Lecture Note 4.1: Volatility as an Asset Class

Introduction:

Today we begin a new phase of our course in which we move beyond

pricing options. Our goal is to study derivative products that allow in-

vestors to gain or hedge exposure to factors that are not literally asset

prices. This exposure may sometimes be embedded in an instrument

that looks like an option (or a forward or a swap). We will also see

examples of unique new payoff structures.

To get started with the subject, we can build on our study of options

pricing with stochastic volatility to think about securities whose payoff

is tied to volatility itself. We’ll discuss why people might be interested

in such products. We’ll also see some amazing results about how we

can price them.

Outline:

I. Why Trade Volatility?

II. Risk-Neutral Expectations

III. Volatility and the Log Contract

IV. Model-Free Implied Volatility

V. VIX

VI. Extensions

VII. Summary

I. Why Trade Volatility?

� The volatility of a financial asset is driven by the rate of infor-

mation flow about the asset, or the degree of uncertainty about

its future value.

� It is not at all uncommon for investors to have opinions – or

feelings – about the level of uncertainty in the economy or in

their personal wealth that is not necessarily connected to an

opinion about whether future news is likely to be good or bad.

� Times of high uncertainty feel very different.

· Prices are likely to be lower due to risk aversion.

· Also importantly, market illiquidity is strongly correlatedwith volatility.

� So, at least for non-diversifiable risk, it’s natural to see why

people might want to “buy volatility” as insurance.

� We have learned that if we are long a delta-hedged option on

an asset, S, that is effectively a bet on volatility.

· If continuously delta hedge it, you will have no exposure to

S and the total amount of money you make will depend on

realized volatility between now and expiration.

� But if this is what you want, then this strategy is not going to

be very efficient.

· The amount of your volatility exposure will move around a

lot. In particular, it will diminsh as S moves away from your

strike price K.

� What if we could write a contract directly on S’s volatility?

· Take the simplest possible bet: a payoff of σT at some future

date T .

Let’s price it for practice.

� Now you might wonder about how we would settle this contract

at T since σ is a parameter of a model – not a physically

observable thing.

· Let’s imagine that you and your counterparty have agreed

on some measure like the standard deviation (or root mean

square) of 1-minute returns in the last hour before T , annu-

alized.

� As I mentioned in an earlier note, the market actually trades

contracts like this, although it is more common to trade claims

to variance than volatility.

· Usually the underlying asset is a major index like the S&P

500. But variances of other portfolios and even single stocks

do get traded in this way.

� Let’s denote the price of our claim to σT as Ft,T.

� Since its payoff makes no reference to ST, we can assert that

Ft,T doesn’t depend on St.

� If σt obeys the stochastic model we used in Lecture 3.6, then

F obeys the PDE

1

2s2σ2

∂2F∂σ2

+ (κ(σ0 − σ) − λsσ)∂F∂σ

− rF +∂F∂t

= 0

subject to FT,T = σT.· Since the payoff is a simple linear function, it is natural to

conjecture that the solution is linear.

· Let’s plug in a trial solution of the form Ft,T = a(τ)+b(τ)σtwhere τ ≡ T − t and a(0) = 0, b(0) = 1.

· The the PDE requires

(κ(σ0−σ)−λsσ) b(τ)−r(a(τ)+b(τ)σ)−(ȧ(τ)+ḃ(τ)σ) = 0or

[κσ0b(τ) − ra(τ) − ȧ(τ)]−[(κ + λs + r)b(τ) + ḃ(τ)

]σ = 0

where the dot signifies the first derivative with respect to τ.

· Now if this solution is going to work for ANY values of σ

it must be the case that all of the coefficients multiplying

σ add up to zero. Hence ḃ(τ) = −(κ + λs + r)b(τ). Thesolution to that ODE with b(0) = 1 is just

b(τ) = e−(κ+λs+r)τ

· With that plugged in to our ODE we now require

κσ0e−(κ+λs+r)τ − ra(τ) − ȧ(τ) = 0.

· This is an ODE for a. I’ll let you verify that its solution, with

a(0) = 0, is

a(τ) =κσ0

κ + λs

(e−rτ − e−(κ+λs+r)τ

)So the conclusion is that

Ft,T = e−r(T−t)(σ̃(1 − e−κ̃(T−t)) + σt e−κ̃(T−t)

)

where I have defined κ̃ ≡ κ + λs and σ̃ ≡ σ0(κ/κ̃).

� The price of the volatility claim today is a weighted average

of today’s volatility and the “adjusted” long run volatility σ̃,

discounted at the riskless rate.

· The weight on today’s volatility is near one for short horizons,

and then declines exponentially at rate κ̃.

� In the previous lecture note, I told you that, for major stock

market indexes, the market price of volatility risk is negative.

Since s > 0, this means κ̃ < κ and σ̃ > σ0.

· If you plot Ft,T for different values of λ (fixing the otherparameters), you will find that it always goes UP when λ

goes up. (You can prove this with a little calculus. I’ll leave

that as an exercise.)

A more negative price of risk means people want to pay more

for volatility insurance.

II. Risk Neutral Expectations

� If we wanted to, we could price call options on σT under the

model we used above.

� If we did that, we could price butterflies.

� Once we had the butterfly prices, we could turn them into a

risk-neutral distibution for σT.

� That tells us that there does exist a set of “probabilities” such

that the price of any claim on any function f(σT) must be equal

to

Bt,T E⋆[f(σT)]

where E⋆ denotes the expectation using these probabilities.

� As a special case, we can put f(σT) = σT and deduce that the

risk neutral expectation of σT is

σ̃(1 − e−κ̃(T−t)) + σt e−κ̃(T−t).

� For our particular model, the true expectation is

σ0(1 − e−κ(T−t)) + σt e−κ(T−t).

This is the same as the risk-neutral expectation when the mar-

ket price of volatility risk is zero.

� Next time we will discuss how to derive the risk-neutral proba-

bilities in any model.

� For now, I just want to note that the implied volatility forecast

embedded in Ft,T is the same as the risk-neutral expectation.

� Also note that our model says that Ft,T as a stochastic processitself is perfectly correlated with σt.

· This means the Sharpe ratio to holding F is λ.

· And this will be also true of any portfolio of such claims.

However, in reality, prices to claims to volatility may also move

around if the market price of volatility risk changes. The model

assumes it is a constant.

� I also want to take the opportunity to remind you again that

there is a different price of volatility risk for each thing.

· Gold volatility and S&P 500 volatility have completely dis-

tinct λs. Knowing one does not tell you the other.

III. Volatility and the Log Contract

� One of the most innovative ideas in financial engineering in the

last 25 years concerns the pricing of a claim to average variance:

1

T

∫ T0

σ2t dt.

� The story begins with an observation due to Anthony Neuberger

in 1994. He asked the following question:

· Suppose there is an asset whose current price is S0 and whose

time-T forward price is F0,T and we are short a derivative

contract that will require us to pay log(ST/F0,T) at time T .

· Further suppose we choose to engage in a dynamic hedging

strategy between now and T in which every ∆t (e.g. every

day) we hold 1/Ft,T forwards.

· How much money will we make?

� This seems like a funny question. But it is pretty easy to answer.

Let’s just do the accounting.

� Here are our trades.

Day 0: Buy 1F0,T

forwards at price F0,T.

Day 1: Buy 1F1,T

− 1F0,T

forwards at price F1,T.

… …

Day T − 1: Buy 1FT−1,T

− 1FT−2,T

forwards at price FT−1,T.

� When we get to time T all of these trades settle. Our profits

from each trade are:

Day 0: [ 1F0,T

] (FT,T − F0,T) +

Day 1: [ 1F1,T

− 1F0,T

] (FT,T − F1,T) +

… …

Day T−1: [ 1FT−1,T

− 1FT−2,T

] (FT,T − FT−1,T).

� Please verify that adding all these terms up givesT−1∑t=0

(Ft+1,TFt,T

− 1)

orT−1∑t=0

(∆Ft+1,TFt,T

).

� Now, what is the difference between the quantity in parenthe-

ses, ∆F/F , and ∆ log F?

· If we let ∆t get very small so that we are talking about

infinitessimals, then, for any diffusion process, we have from

Ito’s lemma:

d log(F) =1

FdF −

1

2σ2 dt

where σ is the volatility of F .

* It is the same as the volatility of S if carry costs are

predictable.

· So, for small time intervals, we can replace ∆F/F by ∆ log F+12σ2 ∆t.

� When we make the substitution for ∆F/F in our hedging profit

expression, our total P&L is

T−1∑t=0

(∆ log Ft+1,T +

1

2σ2t ∆t

)which reduces to

log(FT,T) − log(F0,T) +1

2

T−1∑t=0

σ2t ∆t

since all the intermediate log terms cancel out.

� Now the first terms here are the same as log(FT,T/F0,T) =

log(ST/F0,T). And this is exactly the payout that we are short.

� So the total payoff at T from our position is just the second

term:1

2

T−1∑t=0

σ2t ∆t or1

2

∫ T0

σ2t dt.

� Notice also that the 1/F hedging strategy that we engaged in

was self-financing. In fact, it was costless.

� Thus, a short position in the log contract plus a costless strategy

generates a payoff that is equal to (half) the realized variance

between now and T .

� Neuberger’s conclusion: The market value of the log con-tract must be equal to the minus one half times the market

value of a claim to the realized total variance.

� Or, the value of the log contract must be −T/2 times themarket value of a claim to the realized average variance.

� As we’ll see, this observation is useful when we can’t directly

find the price of the variance claim.

IV. Model Free Implied Volatility

� The really neat thing about Neuberger’s argument is that it

does not depend on any particular model for dσt.

· It only relied on the observation that d log(F) = 1FdF −

12σ2t dt.

� We have been using the model dσt = κ(σ0−σt) dt+s σt dW σt .· And we could use this to derive a price for the average vari-

ance contract, by solving the PDE – as we did above.

· But the answer would definitely be different if we used a

different model.

That’s why it’s remarkable that the conclusion we reached

above is model free.

� Now let us define the model free implied variance as:

MFIV 2 =1

TE⋆[∫ T

0

σ2t dt

].

· The present value of that expectation is the price of contract

that pays the realized average variance at T .

� If L0 denotes the price of the log contract, we have thus de-

duced

L0 = −T

2B0,T MFIV

2.

Note that this conclusion also makes no assumption about the

market price of volatility risk.

V. VIX

� The most important volatility that everyone wants to measure

is that of “the market portfolio”.

· For a well-diversified investor, in fact, this quantity is the

ONLY risk that matters.

� Since the early 1990s, the CBOE had published daily summary

statistics about Black-Scholes implied volatility of options on

the S&P 500.

� But they faced a problem: which options?

· As we’ve discussed, every strike and maturity would give

them a different answer.

· They used the most liquid ones, which are short-dated and

at-the-money. But those don’t capture the beliefs expressed

in out-of-the-money puts, for example.

· More importantly, the fact that implied volatility is not con-

stant means the Black-Scholes model is wrong – which means

there is no coherent way to interpret BSIV in terms of the

market’s expectations about actual volatility of the index.

� So they were very interested when they heard about Neuberger’s

idea about using the value of a (theoretical) log contract to

summarize the market’s beliefs.

� Then came another problem: How can it be implemented?

� The log contracts are not actually traded.

� If we can’t observe prices for the log contract, maybe we’ve just

replaced one measurement problem with another.

� But Neuberger (together with Mark Britten-Jones) had an

answer.

· Butterflies!

� We learned early on in the course that if we have prices for a lot

of ordinary European calls (or puts) we can infer the prices of

butterflies, and then construct arbitrary payoff functions from

them.

· Well, the CBOE does have the for data on calls and puts on

the S&P500.

· So they should be able to synthesize something very close to

a log contract’s payoff by an appropriate portfolio of

butterflies.

· Or, since the butterflies are themselves just made up of calls

and puts, we should be able to construct a weighting scheme

that just sums over all of them.

� That’s what they did. And thus the modern VIX was born.

� Let’s try to understand the weighting scheme for synthesizing

the log contract.

� Consider the idealized case with infinite strike prices available:

· If b(k) denotes the price of a butterfly centered at strike price

k, then we learned that the value of the log contract must

be

L0 =∑k

log(k/F) b(k).

(where I’m writing F for F0,T.)

· Now expand the butterflies into their component options.∑k

log(k/F) b(k) =∑k

log(k/F) [p(k−∆k)−2p(k)+p(k+∆k)]/∆k.

Recall the butterflies are the same whether we use puts or

calls to build them.

· Actually, it will be more convenient to use both calls and

puts, but on different ranges. So write the sum:

F0,T∑k=∆k

log(k/F)p(k − ∆k) − 2p(k) + p(k + ∆k)

∆k

+

∞∑k=F0,T

log(k/F)c(k − ∆k) − 2c(k) + c(k + ∆k)

∆k

· Robert Whaley showed how to rewrite this in terms of the

weight it puts on each individual option.

· Take the first summation. Re-write this by grouping together

terms involving the puts with the same strike.

· (I’ll supress the ∆k and F and just index everything

by integers.)

log(1) (p(0) −2 p(1) + p(2))+ log(2) (p(1) −2 p(2) + p(3))+ log(3) (p(2) −2 p(3) + · · ·

+…

…

= p(1)[log(0)−2 log(1)+log(2)] + p(2)[log(1)−2 log(2)+log(3)] + · · ·

* For example, if ∆k = 0.01 then the terms involving puts

with k = 100.00 are

p(100.00) ∗ [log(99.99) − 2 log(100.00) + log(100.01)]

* Also, to be careful near zero, the formula is assuming

log(1)p(0) − log(0)p(1) goes to zero.

– Mathematically, sufficient conditions for this are thatp(x)d log(x)/dx → 0 and log(x)dp/dx → 0.

– This will be fine as long as the left tail of the risk neutraldistribution goes to zero exponentially, for example.

· Now, bringing back the ∆k that I suppressed, our first sum-

mation isF0,T∑k=∆k

p(k)[log(k − ∆k) − 2 log(k) + log(k + ∆k)]

∆k2∆k

* Inside each of the logs there was a (1/F0,T), but we can

drop them because each term in the summation has

− log(F0,T) + 2 log(F0,T) − log(F0,T) = 0

· Likewise, I can re-arrange the call sum as

∞∑k=F0,T

c(k)[log(k − ∆k) − 2 log(k) + log(k + ∆k)]

∆k2∆k

� As the strikes become infinitely close together, you will recog-

nize that the fractional terms are just going to become second

derivatives.

� And we know from calculus thatd2 log(x)

dx2= − 1

x2.

� So we can conclude that our double sum is equal to

−

(∫ F0,T0

p(k)

k2dk +

∫ ∞F0,T

c(k)

k2dk

).

� So far we have deduced that this integral gives us the value,

L0, of the log contract

� Since we deduced above that L0 = −T2 B0,T MFIV2,

we conclude

MFIV 2 =2

TB−10,T

(∫ F0,T0

p(k)

k2dk +

∫ ∞F0,T

c(k)

k2dk

).

� That equation is in fact what the CBOE’s measure tries to ap-

proximate using finite sums instead of integrals over an infinite

range.

· The squre root of that is VIX.

� You can compute this for any maturity option. And the avail-

able maturities change every day.

· The primary VIX is supposed to always represent options

with 30 days to maturity. Since these don’t usually exist,

the CBOE just linearly interpolates the (squared) VIXes com-

puted from two expirations on either side of 30 days.

· They also do this for 9-day, 3-month, and 6-month horizons.

These are called (in order) VXST, VXV, and VXMT.

� As a practical matter, they are limited by having only finitely

many strikes.

� The first integral is the numerically sensitive part because it

involves both numerator and denominator going to zero as k

goes to zero.

· The exchange has listed some puts with extremely low strike

prices to help in this direction.

· Some researchers have suggested that VIX can be manipu-

lated by relatively small orders for these far out-of-the-money

options. (See Griffin and Shams (2017).)

� Exact details of the biases due to (a) upper and lower limits

not being 0 and infinity and (b) finite ∆k have been analyzed

by Jiang and Tian (2007) and (2009).

· They also suggested some numerical corrections.

· As far as I know, CBOE has not adopted them yet.

� Still it is remarkable how much you can do with butterflies.

· It is fascinating that a contract whose payoff depends on

the realized, random path of volatility can be replicated by

a static portfolio of puts and calls.

VI. Extensions.

� It turns out that a lot of people care about VIX!

� VIX matters not just because financial researchers appreciate

its model-free construction.

· It matters because it takes us one step closer to making

volatility tradeable.

� VIX itself is still just a statistic. We can’t buy it directly.

· Technically, as we saw, V IX2T is itself the price of a partic-

ular portfolio of options on the index.

· But trading that basket would involve very large transactions

costs.

� As I have mentioned, there are variance swap markets, but these

are not available to most investors.

· They also involve potentially tricky measurement issues that

we discussed.

� With VIX, we can write contracts on it, instead of σ.

� As you probably know, the CBOE trades cash-settled futures

on VIX.

· VIX futures prices are obviously highly correlated with VIX

itself.

· These futures are the basis of very popular VIX-based ETFs

and inverse ETFs.

� There are also options on VIX futures. And from those op-

tions….

· CBOE computes VIX on VIX!

· See http://www.cboe.com/micro/VVIX/

� Clearly the market believes volatility is stochastic!

� Since the same construction works for any underlying instru-

ment that has traded options, CBOE also can compute model-

free implied volatilities for the most popular ETF, like ones

based on gold and crude oil and Chinese stocks.

http://www.cboe.com/products/vix-index-volatility/

volatility-on-etfs/cboe-china-etf-volatility-index-vxfxi

� There are also indexes based on futures options for Treasury

bonds and credit indexes.

� Stock exchanges around the world now compute a VIX clone

for their own sets of indexes.

� It won’t surprise you to learn that there are also now crypto-

VIXs:

https://t3index.com/indices/bit-vol/

� Many of these extensions of VIX have themselves had contracts

written on them.

· This could enable investor to make bets on differences be-

tween volatilities of different portfolios or asset classes.

� It is important to recognize, though, that betting on VIX (or

one of its variant indexes) is betting on risk-neutral expected

volatility, not the actual variability of returns.

· The risk-neutral expectation is biased if the market price of

σ risk is substantial.

· The index may move independently of actual vol if the price

of risk changes.

� A lot of recent academic research attempts to measure and

study this bias, which is also called the volatility risk pre-mium.

VII. Summary

� Investors care about volatility risk.

� We can price claims to realized volatility in our no-arbitrage

framework by solving our PDE.

· The answer we get definitely will depend on the stochastic

process that we assume for σ and on the market price of σ

risk.

� It is a very cool fact that we can derive the price of a claim

to average realized variance from butterfly prices embedded in

options on S without ever having to specify what we believe

the true dynamics of volatility are.

· We can interpret this price as the risk-neutral expected cu-

mulative variance – or MFIV .

� Having done so, we can write contracts on it. Contracts on

VIX are very popular.

� We do have to be careful in interpreting what VIX means. It is

not the instantaneous volatility of the market.

� Most important, it is not an unbiased forecast of future market

volatility: it embeds a very substantial negative risk premium.

Lecture Note 4.1: Summary of Notation

Symbol Page Meaning

Ft,T p3 price of a claim to σTκ̃ p4 κ + λs

σ̃ p4 σ0(κ/κ̃)

f() p3 arbitrary payoff function

E⋆[ ] p6 expectation using risk-neutral probabilities

F0,T p8 time-0 forward price of S settling at T

MFIV 2 p12 Risk-neutral expected average variance from 0 to T

L0 p12 time-0 price of a claim to log(ST/F0,T)

b(k) p15 butterfly maturing at T with central strike price k

p(k), c(k) p16 put, call expiring at T with strike price k