Option Pricing via Risk-Neutral
Density Forecasting
Stanislav Khrapov∗
New Economic School
September 15, 2014
Abstract
We propose a novel approach to option pricing. It exploits strong predictability in
option-implied risk-neutral densities. To illustrate the idea we use a mixture of log-normal and
a generalized beta as the candidates for the distribution of underlying stock price under the
risk-neutral measure. Using the closed-form solutions for option prices we extract risk-neutral
densities and forecast them one week ahead. This forecast allows to compute one week ahead
option prices and compare them to the observables. In the empirical exercise we show that the
option pricing performance is on par with state-of-the-art stochastic volatility models.
Keywords: risk-neutral density, option pricing, forecasting
JEL Classification: C58, G13, G17
∗
Address: New Economic School, Nakhimovskiy Prospekt 47, Moscow, Russia, 117418.
Phone: +7 (495) 956 9508. Email: [email protected]
1
Introduction
Black & Scholes (1973) have opened the whole new chapter in the history of quantitative finance.
The field of option pricing has started with this seminal paper. Since then the option pricing
error has become the holy grail for the plethora of financial economists. State-of-the-art option
pricing models include Heston & Nandi (2000), Christoffersen et al. (2006, 2008, 2009), Corsi et al.
(2013), and Majewski et al. (2013) among many others. In this paper we propose a novel approach
to option pricing which even in its simplest form preforms close to above mentioned models in
terms or reducing option pricing errors. The main idea of the model is based on predictability of
option-implied risk-neutral density.
Our paper is related to the literature on option-implied information, including risk-neutral
densities and implied volatility surfaces (see Christoffersen et al. (2012a) for a comprehensive
overview of the field, or Jackwerth (2004) for an earlier survey). The standard approach to option
pricing nowadays is through the no-arbitrage argument. It simply says that the options are the
expectations under the risk-neutral distribution of the future payoff which in the case of call
contract is (S − K)+ . This definition reveals the key to extraction of the risk-neutral density of
the underlying. For example, Ait-Sahalia & Lo (1998) extract this density non-parametrically,
while Bahra (1996, 1997) do it assuming parametric form of the density. Other examples include
Söderlind & Svensson (1997), Melik & Thomas (1997), Jondeau & Rockinger (2000), Jackwerth
(2004), Bu & Hadri (2007), and Figlewski (2010). In particular, we use the approach taken by
Bahra (1996, 1997) to extract parameters of the mixture of log-normal distributions from the
cross-section of option prices available on any trading day. Then we find that these densities are in
fact highly predictable. These predictability patterns allow us to forecast densities one week ahead
and use these forecasts to compute model option prices and volatility surfaces.
Another strand of literature related to our paper is on predictability in the space of implied
volatilities. Option prices are most often quoted in terms of Black-Scholes implied volatilities. The
benefit of doing so is that option prices can then be compared across strikes, maturities, and price
levels of the underlying. It turns out that the volatility surface is also highly predictable across
time. Among papers that observe such predictability in different settings are Dumas et al. (1998),
Cont & da Fonseca (2002), Gonçalves & Guidolin (2006), Fengler et al. (2007), and Homescu (2011).
Note that volatility surface is just a monotone transformation of the option prices. So it becomes
another side of a single big picture, along with risk-neutral density, which says that option-implied
information is persistent. In this paper we exploit this information for the purposes of option
pricing.
To be more specific we do the following. First, similarly to Bahra (1996), Melik & Thomas (1997),
and Söderlind & Svensson (1997), we assume that the stock price under the risk-neutral measure
is distributed as a mixture of log-normal distributions. In particular, this model encompasses
geometric Brownian motion as a special case which makes it our empirical benchmark. Next, we
use the theoretical result that gives us closed-form solutions to option prices that depend only on
a few parameters. The mixture of log-normals is very attractive since it produces asymmetry and
2
fatter tails which is a minimal requirement for successful option pricing model. Similarly to Liu
et al. (2007) we also try the Generalized Beta distribution with some modification relative to the
original in order to make this distribution parameters horizon invariant.
Given a cross-section of daily option prices we find the risk-neutral densities, or equivalently
the set of parameters, that minimize the option pricing errors. This exercise is repeated every day
to collect the time series of density parameters. It turns out that these parameters are highly
predictable. We exploit this predictability by fitting a simplest autoregressive model to each series
of parameters.
Then, we forecast these parameters out-of-sample and compute option prices
corresponding to the forecast of the risk-neutral density. Our empirical analysis shows that this
approach produces uniformly small option pricing errors across moneyness, maturities, and across
time.
The rest of the paper is organized as follows. Section 2 introduces the model for option prices
based on the mixture of log-normal risk-neutral densities of the underlying stock price. Section 3
develops the methodology to extract the risk-neutral densities, forecast them, and use the forecast
to compute model option prices. Section 4 describes the data and empirical results. Section 5
concludes.
2
The Model
2.1
Motivation
We begin this section with a very simplistic set up used in the seminal work of Black & Scholes
(1973). Suppose that under the risk-neutral measure the stock price ST evolves according to the
geometric Brownian motion (GBM) with drift:
dSt
= rdt + σdWt ,
St
where r is the continuously compounded risk-free rate, σ is the volatility parameter, Wt is the
standard Brownian motion. This stochastic differential equation can be solved to show that the
future log price is distributed as normal
√
1
log ST ∼ N log St + r − σ 2 τ, σ τ ,
2
where τ = T − t > 0 is the time interval. This of course implies the log-normal risk-neutral
conditional distribution qt (St+τ ; θ) for the price itself:

2 


1 2


log
(S
/S
)
−
r
−
σ
τ 
t
T
1
1
2

√
√
qt (ST ; σ) =
exp −
.
 2

σ τ
ST 2πσ 2


Given the explicit form of the risk-neutral density we can easily price options using no-arbitrage
3
arguments by evaluating discounted risk-neutral expectation of the (call) option payoff,
−rτ
Ct (τ ; θ) = e
h
+
Et (ST − K)
i
ˆ
−rτ
∞
(ST − K) qt (ST ; σ) dST .
=e
K
Here K is the strike of the contract, (x)+ = max (0, x), and (ST − K)+ is the payoff of the call
option contract. The closed-form solution for the option price is a well-known Black-Scholes (BS)
formula:
Ct (τ ; θ) = St Φ (d1 ) − e−rτ KΦ (d2 ) ,
where
log (St /K) + rτ
1 √
√
+ σ τ,
σ τ
2
d1 =
√
d2 = d1 − σ τ ,
and Φ (·) is the cumulative standard normal probability function. Option price formula can be
somewhat simplified if we normalize it by the current asset price and introduce log-forward
moneyness x = log (K/St ) − rτ = log (K/Ft ):
ct (x, τ, σ) = Ct (τ ; θ) /St = Φ (d1 ) − ex Φ (d2 ) ,
where
2.2
x
1 √
d1 = − √ + σ τ ,
σ τ
2
√
d2 = d1 − σ τ ,
Mixture of log-normals
Now suppose, following Bahra (1996), Melik & Thomas (1997), and Söderlind & Svensson (1997)1 ,
that the risk-neutral density is in fact a mixture of various densities,
qt (ST ; θ) =
M
X
αi qit (ST ; µi , σi ) ,
M
X
αi = 1,
αi ≥ 0,
i=1
i=1
where M is the number of such densities with their own parameter vectors θi . Note that each of
qi (ST ; θi ) is not yet required to be a risk-neutral one. Only the mixture has to possess this property.
It easy to show that the option price is now simply a mixture of option prices each one
corresponding to a single density qi (ST ; θi ):
Ct (τ, r; θ) =
M
X
ˆ
−rτ
∞
(ST − K) qit (ST ; µi , σi ) dST =
αi e
K
i=1
M
X
αi Cit (τ, r; µi , σi ) .
i=1
If we keep the assumption of lognormality of each density in the mixture with parameters θi =
(µi , σi ), then the option price is just a mixture of Black-Scholes prices with a new parameter µi
instead of the risk-free rate r:
Cit (τ, r; µi , σi ) = St Φ (di1 ) − e−rτ KΦ (di2 ) ,
1
see also Jondeau et al. (2007, Ch. 11.3) for a on overview of the topic.
4
where
di1 =
√
di2 = di1 − σi τ .
log (St /K) + µi τ
1 √
√
+ σi τ ,
σi τ
2
BS formula is a specific case with α1 = 1 and µ1 = r. The normalized version is
cit (x, τ, r; µi , σi ) = Φ (di1 ) − ex+(r−µi )τ Φ (di2 ) ,
where
di1 = −
x + (r − µi ) τ
1 √
√
+ σi τ ,
σi τ
2
√
di2 = di1 − σi τ .
The no-arbitrage condition boils down to the following assumption for the underlying stock
price,
St = e−rτ Et [ST ] ,
which after all substitutions can be reduced to
exp (rτ ) =
X
αi exp (µi τ ) .
i
2.3
Generalized Beta distribution
Bookstaber & Mcdonald (1987) originally proposed to use generalized beta distribution of the second
kind (GB2) to characterize stock returns. More recently, Liu et al. (2007) applied this distribution
in the context of risk-neutral pricing. The advantage of this density is that it has enough flexibility
to model four first moments separately, including skewness and kurtosis. The GB2 probability
density function itself is written as
q (x; [a, b, p, q]) =
axap−1
,
bap B (p, q) [1 + (x/b)a ]p+q
X > 0,
where B (p, q) = Γ (p) Γ (q) /Γ (p + q).
Liu et al. (2007) assumed that the stock price itself, ST , is distributed according to GB2
distribution. Here we argue that a much more realistic assumption is that rather total excess
return is distributed as GB2. Denote this random variable as
R̃t,τ =
ST −rτ
e ,
St
and assume that its density is q (x; θ) written above with a replaced by horizon dependent parameter
aτ −1/2 . The moments of order n < qaτ −1/2 are given by
h
n
Et R̃t,τ
i
√
√ bn B p + na τ , q − na τ
=
.
B (p, q)
5
Hence, the martingale restriction is given by
√
bB p +
i
h
1 = Et R̃t,τ =
τ
a ,q
√
−
τ
a
B (p, q)
,
⇒
B (p, q)
b=
√
B p+
τ
a ,q
−
√
τ
a
.
The corresponding option price is given by
ˆ
Ct (τ, r; θ) =e
−rτ
∞
(ST − K) qt (ST ; θ) dST
K
1
1
=St 1 − G K; a, b, p + , q −
a
a
(2.1)
− Ke−rτ [1 − G (K; a, b, p, q)] ,
where G is the cdf of the GB2 distribution. For the proof see Section C in the Appendix.
In empirical implementation this function can be computed through the cdf of beta distribution
using the following relation:
G (x; a, b, p, q) = Gβ (z (x; a, b) ; p, q) ,
h
z (x; a, b) = 1 + (x/b)−a
i−1
.
So the price is actually
Ct (τ, r; θ) = St 1 − Gβ
3
1
1
z (K; a, b) ; p + , q −
a
a
− Ke−rτ [1 − Gβ (z (K; a, b) ; p, q)] .
Option pricing
Now suppose that we observe a large panel of option prices, {Ct (τ, x)}, which are indexed by time
of observation t, maturity τ , and log-forward moneyness2 x = log (K/F ) = log (K/S) − rτ . On each
date t we can find a parametric density which best describes the set observable options. Note that
we already have a few closed-form option pricing formulas for each proposed risk-neutral density.
Hence, we can minimize the sum of squared option pricing errors to find the optimal parameters
for this specific day t. More specifically, the criterion function is
θ̂t = arg min
θ
X
[Ct (τ, x) − Ct (τ, x, rt ; θ)]2 .
(3.1)
x,τ
In case of mixture of log-normals, the option price is given by
Ct (τ, x, r; θ) =
M
X
αk St [Φ (di1 ) − exj Φ (di2 )] ,
i=1
with
di1 =
2
−x + (µi − r) τ
1 √
√
+ σi τ ,
σi τ
2
√
di2 = di1 − σi τ ,
Renault & Touzi (1996) argue for the use of log-forward moneyness for its interesting theoretical properties
6
and the vector of parameters is θ = (αi , µi , σi )M
i=1 . The no-arbitrage condition can be satisfied by
adding the penalty to the objective function:
#2
"
penalty =
X
exp (rt τ ) −
X
τ
.
αi exp (µi τ )
i
After performing the optimization problem in (3.1) we obtain a time series of parametric
n
densities qt St+τ ; θ̂t
oT
t=1
. The key question of this paper is whether there is any predictability
in this object. Predictability in the space of parametric distributions is equivalent, on one hand, to
predictability in ndynamics
of parameters
θ̂t , and on the other hand, to predictability in dynamics
oT
. Actually, option prices are more frequently quoted in terms
of option prices Ct τ, x, r; θ̂t
t=1
of BS implied volatilities which allow us to compare prices across different maturities, strikes, and
current underlying price.
The implied volatility σtimp (τ, x) is defined as the solution to the
following equation:
BS τ, x, r, σtimp (τ, x; θ) = Ct (τ, x, r; θ) ,
where BS (·) is the standard Black-Scholes formula. If, in addition, we convert option prices into the
space of BS implied volatilities, then any predictable dynamics is also equivalent to predictability
in the space of volatility surface σtimp (τ, x).
n oT
Next, given the time series of distribution parameters θ̂t
t=1
, we can fit any time series model to
the vector of parameters. Given the parameters of the time series model we construct the conditional
one
week forecast of the risk-neutral parameters. This gives us the second series
of parameters
n o
n o
θ̂tf
T
t=1
. This series is used to construct the forecast of risk-neutral densities, qt St+τ ; θ̂tf
option prices,
n
Ct τ, x, r; θ̂tf
oT
t=1
, and implied volatility surfaces,
n
σtimp τ, x; θ̂tf
oT
t=1
T
t=1
,
. The last
two objects can be compared to their realized counterparts using the well known measures,
RM SE
IV RM SE
v
u
i2
u1 Xh
= t
,
Ct (τ, x) − Ct τ, x, r; θ̂tf
N
t,x,τ
v
u
i2
u 1 X h imp
= t
σt (τ, x) − σtimp τ, x; θ̂tf
,
N
t,x,τ
where N is the total number of terms in the summation
P
t,x,τ .
The last one was put forward by
Renault (1997) as a scale invariant measure of option pricing performance.
4
Results
In this paper we use the volatility surface data from OptionMetrics. Similar to Heston & Nandi
(2000); Christoffersen et al. (2008); Feunou & Tedongap (2012) we leave only Wednesday data to
avoid any seasonality effects on the weekly frequency. Option premium is computed as a bid-ask
average. Only options with positive bid and asks as well as bid-ask spread were left in the sample.
7
Finally, we limit our sample to options with maturities up to one year. After all the filters we have
87568 option quotes on the time interval from 10 January 1996 to 25 July 2012. Some descriptive
statistics (together with results) is given in Table 2 on page 13, Table 3 on page 13, and Table 4
on page 14. There we sort options into bins by log-forward moneyness, maturity, and current VIX
level.
Given the data we solve optimization problem set in (3.1). To illustrate the point of the paper
without chasing superior quantitative results we choose the simplest time series model possible to
forecast parameters of the option-implied risk-neutral distributions. Specifically, we have chosen
AR(3) for each parameter series separately. All the time series models are fit on the first half of the
sample. Given the set of parameters of all AR(3) models we construct one week ahead forecast, θ̂tf .
The result of this procedure is given in Figure 3 on page 25. Each panel shows time series plots of
implied risk-neutral parameters θ̂t and their one week forecasts θ̂tf . The very top panel corresponds
to the single parameter Black-Scholes model. The rest of the panels correspond to the mixture of
two lognormals with total of five parameters, θ = (α, µ1 , µ2 , σ1 , σ2 ).
The first panel, by construction, is closely related to BS implied volatility. It is different in the
sense that it is calibrated to the whole volatility surface and not only to at-the-money one month
options. Naturally, due to strong persistence in volatility this series is highly predictable. ACF and
PACF plots for all parameters can be found in Figure 4 on page 25. On the same figure we notice
that parameters of log-normal mixture model also demonstrate some degree of predictability.
Coming back to Figure 3 on page 25 we see that on average parameter α is close to 0.9. This
means that the log-normal density with parameters µ1 and σ1 plays the main role in describing
risk-neutral densities. To illustrate the result we draw a single day density on Figure 7 on page
28. This picture shows two risk-neutral densities of the log return, log (St+τ /St ), with τ = 1. The
first density corresponds to the simple Black-Scholes model. Clearly, it is just a normal density
centered at the risk-free rate. The second density is a mixture of two normals. We can see that it
is a bi-modal density with fat tails by construction.
On the same day we compute option prices and corresponding implied volatilities and plot them
along with observables on Figure 8 on page 29. Each panel corresponds to a different maturity
available on this date. In the real data we can clearly see the well known phenomena called volatility
smirk. We can also immediately notice a trivial result that the Black & Scholes (1973) implied
volatilities are flat across moneyness. Finally, the crucial result shown in this figure is that the
log-normal mixture is able to reproduce the volatility smile at least qualitatively on all horizons.
Next, we analyze the results of out-of-sample option pricing on the whole sample. First of all we
need to look at Table 2 on page 13 which shows aggregated statistics across several moneyness bins.
There we can see that the IVRMSE for the BS and mixture model is 3.42% and 3.29%, respectively.
We need to stress that this result is out-of-sample. Analyzing IVRMSE on a more detailed scale we
can see that the mixture model outperforms the simplest BS model especially for out-of-the-money
options. When we sort options into bins according to maturity, Table 3 on page 13, we can see that
the mixture is almost universally better than its specific case. Finally, after sorting options by the
8
current VIX level, Table 4 on page 14, there is also no clear pattern of over performance.
Going even further and performing a double sort by moneyness and maturity we can get even
closer perspective on the results by looking at Figure 10 on page 32. There we see that the BS
model produces universally flat volatility surfaces, while the mixture model matches the observable
option prices much closer.
Next, looking at the pricing error in Figure 11 on page 33 we see that BS model error is mostly
concentrated in out-of-the-money side while the mixture model error is more uniform but minimal
at-the-money. Squaring the errors and then taking the average across several bins gives us Figure
13 on page 35.
Looking from the time perspective on Figure 15 on page 37 (bottom panel) we can see that the
option pricing error is not correlated, at least visually, with the current volatility as measured by
the VIX index. Moreover, there are no noticeable time series patterns in the relation between two
model errors.
To conclude this section we compare our option pricing performance represented by total
IVRMSE to other results in the literature. Feunou & Tedongap (2012) for comparison purposes,
besides their own model with conditional skewness, also estimate discrete-time model by Heston &
Nandi (2000), Christoffersen et al. (2006), Christoffersen et al. (2008), as well as continuous-time
model used by Pan (2002), Andersen et al. (2002), Chernov et al. (2003), and Bates (2006).
Feunou & Tedongap (2012) show that among these models the best IVRMSE is in vicinity of
2.5%. Corsi et al. (2013) in their model which targets long memory in volatility process produce
3.8%.
Majewski et al. (2013) do even better but not uniformly across all option subsets.
Christoffersen et al. (2009) reach 2% in-sample with the multi-factor volatility model. Hence, our
option pricing model although very simple has a potential to compete with the stat-of-the-art
option pricing models.
5
Conclusion
In this paper we propose a novel approach to option pricing. It builds on the well established
empirical fact that option-implied information is highly predictable. Our simple illustrative model
nevertheless produces quite good option pricing performance as measured by IVRMSE across
moneyness, maturity, and time.
At the same time we see several avenues for further improvement. First of all, we only use a
mixture of two log-normal distributions and Generalized Beta distribution. Although they produce
some asymmetry and fatter tails in comparison to a single log-normal, they are far away from
flexibility of other parametric distributions. Besides, the mixture has some computational and
identification issues. To remedy that we plan to try other parametric distributions mentioned
in Jondeau et al. (2007) and Christoffersen et al. (2012a). Among those are parametric forms (i.e.
hypergeometric, Gram-Charlier) and semi-parametric (Edgeworth expansion, Hermite polynomials).
The key criteria here is the availability of closed-form solutions for option prices which is necessary
9
for computational efficiency of the method.
In the empirical exercise all the option pricing errors had the same weight in the objective
function while some suggest to weight them by option vega (Christoffersen et al., 2009), or even
minimize the error in terms of implied volatilities directly (Christoffersen et al., 2010, 2012b). We
also did not go beyond a simple autoregression when forecasting option-implied density parameters.
Naturally, some more sophisticated time series methods would achieve a better quantitative result.
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Application to Crude Oil during the Gulf Crisis. Journal of Financial and Quantitative Analysis,
32(1), 91–115.
Pan, J. (2002). The jump-risk premia implicit in options: evidence from an integrated time-series
study. Journal of Financial Economics, 63(1), 3–50.
Renault, E. (1997). Econometric Models of Option Pricing Errors. In D. M. Kreps & K. F. Wallis
(Eds.), Advances in Economics and Econometrics: Theory and Applications, volume 3 chapter 8,
(pp. 223–278). Cambridge University Press.
Renault, E. & Touzi, N. (1996). Option Hedging and Implied Volatilities in a Stochastic Volatility
Model. Mathematical Finance, 6(3), 279–302.
Söderlind, P. & Svensson, L. (1997). New techniques to extract market expectations from financial
instruments. Journal of Monetary Economics, 40(2), 383–429.
Appendix
A
A.1
Tables
Summary
Table 1: Summary of option pricing
sample
model
Price error
IV error
RMSE
IVRMSE
in
BS
GB2
MBS
-0.02
0.46
-0.53
-0.12
0.31
-0.47
4.53
2.28
2.57
2.65
1.72
1.98
out
BS
GB2
MBS
0.08
-0.30
-0.28
-0.06
-0.06
-0.33
5.45
4.48
4.12
2.94
2.59
2.30
12
A.2
A.2.1
One level sort
Premium
Table 2: Option premiums by moneyness
sample
(−2, +2]
(+2, +4]
(+2, +∞)
count
mean
std
14999
29.23
17.31
9012
35.02
22.32
20895
34.25
20.02
10797
27.01
19.23
18509
24.46
16.60
mean
std
mean
std
mean
std
23.35
15.84
29.30
16.43
28.06
18.43
32.14
20.87
35.58
21.04
34.21
23.36
34.61
19.05
35.27
19.10
33.27
19.91
29.46
18.75
27.59
18.66
26.59
18.33
28.72
16.94
24.46
16.19
25.02
15.89
mean
std
mean
std
mean
std
23.39
15.66
28.43
15.14
28.22
18.10
32.22
20.60
34.74
19.53
34.44
22.88
34.85
18.89
34.88
18.01
33.71
19.61
29.60
18.49
26.97
17.34
26.88
17.98
28.70
16.73
23.34
14.75
25.11
15.53
data
GB2
MBS
BS
out
(−4, −2]
stat
BS
in
(−∞, −4]
model
GB2
MBS
Table 3: Option premiums by maturity
sample
model
stat
(0, 30]
(30, 90]
(90, 180]
(180, 270]
(270, +∞)
data
count
mean
std
10892
13.43
7.53
10691
19.20
10.34
29527
27.97
14.50
8885
36.02
16.97
14217
50.40
21.54
mean
std
mean
std
mean
std
13.64
6.39
14.83
7.06
12.73
5.72
19.36
9.11
20.54
9.89
18.25
8.35
28.10
13.37
28.79
14.21
26.95
12.97
35.90
16.12
36.10
17.00
35.13
16.13
49.83
21.40
48.93
21.74
51.55
22.83
mean
std
mean
std
mean
std
13.66
6.34
14.31
6.46
12.82
5.54
19.39
8.99
19.89
9.02
18.38
8.05
28.19
13.16
28.03
12.98
27.18
12.52
36.03
15.75
35.25
15.36
35.45
15.48
50.07
20.86
48.00
19.58
52.01
21.95
BS
in
GB2
MBS
BS
out
GB2
MBS
13
Table 4: Option premiums by current VIX level
sample
model
stat
(0, 15]
(15, 20]
(20, 25]
(25, 30]
(30, +∞)
data
count
mean
std
14957
19.08
11.29
20920
25.75
15.19
22060
31.73
17.83
8705
36.89
19.60
7570
48.72
26.57
mean
std
mean
std
mean
std
19.48
10.38
19.65
10.20
19.14
11.17
26.20
14.25
26.42
14.02
25.68
15.17
31.94
17.01
32.33
16.80
31.36
17.95
36.50
19.34
37.18
19.21
35.89
20.14
46.37
27.26
48.12
27.55
45.80
27.96
mean
std
mean
std
mean
std
20.36
10.70
20.71
10.54
20.51
11.64
26.78
14.62
26.88
14.30
26.66
15.68
32.05
17.24
31.75
16.74
31.49
18.16
35.90
19.39
34.88
18.62
35.10
20.08
44.40
26.87
41.65
24.79
43.40
26.98
BS
in
GB2
MBS
BS
out
GB2
MBS
A.2.2
Implied volatility
Table 5: Option implied volatilities by moneyness
sample
(−2, +2]
(+2, +4]
(+2, +∞)
count
mean
std
14999
23.25
6.45
9012
21.05
6.35
20895
18.30
5.85
10797
17.79
5.71
18509
18.86
5.63
mean
std
mean
std
mean
std
20.04
5.17
23.38
6.30
22.18
5.42
19.54
5.33
21.51
5.95
20.24
5.52
18.52
5.02
18.96
5.49
17.62
4.96
19.12
5.09
18.17
5.37
17.62
4.82
20.70
5.29
18.81
5.18
19.12
5.09
mean
std
mean
std
mean
std
20.07
5.03
22.92
5.43
22.26
5.14
19.61
5.18
21.15
5.10
20.39
5.19
18.65
4.89
18.77
4.78
17.86
4.70
19.20
4.94
17.88
4.61
17.79
4.58
20.69
5.15
18.27
4.37
19.16
4.85
data
GB2
MBS
BS
out
(−4, −2]
stat
BS
in
(−∞, −4]
model
GB2
MBS
14
Table 6: Option implied volatilities by maturity
sample
model
stat
(0, 30]
(30, 90]
(90, 180]
(180, 270]
(270, +∞)
data
count
mean
std
10892
20.13
7.79
10691
20.01
6.89
29527
19.65
6.02
8885
19.49
5.61
14217
19.38
5.41
mean
std
mean
std
mean
std
20.20
5.50
21.42
6.80
19.41
5.77
20.00
5.30
20.88
6.27
19.33
5.48
19.60
5.12
19.98
5.72
19.09
5.22
19.28
5.10
19.39
5.51
19.02
5.25
18.96
5.19
18.74
5.39
19.44
5.59
mean
std
mean
std
mean
std
20.22
5.35
20.89
5.91
19.52
5.46
20.03
5.15
20.42
5.47
19.44
5.19
19.65
4.98
19.63
5.00
19.23
4.96
19.36
4.96
19.08
4.79
19.18
5.00
19.05
5.05
18.49
4.70
19.61
5.36
BS
in
GB2
MBS
BS
out
GB2
MBS
Table 7: Option implied volatilities by current VIX level
sample
model
stat
(0, 15]
(15, 20]
(20, 25]
(25, 30]
(30, +∞)
data
count
mean
std
14957
12.79
1.99
20920
17.14
2.68
22060
21.00
2.72
8705
24.38
2.94
7570
31.27
7.25
mean
std
mean
std
mean
std
13.09
1.07
13.28
1.81
12.81
1.74
17.39
2.03
17.67
2.76
17.03
2.44
21.07
1.80
21.47
2.90
20.69
2.37
23.94
1.63
24.53
3.14
23.59
2.36
29.16
5.06
30.32
6.58
28.76
5.38
mean
std
mean
std
mean
std
13.48
1.13
13.76
1.83
13.44
1.78
17.65
2.27
17.88
2.91
17.48
2.67
21.11
2.15
21.19
3.03
20.74
2.65
23.65
2.19
23.42
3.30
23.21
2.75
28.38
5.55
27.27
6.07
27.92
5.53
BS
in
GB2
MBS
BS
out
GB2
MBS
15
A.2.3
Premium bias
Table 8: Option premium error by moneyness
sample
(−2, +2]
(+2, +4]
(+2, +∞)
count
14999
9012
20895
10797
18509
mean
std
mean
std
mean
std
-5.89
3.08
0.07
2.14
-1.18
2.88
-2.89
2.82
0.56
2.41
-0.82
3.00
0.37
2.54
1.02
2.29
-0.98
2.31
2.45
2.25
0.59
2.15
-0.42
2.12
4.26
2.55
0.00
2.03
0.56
1.95
mean
std
mean
std
mean
std
-5.84
4.31
-0.80
4.44
-1.01
4.43
-2.81
4.31
-0.28
4.98
-0.58
4.69
0.60
3.79
0.64
4.17
-0.54
3.86
2.59
3.71
-0.04
4.18
-0.13
3.77
4.24
4.10
-1.13
4.51
0.65
3.80
data
GB2
MBS
BS
out
(−4, −2]
stat
BS
in
(−∞, −4]
model
GB2
MBS
Table 9: Option premium error by maturity
sample
model
stat
(0, 30]
(30, 90]
(90, 180]
(180, 270]
(270, +∞)
data
count
10892
10691
29527
8885
14217
mean
std
mean
std
mean
std
0.21
3.58
1.40
2.60
-0.70
2.85
0.16
4.02
1.34
2.32
-0.95
2.83
0.13
4.36
0.83
1.53
-1.02
2.21
-0.12
4.64
0.08
0.93
-0.89
1.39
-0.57
5.64
-1.47
2.45
1.15
2.42
mean
std
mean
std
mean
std
0.23
4.17
0.88
3.95
-0.61
3.68
0.19
4.81
0.69
4.27
-0.82
4.02
0.22
5.34
0.06
4.26
-0.79
3.98
0.02
5.71
-0.77
4.19
-0.56
3.78
-0.34
6.66
-2.41
4.83
1.61
4.38
BS
in
GB2
MBS
BS
out
GB2
MBS
16
Table 10: Option premium error by current VIX level
sample
model
stat
(0, 15]
(15, 20]
(20, 25]
(25, 30]
(30, +∞)
data
count
14957
20920
22060
8705
7570
mean
std
mean
std
mean
std
0.40
3.39
0.56
1.75
0.06
1.05
0.44
3.99
0.67
2.20
-0.08
1.42
0.21
4.35
0.60
2.09
-0.36
1.91
-0.39
4.81
0.29
1.83
-1.00
2.60
-2.35
6.78
-0.59
3.38
-2.92
5.34
mean
std
mean
std
mean
std
1.28
3.55
1.63
1.93
1.43
1.75
1.03
4.53
1.13
2.93
0.90
2.75
0.32
4.97
0.03
3.26
-0.24
3.08
-1.00
5.67
-2.02
3.77
-1.79
3.94
-4.32
8.60
-7.07
7.41
-5.31
7.43
BS
in
GB2
MBS
BS
out
GB2
MBS
A.2.4
Implied volatility bias
Table 11: Option implied volatility error by moneyness
sample
(−2, +2]
(+2, +4]
(+2, +∞)
count
14999
9012
20895
10797
18509
mean
std
mean
std
mean
std
-3.21
2.19
0.12
1.68
-1.08
2.21
-1.51
1.92
0.46
1.89
-0.81
2.09
0.22
1.84
0.66
1.72
-0.68
1.67
1.33
1.74
0.38
1.53
-0.16
1.66
1.84
1.74
-0.05
1.58
0.25
1.72
mean
std
mean
std
mean
std
-3.19
2.59
-0.33
2.60
-0.99
2.55
-1.44
2.31
0.10
2.80
-0.66
2.41
0.35
2.23
0.48
2.48
-0.44
2.15
1.41
2.20
0.09
2.37
0.00
2.14
1.82
2.08
-0.60
2.60
0.30
1.98
data
GB2
MBS
BS
out
(−4, −2]
stat
BS
in
(−∞, −4]
model
GB2
MBS
17
Table 12: Option implied volatility error by maturity
sample
model
stat
(0, 30]
(30, 90]
(90, 180]
(180, 270]
(270, +∞)
data
count
10892
10691
29527
8885
14217
mean
std
mean
std
mean
std
0.07
3.87
1.29
2.87
-0.72
3.26
-0.01
3.07
0.87
1.88
-0.68
2.33
-0.05
2.41
0.33
1.15
-0.56
1.53
-0.21
2.05
-0.10
0.81
-0.47
1.08
-0.42
1.81
-0.64
0.93
0.06
1.03
mean
std
mean
std
mean
std
0.09
4.24
0.76
4.08
-0.61
3.62
0.02
3.42
0.41
3.03
-0.57
2.76
0.00
2.70
-0.02
2.18
-0.42
1.96
-0.13
2.29
-0.41
1.72
-0.31
1.46
-0.33
1.98
-0.89
1.51
0.24
1.23
BS
in
GB2
MBS
BS
out
GB2
MBS
Table 13: Option implied volatility error by current VIX level
sample
model
stat
(0, 15]
(15, 20]
(20, 25]
(25, 30]
(30, +∞)
data
count
14957
20920
22060
8705
7570
mean
std
mean
std
mean
std
0.29
1.67
0.49
1.01
0.02
0.62
0.25
2.07
0.53
1.33
-0.11
0.85
0.07
2.30
0.47
1.30
-0.31
1.12
-0.44
2.64
0.15
1.21
-0.79
1.56
-2.12
4.82
-0.95
3.56
-2.51
4.66
mean
std
mean
std
mean
std
0.69
1.76
0.97
1.17
0.64
0.87
0.52
2.31
0.74
1.66
0.34
1.35
0.11
2.55
0.20
1.71
-0.25
1.60
-0.73
2.97
-0.96
1.84
-1.17
2.09
-2.90
4.92
-4.00
4.75
-3.36
4.31
BS
in
GB2
MBS
BS
out
GB2
MBS
18
A.2.5
RMSE
Table 14: RMSE by moneyness
sample
(−2, +2]
(+2, +4]
(+2, +∞)
count
14999
9012
20895
10797
18509
mean
std
mean
std
mean
std
44.15
52.83
4.57
11.31
9.66
32.43
16.26
38.81
6.10
12.66
9.66
33.10
6.59
20.62
6.29
12.76
6.29
28.81
11.04
19.18
4.95
12.76
4.67
22.32
24.66
28.43
4.10
14.07
4.12
15.55
mean
std
mean
std
mean
std
52.69
91.18
20.34
74.43
20.67
64.37
26.47
78.69
24.88
87.61
22.37
72.37
14.71
43.90
17.78
62.97
15.19
55.03
20.51
45.08
17.51
64.67
14.25
52.81
34.77
57.76
21.61
81.41
14.85
47.31
data
GB2
MBS
BS
out
(−4, −2]
stat
BS
in
(−∞, −4]
model
GB2
MBS
Table 15: RMSE by maturity
sample
model
stat
(0, 30]
(30, 90]
(90, 180]
(180, 270]
(270, +∞)
data
count
10892
10691
29527
8885
14217
mean
std
mean
std
mean
std
12.87
41.01
8.69
19.90
8.60
38.09
16.17
39.00
7.19
13.55
8.92
33.67
19.02
29.92
3.03
6.30
5.92
18.89
21.51
25.51
0.87
2.57
2.74
6.49
32.11
43.99
8.14
17.06
7.17
31.54
mean
std
mean
std
mean
std
17.47
57.55
16.38
59.08
13.89
57.52
23.13
64.37
18.72
70.52
16.85
64.75
28.53
61.57
18.17
72.48
16.48
57.70
32.60
59.56
18.14
70.69
14.62
46.03
44.42
79.33
29.11
88.45
21.79
57.12
BS
in
GB2
MBS
BS
out
GB2
MBS
19
Table 16: RMSE by current VIX level
sample
model
stat
(0, 15]
(15, 20]
(20, 25]
(25, 30]
(30, +∞)
data
count
14957
20920
22060
8705
7570
mean
std
mean
std
mean
std
11.63
15.08
3.38
4.13
1.10
2.18
16.10
23.36
5.30
9.01
2.03
2.68
18.97
25.32
4.72
8.29
3.78
5.34
23.31
26.91
3.43
5.64
7.77
9.88
51.52
83.59
11.77
32.69
36.99
75.77
mean
std
mean
std
mean
std
14.27
16.18
6.38
6.89
5.10
8.52
21.60
38.51
9.87
21.58
8.38
29.73
24.76
37.99
10.66
21.40
9.57
17.46
33.15
48.75
18.29
34.21
18.72
29.12
92.64
160.73
104.85
202.78
83.49
151.09
BS
in
GB2
MBS
BS
out
GB2
MBS
A.2.6
IVRMSE
Table 17: IVRMSE by moneyness
sample
(−2, +2]
(+2, +4]
(+2, +∞)
count
14999
9012
20895
10797
18509
mean
std
mean
std
mean
std
15.08
57.32
2.83
49.81
6.04
67.75
5.98
51.03
3.77
48.06
5.01
63.26
3.44
35.13
3.39
33.85
3.26
42.36
4.82
24.84
2.48
23.82
2.78
31.72
6.41
37.83
2.49
38.68
3.03
43.13
mean
std
mean
std
mean
std
16.89
43.95
6.88
54.54
7.48
33.93
7.40
30.82
7.86
52.81
6.25
29.47
5.08
19.92
6.36
38.18
4.80
22.82
6.82
16.12
5.62
30.30
4.57
18.63
7.65
16.77
7.11
50.04
4.00
17.39
data
GB2
MBS
BS
out
(−4, −2]
stat
BS
in
(−∞, −4]
model
GB2
MBS
20
Table 18: IVRMSE by maturity
sample
model
stat
(0, 30]
(30, 90]
(90, 180]
(180, 270]
(270, +∞)
data
count
10892
10691
29527
8885
14217
mean
std
mean
std
mean
std
14.95
78.60
9.88
71.47
11.14
95.82
9.44
48.32
4.28
45.20
5.89
56.76
5.82
31.80
1.43
30.86
2.65
36.50
4.23
23.56
0.67
23.18
1.39
27.44
3.45
18.61
1.29
18.24
1.07
21.37
mean
std
mean
std
mean
std
17.98
56.79
17.26
86.82
13.46
52.21
11.73
30.49
9.35
52.01
7.93
28.22
7.31
15.56
4.73
33.51
4.02
13.61
5.27
9.43
3.11
24.55
2.22
6.68
4.04
6.92
3.08
19.23
1.57
4.18
BS
in
GB2
MBS
BS
out
GB2
MBS
Table 19: IVRMSE by current VIX level
sample
model
stat
(0, 15]
(15, 20]
(20, 25]
(25, 30]
(30, +∞)
data
count
14957
20920
22060
8705
7570
mean
std
mean
std
mean
std
2.87
3.17
1.25
2.04
0.39
0.83
4.34
5.82
2.07
4.36
0.73
1.39
5.27
7.09
1.92
4.60
1.36
2.38
7.16
10.37
1.50
3.19
3.06
5.13
27.74
129.21
13.57
122.63
28.04
155.25
mean
std
mean
std
mean
std
3.56
3.98
2.31
3.38
1.17
1.92
5.58
8.44
3.31
6.45
1.95
4.36
6.49
9.40
2.97
5.85
2.61
4.47
9.34
14.14
4.29
6.72
5.75
8.77
32.59
76.89
38.64
138.74
29.88
71.47
BS
in
GB2
MBS
BS
out
GB2
MBS
21
A.3
Two-level sort
Table 20: Ratios of RMSE by moneyness and maturity
Moneyness
sample
model
Maturity
(−∞, −4]
(−4, −2]
(−2, +2]
(+2, +4]
(+2, +∞)
GB2
(0, 30]
(30, 90]
(90, 180]
(180, 270]
(270, +∞)
0.24
0.17
0.05
0.01
0.15
0.79
0.55
0.24
0.04
0.39
1.02
1.00
0.80
0.52
1.15
0.54
0.45
0.30
0.16
0.85
0.93
0.39
0.12
0.03
0.19
MBS
(0, 30]
(30, 90]
(90, 180]
(180, 270]
(270, +∞)
0.59
0.45
0.21
0.05
0.16
0.83
0.79
0.48
0.12
0.70
0.73
0.85
1.16
1.35
1.18
0.49
0.40
0.49
0.52
0.14
0.82
0.46
0.19
0.09
0.06
GB2
(0, 30]
(30, 90]
(90, 180]
(180, 270]
(270, +∞)
0.47
0.43
0.36
0.32
0.42
1.01
0.95
0.92
0.88
0.96
1.14
1.17
1.19
1.23
1.37
0.77
0.76
0.79
0.81
1.19
1.69
0.98
0.58
0.48
0.58
MBS
(0, 30]
(30, 90]
(90, 180]
(180, 270]
(270, +∞)
0.68
0.58
0.40
0.25
0.33
0.94
0.96
0.81
0.58
0.91
0.86
0.95
1.09
1.13
1.19
0.65
0.63
0.72
0.74
0.71
0.91
0.66
0.45
0.38
0.34
in
out
22
Table 21: Ratios of IVRMSE by moneyness and maturity
Moneyness
sample
model
Maturity
(−∞, −4]
(−4, −2]
(−2, +2]
(+2, +4]
(+2, +∞)
GB2
(0, 30]
(30, 90]
(90, 180]
(180, 270]
(270, +∞)
0.29
0.20
0.11
0.08
0.22
0.86
0.65
0.46
0.35
0.41
1.02
1.00
0.84
0.72
1.12
0.53
0.51
0.39
0.20
1.17
1.04
0.47
0.22
0.14
0.34
MBS
(0, 30]
(30, 90]
(90, 180]
(180, 270]
(270, +∞)
0.65
0.47
0.31
0.17
0.18
0.93
0.96
0.85
0.64
0.30
0.81
0.98
1.20
1.53
1.25
0.52
0.50
0.71
1.37
0.70
1.02
0.62
0.34
0.25
0.25
GB2
(0, 30]
(30, 90]
(90, 180]
(180, 270]
(270, +∞)
0.46
0.41
0.36
0.34
0.45
1.10
1.07
1.06
1.04
0.94
1.19
1.23
1.30
1.43
1.82
0.74
0.79
0.84
0.91
1.90
2.34
1.07
0.63
0.55
0.76
MBS
(0, 30]
(30, 90]
(90, 180]
(180, 270]
(270, +∞)
0.62
0.53
0.38
0.23
0.22
0.93
0.95
0.82
0.57
0.59
0.85
0.96
1.12
1.17
0.94
0.64
0.62
0.73
0.85
0.79
0.87
0.63
0.45
0.38
0.37
in
out
B
Figures
B.1
Distribution parameters
Figure 1: Calibrated model parameters and out-of-sample forecasts for BS model
0 .5 0
0 .4 5
0 .4 0
0 .3 5
0 .3 0
0 .2 5
0 .2 0
0 .1 5
0 .1 0
s ig m a
IN
OUT
19
97
19
99
20
01
20
05
20
da te
03
23
20
07
20
09
20
11
Figure 2: Calibrated model parameters and out-of-sample forecasts for MBS model
0 .7 5
0 .7 0
0 .6 5
0 .6 0
0 .5 5
0 .5 0
0 .4 5
0 .4 0
0 .0 5
0 .0 0
− 0 .0 5
− 0 .1 0
− 0 .1 5
− 0 .2 0
0 .2 0
a
IN
OUT
m1
IN
OUT
m2
0 .1 5
IN
OUT
0 .1 0
0 .0 5
0 .0 0
0 .7
0 .6
0 .5
0 .4
0 .3
0 .2
0 .1
0 .0
0 .2 5
0 .2 0
0 .1 5
0 .1 0
0 .0 5
0 .0 0
s1
IN
OUT
s2
IN
OUT
19
97
19
99
20
01
20
05
20
da te
03
24
20
07
20
09
20
11
Figure 3: Calibrated model parameters and out-of-sample forecasts for GB2 model
8
7
6
5
4
3
2
1
0 .0 2 5
0 .0 2 0
0 .0 1 5
0 .0 1 0
0 .0 0 5
0 .0 0 0
a
IN
OUT
p
IN
OUT
19
97
19
99
20
01
20
05
20
da te
03
20
07
20
09
20
11
Figure 4: Calibrated model parameters for BS model, ACF and PACF
1.0
0.8
0.6
0.4
0.2
0.0
0.2
0.4
ACF: sigma
0
5
10 15
lags, weeks
PACF: sigma
1.0
0.8
0.6
0.4
0.2
0.0
20
25
25
0
5
10 15
lags, weeks
20
25
Figure 5: Calibrated model parameters for MBS model, ACF and PACF
ACF: a
1.0
0.8
0.6
0.4
0.2
0.0
0.2
ACF: m1
1.0
0.8
0.6
0.4
0.2
0.0
0.2
1.0
0.8
0.6
0.4
0.2
0.0
0.2
5
10
15
lags, weeks
PACF: s1
1.0
0.8
0.6
0.4
0.2
0.0
ACF: s2
0
PACF: m2
1.0
0.8
0.6
0.4
0.2
0.0
ACF: s1
1.0
0.8
0.6
0.4
0.2
0.0
0.2
PACF: m1
1.0
0.8
0.6
0.4
0.2
0.0
ACF: m2
1.0
0.8
0.6
0.4
0.2
0.0
0.2
PACF: a
1.0
0.8
0.6
0.4
0.2
0.0
PACF: s2
1.0
0.8
0.6
0.4
0.2
0.0
20
25
0
26
5
10
15
lags, weeks
20
25
Figure 6: Calibrated model parameters for GB2 model, ACF and PACF
1.0
0.8
0.6
0.4
0.2
0.0
0.2
0.4
1.0
0.8
0.6
0.4
0.2
0.0
0.2
ACF: a
ACF: p
0
5
10 15
lags, weeks
PACF: a
1.0
0.8
0.6
0.4
0.2
0.0
20
25
27
1.0
0.8
0.6
0.4
0.2
0.0
PACF: p
0
5
10 15
lags, weeks
20
25
B.2
One day example
Figure 7: Implied risk-neutral densities of the returns for one day
5
4
3
2
1
0
− 1 .0
− 0 .5
0 .0
Lo g re t u rn
28
0 .5
1 .0
Figure 8: Implied volatility for one day
19
18
17
16
15
14
13
12
19
18
17
16
15
14
13
12
19
18
17
16
15
14
13
12
19
18
17
16
15
14
13
12
19
18
17
16
15
14
13
12
19
18
17
16
15
14
13
12
19
18
17
16
15
14
13
12
19
18
17
16
15
14
13
12
Da ys , 3 0
Da ys , 6 0
Da ys , 9 1
Da ys , 1 2 2
Da ys , 1 5 2
Da ys , 1 8 2
Da ys , 2 7 3
Da ys , 3 6 5
BS
GB2
MBS
im p _vo l_d a t a
−5
0
Mo n e yn e s s , %
29
5
30
B.3
Option pricing results
B.3.1
Absolute values
p re m iu m
p re m iu m
70
60
50
40
30
20
10
0
p re m iu m
70
60
50
40
30
20
10
0
70
60
50
40
30
20
10
0
p re m iu m
70
60
50
40
30
20
10
0
p re m iu m
Figure 9: Premiums, two level sort
Ma t u rit y, (0 , 3 0 ]
Ma t u rit y, (0 , 3 0 ]
Ma t u rit y, (3 0 , 9 0 ]
Ma t u rit y, (3 0 , 9 0 ]
Ma t u rit y, (9 0 , 1 8 0 ]
Ma t u rit y, (9 0 , 1 8 0 ]
Ma t u rit y, (1 8 0 , 2 7 0 ]
Ma t u rit y, (1 8 0 , 2 7 0 ]
Ma t u rit y, (2 7 0 , 3 6 6 ]
Ma t u rit y, (2 7 0 , 3 6 6 ]
70
60
In s a m p le
50
BS
40
GB2
30
MBS
20
p re m iu m _d a t a
10
0
(-1 0 , -4 ] (-4 , -2 ] (-2 , 2 ] (2 , 4 ] (4 , 1 0 ]
Mo n e yn e s s , %
31
Ou t o f s a m p le
BS
GB2
MBS
p re m iu m _d a t a
(-1 0 , -4 ] (-4 , -2 ] (-2 , 2 ] (2 , 4 ]
Mo n e yn e s s , %
(4 , 1 0 ]
im p _vo l
im p _vo l
30
28
26
24
22
20
18
16
im p _vo l
30
28
26
24
22
20
18
16
30
28
26
24
22
20
18
16
im p _vo l
30
28
26
24
22
20
18
16
im p _vo l
Figure 10: Implied volatilities, two level sort
Ma t u rit y, (0 , 3 0 ]
Ma t u rit y, (0 , 3 0 ]
Ma t u rit y, (3 0 , 9 0 ]
Ma t u rit y, (3 0 , 9 0 ]
Ma t u rit y, (9 0 , 1 8 0 ]
Ma t u rit y, (9 0 , 1 8 0 ]
Ma t u rit y, (1 8 0 , 2 7 0 ]
Ma t u rit y, (1 8 0 , 2 7 0 ]
Ma t u rit y, (2 7 0 , 3 6 6 ]
Ma t u rit y, (2 7 0 , 3 6 6 ]
30
28
In s a m p le
26
BS
24
GB2
22
MBS
20
im p _vo l_d a t a
18
16
(-1 0 , -4 ] (-4 , -2 ] (-2 , 2 ] (2 , 4 ] (4 , 1 0 ]
Mo n e yn e s s , %
32
Ou t o f s a m p le
BS
GB2
MBS
im p _vo l_d a t a
(-1 0 , -4 ] (-4 , -2 ] (-2 , 2 ] (2 , 4 ]
Mo n e yn e s s , %
(4 , 1 0 ]
B.3.2
Relative errors
Figure 11: Premium error, two level sort
Ma t u rit y, (0 , 3 0 ]
Ma t u rit y, (0 , 3 0 ]
Ma t u rit y, (3 0 , 9 0 ]
Ma t u rit y, (3 0 , 9 0 ]
Ma t u rit y, (9 0 , 1 8 0 ]
Ma t u rit y, (9 0 , 1 8 0 ]
Ma t u rit y, (1 8 0 , 2 7 0 ]
Ma t u rit y, (1 8 0 , 2 7 0 ]
Ma t u rit y, (2 7 0 , 3 6 6 ]
Ma t u rit y, (2 7 0 , 3 6 6 ]
p e rro r
4
2
0
−2
−4
p e rro r
4
2
0
−2
−4
p e rro r
4
2
0
−2
−4
p e rro r
4
2
0
−2
−4
p e rro r
4
2
0
In s a m p le
Ou t o f s a m p le
BS
GB2
MBS
BS
GB2
MBS
−2
−4
( 1 0 , 4 ] ( 4 , 2 ] ( 2 , 2 ] (2 , 4 ] (4 , 1 0 ]
Mo n e yn e s s , %
33
( 1 0 , 4 ] ( 4 , 2 ] ( 2 , 2 ] (2 , 4 ] (4 , 1 0 ]
Mo n e yn e s s , %
Figure 12: Implied volatility error, two level sort
Ma t u rit y, (0 , 3 0 ]
Ma t u rit y, (0 , 3 0 ]
Ma t u rit y, (3 0 , 9 0 ]
Ma t u rit y, (3 0 , 9 0 ]
Ma t u rit y, (9 0 , 1 8 0 ]
Ma t u rit y, (9 0 , 1 8 0 ]
Ma t u rit y, (1 8 0 , 2 7 0 ]
Ma t u rit y, (1 8 0 , 2 7 0 ]
Ma t u rit y, (2 7 0 , 3 6 6 ]
Ma t u rit y, (2 7 0 , 3 6 6 ]
ve rro r
4
2
0
−2
−4
ve rro r
4
2
0
−2
−4
ve rro r
4
2
0
−2
−4
ve rro r
4
2
0
−2
−4
ve rro r
4
2
0
In s a m p le
Ou t o f s a m p le
BS
GB2
MBS
BS
GB2
MBS
−2
−4
( 1 0 , 4 ] ( 4 , 2 ] ( 2 , 2 ] (2 , 4 ] (4 , 1 0 ]
Mo n e yn e s s , %
34
( 1 0 , 4 ] ( 4 , 2 ] ( 2 , 2 ] (2 , 4 ] (4 , 1 0 ]
Mo n e yn e s s , %
B.3.3
Absolute errors
Figure 13: RMSE, two level sort
10
Ma t u rit y, (0 , 3 0 ]
Ma t u rit y, (0 , 3 0 ]
Ma t u rit y, (3 0 , 9 0 ]
Ma t u rit y, (3 0 , 9 0 ]
Ma t u rit y, (9 0 , 1 8 0 ]
Ma t u rit y, (9 0 , 1 8 0 ]
Ma t u rit y, (1 8 0 , 2 7 0 ]
Ma t u rit y, (1 8 0 , 2 7 0 ]
Ma t u rit y, (2 7 0 , 3 6 6 ]
Ma t u rit y, (2 7 0 , 3 6 6 ]
p e rro r2
8
6
4
2
0
10
p e rro r2
8
6
4
2
0
10
p e rro r2
8
6
4
2
0
10
p e rro r2
8
6
4
2
0
10
p e rro r2
8
6
4
In s a m p le
Ou t o f s a m p le
BS
GB2
MBS
BS
GB2
MBS
2
0
(-1 0 , -4 ] (-4 , -2 ] (-2 , 2 ] (2 , 4 ]
Mo n e yn e s s , %
(4 , 1 0 ]
35
(-1 0 , -4 ] (-4 , -2 ] (-2 , 2 ] (2 , 4 ]
Mo n e yn e s s , %
(4 , 1 0 ]
ve rro r2
8
7
6
5
4
3
2
1
0
ve rro r2
8
7
6
5
4
3
2
1
0
ve rro r2
8
7
6
5
4
3
2
1
0
ve rro r2
Figure 14: IVRMSE, two level sort
8
7
6
5
4
3
2
1
0
Ma t u rit y, (0 , 3 0 ]
Ma t u rit y, (0 , 3 0 ]
Ma t u rit y, (3 0 , 9 0 ]
Ma t u rit y, (3 0 , 9 0 ]
Ma t u rit y, (9 0 , 1 8 0 ]
Ma t u rit y, (9 0 , 1 8 0 ]
Ma t u rit y, (1 8 0 , 2 7 0 ]
Ma t u rit y, (1 8 0 , 2 7 0 ]
Ma t u rit y, (2 7 0 , 3 6 6 ]
Ma t u rit y, (2 7 0 , 3 6 6 ]
ve rro r2
8
7
In s a m p le
6
BS
5
GB2
4
MBS
3
2
1
0
(-1 0 , -4 ] (-4 , -2 ] (-2 , 2 ] (2 , 4 ] (4 , 1 0 ]
Mo n e yn e s s , %
36
Ou t o f s a m p le
BS
GB2
MBS
(-1 0 , -4 ] (-4 , -2 ] (-2 , 2 ] (2 , 4 ]
Mo n e yn e s s , %
(4 , 1 0 ]
B.3.4
Errors across time
Figure 15: Implied volatility over time
VIX
VIX
da te
da te
Im p vo l
Im p vo l
80
70
60
50
40
30
20
10
0
80
70
60
50
m ode l
m ode l
BS
GB2
MBS
im p _vo l_d a t a
BS
GB2
MBS
im p _vo l_d a t a
40
30
20
10
0
80
70
60
50
da te
da te
IVRMSE
IVRMSE
m ode l
m ode l
BS
GB2
MBS
BS
GB2
MBS
40
30
20
10
0
19
97 999 001 003 005 007 009 011
1
2
2
2
2
2
2
da te
19
37
97 999 001 003 005 007 009 011
1
2
2
2
2
2
2
da te
C
C.1
Proofs
Proof of (2.1)
The option price is
+ ct (x, τ ; θ) =Et R̃t,τ − ex
,
ˆ
ˆ ∞
=
uqt (u; θ) du − ex
∞
qt (u; θ) du.
ex
ex
Write the term under the first integral separately:
uqt (u; [a, b, p, q]) =
=
=
auap
bap B (p, q) [1 + (u/b)a ]p+q
aua(p+1/a)−1
ba(p+1/a) b−1 B (p, q) [1 + (u/b)a ](p+1/a)+(q−1/a)
1
aua(p+ a )−1
1
ba(p+ a ) B p + a1 , q −
=qt
1
a
1
1
u; a, b, p + , q −
a
a
1
1
[1 + (u/b)a ](p+ a )+(q− a )
.
The expression under the integral becomes a reparameterized density function, so
ct (x, τ ; θ) =Et
x
R̃t,τ − e
+ ,
1
1
= 1 − Gt ex ; a, b, p + , q −
a
a
38
− ex [1 − Gt (ex ; [a, b, p, q])] .