[Courant Institute, Friz] Valuation of Volatility Derivatives as an Inverse Problem

8/20/2019 [Courant Institute, Friz] Valuation of Volatility Derivatives as an Inverse Problem

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dS t = σ S t dW t ,d S t = σ

2 S 2t dt.

Pricing a nancial contract by computing a risk-neutral expectation re-quires continuous-time hedging and the impact of doing this with a wrongσ is well studied, e.g. (Carr and Madan 1998) and the references therein.Market option prices are quoted as implied Black-Scholes volatilities and theresulting volatility surface is far from constant; the dependence of impliedvolatility on strike and expiration is referred to as the volatility smile. Forbackground on how such implied volatility surfaces look and how they movearound, see Cont and da Fonseca (2002).

Dupire suggested a one factor time-dependent Markov model in which

σ = σ(S t , t )

is determined by the implied volatility surface such that all European optionprices are recovered. It is a pure pricing process and σ represents an effectivevolatility for pricing processes rather than a true model of the dynamics of volatility.

More realistically, stochastic volatility models have been proposed inwhichσ = σ(t, ω)

itself follows a diffusion. These models are indeed able to generate a volatilitysmile. A particularly important measure of the smile, the at-the-money-skew, is known to be proportional to the correlation ρ. Exotic options canbe very sensitive not only the skew but also its dynamics in time in whichcase a stochastic volatility model is a necessity for pricing and hedging, seeGatheral (2003).

If the stock price is assumed to follow a continuous-path diffusion, a semi-static model-independent hedge perfectly replicates the realized variance

T 0 v(s, ω)ds ≡ x T and instruments with this payoff, variance swaps , are actively traded.

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In recent work (Carr and Lee 2003), Carr and Lee showed how to extend

this to instruments with arbitrary payoff

f ( x T )

provided that correlation is zero . For special f of the form exp[λ. ] both pricesand hedges are given explicitly in terms of associated European contractson S T . Consequently, the price of such volatility derivatives is determinedby European puts and calls expiring at T or, equivalently, by the time sliceT of the implied-volatility surface. Formally,

E[f ( x T )] = w

f (k) c(k, T ) dk (1)

where the c(k, T ) are European option prices and the wf (k) are weights tobe found.

In their paper, Carr and Lee use Laplace techniques to decompose anarbitrary f in terms of payoffs of the form exp[λ.] and a formal computationsuggests how to price (and hedge) a claim with payoff f ( x T ). In effect,they show how to compute the weights for a replicating strip of Europeanoptions.

Our main insight is that the weights wf used for the replicating stripof European options do not depend nicely on f and in fact may not bewell-dened for many functions f of interest (such as variance calls).

In our paper we implement a regularization scheme to permit computa-tion of the weights and in so doing, demonstrate that inverting equation (1)directly is a more appealing approach.

We exhibit the problem as an ill-posed one and show how least-squaresoptimization can be used to construct solutions.

We focus on two payoffs of particular nancial interest: the volatility swap

f ( x T ) =

xT

and the call on variance f ( x T ) = ( x T −K )+ .

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2 Laplace transform of quadratic variationWe assume no jumps and zero risk-free rate (for simplicity) so that therisk-neutral stock evolution is given by

dS t = σ(t, ω) S t dW t .

Set xT = log[S T /S 0] and observe that its quadratic variation x T is exactlythe total realized variance T 0 σ2(t, ω)dt.Proposition 1 (Carr,Lee). Assume independence between the Brownian motion W and volatility σ. With p(λ) = 1 / 2

± 1/ 4 + 2λ we have

E [eλ x T ] = E [e p(λ )xT ]. (2)

Proof. Note that xt has a drift −12 x t . Thenyt = xt +

12

x t

is a martingale with associated exponential martingale

exp pyt − 12 p2 y t = exp pxt − 12( p2 − p) x t .

First, assume x t = T 0 σ2(t)dt deterministic. Then taking expectationsyields the required identity. In the stochastic case, we can condition on thevolatility scenario which amounts to xing x t (ω) . Repeat the argumentgiven before and then average over all possible volatility scenarios. Theresult follows.

Remark 2. Either choice of the sign yields a correct expectation, interpreted as the price of an instrument. But the hedging strategy may be different. For instance, λ = 0 leads to p = 0, 1. The fair price 1 can be obtained be holding one dollar or as risk-neutral value of S T /S 0 at time T .

Remark 3. Generalized European-style payoffs, as on the r.h.s. above, can be replicated by an innite strip of (out of the money) calls/puts. Set F = S 0,

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k = log(K/F ) and c(k) = C (K, T ; S 0)/K , p(k) = P (K, T ; S 0)/K . Note that

put-call symmetry holds, p(k) = e− k c(−k).E [eλ x T −1] =

1F p

(E [S pT ]−F p)=

p( p−1)F p F 0 dKP (K )K p− 2 + ∞F dKC (K )K p− 2

= 2 λ 0−∞ dk P (K )K ekp + ∞0 dk C (K )K ekp= 2 λ

∞

0dk c(k) ek e− kp +

∞

0dk c(k) ekp

= 4 λ ∞

0dk c(k) ek/ 2 cosh[k 1/ 4 + 2λ] (3)

where complex λ are not a problem 2.From the second line above observe the ATM-weight

p( p−1)F p

K p− 2|K = F = p( p−1)

F 2 =

2λS 20

Remark 4. For the well-known variance swap,

E[ x T ] = 2

F

0 dK P (K )

K 2 + ∞

F dk C (K )

K 2 (4)

= 2 0−∞ dk p(k) + ∞0 dk c(k) (5)= 2 ∞0 dk c(k) ek + ∞0 dk c(k) using put-call symmetry = 4 ∞0 dk c(k) ek/ 2 cosh[k/ 2] (6)

As a consistency check, the last expression can also be obtained from (3) by differentiating with respect to λ at λ = 0.

For later reference we note the ATM-weight with respect to actual strikes 2/F 2 = 2 /S 20 from (4) and log-strikes 4ek/ 2 cosh[k/ 2]|k=0 = 4 from (6).

2 a) Note that the l.h.s. can become innity for Re[ λ] too large, this will only dependon the pdf of volatility in the considered model. Any stochastic volatility model in whichvolatility stay below some σmax implies nite exponential moments.

b) Note that cosh[ k 1/ 4 + 2 λ] is an entire function in λ .5


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3 Arbitrary functions of realized varianceAssume we want to replicate f ( x T ) and manage to nd a way to represent(or at least approximate) f by a linear combination of Laplace functionalsy →exp[λ y], λ∈C .Example 5. f (y) = √ y. By a well-known formula,

√ y = 12√ π ∞0 1 −e− λyλ3/ 2 dλ. (7)

Example 6. Whenever f (y) has a Laplace-transform F (λ), for a

∈ R on

the right of all singularities of F ,

f (y) = 12πi a+ i∞a− i∞ F (λ) eλ y dλ (8)

.

In general, an innite number of Laplace functionals has to be combinedto recreate a given payoff. However, a nite number will suffice to approxi-mate f . We quote a classical result due to L. Schwartz,(Schwartz 1959)

Theorem 7. The linear span of {e− λ y : λ∈R , λ > 0}is dense in C ([0, ∞), R ).In practical terms, it will suffice to nd an approximation for say y =

x T ∈[0, T ] since we are almost sure that, say, S&P 500 volatility remainsbelow 100%.

3.1 Volatility swapsUsing (7) and taking expectations

E

x T =

12√ π ∞0 1 −E e− λ x T λ3/ 2 dλ. (9)

Substituting the expression for E e− λ x T from equation (3), we obtain

E x T = 2√ π ∞

0

dλ√ λ ∞0 dk ek/ 2 c(k)cosh k 1/ 4 −2 λ

= ∞0 dk c(k) wvol (k)6


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with

wvol (k) ≡ 2√ π ek/ 2 ∞

0dλ√ λ cosh k 1/ 4 −2 λ . (10)Theorem 8. We have

wvol (k) = π2 ek/ 2 I 1(k/ 2) + √ 2π δ (k) (11)To a very good approximation, only the delta-function term counts.

Proof. From (10), the weight of the option with log-strike k in the replicatingstrip is given by

2√ π e

k/ 2 ∞

0

dλ√ λ cosh k ( 1/ 4 −2 λ) = 2√ π ek/ 2 {J + K }

with

J = 1/ 80 dλ√ λ cosh k ( 1/ 4 −2 λ)and

K = ∞1/ 8 dλ√ λ cosh ik ( 2 λ −1/ 4) = ∞

1/ 8

dλ√ λ cos k ( 2 λ −1/ 4)

With the changes of variable u = 1/ 4 −2 λ and v = 2 λ −1/ 4, weobtainJ = √ 2 1/ 20 u du 1/ 4 −u2

cosh(k u) = π2√ 2 L− 1(k/ 2)

where Ln (.) denotes the Struve L-function and

K = √ 2 ∞0 v dv 1/ 4 + v2cos(k v)

The K integrand clearly diverges as v

→ ∞. We may eliminate this singu-

larity by dening

K = √ 2 ∞0 dv v 1/ 4 + v2 −1 cos(k v)

= π2√ 2 {I 1(k/ 2) −L− 1(k/ 2)}

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3.2 Generalized payoffsUsing (8) and taking expectations, a formal computation yields

E [f ( x T )] = 12πi a+ i∞a− i∞ F (λ) E eλ x T dλ

= 12πi a+ i∞a− i∞ F (λ) 1 + 4λ ∞0 dk c(k) ek/ 2 cosh[k 1/ 4 + 2λ] dλ

= f (0) + ∞0 dk c(k) 4 ek/ 2 12πi a+ i∞a− i∞ F (λ) λ cosh[k 1/ 4 + 2λ]dλ .(12)Elegant though the complex integral in (12) might look, its convergence

is not guaranteed; indeed for typical non-smooth payoffs such as calls, itdiverges. The rest of our paper concerns itself with how to deal with suchpayoffs.

Example 11 (Variance call). f (y) = ( y −K )+ with K ≥0 has Laplace-transform F (λ) = e− λK /λ 2. Formally E ( x T −K )+ = ∞0 dk c(k) 4 ek/ 22πi a+ i∞a− i∞ e− λK λ cosh[k 1/ 4 + 2λ]dλ

=: wcall (k)= wcall (k,K )

.

(13)Note that e− λK is a purely oscillating term as λ → a ± i∞. For k > 0 we have the exponentially divergent term

|cosh[k 1/ 4 + 2λ]|∼exp k Im [λ] as Im [λ] →+ ∞.Later we will introduce the mollication technique which will provide a strong enough decay to ensure existence of all integrals.

In anticipation of our discussion of ATM weights in Section 4, note that for k = 0 we observe

wcall (k = 0, K ) = 4 12πi

a+ i∞

a− i∞

eλ (− K )

λ dλ = 4 θ(−K )

using the fact that the Heaviside-function θ(.) has Laplace-transform 1/λ. In particular,

wcall (k = 0, K ) = 0 for all K > 0

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where as from the discussion of the variance swap

E [( x T −0)+ ] = E [( x T ] = 4 ∞

0dk c(k) ek/ 2 cosh[k/ 2]

and we observe the discontinuity

wcall (k = 0, K = 0) = 4 .

4 The structure of ATM weightsIn this section only, we revert to dollar call and put prices ( C, P ) and dollarstrikes K and assume that S t is one component of a continuous-path (notnecessarily time-homogeneous) Markovian diffusion.

The following result does not require a zero-correlation assump-tion. In particular, local volatility and arbitrary stochastic volatility under-lying dynamics are allowed.

Theorem 12. Assume zero risk-free rate, no dividends and a decomposition

E [f ( x T )] = ∞

0 dK w(K ; S 0) P (T, S 0; K ) if K < S 0C (T, S 0; K ) if K ≥S 0 .

Note that for this decomposition in terms of out-of-the-money options to hold we necessarily have f (0) = 0 .

Then the ATM-weight w(S 0, S 0) is given by

2f (0)S 20

.

Proof. We only discuss two special cases, the general case is an obviousextension.

Case 1 (Local Volatility)

dS t = σloc(S t , t )S t dW tdyt = σ2loc(S t , t ) dt

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Note

x T

≡ yT . As a 2-dimensional time-inhomogeneous Markov process,

its generator is written as

L= 12

σ2loc(S t , t ) S 2 ∂ SS + σ2loc(S t , t ) ∂ y

Observe thatE [f ( x T )] = E [f (yT )]

= E [f ( x T )|S 0, y0 = 0]= u(T, S 0, 0)

where u(T,S,y ) is a solution of the PDE

∂ T u = L[u]with initial condition

u(0,S ,y) = f (y).Pricing standard (out-of-the-money) Europeans calls and puts is based onthe same PDE but uses different inital conditions,

gK (S ) := ( K −S )+ when K < S 0 and gK (S ) := ( S −K )+ otherwise.The resulting PDE solutions ( e.g. the prices) at time 0 are

P (T, S 0; K ) resp. C (T, S 0; K ).

By put-call parity,P (T, S 0; S 0) = C (T, S 0; S 0).

By assumption,

u(T ; S 0, 0) = ∞0 dK w(K ; S 0) P (T, S 0; K ) if K < S 0C (T, S 0; K ) if K ≥S 0 .Now differentiate w.r.t. T and evaluate at T = 0. From the PDE

[Lf ](S 0, 0) = ∞0 dK w(K ; S 0) [LgK ](S 0, 0).Note that [ LgK ](S, 0) =

12 σ

2loc(S, 0) S

2

δ K (S ) and [Lf ](S, y) = σ2loc(S 0, 0) f (y).Hence

σ2loc(S 0, 0)f (0) = 1

2 ∞0 dK w(K ; S 0) σ2loc(S 0, 0) S 20 δ K (S 0)=

12

σ2loc(S 0, 0) S 20 w(S 0; S 0).

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Case 2 (Stochastic Volatility)

The modications are minor. The underlying Markovian diffusion now con-tains three components: the stock price process S t , its instantaneous varianceprocess vt and the accumulated total variance yt . As before

dyt = vt dt

and

L= 12

v S 2∂ SS + (terms involving ∂ v , ∂ vv , ∂ Sv ) + v∂ y .

The derivation goes through line by line as in the local volatility case. It

suffices to note that all the payoffs under consideration, namely f (y), (K −S )+ and (S −K )+ are insensitive to v so differentiating w.r.t v yields a zerocontribution.Example 13 (Variance Swap). f (z ) = z. We nd ATM-weight 2/S 20 in agreement with the well-known result (see Remark 4).

Example 14 (Volatility Swap). f (z ) = √ z leads to f (0) = ∞. This implies mass-concentration of k →w(k, S 0) at the money ( k = S 0) in agree-ment with Section 3.1.Example 15 (Laplace payoff). f (z ) = exp[λz ]

− 1. The ATM-weight

equals 2λ/S 20 in agreement with equation (3).

Example 16 (Variance Call). f (z ) = ( z −K )+ . We have f (0) = 0implying zero ATM weight in agreement with Example 11.The last example provides a different proof of the discontinuity of ATM-

weights for variance calls as K →0 which we found earlier.

5 Pdf of variance

Fix some strike K > 0. Assume P [ x T ∈ dy] = g(y)dy. Crystallizing theessence of the Carr-Lee result, we propose to recover the pdf of realizedquadratic variation from a strip of European calls.

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Let δ (h)K denote the Gaussian density with mean K and standard devia-

tion √ h 1. Then, using (3) we obtaing(K ) = E [δ K ( x T )] ≈E [δ

(h)K ( x T )]

= E 12π ∞−∞ dp e− ip x T ei p K e− hp 2 / 2 (Fourier-inversion)

= 12π ∞−∞ dp ei pK e− hp 2 / 2 E e− ip x T

= 12π ∞−∞ dp ei pK e− hp 2 / 2 1 −4ip ∞0 dk c(k) ek/ 2 cosh k 1/ 4 −2ip

= δ (h)K (0)

→0 as h →0+

∞

0 c(k) w pdf (k, K ; h) dk

by application of Fubini’s Theorem 3 and with call option weights

w pdf (k, K ; h) = 4 ek/ 2 12π ∞−∞ dp e− hp 2 / 2 (−ip) eipK cosh k 1/ 4 −2ip .

3 We sketch how to justify the use of Fubini’s theorem above as we don’t believe it istrivial. It suffices to check that the iterated integral

= 1

2π ∞

−∞

dp ei pK e− hp2 / 2 1

−4ip

∞

0

dk c(k) ek/ 2 cosh k

1/ 4

−2ip

(which we know to be nite) exists as absolutely convergent iterated integral. Noting that

cosh k 1/ 4 −2ip ∼exp k | p|1 / 2the essence is

∞

−∞

dp e− hp2 / 2 p

∞

0dk c(k) exp k( 1/ 2 + | p|

1 / 2 ) .

W.l.o.g. we may consider BS-calls where c(k)∼exp(−ηk2 ) and we estimate

∞

−∞

dp e− hp2 / 2 p

∞

0dk exp(−ηk

2 ) exp k( 1/ 2 + | p|1 / 2 )

∼ ∞

−∞

dp e− hp 2 / 2 ∞

0dk exp(−ηk2 + k | p|1 / 2 )

≤ ∞

−∞

dp e− hp2 / 2 e p/ 4 √ π

< ∞.

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Then

E I [K, ∞ )( x T )

≈ E I(h)[K, ∞ )( x T )

= ∞K E [δ (h)y ( x T )] dy.From Section 5, this equals

∞K dy δ (h)y (0) + ∞0 c(k) 4 ek/ 2 12π ∞−∞ dp e− hp 2 / 2 (−ip) eipy cosh k

1/ 4 −2ip dk= I (h)[K, ∞ )(0)

→0 as h→0+

∞

0c(k) 4 ek/ 2 1

2π ∞

−∞dp e− hp

2 / 2 eipK cosh k 1/ 4 −2ip weights wdig = wdig (k,K ;h)dk.

The last equality is based on the fact that M K dy (−ip) eipy = eipK −eipM and that eipM tends weakly to zero as M → ∞4.We note the ATM-weight ( k = 0) of 4

12π ∞−∞ dp e− hp 2 / 2 eipK = 4 δ (h)K (0) ≈0 (14)

for h small enough relative to variance-strike K (a harmless and realisticassumption). Note also that with

f (z ) := I (h)[K, ∞ )(z ) −I(h)[K, ∞ )(0) ≈ I

(h)[K, ∞ )(z ) ≈ I [K, ∞ )(z )

we have a decomposition

E [f ( x T )] = ∞0 c(k) wdig (k, K ; h) dkto which our Theorem 12 applies:

wdig (0, K ; h)∼f (0) ≈0.which conrms (14).

4 Formally set M = ∞. The oscillations of eip ∞ will completely cancel the remainingsmooth integrand w.r.t. p

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7 Variance callsFix a strike K > 0. Once again, we propose to compute E [( x T −K )+ ] froma strip of European calls. We proceed by piling up digital payoffs,

gK (z ) := ( z −K )+ = ∞K I [y, ∞ ) (z ) dy.It is easy to check that

g(h)K (z ) := ( gK ∗δ (h)0 )(z ) = ∞K I (h)[y, ∞ )(z )dy.

As before, g(h)K (0) ≈ 0 for K > 0 and h small enough. HenceE ( x T −K )+ ≈ E g(h)K ( x T ) −g(h)K (0)

= E ∞K I (h)[y, ∞ )( x T ) −I (h)[y, ∞ )(0) dy= ∞K dy ∞0 c(k) wdig (k, y; h) dk= ∞0 dk c(k) ∞K wdig (k, y; h) dy=:

∞

0dk c(k) wcall (k, K ; h)

Proposition 18. Let a > 0. We have

wcall (k, K ; h) = 4ek/ 2

2πi a+ i∞a− i∞ ehλ 2 / 2 e− λK λ cosh k 1/ 4 + 2λ dλ (15)and this integral converges absolutely due to the exponential damping factor ehλ

2 / 2. Setting h = 0 we recover the (divergent) weight wcall (k, K ) which was obtained by a formal computation earlier (Example 11).Proof. Up to a factor 4 ek/ 2 the weights wdig (k, y; h) for the (mollied) digitalpayoff are

12π

∞

−∞dp e− hp

2

/ 2 eipy cosh k 1/ 4 −2ip= 12πi 0+ i∞0− i∞ dλ ehλ 2 / 2 e− λy cosh k 1/ 4 + 2λ (set λ = −ip)

= 12πi a+ i∞a − i∞ dλ ehλ 2 / 2 e− λy cosh k 1/ 4 + 2λ for any a∈R .

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The last equality follows by change of contour noting that the integrand

decays uniformly with Im (z ) → ±∞.In order to synthesize a variance call payoff we integrate as before

M K e− λy dy = 1λ (e− λK −e− λM )and get

12πi a+ i∞a− i∞ dλ e λ 2 / 2 1λ (e− λK −e− λM ) cosh k 1/ 4 + 2λ =: F (K, a )−F (M, a )Then lim M →∞ F (M, a ) = 0 since a > 0 and

ehλ2 / 2 1

λ e− λM cosh k 1/ 4 + 2λ ≤e− aM ehλ

2 / 2 1λ

cosh k 1/ 4 + 2λ integrable over a ± i∞

7.1 Visualization of the weightsNote rst that as h

→0, the integrand in (15) becomes highly oscillatory and

so direct integration is not obviously the right way to compute the weights.To see what this means in practice, consider the examples presented inFigures 1 and 2.

It is clear from the form of the regularization scheme that increasing h ing(h)K (z ) effectively smoothes the payoff gK (z ) of a variance call. For reference,in Figure 3 we graph g(h)K (z ) for h = .0001 and h = 0.00001 respectively withK = 0.04 in both cases.

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1 2 3 4 5k

-20000

-15000

-10000

-5000

5000

10000

15000

20000

Weight

Figure 1: Here we see a plot of the weights wcall (k, K ; h) as a function of log-strike k with K = 0.04 and h = 0.0001. Note that the scale has beencarefully chosen so the rst peak (at +10 , 000 or so) can be seen. It wouldbe an understatement to say that the weights are oscillatory.

1 2 3 4 5k

-1·1013

-7.5·1012

-5·1012

-2.5·1012

2.5·1012

5·10 12

7.5·1012

1·1013

Weight

Figure 2: Now we see a plot of the weights wcall (k, K ; h) as a function of log-strike k with K = 0.04 and h = 0.00001 (ten times smaller than in thegure above. The weights are even more oscillatory with this smaller valueof h.

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0.02 0.04 0.06 0.08

0.01

0.02

0.03

0.04

Payoff

Figure 3: The solid blue line is the regularized payoff g(h)K (z ) with h =0.00001 plotted as a function of total realized variance z and the dashed redline, the payoff with h = 0.0001 both with K = 0.04 = 20%2.

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7.2 Computing the weights: an ill-posed PDEAdopting the same notation as in the last section, consider the reducedweight

wred (k, K ) = wcall4ek/ 2

= 12πi a+ i∞a− i∞ ehλ 2 / 2 e− λK λ cosh k 1/ 4 + 2λ dλ.

Then

∂ 2

∂k 2 wred =

12πi a+ i∞a− i∞ ehλ 2 / 2 e− λK λ (1/ 4 + 2λ) cosh k 1/ 4 + 2λ dλ

= 1

4 wred −2 ∂ ∂K wred . (16)

This PDE lives on the domain ( k, K ) ∈ [0, ∞) × [0, ∞) with boundaryconditions (as h →0)wred (k = 0, K > 0) = 0, wred (k > 0, K = 0) = cosh[ k/ 2].

The rst boundary condition follows from Example 16 and the second onefrom the known variance swap weights, see (4). Note however the disconti-nuity at k = 0, K = 0.

At rst sight, equation (16) is a standard parabolic PDE with time vari-able K and space variable k. It is the negative (wrong) sign of the K -derivative that leads us to identify it as ill-posed. To solve it, we wouldeffectively have to solve a backward equation forward in time given an ini-tial condition 5.

8 Dealing with ill-posednessWe have accumulated evidence that the map

f (.)

→“weights”

is ill-posed in the sense that small perturbations of f in sup-topology mayresult in dramatic changes in the weights. The most that we can hope forin that case is a solution in some least-squares sense.

5 In Fourier space, a symptom of this problem would be exponential blowup of themodes.

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To this end, recall Remark 17 that given the law of realized variance

g(y) and under the standing zero-correlation assumption, calls are priced byaveraging Black and Scholes prices,

c(k) = ∞0 cBS (k, y) g(y) dy =: [L g(.)](k).Formally then, we may obtain the pdf of variance by inverting the linear

integral operator L . In terms of the regularization scheme {W h : h > 0} of Section 17,L − 1 = lim

h→0W h .

8.1 Matrix implementation

We pick a number of (log) strikes ( e.g. from the market)

{ki}, i = 1,...,nand a number of variance levels

{v j}, j = 1,...,m.To capture the shape of total variance, {v j }should be ne enough and cover arealistically large range of variances. Setting ci = c(ki) and Aij = cBS (ki , v j )we obtain

ci =m

j =1

Aij g j

where g = {g j} is a probability vector. From the injectivity of the BS-integral operator, we see that A : R m →R n is injective.If m > n , we have an under-determined linear system, whose minimal so-lution ( |g|2 = g2 j → min!) is computed using the Moore-Penrose pseudo-inverse,

g = A T (A A T )− 1 c =: M c .

The numerical quality of this procedure is determined by the conditioningnumber γ , the ratio of the largest to the smallest eigenvalue of A A T (theseare the singular values of A). Observe that at no point we did we imposeon g that it should be an actual probability vector, that is

g j ≥0, j

g j = 1.

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In fact, from the consistency of the European call option prices used as in-

put (assuming they come from a zero-correlation world so that Hull-Whiteapplies), these conditions are automatically satised. In practice, althoughin principle we can nd a discretization with sufficiently many points suf-ciently nely spaced that these conditions are approximately satised, itmay make sense to add these conditions as explicit constraints.

Remark 19. One might wonder whether or not the Moore-Penrose pseudoinverse procedure generates a result that is stable with respect to small per-turbations in the input option prices. In fact, niteness of the matrix norm of M guarantees stability of the solution with respect to such small pertur-bations.

8.2 Variance call options revisitedWith the same notation as before, pricing a variance-call struck at K nowamounts to computing

i

(vi −K )+ gi =: C var (K )

On the other hand, in the context of a stochastic volatility model, variance-calls may be priced by Monte Carlo simulation of

E T 0 σ2(s, ω) ds −K +

=: C̃ var (K )

or by solving a PDE using the ideas of Section 4.Alternatively, if the characteristic function of quadratic variation (total

realized variance) is known in closed-form (as it is for many commonly-usedmodels), we may apply the Fourier transform techniques of Carr and Madan(1999).

Example 20. Consider the Heston model with the well-known Bakshi, Caoand Chen parameters (see Bakshi, Cao, and Chen (1997) and the references therein). Heston volatility of volatility η is 0.39. We price n one-year Euro-pean calls with reasonably spread out (log-)strikes in increments of ∆ k. We also pick m levels of total variance in increments of ∆ v. We choose

n = 5, ∆ k = 0.14, m = 45, ∆ v = 0.005

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(this choice actually minimizes the conditioning number γ from the last sec-

tion). This allows us to compute C var (K ) as explained above.On the other hand, by applying the bond pricing formula of Cox, In-

gersoll, and Ross (1985), we obtain the characteristic function of quadratic variation in the Heston model.

φT (u) = E [eiu x T ] = E [eiu T 0 vt dt ]

As noted earlier, we can then apply the Fourier transform methods of Carr and Madan (1999) to compute the fair value of a variance call. With the same choice of parameters, we obtain the nearly perfect t shown in Figure 4. The maximum numerical error in our regularization procedure in this case was 0.00043 which is around 1% of the variance swap value.

0.05 0.1 0.15 0.2K0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Variance Call

Figure 4: Value of one-year variance call vs variance strike K with Bakshi,Cao and Chen Heston model parameters. The solid blue line is the Fouriertransform computation. The dashed red line comes from our regularizationprocedure.

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8.3 Consistency with variance swapsFinally, we connect our regularization scheme results to the variance call“weights” studied earlier. Specically,

C var (K ) =m

i=1

(vi −K )+ gi .

=m

i=1

n

j =1

(vi −K )+ M ij c j

=:n

j =1

c j w j [K ]

yields a discretization w j [K ] of the continuous (but not dened) weightswcall (k, K ).

Setting K = 0, we retrieve the (well-dened) variance swap weights w j [0].Then we have

C var (0) = E [ x T ]

= 4 ∞0 c(k) ek/ 2 cosh[k/ 2]dkn

j =1c j 4 e

kj / 2cosh[k j / 2](∆ k) j .

Now we can enforce consistency between calls on variance and variance swapsby requiring that

4 ekj / 2 cosh[k j / 2](∆ k) j =m

i=1

vi M ij for all j = 1,...,n

which generates n constraints on our choice of {vi , k j} pairs. Of course, theset of log-strikes k j will be dictated in practice by what is available in the(zero-correlation) market.

9 Relaxing the zero-correlation assumptionThroughout this paper, we have depended on the assumption of zero-correlationbetween quadratic variation and underlying returns. However, in the con-

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text of stochastic volatility for example, the prices of volatility derivatives

clearly do not depend on the correlation assumption.To make this observation concrete, suppose we were to t a stochastic

volatility model such as the Heston model to European option prices. Wewould obtain values for all the parameters of the model including the corre-lation ρ. Suppose we were to take that same model and regenerate optionprices with a different value of ρ (e.g. ρ = 0): clearly the fair values of volatility derivatives would not change.

We further observe that as noted in Lee (2001) for example the volatilitysmile must be symmetric around the at-the-money forward strike ( k = 0).This leads us to propose a recipe 6 for computing the fair value of volatility

derivatives under the assumption of continuous paths of the underlying butwith non-zero correlation between quadratic variation and returns:

• Fit a model to European option prices of a given expiration. Thismodel could either be a proper model or merely a functional form forimplied volatility as a function of strike.

• Compute the fair value of a variance swap from the t.• Change the parameters of the model in such a way that the volatilitysmile retains its shape and the fair value of the variance swap is pre-

served but the at-the-money volatility skew is zero7

. Effectively, rotatethe smile while retaining its overall level. For example, in a stochasticvolatility model, set the correlation ρ to zero.

• Apply the results of this paper to compute the values of other volatilityderivatives under the zero-correlation assumption.

10 Concluding remarksIn this paper, we have studied in detail the formal expression for the value

of volatility derivatives in terms of the prices of standard European optionsproposed by Carr and Lee under the zero-correlation assumption. Aftershowing that the formal expression for the weights that they derive diverges

6 The idea of rotating the smile is due to Jining Han.7 We will show how to make this more precise in a forthcoming paper.

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Cox, John C., Jonathan E. Ingersoll, and Steven A. Ross, 1985, A Theory

of the Term Structure of Interest Rates, Econometrica 53, 385–407.Gatheral, Jim, 2003, Case Studies in Financial Modeling Lecture Notes,

http://www.math.nyu.edu/fellows n math/gatheral/case studies.html.

Gradshteyn, I. S., and I. M. Ryzhik, 1980, Table of Integrals, Series and Products . (Academic Press New York, NY, USA).

Hull, John, and Alan White, 1987, The Pricing of Options with StochasticVolatilities, The Journal of Finance 19, 281–300.

Lee, Roger W., 2001, Implied and Local Volatilities under Stochastic Volatil-ity, International Journal of Theoretical and Applied Finance 4, 45–89.

Monk, P., 2003, Basic Theory of Linear Ill-posed Problems,http://www.inria.fr/actualites/colloques/2003/pnla/monk1.pdf.

Schwartz, Laurent, 1959, ´ Etude des sommes d’exponentielles . (HermannParis).

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[Courant Institute, Friz] Valuation of Volatility Derivatives as an Inverse Problem