Mixed Log-Normal Volatility Model

Richard White∗

First version: 2 August 2012; this version†August 17, 2012

Abstract

We present two versions of a Mixed Log-Normal Volatility Model. The first is applicableto produce a single, arbitrage free, volatility smile (i.e. it produces a single terminal Proba-bility Density Function (PDF) for some expiry) but not a consistent volatility surface. Thesecond version can produce an arbitrage free implied volatility surface and the correspondinglocal volatility surface via simple analytic expressions. These surfaces are useful for testingother numerical methods, in particular methods to calculate the fair value of variance swapsas the true value is known from the parameters of the model via another simple expression.

1 Single Horizon Model

Let the terminal stock price ST be given by

ST = FT eX (1)

where FT is the forward and X is a random variable with density ρ(x) given by a mixture ofnormal distributions:

ρ(x) =

N∑i=1

wiφ

[x;

(µi −

σ2i

2

)T, σ2

i T

] N∑i=1

wi = 1 wi ≥ 0 ∀ i (2)

where φ(x;µ, σ2) denotes a normal distribution with mean µ and variance σ2. In order thatE[ST ] = FT , we we require E[eX ] = 1, hence

N∑i=1

wieµiT = 1 (3)

Except in the special case of µi = 0 ∀ i, equation 3 can only be satisfied at one expiry T - this iswhy this setup can only produce a single smile not a full volatility surface.

The price of a European call is given by

C(T, k) = P (0, T )

∫ ∞−∞

(FT ex − k)+ρ(x)dx

= P (0, T )

N∑i=1

wiB(F ∗i , k, T, σi)

(4)

∗richard@opengamma.com†Version 1.1

1

where F ∗i = FT eµiT and B(·) is the Black formula for the forward price

B(F, k, T, σ) = FΦ(d1)− kΦ(d2)

d1 =ln(FK

)+ σ2T

2

σ√T

d2 = d1 − σ√T

(5)

Where Φ(·) is the cumulative Normal distribution. Hence options prices in this model aregiven simply by a weighted sum of Black prices. Implied volatilities are found by numericallyinverting (root finding) the Black formula in the the usual way. Clearly Greeks are also given bythe weighted sum of Black greeks.

1.1 Calibration

If the drifts µ are all set to zero, then the model has 2N − 1 free parameters (one free parameter

is absorbed by the weights constraint∑Ni=1 wi = 1) - the N volatilities and the N−1 parameters

that form the weights. Call these N − 1 parameters the vector y ∈ RN−1, then we require amapping

y→ w such that

N∑i=1

wi = 1 and wi ≥ 0 ∀ i (6)

Various non-linear mappings (normally trigonometric functions) exist and are discussed in theappendix. Calibration can then proceed by finding the 2N − 1 parameters that minimised thesquared difference between the model and market prices.1

In the case of non-zero drifts, there are 3N − 2 free parameters. Let ai ≡ wieµiT , then

the second constraint is∑Ni=1 ai = 1 ai ≥ 0 ∀ i, which is exactly the same form as the

weight constraint. Having found the ai from an additional N − 1 free parameters, the drifts are

µi = 1T ln

(aiwi

). So once again calibration can then proceed by finding the 3N − 2 parameters

that minimised the squared difference between the model and market prices.

1.2 Variation of the Model

The constraint of equation 3 can be rewritten as

N∑i=1

wiFi = 1 where Fi =F ∗iFT

(7)

and the call price as

C(T, k) = P (0, T )FT

N∑i=1

wiB(Fi, x, T, σi) (8)

where x = kFT

is the moneyness. If the relative forward terms, Fi, are treated as constantacross expiries, then equation 8 gives European option prices for all T and k while preservingthe forward. However the interpretation is thus: at time zero, the stock price, S0, jumps to oneof N levels FiS0, it then follows a geometrical Brownian motion with volatility σi such that

E[St|statei] = FiFt

and E[St] =

N∑i=1

wiE[St|statei] = Ft(9)

1It is best to do this between market and model implied volatilities or prices weighted by vega.

2

This jump at time zero is somewhat unphysical and means it is not possible to delta hedge.

2 Multi Horizon Model

We make small change from equation 1 and write the stock price as

St = FteXt−Ωt (10)

where Ωt is the drift compensator, which ensures that E[St] = Ft ∀ t. Its value is given by

Ωt = ln

[N∑i=1

wieµit

](11)

Writing the relative forward, Fi(t) as

Fi(t) ≡ eµit−Ωt ≡ eµit∑Ni=1 wie

µit(12)

then the call price is again given by

C(T, k) = P (0, T )FT

N∑i=1

wiB(Fi(T ), x, T, σi) (13)

however the interpretation is different: at time zero the stock goes into one of N states, but thestock price does not jump, instead it follows a geometrical Brownian motion with volatility σisuch that

E[St|statei] = Fteµit−Ωt

and E[St] =

N∑i=1

wiE[St|statei] = Ft(14)

Since we can find an arbitrage free price at any expiry, T , and any strike, k, we can find thecorresponding implied volatility, hence we have a complete volatility surface.

2.0.1 Implied Volatility for T → 0

For an ATM option we have

C(T, F ) = F

(2Φ

(σ√T

2

)− 1

)(15)

so for small times2 (or zero drifts) the ATM implied volatility is

σ(T, F ) =2√T

Φ−1

(N∑i=1

wiΦ

(σi√T

2

))(16)

2small times mean when diffusion dominates, i.e. σ√t µt → t

(σµ

)2, and since σ is nearly always

numerically greater than µ, this means any time significantly less than a year.

3

In the limit of T → 0 this becomes3

σ(0, F ) =

N∑i=1

wiσi (17)

This should be very accurate for expiries less than a few days.

2.1 Local Volatility

Defining the fractional call price[Whi12b] as

C(T, x) =C(T, xFT )

P (0, T )FT(18)

we have

C(T, x) =

N∑i=1

wiB(Fi(T ), x, T, σi) (19)

The local volatility can be found by a slight modification of the Dupire formula [Dup94,Whi12b]

σ(T, x)2 = 2∂C∂T

x2 ∂2C∂x2

(20)

The local volatility given here is parameterised by moneyness, x - the local volatility parame-

terised by strike, k, is given by σ(T, k) = σ(T, k

FT

).

The term in the denominator of equation 20, ∂2C∂x2 , is the (modified) dual gamma. It is given

by

∂2C

∂x2≡ Γx =

N∑i=1

wiΓx(Fi(T ), x, T, σi)

where Γx(F, x, T, σ) =φ(d2)

xσ√T

=Fφ(d1)

x2σ√T

(21)

φ(·) is the Normal PDF.The numerator is the modified theta4, it is given by

∂C

∂T≡ Θ =

N∑i=1

wi

[Θ(Fi(T ), x, T, σi) + ∆(Fi(T ), x, T, σi)

∂Fi(T )

∂T

]

where Θ(F, x, T, σ) =Fφ(d1)σ

2√T

, ∆(F, x, T, σ) = Φ(d1)

and∂Fi(T )

∂T= Fi(T ) (µi − µ∗T ) where µ∗T =

∑Nj=1 wjµje

µjT∑Nj=1 wje

µjT

(22)

3We have used the well know result that Φ(x) ≈ 12

+ x√2π

for x→ 04theta is normally defined as an options time decay - the change in price with respect to calendar time. The

theta here is the sensitivity of (fractional) option price to expiry.

4

Putting this all together, we have an expression for the local volatility

σ(T, x)2 =

∑Ni=1 wiFi(T )

[φ(d1,i)σi + 2

√TΦ(d1,i)(µi − µ∗T )

]∑Ni=1 wiFi(T )

φ(d1,i)σi

where d1,i =ln(Fi(T )x

)+

σ2i T2

σi√T

(23)

It should be noted that switching from call delta, Φ(d1,i), to put delta, −Φ(−d1,i) ≡ Φ(d1,i)− 1,gives the same local volatility (as it must) by virtue of the definition of µ∗T , and one should usethe one for out of the money options (i.e. call for strike greater than forward).

2.1.1 Calendar Arbitrage

While the denominator of equation 20 is always positive, the numerator is not. If the numeratordoes become negative, the local volatility does not exist at that point and the model is not validfor times at and beyond that point. The condition

N∑i=1

wiFi(T )[φ(d1,i)σi + 2

√TΦ(d1,i)(µi − µ∗T )

]≥ 0 ∀T, x (24)

cannot be satisfied for arbitrary µ - this limits its applications as care must be taken in thechoice of parameters.

2.1.2 Regularisation

At extreme strikes, the values |d1,i| can be quite large and thus the φ(d1,i) terms are very small -this division of two very small numbers can give spurious results (and not-a-number in extremecases). If j is the index of the smallest value of |d1,i| and we dividend top and bottom by φ(d1,j),we rewrite the expression as as

σ(T, x)2 =

∑Ni=1 wiFi(T )

[exp

(d21,j−d

21,i

2

)σi + 2

√2πT exp

(d21,j

2

)Φ(d1,i)(µi − µ∗T )

]∑Ni=1 wi

Fi(T )σi

exp(d21,j−d21,i

2

) (25)

The term exp(d21,j

2

)Φ(d1,i) can still be problematic, however for very large negative d1,i we can

use the approximation

Φ(x) ≈ φ(x)√1 + x2

for x 0 (26)

and write the term as

exp

(d2

1,j

2

)Φ(d1,i) ≈

exp(d21,j−d

21,i

2

)√

2π√

1 + d21,i

(27)

2.1.3 Local Volatility for T → 0

In the limit of T → 0, the ATM (i.e. x = 1) local volatility is given by

σ(0, 1)2 =

∑Ni=1 wiσi∑Ni=1

wi

σi

(28)

5

If we define y = ln(x)√T

then the local volatility for T → 0 is

σ(T, x)2 =

∑Ni=1 wi exp

(− y2

2σ2i

)σi∑N

i=1 wi exp(− y2

2σ2i

)1σi

(29)

That is, it is a function of y only. For T < T0 where T0 is some small expiry cut-off5, we have

σ(T, x) = σ

(T0, x

√T0T

)(30)

In this way we can extrapolate the local volatility surface to T = 0 without numerical problems.

2.2 Variance Swaps

The pricing of variance swaps from implied and local volatility surfaces is discussed here [Whi12a].When the underlying is a pure diffusion process (i.e. no jumps), the expected value of theannualised realised variance is given by [DDKZ99, Gat06]

EV (T ) = − 2

TE[ln

(STFT

)](31)

For the mixed log-normal model this becomes

EV (T ) = − 2

T(E[XT ]− ΩT )

= − 2

T

(N∑i=1

wi

(µi −

σ2i

2

)T − ln

[N∑i=1

wieµiT

])

= 2

(1

Tln

[N∑i=1

wieµiT

]−

N∑i=1

wi

(µi −

σ2i

2

)) (32)

In the limit of T → 0 (or µi = 0 ∀ i) this becomes simply

EV (T → 0) =

N∑i=1

wiσ2i (33)

So we have a very simple formula for the exact value of a variance swap under a mixed log-normalmodel. The volatility surfaces (implied and local) that this model produces can be treated asextraneously given, and use in numerical procedures to calculate the value of variance swaps -thus providing an excellent test of the numerical procedures.

2.3 Examples

Figures 1 and 2 show the implied and local volatility surfaces from a model with N = 3, w =0.7, 0.25, 0.05, σ = 0.3, 0.6, 1.0 and µ = 0.0, 0.3,−0.5. The spot is 100 and drift is 5%.

5T0 = 10−4 is general suitable

6

0.01

0.08

960.16

920.24

880.32

840.40

80.48

760.56

720.64

680.72

640.80

60.88

560.96

521.04

481.12

441.20

4

1.28

36

1.36

32

1.44

28

1.52

24

1.60

2

1.68

16

1.76

12

1.84

08

1.92

04 20.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

101927364554627180889710

611

412

313

214

114

915

816

717

518

4

193

201

210

219

228

236

245

254

262

271

280

288

297

Expiry strike

Implied Volatility

Figure 1: The Implied Volatility Surface from a mixed log-normal model.

00.08

0.16

0.24

0.32

0.4

0.48

0.56

0.64

0.72

0.8

0.88

0.96

1.04

1.12

1.2

1.28

1.36

1.44

1.52

1.6

1.68

1.76

1.84

1.92 2

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

1.10

1.20

1.30

101927364554627180889710

611

412

313

214

114

915

816

717

518

4

193

201

210

219

228

236

245

254

262

271

280

288

297

time spot

Local Volatility

Figure 2: The LocalVolatility Surface from a mixed log-normal model.

7

A Sum to a Fixed Number Constraint

The constraintN∑i=1

α2i = C C > 0 (34)

where C is some constant, occurs frequently We can rescale to C = 1 without any loss ofgenerality. The constraint

N∑i=1

wi = 1 wi ≥ 0 ∀ i (35)

is equivalent by setting wi = α2i . For N = 2 the obvious solution is

α1 = cos θ

α2 = sin θ(36)

where the angle θ is unconstrained6. The vector α = (α1, α2)T , lies on the unit circle. This canbe extended to higher dimensions by describing a point on the N -dimensional unit hypersphereusing N − 1 angles. One such scheme [Reb04] is to build the N scheme from the N − 1 scheme.If the vector αN−1 satisfies equation 34 then

αNi =

c1 for i = i

s1αN−1i for i > 1

where si ≡ sin θi & ci ≡ cos θi

(37)

will also satisfy the equation. Starting from the N = 2 case, the general expression is

αi =

(i−2∏k=1

sk

)×

ci for i < N

sN−1 for i = N − 1(38)

where the product term is set to 1 if it contains no terms (i.e. i < 3).This does not evenly distribute the number of trigonometric terms among the αs - αN has

N − 1 terms while α1 has only a single cosine. This means that a uniformly distributed set ofpoints in the N − 1 hypercube, will map to a set of points heavily concentrated around the polesof the hypersphere.7 The effect grows with N and can affect search algorithms.

In order to democratise the distribution of trigonometric terms, we prepose the followingscheme. Firstly assume that we have two schemes of size N1 and N2 with vectors αN1 and αN2

both satisfying equation 34. The scheme for N = N1 +N2 is then

αi =

sN−1α

N1i for i ≤ N1

cN−1αN2

i−N1for i > N1

(39)

and it is easy to verify that this also satisfies equation 34. So by partitioning N into two nearequal sized parts N1 and N2 and repeating until N = 1, we can build up the scheme starting

6of course only some 2π range is unique.7This is the opposite of what happens when a global is projected onto a plane (i.e. a map), making Greenland

disproportionally large.

8

from the trivial αN=11 = 1. For the case of N = 7 the following scheme results:

α1 = s1s2s4

α2 = s1s2c4

α3 = s1c2s5

α4 = s1c2c5

α5 = c1s3s6

α6 = c1s3c6

α7 = c1c3 1

(40)

We have included the multiply by 1 term in α7 for clarity. The maximum number of trigono-metric terms for any α is dlog2Ne and the minimum is one less than this. Also the number of

multiplication has been reduced from N(N−1)2 to N log2N .

References

[DDKZ99] Demeterfi, Derman, Kamal, and Zou. More than you ever wanted to know aboutvolatility swaps. Technical report, Goldman Sachs Quantitative Strategies ResearchNotes, 1999.

[Dup94] Bruno Dupire. Pricing with a smile. Risk, 7(1):18–20, 1994. Reprinted in DerivativePricing:The Classical Collection, Risk Books (2004).

[Gat06] J. Gatheral. The Volatility Surface: a Practitioner’s Guide. Finance Series. Wiley,2006.

[Reb04] Riccardo Rebonato. Volatility and Correlation. John Wiley & Sons, 2nd edition,2004.

[Whi12a] Richard White. Equity variance swap with dividends. Technical report,OpenGamma working paper, 2012. http://developers.opengamma.com/

quantitative-research/Equity-Variance-Swaps-with-Dividends-OpenGamma.

pdf.

[Whi12b] Richard White. Local volatility. OG notes, OpenGamma, 2012. http://developers.opengamma.com/quantitative-research/Local-Volatility-OpenGamma.pdf.

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