1. Option valuation in a general stochastic volatility
model
Kai Zhang and Nick Webber
University of Warwick
10 November 2010
Zhang and Webber: General stochastic volatility 1 / 48

2. Abstract
Stochastic volatility models are frequently used in the markets to model the implied
volatility surface. These models have several failings. Firstly, although improvements
on a basic Black-Scholes model, they nevertheless fail to fit the entire surface
adequately. Secondly, the improvements they offer are usually at the cost of greatly
reduced tractability. Thirdly, these models still fail to fit to market prices of
non-vanilla securities.
This paper addresses the second of these three issues. A general stochastic volatility
model is described, nesting both the Heston models and Sabr-related models. A
control variate Monte Carlo valuation method for this model is presented that, when
it can be applied, is shown to be a significant improvement over existing simulation
methods; when applied to barrier option pricing, it out-performs importance
sampling methods.
By providing a plausible simulation method for this general model, the paper opens
the possibility of exploring calibration to non-vanilla, as well as vanilla, instruments.
Zhang and Webber: General stochastic volatility 2 / 48

3. 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 surface
Do (2): Fit to instruments whose values are path-dependent
Zhang and Webber: General stochastic volatility 3 / 48

4. 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.
Zhang and Webber: General stochastic volatility 4 / 48

5. 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).
Zhang and Webber: General stochastic volatility 5 / 48

6. 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 these
Barrier options.
4. Explore variations in barrier option pricing, not discussed
consistent with the implied volatility surface.
Zhang and Webber: General stochastic volatility 6 / 48

7. A general stochastic volatility model
Let
(S)t 0 be an asset price process,
(V)t 0 be a volatility process,
(1)
with SDEs
dSt = rStdt + σf (Vt) S
β
t dWS
t , (2)
dVt = α(µ Vt)dt + ηV
γ
t dWV
t , (3)
dWS
t 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 /2 Z,
σ > 0, included for generality,
α, µ 2 R, are not required to be positive.
(5)
When ξ /2 Z, Vt is required to remain positive.
When ξ 2 Z, Vt is permitted to become negative.
Zhang and Webber: General stochastic volatility 7 / 48

8. Characteristics of the processes
β and the process for St:
β = 0, SV absolute diffusion process
β 2 (0, 1) , SV CEV process
β = 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.
Zhang and Webber: General stochastic volatility 8 / 48

9. Nesting other models
β = 0 β 2 (0, 1) β = 1
Model ξ γ Model ξ γ Model ξ γ
1 1 Sabr 1 1 H&W 1 1
Absolute 1 1
2 Sabr- 1 1
2 J&S 1 1
2
diffusion 1 0 like 1 0 S&Z, S&S 1 0
models 1
2 1 LKD 1
2 1 Garch 1
2 1
1
2
1
2
1
2
1
2 Heston 1
2
1
2
n/a 1
2 0 n/a 1
2 0 n/a 1
2 0
(β, 1, γ): Johnson and Shanno (1987).
(β, 1, 1): Sabr, Hagan et al. (2002).
β, 1
2 , 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, 1
2 , 1 : Nelson (1990); Barone-Adesi et al. (2003) (ρ = 0); (and H & W, as µ = 0).
1, 1
2 , 1
2 : Heston (1993); Ball & Roma (1994).
1, 1
2 , 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

10. Not nested
More general specifications:
(β, f (v) , g (v)): Jourdain & Sbai (2010) (including (1, 2, 0)),
(h (S) , ξ, γ): Bourgade & Croissant (2005).
Log-specifications: β, ˆξ, γ = (1, 1, 0), with f (v) = exp v
ˆξ :
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.
Zhang and Webber: General stochastic volatility 10 / 48

11. 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).)
Zhang and Webber: General stochastic volatility 11 / 48

12. 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.
Zhang and Webber: General stochastic volatility 12 / 48

13. 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.
Zhang and Webber: General stochastic volatility 13 / 48

14. Control variates and Monte Carlo
Plain Monte Carlo:
Generate M sample paths.
Get discounted payoff cj along each sample path, j = 1, . . . , M.
Plain MC option value ˆc is
ˆc =
1
M
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 then
CV corrected option value ˆcCV is
ˆcCV =
1
M
M
∑
j=1
cj βdj . (7)
Can extend to have multiple CVs.
Zhang and Webber: General stochastic volatility 14 / 48

15. Efficiency gain
Suppose ˆc computed in time τ with standard error σ,
Suppose ˆcCV computed in time τCV with standard error σCV.
Efficiency gain E is
E =
τσ2
τCVσ2
CV
. (8)
Proportional reduction in time by CV method
to get same standard error as plain method.
If ρ = corr cj, dj then
E =
τ
τCV
1
1 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”.
Zhang and Webber: General stochastic volatility 15 / 48

16. Auxiliary model CVs
Auxiliary 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 ca
j for discounted payoff on path j under Ma
(so that ca ˆca = 1
M ∑M
j=1 ca
j .)
Then dj = ca
j ca is an auxiliary model CV.
Zhang and Webber: General stochastic volatility 16 / 48

17. The SV model: Auxiliary model and instrument CV
Find suitable:
Auxiliary model Ma
j (effectively sample path dependent),
Auxiliary instrument p, so that p has an explicit value pe
j in Ma
j .
Ma
j 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 j Vt .)
Along each sample path compute
cj in model M,
pj and pe
j in model Ma
j .
Correlation CV is dj = pj pe
j .
CV corrected option value ˆcCV is, as usual,
ˆcCV =
1
M
M
∑
j=1
cj βdj . (10)
Zhang and Webber: General stochastic volatility 17 / 48

18. Preliminary re-write
Set ˜ρ =
p
1 ρ2.
Write processes in the form
dSt = rStdt + V
ξ
t S
β
t ρdWV
t + ˜ρdW2
t , (11)
dVt = α(µ Vt)dt + ηV
γ
t dWV
t , (12)
dWV
t dW2
t = 0, (13)
where
WS
t = ρWV
t + ˜ρW2
t and
σ has been absorbed into Vt.
Zhang and Webber: General stochastic volatility 18 / 48

19. The SV model: Auxiliary model, 1
First step. Transform SDE of St to make volatility independent of St.
Two cases: β = 1 and β 2 (0, 1).
β = 1 case. Set Yt = ln St, then
dYt = r
1
2
V
2ξ
t dt + V
ξ
t dWS
t (14)
β 2 (0, 1) case. Set Yt = 1
1 β S
1 β
t , then
dYt = r(1 β)Yt
β
2(1 β)Yt
V
2ξ
t dt
+V
ξ
t ρdW1
t + ˜ρdW2
t , (15)
Zhang and Webber: General stochastic volatility 19 / 48

20. The SV model: Auxiliary model, 2
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 εY
i N (0, 1) normal iid.
The discretization of Yt determines the auxiliary model Ma
j .
˜Yi has normal increments (conditional on Vt)?
More likely to get explicit solutions in Ma
j .
Zhang and Webber: General stochastic volatility 20 / 48

21. The SV model: Auxiliary model, 3
Need some definitions. Set
Ii =
Z ti+1
ti
V
2ξ
s ds, (17)
Ji =
Z ti+1
ti
V
ξ γ
s α(µ Vs) +
1
2
(ξ γ)η2
V
2γ 1
s ds. (18)
Models determine the form of Ii and Ji, eg:
Model Ii Ji
Heston
R ti+1
ti
Vsds, α (µ∆t Ii) ,
Garch
R ti+1
ti
Vsds,
R ti+1
ti
V
1
2
s αµ α + 1
4 η2 Vs ds,
Sabr
R ti+1
ti
V2
s ds, αµ∆t α
R ti+1
ti
Vsds
J&S
R ti+1
ti
V2
s ds, 1
4 η2∆t +
R ti+1
ti
V
1
2
s α (µ Vs) ds
Zhang and Webber: General stochastic volatility 21 / 48

22. The SV model: Auxiliary model, 4
β = 1 case. Integrating 14, obtain
Yti+1
= Yti
+ r∆t
1
2
Z ti+1
ti
V
2ξ
s ds + ˜ρ
Z ti+1
ti
V
ξ
s dW2
s
+
ρ
η
8
<
:
V
ξ γ+1
ti+1
V
ξ γ+1
ti
ξ γ + 1
Z ti+1
ti
V
ξ γ
s α(µ Vs) +
1
2
(ξ γ)η2
V
2γ 1
s ds
9
=
;
,
(19)
so
Yti+1
= Yti
+ r∆t +
ρ
η
8
<
:
V
ξ γ+1
ti+1
V
ξ γ+1
ti
ξ γ + 1
Ji
9
=
;
1
2
Ii + ˜ρ
p
IiεY
i . (20)
Hence
µY ˜Yi, ˜Vi = r∆t +
ρ
η
8
<
:
˜V
ξ γ+1
ti+1
˜V
ξ γ+1
ti
ξ γ + 1
Ji
9
=
;
1
2
Ii, (21)
σY ˜Yi, ˜Vi = ˜ρ
p
Ii. (22)
Zhang and Webber: General stochastic volatility 22 / 48

23. The SV model: Auxiliary model, 5
β 2 (0, 1) case. Integrating 15,
Yti+1
= Yti
+ r(1 β)
Z ti+1
ti
Ysds
β
2(1 β)
Z ti+1
ti
V
2ξ
s
Ys
ds + ˜ρ
Z ti+1
ti
V
ξ
s dW2
s
+
ρ
η
8
<
:
V
ξ γ+1
ti+1
V
ξ γ+1
ti
ξ γ + 1
Z ti+1
ti
V
ξ γ
s α(µ Vs) +
1
2
(ξ γ)η2
V
2γ 1
s ds
9
=
;
.
(23)
Freezing Ys (eg at initial value Y0) and integrating,
˜Yi+1 = ˜Yi + r(1 β)Y0∆t
β
2(1 β)Y0
Ii +
ρ
η
˜V
ξ γ+1
i+1
˜V
ξ γ+1
i
ξ γ + 1
Ji
!
+˜ρ
p
IiεY
i , (24)
so
µY ˜Yi, ˜Vi = r(1 β)Y0∆t
β
2(1 β)Y0
Ii +
ρ
η
˜V
ξ γ+1
i+1
˜V
ξ γ+1
i
ξ γ + 1
Ji
!
,(25)
σY ˜Yi, ˜Vi = ˜ρ
p
Ii. (26)
Zhang and Webber: General stochastic volatility 23 / 48

24. The SV model: Auxiliary instrument, 1
Example: 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 = ˜S0
j , . . . , ˜SN
j , ˜S0
j = S0, for a sample path of St.
Discounted payoff aong ˜Sj is
cj = e rT
AA
j X
+
(27)
where X is strike, AA
j is the arithmetic average along ˜Sj,
AA
j =
1
K ∑
k2κ
˜Sk. (28)
Zhang and Webber: General stochastic volatility 24 / 48

25. Auxiliary instrument, 2
Auxiliary instruments:
β = 1 case. Option on discrete geometric average, AG,
AG
= ∏
k2κ
˜Sk
! 1
K
. (29)
Discounted payoff is pj = e rT AG
j X
+
.
β 2 (0, 1) case. Option on discrete β-average, Aβ,
Aβ
=
1
K ∑
k2κ
˜S
1 β
k
! 1
1 β
. (30)
Discounted payoff is pj = e rT A
β
j X
+
.
Need to compute pe
j in each case.
Zhang and Webber: General stochastic volatility 25 / 48

26. Auxiliary instrument, 3.
Computing pe
j , β = 1 case.
Set g = 1
K ∑k2κ
˜Yk so that AG = exp (g). From 20,
Yti
= Y0 + rti
1
2
i 1
∑
j=0
Ij +
ρ
η
0
@
V
ξ γ+1
ti
V
ξ γ+1
0
ξ γ + 1
i 1
∑
j=0
Jj
1
A + ˜ρ
i 1
∑
j=0
εj
q
Ij, εj N(0, 1).
(31)
Set ¯T = 1
K ∑k2κ tk and
ν =
1
η (ξ γ + 1)
1
K ∑
k2κ
˜V
ξ γ+1
k
˜V
ξ γ+1
0 . (32)
Then
g = ˜Y0 + r ¯T
1
2
H1 + ε
p
H2, ε N(0, 1), (33)
where
H1 =
1
K
K 1
∑
k=0
(K k)
ik+1 1
∑
i=ik
˜Ii +
2ρ
η
˜Ji 2ρν, (34)
H2 =
˜ρ2
K2
K 1
∑
k=0
(K k)2
ik+1 1
∑
i=ik
˜Ii. (35)
Zhang and Webber: General stochastic volatility 26 / 48

27. Auxiliary instrument, 4
Hence g is normally distributed and
pe
j = e rT
E
h
(eg
X)+
j V, I, J
i
(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 with
strike 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 =
1
K ∑
k2κ
tk, ¯σ2
=
H2
¯T
, y =
1
2 ¯T
(H1 H2). (38)
Zhang and Webber: General stochastic volatility 27 / 48

28. Auxiliary instrument, 5.
Computing pe
j , β 2 (0, 1) case.
Set g = 1 β
K ∑k2κ
˜Yk so that Aβ = g
1
1 β . From 24, g is normal,
g = (1 β) m + (1 β) sε, ε N(0, 1), (39)
where
m = ˜Y0 + r(1 β) ˜Y0
¯T + ρν
1
K
K 1
∑
k=0
(K k)
ik+1 1
∑
i=ik
β
2(1 β) ˜Y0
˜Ii +
ρ
η
˜Ji , (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
pe
j = e rT
E g+
1
1 β
X
+
j V, I, J . (42)
Zhang and Webber: General stochastic volatility 28 / 48

29. Auxiliary instrument, 6
Write λ = 1 β and set b = Kλ λm /λs. Have
E g+
1
1 β
K
+
= E
h
g
1
λ j g > Kλ
i
P
h
g > Kλ
i
. (43)
P g > Kλ = n (b) is known; get series expansion for E
h
g
1
λ j g > Kλ
i
.
Let Mi = E Zi j Z > b where Z N (0, 1) is normal. Then
E
h
g
1
λ j g > Kλ
i
= (λm)
1
λ +
∞
∑
i=1
1
i!
(λm)
1
λ i
si
Mi
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 1
φ + (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

30. Special case: Auxiliary model, zero-correlation CV, 1
Obtain Ma,0
j 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 better
than standard correlation CV.
Zero correlation CV:
Can also be used in the f (V) = eV case.
Zhang and Webber: General stochastic volatility 30 / 48

31. Special case: Auxiliary model, zero-correlation CV, 2
β = 1 case. In 31, 33, 34 and 35, set ρ = 0, then
˜Yi = ˜Y0 + rti
1
2
i 1
∑
j=0
Ij +
i 1
∑
j=0
εj
q
Ij, εj N(0, 1) (48)
and
g = ˜Y0 + r ¯T
1
2
h1 + ε
p
h2, ε N(0, 1), (49)
where
h1 =
1
K
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
pe
j = e rT
E
h
(eg
X)+
j V, I, J
i
(52)
= er( ¯T T)
cBS (K, ¯T, S0, y, r, ¯σ) , (53)
with ¯σ2 = 1
¯T
h2, y = 1
2 ¯T
(h1 h2).
Zhang and Webber: General stochastic volatility 31 / 48

32. Special case: Auxiliary model, zero-correlation CV, 3
β 2 (0, 1) case.
In 40 and 41 set ρ = 0, then
m = ˜Y0 + r(1 β) ˜Y0
¯T
1
K
K 1
∑
k=0
(K k)
ik+1 1
∑
i=ik
β
2(1 β) ˜Y0
˜Ii,
s2
= H2 =
˜ρ2
K2
K 1
∑
k=0
(K k)2
ik+1 1
∑
i=ik
˜Ii, (54)
and formula for pe
j follows from 44, as before.
Zhang and Webber: General stochastic volatility 32 / 48

33. Approximating volatility integrals
Computing values for ˜Ii and ˜Ji.
Approximate the integrals by a single-step trapizium, eg,
Z ti+1
ti
V2
s ds
1
2
V2
i+1 + V2
i . (55)
Take care:
discretizations of Vt must not allow ˜Vi to become negative.
Zhang and Webber: General stochastic volatility 33 / 48

34. Numerical results
Apply 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 < 1
2 ).
Zhang and Webber: General stochastic volatility 34 / 48

35. 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, 1
2 , 1
2 100 0.0175 0.025 1.5768 0.0398 0.5751 0.5711
Case 2: 1, 1
2 , 1
2 100 0.04 0.05 0.2 0.05 0.1 0.5
Garch Case 1: 1, 1
2 , 1 100 0.0175 0.025 4 0.0225 1.2 0.5
Case 2: 1, 1
2 , 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 0.5
Case 2: (0.6, 1, 1) 100 2 0.05 0 0 0.4 0.5
J&S Case 1: 0.4, 1, 1
2 100 2 0.05 2 2 0.1 0.5
Case 2: 0.6, 1, 1
2 100 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%.
Zhang and Webber: General stochastic volatility 35 / 48

36. Results, plain Monte Carlo
Value options with plain Monte Carlo, M = 106 sample paths.
Heston Garch Sabr J&S
Case 1 Case 2 Case 1 Case 2 Case 1 Case 2 Case 1 Case 2
itm
20.91
(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]
atm
3.93
(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]
otm
0.039
(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.
Zhang and Webber: General stochastic volatility 36 / 48

37. 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&S
Case 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]
Zhang and Webber: General stochastic volatility 37 / 48

38. Empirical correlations
Correlations, corr cj, dj , for K = 64 average rate options.
Correlation CV: correlations increase as options go OTM
ρ 6= 0 Heston Garch Sabr J&S
CV Case 1 Case 2 Case 1 Case 2 Case 1 Case 2 Case 1 Case 2
ITM 0.84 0.87 0.86 0.87 0.86 0.89 0.87 0.88
ATM 0.96 0.91 0.92 0.91 0.92 0.93 0.89 0.90
OTM 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&S
CV Case 1 Case 2 Case 1 Case 2 Case 1 Case 2 Case 1 Case 2
ITM 0.94 0.9990 0.997 0.9988 0.995 0.997 0.99996 0.99988
ATM 0.89 0.997 0.994 0.997 0.993 0.992 0.9994 0.9990
OTM 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.
Zhang and Webber: General stochastic volatility 38 / 48

39. Efficiency gains, K = 64 average rate options, 1
Zero correlation CV
ρ = 0 Heston Garch Sabr J&S
CV Case 1 Case 2 Case 1 Case 2 Case 1 Case 2 Case 1 Case 2
ITM 5.6 340 140 340 84 110 9000 3300
ATM 3.3 130 71 120 55 47 4100 1500
OTM 1.2 81 19 62 7.5 35 640 370
Correlation CV
ρ 6= 0 Heston Garch Sabr J&S
CV Case 1 Case 2 Case 1 Case 2 Case 1 Case 2 Case 1 Case 2
ITM 3.1 3.7 2.9 3.1 3.0 3.5 2.6 2.7
ATM 11 5.3 4.7 4.4 4.6 5.2 3.0 3.4
OTM 48 21 45 11 200 14 21 5.6
Both CVs combined
Both Heston Garch Sabr J&S
CVs Case 1 Case 2 Case 1 Case 2 Case 1 Case 2 Case 1 Case 2
ITM 7.1 330 110 280 70 73 6600 2600
ATM 14 170 83 120 74 63 3700 1100
OTM 38 140 66 68 250 64 510 240
Zhang and Webber: General stochastic volatility 39 / 48

40. Efficiency gains, K = 64 average rate options, 2
Efficiency 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 model
delta: 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.
Zhang and Webber: General stochastic volatility 40 / 48

41. Efficiency gains, ITM K = 64 average rate option, 3
CV combinations Heston Garch Sabr J&S
ITM, K = 64 C1 C2 C1 C2 C1 C2 C1 C2
ρ = 0 5.6 340 140 340 84 110 9000 3300
ρ 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 2600
GBM 3.0 74 34 37 23 24 450 310
delta 120 360 520 140 450 82 800 180
GBM + delta 120 370 530 150 530 81 1500 350
ρ = 0 5.9 390 150 400 85 110 9500 3100
GBM + ρ 6= 0 5.2 75 30 34 20 24 290 220
both 7.7 370 120 350 67 100 6400 2300
ρ = 0 120 690 880 350 810 140 6000 1400
delta + ρ 6= 0 160 350 490 140 430 95 660 160
both 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.
Zhang and Webber: General stochastic volatility 41 / 48

42. Efficiency gains, ATM K = 64 average rate option, 4
CV combinations Heston Garch Sabr J&S
ATM, K = 64 C1 C2 C1 C2 C1 C2 C1 C2
ρ = 0 3.3 130 71 120 55 47 4100 1500
ρ 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 1100
GBM 2.1 55 26 27 19 15 390 160
delta 3.6 65 43 48 31 24 170 100
GBM + delta 4.0 75 43 49 31 18 310 140
ρ = 0 3.2 130 74 120 54 43 4400 1600
GBM + ρ 6= 0 14 63 35 32 27 21 260 100
both 14 180 92 140 76 66 4300 1300
ρ = 0 3.4 85 48 61 33 23 1600 530
delta + ρ 6= 0 25 120 83 67 74 36 150 88
both 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.
Zhang and Webber: General stochastic volatility 42 / 48

43. Efficiency gains, OTM K = 64 average rate option, 5
CV combinations Heston Garch Sabr J&S
OTM, K = 64 C1 C2 C1 C2 C1 C2 C1 C2
ρ = 0 1.2 82 19 62 7.5 35 640 370
ρ 6= 0 48 22 45 11 200 14 21 5.6
ρ = 0 + ρ 6= 0 38 140 66 68 250 64 510 240
GBM 1.2 9.6 2.7 6.6 1.3 6.8 53 110
delta 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 49
ρ = 0 1.7 86 20 62 6.7 30 1400 400
GBM + ρ 6= 0 38 38 39 17 150 23 79 83
both 28 150 82 69 240 63 1200 320
ρ = 0 0.9 35 9.9 20 3.8 12 350 170
delta + ρ 6= 0 18 17 17 17 56 21 14 17
both 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.
Zhang and Webber: General stochastic volatility 43 / 48

44. Efficiency gains, ATM K = 64 average rate option, 6
High correlation case: ρ = 0.9
CV combinations Heston Garch Sabr J&S
ATM, K = 64 C1 C2 C1 C2 C1 C2 C1 C2
ρ = 0 2.1 36 19 17 19 16 960 470
ρ 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 13 650 320
GBM 1.9 60 18 15 29 26 1700 360
delta 22 110 80 46 110 42 180 100
GBM + delta 24 110 84 55 100 47 670 160
ρ = 0 3.1 50 21 23 31 29 3300 760
GBM + ρ 6= 0 3.4 46 15 10 23 26 1200 260
both 2.9 43 19 20 28 19 2400 540
ρ = 0 20 100 79 56 100 46 440 190
delta + ρ 6= 0 28 130 86 56 120 47 160 82
both 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

45. Efficiency gains, ATM K = 64 average rate option, 7
High negative correlation case: ρ = 0.9
CV combinations Heston Garch Sabr J&S
ATM, K = 64 C1 C2 C1 C2 C1 C2 C1 C2
ρ = 0 1.5 41 24 36 22 17 1200 820
ρ 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 15 840 570
GBM 2.0 55 35 41 24 16 320 140
delta 5.4 110 62 64 53 40 170 100
GBM + delta 5.5 125 61 61 61 46 320 140
ρ = 0 1.7 54 37 54 26 18 1200 930
GBM + ρ 6= 0 3.3 63 26 28 21 15 220 99
both 2.5 55 30 48 21 16 840 690
ρ = 0 5.0 120 66 75 59 47 660 350
delta + ρ 6= 0 17 120 66 63 66 52 150 84
both 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

46. Efficiency gains, ATM K = 64 average rate option, 8
High volatility cases:
CV combinations Heston Garch Sabr J&S
ATM, K = 64 C1 C2 C1 C2 C1 C2 C1 C2
ρ = 0 23 190 60 82 45 45 2200 930
ρ 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 700
GBM 10 120 22 30 9.5 13 37 82
delta 28 27 31 36 18 20 59 63
GBM + delta 26 65 31 35 19 21 79 87
ρ = 0 23 190 58 82 44 45 2400 980
GBM + ρ 6= 0 19 140 27 34 15 19 28 56
both 30 180 64 84 66 56 1700 710
ρ = 0 29 83 35 45 24 26 950 380
delta + ρ 6= 0 49 61 43 49 40 39 53 55
both 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

47. Comparisons of efficiency gains
Relative improvement over existing methods:
Best gain including new CVs / Best gain with old CVs alone.
Relative Heston Garch Sabr J&S
performance C1 C2 C1 C2 C1 C2 C1 C2
ITM 1.7 2.4 1.7 2.7 1.9 2.7 6.3 9.4
ATM 6.9 2.5 2.3 2.9 3.1 2.8 11 10
OTM 34 10.7 14 5.8 92 8.4 26 3.6
ρ " 1.2 1.2 1.0 1.0 1.3 1.2 1.9 2.1
ρ # 3.1 1.4 1.3 1.2 1.6 1.7 3.8 6.6
σ " 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-ups
Extreme volatility: Still give very reasonable speed-ups
Zhang and Webber: General stochastic volatility 47 / 48

48. 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.
Zhang and Webber: General stochastic volatility 48 / 48