ImperialMathFinance: Finance and Stochastics Seminar

Option valuation in a general stochastic volatilitymodel

Kai Zhang and Nick Webber

University of Warwick

10 November 2010

Zhang and Webber: General stochastic volatility 1 / 48

Abstract

Stochastic volatility models are frequently used in the markets to model the impliedvolatility surface. These models have several failings. Firstly, although improvementson a basic Black-Scholes model, they nevertheless fail to fit the entire surfaceadequately. Secondly, the improvements they offer are usually at the cost of greatlyreduced tractability. Thirdly, these models still fail to fit to market prices ofnon-vanilla securities.

This paper addresses the second of these three issues. A general stochastic volatilitymodel is described, nesting both the Heston models and Sabr-related models. Acontrol variate Monte Carlo valuation method for this model is presented that, whenit can be applied, is shown to be a significant improvement over existing simulationmethods; when applied to barrier option pricing, it out-performs importancesampling methods.

By providing a plausible simulation method for this general model, the paper opensthe possibility of exploring calibration to non-vanilla, as well as vanilla, instruments.


Pricing in financial markets

Need to price and hedge financial instruments.

Issues in effective hedging:Need to

1. Recover prices of hedging instruments;2. Model evolution of prices of hedging instruments closely,

ie, match hedge deltas and gammas as closely as possibile.

Do (1): Fit to implied volatility surfaceDo (2): Fit to instruments whose values are path-dependent


Calibration to vanillas

Calibration to market prices of vanilla options.

1. Necessary.Must correctly price one’s hedging instruments.

2. Insufficient.Can calibrate to vanillas but still misprice other instruments.

Vanillas are priced off asset density at their maturity time.Do not depend upon sample path of asset process.

Barrier options, Americans/Bermudan, average rate options, etc,Depend upon (full) sample path.


Fitting to the implied volatility surface

Various attempts:Jump-diffusion models (eg Merton, Bates, etc);Lévy models (eg VG, NIG, CGMY, etc);Stochastic volatility (SV) models (eg Heston, Sabr, etc).

Natural to investigate SV models:Naively attractive as implied volatility is clearly stochastic;Allows for (relatively) persistent smiles;Generates volatility clustering(and other empirically observed times series properties).


Contribution

1. Present a general SV model,nesting Heston, Sabr-clone, et cetera.

2. Construct a correlation-control variate for the model.

3. Apply the model to pricing:Average rate options, ←− focus on theseBarrier options.

4. Explore variations in barrier option pricing, ←− not discussedconsistent with the implied volatility surface.


A general stochastic volatility modelLet

(S)t≥0 be an asset price process,(V)t≥0 be a volatility process,

(1)

with SDEs

dSt = rStdt+ σf (Vt) Sβt dWS

t , (2)

dVt = α(µ−Vt)dt+ ηVγt dWV

t , (3)

dWSt dWV

t = ρdt, (4)

where f (v) = vξ and

ξ, β ≥ 0, determine the structure of volatility for S,γ ≥ 0, determines the process for V,γ > 0, when ξ /∈ Z,σ > 0, included for generality,α, µ ∈ R, are not required to be positive.

(5)

When ξ /∈ Z, Vt is required to remain positive.When ξ ∈ Z, Vt is permitted to become negative.


Characteristics of the processes

β and the process for St:

β = 0, SV absolute diffusion processβ ∈ (0, 1) , SV CEV processβ = 1 geometric SV process

ξ and the processes for St and Vt:

ξ = 0, St is non-stochastic GBM,ξ = 1

2 , Vt is a variance process,ξ = 1 Vt is a volatility process.

When ξ = 1, do not require Vt to remain positive.When β < 1

2 impose an absorbing boundary at 0.


Nesting other modelsβ = 0 β ∈ (0, 1) β = 1

Model ξ γ Model ξ γ Model ξ γ

1 1 Sabr 1 1 H&W 1 1Absolute 1 1

2 Sabr- 1 12 J&S 1 1

2diffusion 1 0 like 1 0 S&Z, S&S 1 0models 1

2 1 LKD 12 1 Garch 1

2 112

12

12

12 Heston 1

212

n/a 12 0 n/a 1

2 0 n/a 12 0

(β, 1, γ): Johnson and Shanno (1987).(β, 1, 1): Sabr, Hagan et al. (2002).(

β, 12 , γ

): Lord, Koekkoek & Dijk (2008).

(1, 1, 1): Wiggins (1987); Hull &White (1987) (µ = 0, ρ = 0).(1, 1, 0): Stein & Stein (1991) (ρ = 0); Schöbel & Zhu (1999) (ρ 6= 0).(

1, 12 , 1)

: Nelson (1990); Barone-Adesi et al. (2003) (ρ = 0); (and H & W, as µ = 0).(1, 1

2 , 12

): Heston (1993); Ball & Roma (1994).(

1, 12 , γ

): Andersen & Piterbarg (2007); Lewis (2000); Ait-Sahalia & Kimmel (2007).

(1, 2, 0): Jourdain & Sbai (2010) (special case).Zhang and Webber: General stochastic volatility 9 / 48

Not nestedMore general specifications:

(β, f (v) , g (v)): Jourdain & Sbai (2010) (including (1, 2, 0)),(h (S) , ξ, γ): Bourgade & Croissant (2005).

Log-specifications:(

β, ξ, γ)= (1, 1, 0), with f (v) = exp

(vξ)

:Scott (1987), Chesney and Scott (1989),Melino and Turnbull (1990).

Jump-diffusion models:Bates (1998).

General Lévy models:VG, NIG, CGMY, etc.

Stochastic interest rate and higher factor models:van Haastrecht, Lord, Pelsser & Schrager (2009), etc.


Valuation issues

Ackward to get prices out.

PDE and lattice methods:Two-factor PDEs are tricky;Difficult to get accurate prices quickly.

Monte Carlo methods:Issues with bias and convergence.

State dependent volatility is likely to be problematical, eg CIR.No strong solution to SDE,Zero is accessible (in equity calibrated models),Exact simulation is possible (but expensive. Scott (1996).)


Issues with Heston

Valuation:European options: direct numerical integration possible.Path-dependent options: can use only Monte Carlo.

Simulation:Exact simulation is possible but too expensive.

(Broadie and Kaya (2006), Glasserman and Kim (2008)).OK for long-step Monte Carlo, infeasible for short step.

Approximate solutions are poor(particulary when zero accessible for vt).

Need fast simulation methods.


Using Monte Carlo

Evolve from time t to time t+ ∆t.Short step: ∆t small.Usually required if:1. No exact solution to SDE.2. Option is continuously monitored.

Long step: ∆t equal to time between reset dates.Usually possible if:1. Option is discretely monitored.2. Exact solution to SDE, or a good approximation, is known.


Control variates and Monte CarloPlain Monte Carlo:

Generate M sample paths.Get discounted payoff cj along each sample path, j = 1, . . . , M.Plain MC option value c is

c =1M

M

∑j=1

cj. (6)

Control variate Monte Carlo:Along each sample path also generate CV value dj st E

[dj]= 0.

Let β = cov(cj, dj

)/ var

(dj)

thenCV corrected option value cCV is

cCV =1M

M

∑j=1

(cj − βdj

). (7)

Can extend to have multiple CVs.Zhang and Webber: General stochastic volatility 14 / 48

Efficiency gain

Suppose c computed in time τ with standard error σ,Suppose cCV computed in time τCV with standard error σCV.

Efficiency gain E is

E =τσ2

τCVσ2CV

. (8)

Proportional reduction in time by CV methodto get same standard error as plain method.

If ρ = corr(cj, dj

)then

E =τ

τCV

11− ρ2 . (9)

If τCV/τ ∼ 5 and ρ = 0.99 then E ∼ 10; if ρ = 0.999 then E ∼ 100.τCV/τ large? “Speed-up” is a better term than “variance reduction”.


Auxiliary model CVsAuxiliary instument.Suppose that have an option p “similar” to c such that

1. Along sample paths, corr(cj, pj

)is close to 1;

2. An explict solution p is known.

Then dj = pj − p is an auxiliary instrument CV.

Auxiliary model.Write M for the pricing model.Suppose have an auxiliary model Ma “similar” to M, such that

1. Same set of Wiener sample paths can be used for each model;2. c has an explicit solution ca in Ma.

Write caj for discounted payoff on path j under Ma

(so that ca ∼ ca = 1M ∑M

j=1 caj .)

Then dj = caj − ca is an auxiliary model CV.


The SV model: Auxiliary model and instrument CVFind suitable:

Auxiliary model Maj (effectively sample path dependent),

Auxiliary instrument p, so that p has an explicit value pej in Ma

j .

Maj is conditioned on a realisation of a volatility sample path:

Evolve Vt to get sample path Vt,then evolve St as if Vt were piece-wise constant with values Vt.(Works since c = E

[cj]= E

[E[cj | Vt

]].)

Along each sample path computecj in model M,pj and pe

j in model Maj .

Correlation CV is dj = pj − pej .

CV corrected option value cCV is, as usual,

cCV =1M

M

∑j=1

(cj − βdj

). (10)


Preliminary re-write

Set ρ =√

1− ρ2.Write processes in the form

dSt = rStdt+Vξt Sβ

t

(ρdWV

t + ρdW2t

), (11)

dVt = α(µ−Vt)dt+ ηVγt dWV

t , (12)

dWVt dW2

t = 0, (13)

whereWS

t = ρWVt + ρW2

t andσ has been absorbed into Vt.


The SV model: Auxiliary model, 1

First step. Transform SDE of St to make volatility independent of St.Two cases: β = 1 and β ∈ (0, 1).

β = 1 case. Set Yt = ln St, then

dYt =

(r− 1

2V2ξ

t

)dt+Vξ

t dWSt (14)

β ∈ (0, 1) case. Set Yt =1

1−β S1−βt , then

dYt =

[r(1− β)Yt −

β

2(1− β)YtV2ξ

t

]dt

+Vξt

(ρdW1

t + ρdW2t

), (15)



Second step. Discretize the transformed process.

Discretize the process for Vt, with an approximation Vt, as you like.Discretize the process for Yt, conditional on Vt,with an approximation Yt, so that increments are normal.

Set Yi = Yti , Vi = Vti , then have µY (Yi, Vi)

and σY (Yi, Vi)

such that

Yi+1 = Yi + µY (Yi, Vi)+ σY (Yi, Vi

)εY

i (16)

for εYi ∼ N (0, 1) normal iid.

The discretization of Yt determines the auxiliary model Maj .

Yi has normal increments (conditional on Vt)?More likely to get explicit solutions in Ma

j .



Need some definitions. Set

Ii =∫ ti+1

ti

V2ξs ds, (17)

Ji =∫ ti+1

ti

Vξ−γs

[α(µ−Vs) +

12(ξ − γ)η2V2γ−1

s

]ds. (18)

Models determine the form of Ii and Ji, eg:

Model Ii Ji

Heston∫ ti+1

tiVsds, α (µ∆t− Ii) ,

Garch∫ ti+1

tiVsds,

∫ ti+1ti

V−12

s[αµ−

(α+ 1

4 η2)Vs]

ds,

Sabr∫ ti+1

tiV2

s ds, αµ∆t− α∫ ti+1

tiVsds

J&S∫ ti+1

tiV2

s ds, 14 η2∆t+

∫ ti+1ti

V12s α (µ−Vs)ds


The SV model: Auxiliary model, 4β = 1 case. Integrating 14, obtain

Yti+1 = Yti + r∆t− 12

∫ ti+1

ti

V2ξs ds+ ρ

∫ ti+1

ti

Vξs dW2

s

+ρ

η

Vξ−γ+1ti+1

−Vξ−γ+1ti

ξ − γ+ 1−∫ ti+1

ti

Vξ−γs

[α(µ−Vs) +

12(ξ − γ)η2V2γ−1

s

]ds

,

(19)

so

Yti+1 = Yti + r∆t+ρ

η

Vξ−γ+1ti+1

−Vξ−γ+1ti

ξ − γ+ 1− Ji

− 12

Ii + ρ√

IiεYi . (20)

Hence

µY (Yi, Vi)

= r∆t+ρ

η

Vξ−γ+1ti+1

− Vξ−γ+1ti

ξ − γ+ 1− Ji

− 12

Ii, (21)

σY (Yi, Vi)

= ρ√

Ii. (22)


The SV model: Auxiliary model, 5β ∈ (0, 1) case. Integrating 15,

Yti+1 = Yti + r(1− β)∫ ti+1

ti

Ysds− β

2(1− β)

∫ ti+1

ti

V2ξs

Ysds+ ρ

∫ ti+1

ti

Vξs dW2

s

+ρ

η

Vξ−γ+1ti+1

−Vξ−γ+1ti

ξ − γ+ 1−∫ ti+1

ti

Vξ−γs

[α(µ−Vs) +

12(ξ − γ)η2V2γ−1

s

]ds

.

(23)

Freezing Ys (eg at initial value Y0) and integrating,

Yi+1 = Yi + r(1− β)Y0∆t− β

2(1− β)Y0Ii +

ρ

η

(Vξ−γ+1

i+1 − Vξ−γ+1i

ξ − γ+ 1− Ji

)+ρ√

IiεYi , (24)

so

µY (Yi, Vi)

= r(1− β)Y0∆t− β

2(1− β)Y0Ii +

ρ

η

(Vξ−γ+1

i+1 − Vξ−γ+1i

ξ − γ+ 1− Ji

),(25)

σY (Yi, Vi)

= ρ√

Ii. (26)


The SV model: Auxiliary instrument, 1Example: arithmetic average rate option.

K Reset dates at times T1 < · · · < TK = T, final maturity date,Tk − Tk−1 = ∆T constant, i = 1, · · · , K.

Discretization times 0 = t0 < t1 < · · · < tN = TK,ti − ti−1 = ∆t constant, i = 1, · · · , N.

Assume that ∆T = δ∆t, so Tk = tik for some index ik.Index vector κ = (i1, · · · , iK) is indexes of reset dates.

Write Sj =(

S0j , . . . , SN

j

), S0

j = S0, for a sample path of St.

Discounted payoff aong Sj is

cj = e−rT(

AAj −X

)+(27)

where X is strike, AAj is the arithmetic average along Sj,

AAj =

1K ∑

k∈κ

Sk. (28)


Auxiliary instrument, 2Auxiliary instruments:β = 1 case. Option on discrete geometric average, AG,

AG =

(∏k∈κ

Sk

) 1K

. (29)

Discounted payoff is pj = e−rT(

AGj −X

)+.

β ∈ (0, 1) case. Option on discrete β-average, Aβ,

Aβ =

(1K ∑

k∈κ

S1−βk

) 11−β

. (30)

Discounted payoff is pj = e−rT(

Aβj −X

)+.

Need to compute pej in each case.


Auxiliary instrument, 3.Computing pe

j , β = 1 case.Set g = 1

K ∑k∈κ Yk so that AG = exp (g). From 20,

Yti = Y0 + rti −12

i−1

∑j=0

Ij +ρ

η

Vξ−γ+1ti

−Vξ−γ+10

ξ − γ+ 1−

i−1

∑j=0

Jj

+ ρi−1

∑j=0

εj

√Ij, εj ∼ N(0, 1).

(31)Set T = 1

K ∑k∈κ tk and

ν =1

η (ξ − γ+ 1)1K ∑

k∈κ

(Vξ−γ+1

k − Vξ−γ+10

). (32)

Theng = Y0 + rT− 1

2H1 + ε

√H2, ε ∼ N(0, 1), (33)

where

H1 =1K

K−1

∑k=0(K− k)

ik+1−1

∑i=ik

(Ii +

2ρ

ηJi

)− 2ρν, (34)

H2 =ρ2

K2

K−1

∑k=0(K− k)2

ik+1−1

∑i=ik

Ii. (35)


Auxiliary instrument, 4

Hence g is normally distributed and

pej = e−rTE

[(eg −X)+ | V, I, J

](36)

= er(T−T)cBS (K, T, S0, y, r, σ) , (37)

where cBS(K, T, S0, y, r, σ) is Black-Scholes European call value withstrike K, maturity time T,on an asset with initial value S0, volatility σ, and dividend yield y,when the riskless rate is r, and

T =1K ∑

k∈κ

tk, σ2 =H2

T, y =

12T(H1 −H2). (38)


Auxiliary instrument, 5.Computing pe

j , β ∈ (0, 1) case.

Set g = 1−βK ∑k∈κ Yk so that Aβ = g

11−β . From 24, g is normal,

g = (1− β)m+ (1− β) sε, ε ∼ N(0, 1), (39)

where

m = Y0 + r(1− β)Y0T+ ρν− 1K

K−1

∑k=0(K− k)

ik+1−1

∑i=ik

(β

2(1− β)Y0Ii +

ρ

ηJi

), (40)

s2 = H2 =ρ2

K2

K−1

∑k=0(K− k)2

ik+1−1

∑i=ik

Ii. (41)

Set g+ = (g)+, so that (g+)1

1−β is well defined.

When g is normal can compute

pej = e−rTE

[((g+) 1

1−β −X)+| V, I, J

]. (42)


Auxiliary instrument, 6Write λ = 1− β and set b =

(Kλ − λm

)/λs. Have

E

[((g+) 1

1−β − K)+]

= E[g

1λ | g > Kλ

]·P[g > Kλ

]. (43)

P[g > Kλ

]= n (b) is known; get series expansion for E

[g

1λ | g > Kλ

].

Let Mi = E[Zi | Z > b

]where Z ∼ N (0, 1) is normal. Then

E[g

1λ | g > Kλ

]= (λm)

1λ +

∞

∑i=1

1i!(λm)

1λ−i siMi

i−1

∏j=0(1− jλ). (44)

Can compute Mi rapidly, with only a single evaluation of N (b).Let φ = n(b)

1−N(b) then (Dhrymes (2005))

M0 = 1, (45)M1 = φ, (46)Mi = bi−1φ+ (i− 1)Mi−2, (47)

Can truncate 44 at a level Nmax where Nmax = 10 is small.Zhang and Webber: General stochastic volatility 29 / 48

Special case: Auxiliary model, zero-correlation CV, 1

Obtain Ma,0j auxiliary model from Ma

j auxiliary model by setting ρ = 0.

Why?Is cheaper to compute,May yield higher value of corr

(cj, dj

).

Verified empirically:Zero correlation CV often performs betterthan standard correlation CV.

Zero correlation CV:Can also be used in the f (V) = eV case.


Special case: Auxiliary model, zero-correlation CV, 2β = 1 case. In 31, 33, 34 and 35, set ρ = 0, then

Yi = Y0 + rti −12

i−1

∑j=0

Ij +i−1

∑j=0

εj

√Ij, εj ∼ N(0, 1) (48)

andg = Y0 + rT− 1

2h1 + ε

√h2, ε ∼ N(0, 1), (49)

where

h1 =1K

K−1

∑k=0(K− k)

ik+1−1

∑i=ik

Ii, (50)

h2 = H2 =ρ2

K2

K−1

∑k=0(K− k)2

ik+1−1

∑i=ik

Ii. (51)

Hence g is normally distributed and

pej = e−rTE

[(eg −X)+ | V, I, J

](52)

= er(T−T)cBS (K, T, S0, y, r, σ) , (53)

with σ2 = 1T h2, y = 1

2T (h1 − h2).Zhang and Webber: General stochastic volatility 31 / 48

Special case: Auxiliary model, zero-correlation CV, 3

β ∈ (0, 1) case.

In 40 and 41 set ρ = 0, then

m = Y0 + r(1− β)Y0T− 1K

K−1

∑k=0(K− k)

ik+1−1

∑i=ik

β

2(1− β)Y0Ii,

s2 = H2 =ρ2

K2

K−1

∑k=0(K− k)2

ik+1−1

∑i=ik

Ii, (54)

and formula for pej follows from 44, as before.


Approximating volatility integrals

Computing values for Ii and Ji.

Approximate the integrals by a single-step trapizium, eg,∫ ti+1

ti

V2s ds ∼ 1

2

(V2

i+1 +V2i

). (55)

Take care:discretizations of Vt must not allow Vi to become negative.


Numerical resultsApply the two new correlation CVs

(correlation CV (ρ 6= 0); zero correlation CV (ρ = 0))to average rate options.

Compare with 3 “old” CVs.GBM auxiliary CV; GBM delta CV;

European call (where explicit solution exists).

Apply to average rate options: 4, 16, 64 resets.Maturity T = 1;Three cases: ITM, X = 80; ATM, X = 100; OTM, X = 120;

Use M = 106 sample paths for plain MC, M = 104 for CV MC,N = 320 times steps.

Evolving Vt: Model dependent.Log-normal approximation, exact or Milstein.

Evolving Yt: Can use Euler (absorbed at zero when β < 12 ).


Choice of parameters

4 models: Heston, Garch, Sabr, Johnson & Shanno (J&S),Two cases each.

Parameters (β, ξ, γ) S0 V0 r α µ η ρ

Heston Case 1:(

1, 12 , 1

2

)100 0.0175 0.025 1.5768 0.0398 0.5751 −0.5711

Case 2:(

1, 12 , 1

2

)100 0.04 0.05 0.2 0.05 0.1 −0.5

Garch Case 1:(

1, 12 , 1)

100 0.0175 0.025 4 0.0225 1.2 −0.5

Case 2:(

1, 12 , 1)

100 0.04 0.05 2 0.09 0.8 −0.5

Sabr Case 1: (0.4, 1, 1) 100 2 0.05 0 0 0.4 −0.5Case 2: (0.6, 1, 1) 100 2 0.05 0 0 0.4 −0.5

J&S Case 1:(

0.4, 1, 12

)100 2 0.05 2 2 0.1 −0.5

Case 2:(

0.6, 1, 12

)100 2 0.05 2 2 0.1 −0.5

Zero is accessible in: Heston, Case 1; Sabr, Case 1; J&S, Case 1.Heston, case 1, is Albrecher et al. (2007); case 2 is Webber (2010)Garch cases: Vt parameters are Lewis (2000)Sabr and J&S: Vt parameters chosen to give IVs of ∼ 15% and ∼ 30%.


Results, plain Monte CarloValue options with plain Monte Carlo, M = 106 sample paths.

Heston Garch Sabr J&SCase 1 Case 2 Case 1 Case 2 Case 1 Case 2 Case 1 Case 2

itm20.91(0.008)[740]

21.57(0.01)[730]

20.76(0.008)[600]

21.72(0.01)[600]

21.48(0.007)[610]

22.46(0.02)[610]

21.49(0.007)[710]

22.25(0.02)[700]

atm3.93(0.005)[740]

5.87(0.008)[730]

3.90(0.005)[610]

6.70(0.009)[600]

4.27(0.005)[600]

8.39(0.01)[600]

4.25(0.005)[710]

8.43(0.01)[710]

otm0.039(0.0006)[740]

0.446(0.002)[740]

0.039(0.0005)[600]

0.88(0.003)[600]

0.012(0.0002)[610]

1.70(0.005)[610]

0.030(0.0004)[700]

2.00(0.006)[710]

In each box:Top number is MC option value;Middle number (round brackets) is standard error;Bottom number (square brackets) is time in seconds.

Results are poor.Takes 600 - 700 seconds to achieve prices accurate to ∼ 1 - 2 bp,ie, in 6 - 7 seconds prices accurate only to ∼ 10 - 20 bp.


Extreme cases

In each case also look at:Correlation extremes: ρ = +0.9; ρ = −0.9.High volatility extreme:

Heston and Garch: Set V0 = µ = 0.25;Sabr: Set V0 = 8 (case 1), V0 = 3 (case 2)J&S: Same as Sabr, but also set µ = V0.

Value options with plain Monte Carlo, M = 106 sample paths.

Heston Garch Sabr J&SCase 1 Case 2 Case 1 Case 2 Case 1 Case 2 Case 1 Case 2

ρ ↑3.65(0.009)[750]

5.79(0.009)[740]

3.83(0.006)[600]

6.64(0.01)[600]

4.07(0.006)[610]

8.24(0.01)[610]

4.23(0.005)[710]

8.46(0.01)[710]

ρ ↓3.88(0.004)[740]

5.89(0.007)[740]

3.88(0.005)[600]

6.70(0.009)[600]

4.29(0.004)[610]

8.40(0.01)[600]

4.25(0.005)[700]

8.44(0.01)[710]

σ ↑11.73(0.02)[740]

12.44(0.02)[740]

11.83(0.02)[600]

12.33(0.02)[610]

12.45(0.02)[610]

11.84(0.02)[610]

12.61(0.02)[700]

11.99(0.02)[700]


Empirical correlationsCorrelations, corr

(cj, dj

), for K = 64 average rate options.

Correlation CV: correlations increase as options go OTM

ρ 6= 0 Heston Garch Sabr J&SCV Case 1 Case 2 Case 1 Case 2 Case 1 Case 2 Case 1 Case 2ITM 0.84 0.87 0.86 0.87 0.86 0.89 0.87 0.88ATM 0.96 0.91 0.92 0.91 0.92 0.93 0.89 0.90OTM 0.991 0.98 0.991 0.97 0.997 0.98 0.98 0.94

Zero correlation CV: correlations increase as options go ITM

ρ = 0 Heston Garch Sabr J&SCV Case 1 Case 2 Case 1 Case 2 Case 1 Case 2 Case 1 Case 2ITM 0.94 0.9990 0.997 0.9988 0.995 0.997 0.99996 0.99988ATM 0.89 0.997 0.994 0.997 0.993 0.992 0.9994 0.9990OTM 0.67 0.997 0.98 0.994 0.92 0.989 0.9994 0.9990

Correlations are almost always higher with zero-correlation CV.Exception: OTM options, when zero is accessible.


Efficiency gains, K = 64 average rate options, 1Zero correlation CV

ρ = 0 Heston Garch Sabr J&SCV Case 1 Case 2 Case 1 Case 2 Case 1 Case 2 Case 1 Case 2ITM 5.6 340 140 340 84 110 9000 3300ATM 3.3 130 71 120 55 47 4100 1500OTM 1.2 81 19 62 7.5 35 640 370

Correlation CVρ 6= 0 Heston Garch Sabr J&SCV Case 1 Case 2 Case 1 Case 2 Case 1 Case 2 Case 1 Case 2ITM 3.1 3.7 2.9 3.1 3.0 3.5 2.6 2.7ATM 11 5.3 4.7 4.4 4.6 5.2 3.0 3.4OTM 48 21 45 11 200 14 21 5.6

Both CVs combinedBoth Heston Garch Sabr J&SCVs Case 1 Case 2 Case 1 Case 2 Case 1 Case 2 Case 1 Case 2ITM 7.1 330 110 280 70 73 6600 2600ATM 14 170 83 120 74 63 3700 1100OTM 38 140 66 68 250 64 510 240


Efficiency gains, K = 64 average rate options, 2Efficiency gain of 6000? Get same se as plain MC in ∼ 0.1 seconds.

Gains for ρ = 0 CV decrease as options go from ITM to OTM.Gains for ρ 6= 0 CV decrease as options go from ITM to OTM.

Gains for ρ 6= 0 CV usually much less than ρ = 0 CV.Exception: ATM/OTM in Heston and Sabr, when zero accessible.

Usually better to use both CVs together (except for J&S cases).

Using correlation CVs with other CVs

GBM: GBM auxiliary modeldelta: delta CV in GBM auxiliary model

Do not compare with Heston or Sabr explicit call CV.Gains with these CVs, for average rate call option, are not large.


Efficiency gains, ITM K = 64 average rate option, 3

CV combinations Heston Garch Sabr J&SITM, K = 64 C1 C2 C1 C2 C1 C2 C1 C2

ρ = 0 5.6 340 140 340 84 110 9000 3300ρ 6= 0 3.1 3.7 2.9 3.1 3.0 3.5 2.6 2.7

ρ = 0 + ρ 6= 0 7.1 330 110 280 70 73 6600 2600GBM 3.0 74 34 37 23 24 450 310delta 120 360 520 140 450 82 800 180

GBM + delta 120 370 530 150 530 81 1500 350ρ = 0 5.9 390 150 400 85 110 9500 3100

GBM + ρ 6= 0 5.2 75 30 34 20 24 290 220both 7.7 370 120 350 67 100 6400 2300ρ = 0 120 690 880 350 810 140 6000 1400

delta + ρ 6= 0 160 350 490 140 430 95 660 160both 160 700 850 340 770 190 5100 1300

all 200 900 870 350 1000 220 5300 1200

(Best gains in bold.)Usually best to use all 4 CVs.In J&S cases, best to use only GBM + ρ = 0 CVs.


Efficiency gains, ATM K = 64 average rate option, 4CV combinations Heston Garch Sabr J&SATM, K = 64 C1 C2 C1 C2 C1 C2 C1 C2

ρ = 0 3.3 130 71 120 55 47 4100 1500ρ 6= 0 11 5.3 4.7 4.4 4.6 5.2 3.0 3.4

ρ = 0 + ρ 6= 0 14 170 84 120 75 63 3700 1100GBM 2.1 55 26 27 19 15 390 160delta 3.6 65 43 48 31 24 170 100

GBM + delta 4.0 75 43 49 31 18 310 140ρ = 0 3.2 130 74 120 54 43 4400 1600

GBM + ρ 6= 0 14 63 35 32 27 21 260 100both 14 180 92 140 76 66 4300 1300ρ = 0 3.4 85 48 61 33 23 1600 530

delta + ρ 6= 0 25 120 83 67 74 36 150 88both 23 160 100 89 86 52 1700 530

all 25 190 98 89 97 67 2200 690

Often best to use all 4 CVs.Main contribution is usually GBM + ρ = 0 + ρ 6= 0 CVs.In Garch, C2, and J&S cases, best to GBM + ρ = 0 CVs.In Heston, C1, main contribution is from delta + ρ 6= 0 CVs.


Efficiency gains, OTM K = 64 average rate option, 5

CV combinations Heston Garch Sabr J&SOTM, K = 64 C1 C2 C1 C2 C1 C2 C1 C2

ρ = 0 1.2 82 19 62 7.5 35 640 370ρ 6= 0 48 22 45 11 200 14 21 5.6

ρ = 0 + ρ 6= 0 38 140 66 68 250 64 510 240GBM 1.2 9.6 2.7 6.6 1.3 6.8 53 110delta 1.3 14 4.3 11 2.5 7.6 10 31

GBM + delta 1.4 13 5.9 12 2.6 7.6 36 49ρ = 0 1.7 86 20 62 6.7 30 1400 400

GBM + ρ 6= 0 38 38 39 17 150 23 79 83both 28 150 82 69 240 63 1200 320ρ = 0 0.9 35 9.9 20 3.8 12 350 170

delta + ρ 6= 0 18 17 17 17 56 21 14 17both 9.7 67 31 24 90 28 260 140

all 17 87 40 28 120 37 510 170

Usually best to use GBM + ρ = 0 + ρ 6= 0 CVs,Exception: J&S, when best to use only GBM + ρ = 0 CVs.


Efficiency gains, ATM K = 64 average rate option, 6High correlation case: ρ = 0.9

CV combinations Heston Garch Sabr J&SATM, K = 64 C1 C2 C1 C2 C1 C2 C1 C2

ρ = 0 2.1 36 19 17 19 16 960 470ρ 6= 0 1.1 1.2 0.9 0.9 0.9 0.9 0.8 0.8

ρ = 0 + ρ 6= 0 2.1 37 14 18 16 13 650 320GBM 1.9 60 18 15 29 26 1700 360delta 22 110 80 46 110 42 180 100

GBM + delta 24 110 84 55 100 47 670 160ρ = 0 3.1 50 21 23 31 29 3300 760

GBM + ρ 6= 0 3.4 46 15 10 23 26 1200 260both 2.9 43 19 20 28 19 2400 540ρ = 0 20 100 79 56 100 46 440 190

delta + ρ 6= 0 28 130 86 56 120 47 160 82both 22 110 83 49 110 41 390 150

all 29 150 88 54 140 58 1200 260

Best to use all CVs (except for J&S).Main contribution is from delta + ρ = 0 + ρ 6= 0.

J&S: main contribution is from ρ = 0 CV.Zhang and Webber: General stochastic volatility 44 / 48

Efficiency gains, ATM K = 64 average rate option, 7High negative correlation case: ρ = −0.9


ρ = 0 1.5 41 24 36 22 17 1200 820ρ 6= 0 1.7 1.2 1.0 1.0 1.0 1.1 0.8 0.8

ρ = 0 + ρ 6= 0 2.1 40 18 29 17 15 840 570GBM 2.0 55 35 41 24 16 320 140delta 5.4 110 62 64 53 40 170 100

GBM + delta 5.5 125 61 61 61 46 320 140ρ = 0 1.7 54 37 54 26 18 1200 930

GBM + ρ 6= 0 3.3 63 26 28 21 15 220 99both 2.5 55 30 48 21 16 840 690ρ = 0 5.0 120 66 75 59 47 660 350

delta + ρ 6= 0 17 120 66 63 66 52 150 84both 16 140 73 75 78 58 590 300

all 17 180 77 78 100 78 580 330

Best to use all CVs (except for J&S).Main contribution is from delta + ρ 6= 0.

J&S: main contribution is from ρ = 0 CV.Zhang and Webber: General stochastic volatility 45 / 48

Efficiency gains, ATM K = 64 average rate option, 8High volatility cases:


ρ = 0 23 190 60 82 45 45 2200 930ρ 6= 0 4.9 5.2 5.0 4.8 5.4 5.3 3.4 3.5

ρ = 0 + ρ 6= 0 34 160 58 78 63 56 1600 700GBM 10 120 22 30 9.5 13 37 82delta 28 27 31 36 18 20 59 63

GBM + delta 26 65 31 35 19 21 79 87ρ = 0 23 190 58 82 44 45 2400 980

GBM + ρ 6= 0 19 140 27 34 15 19 28 56both 30 180 64 84 66 56 1700 710ρ = 0 29 83 35 45 24 26 950 380

delta + ρ 6= 0 49 61 43 49 40 39 53 55both 54 77 50 54 55 51 830 330

all 66 110 47 52 65 63 1200 410

Best to use GBM + ρ = 0 + ρ 6= 0.Except: J&S: Best is GBM + ρ = 0 (ρ = 0 is main contributor),

Heston, c1: Best to use all CVs (delta is main contributor).Zhang and Webber: General stochastic volatility 46 / 48

Comparisons of efficiency gainsRelative improvement over existing methods:

Best gain including new CVs / Best gain with old CVs alone.

Relative Heston Garch Sabr J&Sperformance C1 C2 C1 C2 C1 C2 C1 C2ITM 1.7 2.4 1.7 2.7 1.9 2.7 6.3 9.4ATM 6.9 2.5 2.3 2.9 3.1 2.8 11 10OTM 34 10.7 14 5.8 92 8.4 26 3.6ρ ↑ 1.2 1.2 1.0 1.0 1.3 1.2 1.9 2.1ρ ↓ 3.1 1.4 1.3 1.2 1.6 1.7 3.8 6.6σ ↑ 2.4 1.6 2.1 2.3 3.5 3.0 30 11

.

Ordinary parameter values:New CVs enhance performance by sizable factors.Generally improve as options go OTM.

Extreme correlations: Perform less well but still give speed-upsExtreme volatility: Still give very reasonable speed-ups


Summary

Have priced average rate options in a general SV model,nesting Heston, Sabr, et cetera.

Have derived a pair of correlation CVs.Have demonstrated their effectiveness compared to existing CVs.

The new CVs apply more generally to other options,incuding barrier options.

Can attempt to calibrate to options other than vanillas.


Documents

ImperialMathFinance: Finance and Stochastics Seminar