strike and short a call at the higher strike - does that make you short the RR?

Vega profile should be positive around the downside strike, dip to 0 at ATM strike, and be negative around the topside strike?

As spot goes down you'd expect vol to go up?

You would expect to have to pay out some premium on net because the put you bought would be more expensive (higher vol) than the call you sold?

What i'm not sure is how all this ties together in a "grand unified theory" kind of deal, so if someone could lay it out step by step and explain how everything connects that'd be awesome haha

Sep 4, 2012 - 10:29pm
whalesquid123:
Ok so you are long a put at the lower strike and short a call at the higher strike - does that make you short the RR?

In equities, puts are priced on a higher vol. So if you had a put and a call equidistant from ATM strike, the put would be worth more. So you would be long the RR if you bought it.

Sep 4, 2012 - 11:24pm

http://en.wikipedia.org/wiki/Risk_reversal

wikipedia says risk reversal is vol of call minus vol of put. so if call is more expensive, RR is positive, and if put is more expensive, RR is negative. here the put is more expensive, so RR should be negative - why do you say long the RR?

if you mean RR as in the option structure itself, wikipedia also says being long the RR is the same as selling a 25d put and buying a 25d call, to make a "synthetic long". but here you say being long the RR means being long whichever strike (25d topside or 25d downside) is worth more?

Sep 5, 2012 - 1:19pm
whalesquid123:
http://en.wikipedia.org/wiki/Risk_reversal

wikipedia says risk reversal is vol of call minus vol of put. so if call is more expensive, RR is positive, and if put is more expensive, RR is negative. here the put is more expensive, so RR should be negative - why do you say long the RR?

if you mean RR as in the option structure itself, wikipedia also says being long the RR is the same as selling a 25d put and buying a 25d call, to make a "synthetic long". but here you say being long the RR means being long whichever strike (25d topside or 25d downside) is worth more?

In market speak, whenever you buy a RR it means you are buying the side with the higher vol, and drop any negative sign. So 1m AUDUSD 25d RR, which is bid to puts, might be 1.75 in the mkt, even though it's technically -1.75 (call vol - put vol).

Jack: They’re all former investment bankers who were laid off from that economic crisis that Nancy Pelosi caused. They have zero real world skills, but God they work hard. -30 Rock
Sep 5, 2012 - 4:43pm
Revsly:
whalesquid123:
http://en.wikipedia.org/wiki/Risk_reversal

wikipedia says risk reversal is vol of call minus vol of put. so if call is more expensive, RR is positive, and if put is more expensive, RR is negative. here the put is more expensive, so RR should be negative - why do you say long the RR?

if you mean RR as in the option structure itself, wikipedia also says being long the RR is the same as selling a 25d put and buying a 25d call, to make a "synthetic long". but here you say being long the RR means being long whichever strike (25d topside or 25d downside) is worth more?

In market speak, whenever you buy a RR it means you are buying the side with the higher vol, and drop any negative sign. So 1m AUDUSD 25d RR, which is bid to puts, might be 1.75 in the mkt, even though it's technically -1.75 (call vol - put vol).

So this is how the process works?
1. See whether 25d calls or puts are bid (relative to the other)
2. A 25d RR (as a measure of vol skew) is the difference (absolute value) between call and put vol
3. Long the RR option structure means long whichever strike is bid (topside for 25d calls, downside for 25d puts) and short the other

How does plotting the Vega profile fit into this? In the case of AUDUSD, I'd only know the Vega profile of the 25d RR because you told me puts were bid, so long RR here means long put short call, so Vega is positive around the downside strike and negative around the topside strike. You have to see which side is bid to figure out the Vega profile, so why did you initially suggest plotting the Vega profile to figure out vol skew?

I know I'm missing something here. Thanks...

Start Discussion

Total Avg Compensation

April 2021 Investment Banking

  • Director/MD (9) $911
  • Vice President (35) $364
  • Associates (188) $234
  • 2nd Year Analyst (107) $150
  • Intern/Summer Associate (96) $145
  • 3rd+ Year Analyst (26) $145
  • 1st Year Analyst (392) $132
  • Intern/Summer Analyst (323) $82