# Posterior Cramer-Rao Lower Bound based Adaptive State Estimation for Option
Price Forecasting

Kumar Yashaswi1 *This work was supported by Department of Mathematics, Indian
Institute of Technology Kharagpur1K. Yashaswi- Department of Mathematics,
Indian Institute of Technology Kharagpur

###### Abstract

The use of Bayesian filtering has been widely used in mathematical finance,
primarily in Stochastic Volatility models. They help in estimating unobserved
latent variables from observed market data. This field saw huge developments
in recent years, because of the increased computational power and increased
research in the model parameter estimation and implied volatility theory. In
this paper, we design a novel method to estimate underlying states (volatility
and risk) from option prices using Bayesian filtering theory and Posterior
Cramer-Rao Lower Bound (PCRLB), further using it for option price prediction.
Several Bayesian filters like Extended Kalman Filter (EKF), Unscented Kalman
Filter (UKF), Particle Filter (PF) are used for latent state estimation of
Black-Scholes model under a GARCH model dynamics. We employ an Average and
Best case switching strategy for adaptive state estimation of a non-linear,
discrete-time state space model (SSM) like Black-Scholes, using PCRLB based
performance measure to judge the best filter at each time step [1]. Since
estimating closed-form solution of PCRLB is non-trivial, we employ a particle
filter based approximation of PCRLB based on [2]. We test our proposed
framework on option data from S$\&$P 500, estimating the underlying state from
the real option price, and using it to estimate theoretical price of the
option and forecasting future prices. Our proposed method performs much better
than the individual applied filter used for estimating the underlying state
and substantially improve forecasting capabilities.

Keywords- Option Theory; Stochastic volatility; Bayesian Filtering; Particle
Filter; Posterior Cramer-Rao Lower Bound (PCRLB)

##  I INTRODUCTION

Option pricing theory involves obtaining a theoretical price of an option
contract on an asset using several factors like asset price, strike price,
volatility, time to expiration, interest rate, etc. These estimated prices are
widely used for improving investment yield and hedging purposes for
portfolios. Black-Scholes equation derived in 1973 [3] laid the foundation for
pricing option’s with the following works improving upon it. With a lot of
factors governing option prices, an effective option price forecasting tool is
important for an investor to identify profitable trades. Stochastic volatility
models are one such set of models that model the movement of option prices.
GARCH model is one such class of volatility model that is used significantly
in option pricing. Though it does not come under the stochastic volatility
models due to computing deterministic value, we add a noise term to it to make
it a stochastic quantity. GARCH model dynamics is used to model volatility
along with Black-Scholes (BS) model for estimating option prices and make a
forecasting framework of option prices.

BS model requires calibration of its parameters (volatility, risk-rate) based
on observed market data like asset prices and option prices. A popular
technique for calibration of parameters is non-linear filtering which has been
widely used in finance for yield curve modelling, time-series filtering and
stochastic volatility modelling [4], [5]. In Bayesian filtering theory,
specific model dynamics is defined for a given system, and the underlying
state or signal is estimated using concepts of recursive Bayesian estimation.
Bayesian methods in the industry have been described thoroughly in [6]. This
is used to determine unobserved latent variables from observed market data
under given model dynamics like GARCH and BS. Volatility and risk which are
the parameter’s of BS model are taken as the state-space for filters.

Since BS model is a non-linear function of underlying asset prices, Kalman
filter can’t be used since it requires linear and Gaussian conditions of
measurement and state space. Hence we use three non-linear filtering
techniques EKF, PF, UKF for latent state estimation. One major drawback with
using individual filters is that no single filter will be optimal for all time
steps. At different time step, the observed option prices may be more
accurately be modelled by different filters under given model dynamics.

In this paper, we use a framework for adaptive state estimation using filter
switching strategy based on Posterior Cramer-Rao Lower Bound (PCRLB) measure.
PCRLB gives a theoretical lower bound of mean square error (MSE) of a non-
linear Bayesian filter. [1] proposed an efficient strategy to select the most
optimal filter at each time step from a set of Bayesian filters for a general
non-linear, stochastic, discrete-time state-space models (SSMs). Their idea
was based on the fact that there is no single Bayesian estimator that will
perform optimally for a given non-linear system under all operating
conditions. The switching strategy used for state estimation was to switch
between a set of Bayesian filters at each time step based on their
performance. The performance metric was derived using PCRLB, which is defined
for a class of non-linear SSMs.

The computation of a closed-form solution of PCRLB itself is a very difficult
task and is not possible unless a true state is known, which for the option
pricing theory is missing in form of volatility and risk, a non-observable
quantity. Hence we use a particle filter based approximation of PCRLB based on
the work of [2]. The computation of PCRLB along with the use of adaptive state
estimation strategy from [1] gives a stable framework for estimating the
underlying states of BS model. [1] proved the stability of the adaptive
switching strategy and [2] justified the accuracy of particle filter based
approximation of PCRLB as it ensured that the proposed numerical solution does
not result in accumulation of errors.

We largely modify work done by [7], [8], and [9]. The major addition to their
work being the adaptive state estimation strategy employed which we shall
compare with individual filters. We test the method on S$\&$P 500 options data
which provide a large amount of highly liquid data for training and test
dataset. We use a 1 time step forecasting technique on test data using the
filter estimated in previous time step from the proposed strategy.

To the best of our knowledge, this is the first of many work that uses an
adaptive state estimation technique for inferring underlying state from
observed option prices movement, and further using it for option price
forecasting. The main aim of our work is to investigate the efficiency of
using the switching strategy to optimally and sequentially select a different
Bayesian filter at each time step for stochastic volatility models and compare
its performance with respect to individual filters throughout the training
data. Such a framework can be used to design an effective option trading
strategy.

###  I-A Related Work

With the increased research being done on Bayesian filtering theory for
several applications, their use in state estimation of stochastic volatility
models has increased widely over the past years. [9], [10] were one of the
major works in this domain. The usage of filtering theory was aided using
their research. [8] used a hybrid model of EKF and SVM for option-price
prediction, built on Black-Scholes formula. [7] further improved upon it using
UKF. [4] and [5] gives a basic overview of filtering theory in finance
including in stochastic volatility model.

Several works have been done for estimating PCRLB, which is a vital component
computed for our proposed method. [11] used a similar approximation to [2] for
PCRLB and tested it on non-linear tracking applications. Additionally [12]
evaluated the bound for three cases: recursive estimation of slowly varying
parameters of an autoregressive process, tracking a slowly varying frequency
of a single cisoid in noise and tracking parameters of a sinusoidal frequency
with sinusoidal phase modulation. An alternate Bayesian filter switching
strategy was used by [13], using a performance-based filter switching strategy
for state estimation under known model parameters case.

##  II Background

We discuss computation of PCRLB which is the main concept on which the
switching strategy is based. For theory on Filtering techniques used, readers
are referred to [14], [15], [16].

###  II-A Posterior Cramér-Rao lower bound

Cramer–Rao bound (CRB) is a commonly used statistical lower bound in time-
invariant models. It is used to indicate the lower theoretical bound of the
mean square error (MSE) of a state-estimator. In time-varying systems such as
the option pricing framework, the estimated price movement is considered
random since it corresponds to an underlying nonlinear, randomly driven state
process. For such systems a Posterior Cramer–Rao bound (PCRLB) is computed.
Since option pricing is defined in non-linear terms, we are concerned with
using PCRLB for lower-bound analysis of the MSE of our SSM and helps in the
algorithm for adaptive state estimation of options price movement according to
BS and GARCH model formulation. Hence it’s an important statistic to compute
for the state estimation system. Two of the most vital uses of PCRLB is

  * •

gives a benchmark measure of fit for different non-linear filters

  * •

performance comparison of several non-linear filters against one another and
to that of an optimal filter

For a system of non-linear model:

$X_{t+1}=f_{t}(X_{t})+V_{t},\\\ \hskip 2.84544ptV_{t}\sim N(0,Q_{t})$ (1)
$Y_{t}=g_{t}(X_{t})+W_{t},\\\ \hskip 2.84544ptW_{t}\sim N(0,R_{t})$ (2)

the PCRLB inequality is described as:

$P_{t|t}\equiv
E_{p(X_{0:t},Y_{1:t})}[(X_{t}-{\widehat{X}}_{t|t})(X_{t}-{\widehat{X}}_{t|t})^{T}]\geq
J_{t}^{-1}$ (3)

For a randomly sampled measurement sequence $\\{Y_{1:t}\\}_{t\epsilon N}$ from
Eq (1-2), the MSE of any Bayesian filter at ${t\in N}$ is bounded where
$P_{t|t}$ is a $s\times s$ sized MSE matrix which belongs to the set of
symmetric positive definite matrices $S_{++}^{s}$, $X_{t|t}\equiv
X_{t}(Y_{1:t}):R^{tm}\rightarrow R^{s}$ is point estimate of state $X_{t}$ at
time $t$. The $s\times s$ matrix $J_{t}$ is known as the posterior Fisher
information matrix (PFIM) is a way of measuring the information present in a
state $X$ using a distribution that models $X$. It’s inverse is the PCRLB
matrix. The proof of the above equation can be seen in works like [17]. Eq (3)
can also be written in the form

$P_{t|t}^{S}\equiv
E_{p(X_{0:t},Y_{1:t})}[\|X_{t}-{\widehat{X}}_{t|t}\|^{2}]\geq Tr[J_{t}^{-1}]$
(4)

where $Tr$ is the trace of matrix and $\|.\|$ is the 2-norm. Eq (42) can be
attributed to the fact that $P_{t|t}-J_{t}^{-1}\geq 0$ is a positive definite
matrix for all state estimates at time $t\in N$. The model in Eq (1-2),
following set of assumptions (Assumption (3.2-3.5) as described in [1]), gives
way to the recursive computation of the PFIM as:

$J_{t+1}=D_{t}^{22}-[D_{t}^{12}]^{T}(J_{t}+D_{t}^{11})^{-1}D_{t}^{12}$ (5)

The individual components are defined as in Eq (45-47) below:

$D_{t}^{11}=E_{p(X_{0:t+1},Y_{1:t+1})}[-\triangle_{X_{t}}^{X_{t}}log(p(X_{t+1}|X_{t}))]$
(6)
$D_{t}^{12}=E_{p(X_{0:t+1},Y_{1:t+1})}[-\triangle_{X_{t}}^{X_{t+1}}log(p(X_{t+1}|X_{t}))]$
(7)
$\begin{split}D_{t}^{22}=E_{p(X_{0:t+1},Y_{1:t+1})}[-\triangle_{X_{t+1}}^{X_{t+1}}log(p(X_{t+1}|X_{t}))\\\
-\triangle_{X_{t+1}}^{X_{t+1}}log(p(Y_{t+1}|X_{t+1}))]\end{split}$ (8)

The gradients using the Laplacian operator $\triangle$ is calculated at the
true states. The PFIM matrix is defined for initial states as:

$J_{0}=E_{p(X_{0})}[-\triangle_{X_{0}}^{X_{0}}log(p(X_{0})]$ (9)

Any Bayesian estimator of SSM model that solves for $p(X_{t}|Y_{1:t})$ is
valid to obtain a PCRLB inequality. Getting a closed-form solution to the PFIM
or PCRLB is non-trivial, due to the complex integrals involved in Eq (3). To
solve this issue, [2] used particle filter to give approximate solution to
PFIM, hence PCRLB solving the set of Eq (6-8) using particle filter. We add a
modification to the approach used in [2] by adding the information present in
state estimated through different filters.

###  II-B Particle Filter Approximation of PCRLB

Since Black-Scholes is a non-linear SSM with additive Gaussian noise, we work
with the model represented by Model 4.15 in [2], which gives an appropriate
approximation for the set of systems. This technique was inspired as an
alternative to the Monte-Carlo methods which required an ensemble of true
state’s and measurements which are not available in many defined systems. We
shall explain the general approach and result without jumping to the
derivation, which is rigorously explained in [2]. Based on the approach
mentioned in [2], approximation of Eq (6-8) was derived as:

$D_{t}^{11}=\frac{1}{MN}\sum_{j=1}^{M}\sum_{i=1}^{N}[\nabla_{X_{t}}f_{t}^{T}(X_{t|t+1}^{i,j})]Q_{t}^{-1}[\nabla_{X_{t}}f_{t}(X_{t|t+1}^{i,j})]$
(10)
$D_{t}^{12}=\frac{1}{MN}\sum_{j=1}^{M}\sum_{i=1}^{N}-[\nabla_{X_{t}}f_{t}^{T}(X_{t|t+1}^{i,j})]Q_{t}^{-1}$
(11)
$\begin{split}D_{t}^{22}=Q_{t}^{-1}+\frac{1}{MN}\sum_{j=1}^{M}\sum_{i=1}^{N}([\nabla_{X_{t+1}}g_{t+1}^{T}(X_{t+1|t}^{i,j})]\\\
R_{t+1}^{-1}[\nabla_{X_{t+1}}g_{t+1}(X_{t+1|t}^{i,j})])\end{split}$ (12)

The particles are N-samples following $\\{X_{t+1|t}^{i,j}\\}_{i=1}^{N}\sim
p(x_{t+1}|y_{1:t}^{j})$ and $\\{X_{t|t+1}^{i,j}\\}_{i=1}^{N}\sim
p(x_{t}|y_{1:t+1}^{j})$, with M measurement sequence being
$\\{Y_{1:t+1}=y_{1:t+1}\\}_{i=1}^{M}$ obtained from historical data. For the
problem of option pricing there is only 1 measurement sequence.

The distribution $p(x_{t+1}|y_{1:t}^{j})$ and $p(x_{t}|y_{1:t+1}^{j})$ are
obtained through distribution $p(dx_{t:t+1}|y_{1:t+1})$, which can be computed
through $p(dx_{t}|y_{1:t})$ and $p(dx_{t+1}|y_{1:t+1})$ as explained in [2].
Firstly, $p(dx_{t}|y_{1:t})$ and $p(dx_{t+1}|y_{1:t+1})$ are obtained by
Algorithm (1).

Result: Particles from approximated probability distribution’s
$p(dx_{t}|y_{1:t})$ and $p(dx_{t+1}|y_{1:t+1})$

Initialization: Estimated State $x_{t|t}$ and state Covariance $P_{t|t}$
through filter $f\in B$ where $B$ is set of Bayesian Filters;

if t<T ;

Sampling Step-;

1) For i=1,…N sample $(X^{(i)}_{t|t})\sim p(x_{t|t},P_{t|t})$;

2) For i=1,…N compute
$(X^{(i)}_{t+1|t})\sim\frac{1}{N}\sum_{i=1}^{N}p(X_{t+1|t}|X^{(i)}_{t|t})$\-
according to state equation (56);

3) For i=1,…N set particle weight’s $w^{(i)}_{t+1|t}=\frac{1}{N}$ ;

4) For i=1,…N, $\\{Y_{t+1}=y_{t+1}\\}$ set particle weight’s
$w^{(i)}_{t+1|t+1}=w^{(i)}_{t+1|t}\frac{p(Y_{t+1}|X^{(i)}_{t+1|t})}{\sum_{j=1}^{N}p(Y_{t+1}|X^{(j)}_{t+1|t})}$
;

5) Resample particle set $\\{X^{(j)}_{t+1|t+1}\\}$ for j=1,..N with
replacement from $\\{X^{(i)}_{t+1|t}\\}$ using resampling technique such that
$Pr(X^{(j)}_{t+1|t+1}=X^{(i)}_{t+1|t})=w^{(i)}_{t+1|t+1}$, where Pr is
probability measure. Set {$W^{i}_{t+1|t+1}=\frac{1}{N}$} ;

Output: Particle Filter Approxmated probability distribution\-
$X^{(i)}_{t|t}\sim p(dx_{t}|y_{1:t})$ and $X^{(i)}_{t+1|t+1}\sim
p(dx_{t+1}|y_{1:t+1})$ for i=1,2,..N;

Algorithm 1 Particle Filter Algorithm for generating PCRLB distributions

Most of the algorithm is similar to the one used in [2]. The major variation
was the use of state and covariance matrix obtained from the set of individual
Bayesian filters as opposed to only particle filter. Based on the output, the
SMC approximation of target distribution $p(dx_{t:t+1}|y_{1:t+1})$ can be
defined as:

$\widetilde{p}(dx_{t:t+1}|y_{1:t+1})=\sum_{i=1}^{N}W^{i}_{t|t,t+1|t+1}\delta_{X^{i}_{t|t},X^{i}_{t+1|t+1}}(dx_{t:t+1})$
(13)

where

$W^{i}_{t|t,t+1|t+1}=\frac{\zeta^{i}_{t|t,t+1|t+1}}{N\sum_{j=1}^{N}\zeta^{j}_{t|t,t+1|t+1}}$
(14)
$\zeta^{i}_{t|t,t+1|t+1}=\frac{p(X^{i}_{t+1|t+1}|X^{i}_{t|t})}{N\sum_{m=1}^{N}p(X^{i}_{t+1|t+1}|X^{m}_{t|t})}$
(15)

and $\delta_{X^{i}_{t|t},X^{i}_{t+1|t+1}}(.)$ is marginalized Dirac delta
function in $dx_{t:t+1}$, centred around the random particle’s
$X^{i}_{t|t},X^{i}_{t+1|t+1}$. The particles are resampled according to:

$Pr(X^{(j)}_{t:t+1|t+1}=\\{X^{(i)}_{t|t};X^{(i)}_{t+1|t+1}\\})=W^{(i)}_{t|t,t+1|t+1}$
(16)

where $\\{X^{i}_{t:t+1|t+1}\\}^{N}_{i=1}$ are resampled i.i.d particles. After
resampling target distribution is approximated as:

$\widetilde{p}(dx_{t:t+1}|y_{1:t+1})=\frac{1}{N}\sum_{i=1}^{N}\delta_{X^{i}_{t:t+1|t+1}}(dx_{t:t+1})$
(17)

According to Lemma (4.13) in [2], the SMC based approximation of
$p(dx_{t}|y_{1:t+1})$ is given by :

$\widetilde{p}(dx_{t}|y_{1:t+1})=\frac{1}{N}\sum_{i=1}^{N}\delta_{X^{i}_{t|t+1}}(dx_{t})$
(18)

$\delta_{X^{i}_{t|t+1}}(.)$ is marginalized Dirac delta function in $dx_{t}$,
centred around the random particle $X^{i}_{t|t+1}$. Distribution
$\widetilde{p}(x_{t+1}|y_{1:t})$ is approximated similarly as:

$\begin{split}\widetilde{p}(dx_{t+1}|y_{1:t})=\int_{\chi}p(x_{t+1}|x_{t})\widetilde{p}(dx_{t}|y_{1:t})=\\\
\frac{1}{N}\int_{\chi}p(x_{t+1}|x_{t})\sum_{i=1}^{N}\delta_{X^{i}_{t|t}}(dx_{t})\\\
=\frac{1}{N}\sum_{i=1}^{N}p(X_{t+1|t+1}|X^{m}_{t|t})\end{split}$ (19)

The particle filter approximation of both PFIM and PCRLB using Eq (10-12) and
matrix inversion lemma are given by:

$J_{t}=D_{t}^{22}-[D_{t}^{12}]^{T}(J_{t}+D_{t}^{11})^{-1}D_{t}^{12}$ (20)
$\begin{split}J_{t}^{-1}=[D_{t}^{22}]^{-1}-[D_{t}^{22}]^{-1}[D_{t}^{12}]^{T}\times[D_{t}^{12}[D_{t}^{22}]^{-1}[D_{t}^{12}]^{T}\\\
-(J_{t}+D_{t}^{11})]^{-1}D_{t}^{12}[D_{t}^{22}]^{-1}\end{split}$ (21)

where $J_{t}$ and $J_{t}^{-1}$ are particle filter approximated PFIM and PCRLB
respectively. The convergence of particle filter approximation to true PCRLB
can be shown. It is not necessarily implied about the convergence of the SMC
based PCRLB and MSE to its theoretical values, it does provide a good
theoretical basis for the numerous approximations.

###  II-C Adaptive State-Estimation Switching Strategy

We have discussed the relevant concepts that would be required for designing a
strategy to estimate state-space from stochastic volatility models. It will
select different Bayesian estimators to be used in each step, depending on the
optimality of each filter. PCRLB judges the MSE performance of a Bayesian
estimator, with the estimator with the closest MSE to PCRLB is the best
performing. MSE matrix is computed for each filter and PCRLB through particle
filter approximation explained in the last subsection. The strategy is
initiated with a set of Bayesian estimators, and switch between them as and
when required based on their performance. To get a scalable metric we define
performance metric as:

$\Phi_{t}=J_{t}^{-1}oP_{t|t}^{-1}$ (22)

$\Phi_{t}\in R^{s\times s}$ is the new performance matrix to assess the
performance of the Bayesian state estimator used, where $s$ is the number of
states to be estimated. An alternative definition of equation (22) is:

$\Phi_{t}(i,j)=\begin{cases}J_{t}(j,j)^{-1}[P_{t|t}(j,j)]^{-1},&\text{if
i=j;}\\\ 0,&\text{if i $\neq$ j.}\\\ \end{cases}$

where $i,j\in 1,2,..s$. We state some results from [1] with given proof in the
same-

$0<\Phi_{t}(j,j)\leq 1$ (23)

This lead’s to the condition $Trace[\Phi_{t}]\in(0,s]$ for all Bayesian
estimators at any time. This puts a bound on the performance measure as
compared to PCRLB which may be unbounded in many applications. For the
strategy, the best point estimate at each time step is decided based on the
maximum value of performance metric. The adaptive state estimation strategy is
based on using (22) to judge the best state estimator $f\in B$ at time step
$t$. Our initial process is computing performance metrics using MMSE and PCRLB
matrix for all Bayesian estimators from $B$. We then switch between them for
each time step based on which gave the best performance. We employ the Average
and Best case switching strategy from [1] for our state estimation. The
average-case strategy can be described as:

  * •

Input: Set of measurements upto time period t, set of Bayesian filters $B$

  * •

Step 1: Compute MMSE matrix, PCRLB and Performance metric for all Bayesian
estimator $f\in B$

  * •

Step 2: Compute $Trace[\Phi_{t}^{f}]$ using Performance metric for all $f\in
B$

  * •

Step 3: Solve for f such that $\widehat{f}=argmax_{f}(Trace[\Phi_{t}^{f}])$

  * •

Output: $\widehat{X}_{t|t}^{(\widehat{f})}$ is the point estimate to be
selected at time $t$

and the best-case switching strategy as:

  * •

Input: Set of measurements upto time period t, set of Bayesian filters $B$

  * •

Step1: Compute MMSE matrix, PCRLB and Performance metric for all Bayesian
estimator $i\in B$

  * •

Step 2: Compute $Trace[\Phi_{t}^{f}]$ using Performance metric for all $f\in
B$

  * •

Step 3: Set j <\- 1, While j $\leq$ s, repeat,
1) Solve for f such that
$\widehat{f}=argmax_{f}(Trace[\Phi_{t}^{f}](j,j))$
2) Select $\widehat{X}_{t|t}^{(\widehat{f})}(j)$ as point estimate
3) j <\- j+1

  * •

Output: $\widehat{X}_{t|t}$ is the point estimate to be selected at time $t$

The major difference to average-case, which is a one-dimensional switching
strategy, best-case strategy provides an s dimensional strategy, where each
state has its switching strategy defined.

All the estimators in the set estimate all the states of the system at each
sampling time parallelly, but only the ones suggested by the switching
strategy are selected as the final estimate. This technique gives the
advantage of parallel computation for increasing computation of the sub-
optimal state. The numerical stability of the algorithm is explained
thoroughly in [1] by applying the boundedness of stochastic processes.

##  III Problem Formulation

###  III-A State Space Modelling of Stochastic Volatility Model

We model the option prices using Black-Scholes model. The initial formulation
of BS model assumed constant volatility structure. A non-linear behaviour was
observed in volatility that came to be known as volatility smile, which has
since been observed for all asset classes.

To get a structure of volatility, we combined GARCH(1,1) with Black-Scholes
model and formulate it in the dynamics of the state-space model. The Black-
Scholes price for call and put option prices ($C$ and $P$ respectively) are
represented as:

$C=SN(d_{1})-Ke^{-r\tau}N(d_{2})$ (24) $P=-SN(-d_{1})+Ke^{-r\tau}N(-d_{2})$
(25)

where $S$ is the stock price, $K$ is strike price, $r$ is the risk-free rate,
$\tau$ is time left to expiration. $d_{1}$ and $d_{2}$ are defined as:

$d_{1}=\frac{ln(S/K)+(r+\sigma^{2}/2)\tau}{\sigma\sqrt{\tau}}$ (26)
$d_{2}=d_{1}-\sigma\sqrt{\tau}$ (27)

The GARCH model assumes the underlying stock price $S_{t}$ has a stochastic
variance $v_{t}$, which is estimated using predicted variance at $t-1$ step,
long-run average variance rate $V_{L}$ and stock returns up to $t$ ($u_{t}$).
The dynamics are represented as:

$v_{t}=\omega+\alpha u_{t}^{2}+\beta v_{t-1}+w_{t}$ (28) $r_{t}=r_{t-1}+v_{t}$
(29) $c_{t}=BS(v_{t},r_{t})+z_{t}$ (30)

where Eq (28) is the GARCH(1,1) formulation with added Gaussian noise $w_{t}$.
We add noise variation’s in risk-free rate as well, working with a combination
of formulation in both [8] and [9]. Eq (28-29) represents the state equations,
and Eq (30), which is the BS prices with a noise term $z_{t}$, represents the
measurement equation. This will help us to employ the adaptive state
estimation technique to probabilistically estimate the volatility state of the
dynamic system from the noisy option price observations.

Algorithm (2) explains the process of state estimation. The $filterUpdate$ is
the prediction and update step of each individual filter. It is then followed
by the PCRLB computation in $PCRLB$ for all $f\in B$. These computations are
then used in the switching strategy function $SwitchingStrategy$ for selecting
the most optimal state estimated from all filter. The additional function
parameters are represented as $\theta$ and $\Theta$.

Result: Estimated underlying states $\widehat{x}_{t}$ using option prices
$c_{t}$ for $t\in\\{1,2,3,..T\\}$

Initialization: Initial State $x_{0}=\\{v_{0},r_{0}\\}$, state covariance
$P_{0}$ and $J^{f}_{0}=P^{-1}_{0}$ for filters $f\in B$ where $B$ is set of
Bayesian filters;

while  _$t <T$_ do

while  _$f\in B$_ do

$({x}^{f}_{t},P^{f}_{t})\leftarrow
filterUpdate^{f}(c_{t},{x}_{t-1},P_{t-1},\theta_{t})$;
$(J^{f}_{t+1},(J^{f}_{t+1})^{-1})\leftarrow
PCRLB(J^{f}_{t},c_{t+1},{x}^{f}_{t},P^{f}_{t},\Theta_{t+1})$;

end while

$({x}_{t},P_{t})\leftarrow
SwitchingStrategy(({J}^{B}_{t})^{-1},{x}^{B}_{t},P^{B}_{t})$

end while

Output: Volatility and Risk state estimated through the switching strategy\-
$X^{(i)}_{t|t}\sim p(dx_{t}|y_{1:t})$ and $X^{(i)}_{t+1|t+1}\sim
p(dx_{t+1}|y_{1:t+1})$ for i=1,2,..N

Algorithm 2 Adaptive State Estimation Strategy

###  III-B Option Data Set

In the previous sections, we had described our proposed technique based on
PCRLB measure and the filtering techniques for estimation of underlying
volatility and risk state under a Black-Scholes and GARCH(1,1) modelling
framework. We will use 3 different filters (EKF, UKF, PF) for state estimation
and particle filter for approximate computation of PCRLB.

The data used for backtesting the proposed method is the S$\&$P 500 index
option [18] from the year 2019 to 2020. For each day, it has option quotes for
both call and put options with various expiration and strike price data.

For backtesting forecasting accuracy we consider roughly 150 daily call option
prices. We took a single expiration date of December 2020 based on the
liquidity of options with strike prices of 2000, 2500 and 3000. The
predictions are made for 1 day ahead and as the new option prices come in, the
predicted values are corrected and PCRLB is computed according to the new
prices. This ensures the error in forward prediction by the filters do not add
up.

We further use it to obtain volatility structure using option data and compare
it with VIX index, 30-day historical volatility measure and close of day
implied volatility of the options. The dataset for volatility structure is
obtained by taking the call-options with maximum traded volume on a given
quote date. Thereafter the adaptive state estimation strategy is used similar
to option pricing forecasting framework.

Though this procedure is computationally expensive, hence will be difficult to
use on high-frequency data, it is feasible for online estimation in intraday
data. For the prediction step of filters, the underlying price in BS model is
modelled as geometric Brownian motion.

##  IV Experimental Results and Analysis

We test the accuracy of the 3 filters: EKF, UKF, PF on the training dataset.
We compare the result’s of the 3 filters to the average and best-case
switching strategy based adaptive state estimation technique proposed in our
research. They will be referred to as AAF and ABF respectively hereafter. The
measure of performance used is the deviation of estimated signal from the
observed market option prices. The measure is represented as:

${RMSE}=(\frac{1}{T}\sum_{t=1}^{T}\frac{(c_{t}-BS(\widehat{v_{t}},\widehat{r_{t}}))^{2}}{K})^{\frac{1}{2}}$
(31)

where $\widehat{v_{t}},\widehat{r_{t}}$ are the underlying volatility and risk
states at time $t$. $c_{t}$ is the real option price.

Figure 1: Price Prediction for various filters for several Quote Dates Figure
2: Maximum Performance Measure computed for AAF

Figure 3: Volatility Cmparison of AAF estimated volatility with VIX and
Historical Volatility Forecasting RMSE
---
|  K=2000 |  K=2500 |  K=3000
EKF |  0.040 |  0.036 |  0.020
UKF |  0.033 |  0.027 |  0.019
PF |  0.031 |  0.031 |  0.017
AAF |  0.028 |  0.020 |  0.019
ABF |  0.030 |  0.024 |  0.018
TABLE I: RMSE for Price Forecasting on Test Set Filter Frequency
---
|  EKF |  UKF |  PF
AAF |  56 |  164 |  242
ABF Volatility |  94 |  148 |  220
ABF Risk |  121 |  72 |  269
TABLE II: Frequency of filters used for individual methods

As mentioned previously, one time step ahead forecasts are considered as they
prevent the problem of cumulative errors from the previous period for out-of-
sample forecasting. Since the proposed strategy considers the most optimal
filter at each time step based on performance measure derived from PCRLB, it
gives much better results than individual filters in terms of RMSE measure.

Table (1) displays the forecasting performance of the individual EKF model,
UKF model and PF model and the proposed AAF and ABF filter. Performance was
evaluated on the 3 call options. Figures (1) plots the predicted results of
the various filters on option of K=2500. The results in Table (1) and Figure
(1) indicate that the proposed adaptive state estimation technique perform
better than the individual filter on option price forecasting. The technique
takes the advantage of each filter and selects it adaptively according to
price and underlying state movement. Fig (2) shows the maximum Performance
metric $(\phi)$ computed for AAF. Table (2) gives the frequency of filter used
for each strategy. Analysis shows that PF was the most used filter followed by
UKF and lastly EKF for overall state and individual volatility movement. For
risk, EKF was selected more than UKF, providing an insight that EKF can be
suitable when a risk-free movement for an option contract is studied.

The technique also replicates the volatility structure of $S\&P500$ well. As
seen in Fig (3) having similar structure to VIX, historical volatility and
implied volatility. The volatility estimated is mostly lower than other
volatility sources.

These results indicated that adaptive state estimation technique capture the
nonlinear pattern of option prices in a much better way than individual
filters. The technique has the flexibility to use as many Bayesian filters
with the advantage of parallel computation.

##  V CONCLUSIONS

In this paper, we use an adaptive state estimation technique for option price
forecasting which takes into account the benefits of a combination of sub-
optimal Bayesian estimators under appropriate non-linear model dynamics. The
most optimal state estimated individually for the various filters is selected
at each time step based on maximum PCRLB based performance measure, computed
through a particle filter based approach. The volatility and risk were taken
as the state and the option prices were modelled through Black-Scholes formula
with GARCH(1,1) formulation for volatility structure.

We compared the performances of the proposed technique with the individual
filters in the set for estimating underlying option prices forecasting
performance. It performed much better as compared to the individual filters in
error metrics like RMSE. Though both measures of state covariance matrix and
PCRLB are approximations, the work done provides a flexible and efficient
modelling framework for estimating option prices hidden state and forecasting
future prices.

##  VI Future Work

The framework described above have a large scope of modifications and
extensions to other field’s of finance and otherwise. With respect to the
problem of volatility modelling and estimation of the underlying state of
option prices for price forecasting, more advanced stochastic volatility
models can be considered. Additional non-linear filtering approaches can also
be added for switching strategy for adaptive state estimation. Using technique
used in [7], [8], the filtering results can also be added with non-linear
residuals prediction by regression techniques like SVM, Neural Networks or
Boosting techniques. Lastly, our strategy has a major drawback of high time
complexity, preventing its use on high-frequency trading and hedging methods.
Several mathematical optimizations has to be made to reduce this drawback.

##  VII Acknowledgement

This work was supported by Professor Swanand Khare, Department of Mathematics,
Indian Institute of Technology, Kharagpur. I would like to thank him for his
helpful insights on concepts of Bayesian Inference, which was essential in the
progress of the project.

## References

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††thanks: Corresponding author<EMAIL_ADDRESS>Corresponding author.
Email address<EMAIL_ADDRESS>Present address: GSI Helmholtzzentrum für
Schwerionenforschung GmbH, 64291 Darmstadt, Germany††thanks: Corresponding
author<EMAIL_ADDRESS>
# Observation of an ultra-low $Q$-value electron-capture channel decaying to
75As via high-precision mass measurement

M. Ramalho  Department of Physics, University of Jyväskylä, P.O. Box 35,
FI-40014 Jyväskylä, Finland  Z. Ge  Department of Physics, University of
Jyväskylä, P.O. Box 35, FI-40014 Jyväskylä, Finland  T. Eronen  Department of
Physics, University of Jyväskylä, P.O. Box 35, FI-40014 Jyväskylä, Finland  D.
A. Nesterenko  Department of Physics, University of Jyväskylä, P.O. Box 35,
FI-40014 Jyväskylä, Finland  J. Jaatinen  Department of Physics, University of
Jyväskylä, P.O. Box 35, FI-40014 Jyväskylä, Finland  A. Jokinen  Department of
Physics, University of Jyväskylä, P.O. Box 35, FI-40014 Jyväskylä, Finland  A.
Kankainen  Department of Physics, University of Jyväskylä, P.O. Box 35,
FI-40014 Jyväskylä, Finland  J. Kostensalo  Department of Physics, University
of Jyväskylä, P.O. Box 35, FI-40014 Jyväskylä, Finland  J. Kotila  Finnish
Institute for Educational Research, University of Jyväskylä, P.O. Box 35,
FI-40014 Jyväskylä, Finland  Center for Theoretical Physics, Sloane Physics
Laboratory Yale University, New Haven, Connecticut 06520-8120, USA  M. I.
Krivoruchenko  National Research Centre “Kurchatov Institute”, Ploschad’
Akademika Kurchatova 1, 123182 Moscow, Russia  Institute for Theoretical and
Experimental Physics, NRC “Kurchatov Institute”, B. Cheremushkinskaya 25,
117218 Moscow, Russia  J. Suhonen  Department of Physics, University of
Jyväskylä, P.O. Box 35, FI-40014 Jyväskylä, Finland  K. S. Tyrin  National
Research Centre “Kurchatov Institute”, Ploschad’ Akademika Kurchatova 1,
123182 Moscow, Russia  V. Virtanen  Department of Physics, University of
Jyväskylä, P.O. Box 35, FI-40014 Jyväskylä, Finland

###### Abstract

A precise determination of the atomic mass of 75As has been performed
utilizing the double Penning trap mass spectrometer, JYFLTRAP. The mass excess
is measured to be -73035.519(42) keV/c2, which is a factor of 21 more precise
and 1.3(9) keV/c2 lower than the adopted value in the newest Atomic Mass
Evaluation (AME2020). This value has been used to determine the ground-state-
to-ground-state electron-capture decay $Q$ value of 75Se and $\beta^{-}$ decay
$Q$ value of 75Ge, which are derived to be 866.041(81) keV and 1178.561(65)
keV, respectively. Using the nuclear energy-level data of 860.00(40) keV,
865.40(50) keV (final states of electron capture) and 1172.00(60) keV (final
state of $\beta^{-}$ decay) for the excited states of 75As∗, we have
determined the ground-state-to-excited-state $Q$ values for two transitions of
75Se $\rightarrow$ 75As∗ and one transition of 75Ge $\rightarrow$ 75As∗. The
ground-state-to-excited-state $Q$ values are determined to be 6.04(41) keV,
0.64(51) keV and 6.56(60) keV, respectively, thus confirming that the three
low $Q$-value transitions are all energetically valid and one of them is a
possible candidate channel for antineutrino mass determination. Furthermore,
the ground-state-to-excited-state $Q$ value of transition 75Se $\rightarrow$
75As∗ (865.40(50) keV) is revealed to be ultra-low (< 1 keV) and the first-
ever confirmed EC transition possessing an ultra-low $Q$ value from direct
measurements.

##  I Introduction

The discovery of flavor oscillations of atmospheric, solar, and reactor
neutrinos confirms that neutrinos have mass Fukuda et al. (1998); SNO
Collaboration (2002); Gerbino and Lattanzi (2018). The standard model (SM)
contradictorily predicts that the neutrino mass is zero. How neutrinos acquire
their small masses is consequently a matter of great theoretical interest and
may be an evidence of new physics beyond the SM. Oscillation data provide only
the differences of the squared neutrino masses but not their absolute values.
In order to solve the open problem of the absolute values of the neutrino
masses, laboratory measurements are necessary. Among these experiments, the
ones dedicated to measurements of the neutrinoless double beta decay aim to
determine if neutrinos are of Dirac or Majorana nature and to measure the
effective Majorana neutrino mass Suhonen and Civitarese (1998); Avignone et
al. (2008); Ejiri et al. (2019); Blaum et al. (2020). Unfortunately, this
method is model-dependent and highly relies on the calculated values of the
involved transition matrix elements. A direct and model-independent method for
determining (anti)neutrino mass is based on single beta decay or electron
capture (EC) Gastaldo et al. (2017); Aker et al. (2021). From the $\beta^{-}$
decay of tritium, one can determine the electron-antineutrino mass by zooming
in the slight distortion of the shape of the electron spectrum near the
endpoint, determined by the decay $Q$ value minus the antineutrino mass. In a
similar vein, in the EC of 163Ho Gastaldo et al. (2017), one can determine the
electron-neutrino mass from the endpoint of the measured de-excitation
spectrum, which is shifted below $Q$ by the non-zero neutrino mass. In both
beta decay and EC, the sensitivity to (anti)neutrino mass is increased by a
small $Q$ value. Therefore, as small as possible $Q$ value is desired in these
(anti)neutrino-mass determination experiments. Due to this reason, tritium
with low $\beta^{-}$-decay $Q$ value of 18.59201(7) keV Myers et al. (2015) in
the KATRIN (KArlsruhe TRitium Neutrino) experiment Drexlin et al. (2013); Aker
et al. (2019), and 163Ho with low EC $Q$ value of 2.833(30)sta(15)sys keV
Eliseev et al. (2015) in the ECHo (Electron Capture in 163Ho) Gastaldo et al.
(2014, 2017) HOLMES Alpert et al. (2015); Faverzani et al. (2016) experiments
are used. An upper limit of 0.8 eV/c2 (90% Confidence Level (C.L.)) for the
electron-antineutrino mass is achieved in the tritium decay by KATRIN Aker et
al. (2021), and for electron-neutrino mass an upper limit of 150 eV/c2 (95%
C.L.) was obtained exploiting the EC of 163Ho in the ECHo experiment Velte et
al. (2019).

Other isotopes with low $Q$-value decay transition, especially ultra-low (< 1
keV), are of significant interest for future neutrino-mass absolute-scale
determination experiments Mustonen and Suhonen (2010, 2011); Suhonen (2014);
De Roubin et al. (2020); Ge et al. (2021a, b). The $Q$ values determined with
indirect methods may include unknown systematic uncertainties. This can make
these $Q$ values deviate by more than 10 keV Ge et al. (2021a); Fink et al.
(2012); Nesterenko et al. (2019) from those directly determined through high-
precision Penning-trap mass spectrometry (PTMS). Direct measurements of the
masses or $Q$ values through PTMS, which is to date the only direct method to
have achieved $\sim$ 100 eV precision or better, are indispensable in the
searches for ultra-low $Q$-value transitions. The first ultra-low $Q$-value
decay, 115In (9/2+, ground state) $\rightarrow$ 115Sn (9/2+, excited state),
was discovered by Cattadori _et al._ Cattadori et al. (2005) and the ultra-low
$Q$ value was confirmed by the JYFLTRAP Penning trap Wieslander et al. (2009)
and the FSU Penning trap Mount et al. (2009). The second ultra-low $Q$-value
case of 135Cs (7/2+, ground state) decaying to 135Ba (11/2-, second excited
state) was discovered at JYFLTRAP De Roubin et al. (2020); Kankainen et al.
(2020) recently. These are the only two cases that are confirmed to possess
ultra-low $Q$ values from direct measurements, and they both belong to the
category of $\beta^{-}$ decay.

In this paper, we report on the high-precision mass determination of 75As from
cyclotron frequency measurements with high-precision PTMS. The ground-state-
to-ground-state (gs-to-gs) EC $Q$ value of the transition 75Se $\to\,^{75}$As
and $\beta^{-}$-decay $Q$ value of the transition 75Ge $\to\,^{75}$As are
determined. The high-precision gs-to-gs $Q$ values from this work, combined
with the nuclear energy-level data for three excited states of 75As, are used
to determine their ground-state-to-excited-state (gs-to-es) $Q$ values. Three
low $Q$-value transitions are confirmed to be energetically possible and the
first to date ultra-low $Q$-value EC transition has been discovered in this
experiment.

Figure 1: (Color online). (a) Schematic view of ion production and mass
measurements with PI-ICR technique. The stable 75As+ and 76Ge+ ions were
simultaneously produced with an offline glow-discharge ion source, where the
ions were produced and transported with an He gas flow and electric fields.
Ions having mass number of 75 or 76 were selected with a dipole magnet and
transported to the JYFLTRAP PTMS for final ion species selection in the
preparation trap by means of a buffer-gas cooling and cyclotron frequency
determination using the phase-imaging technique at the measurement trap. A
position-sensitive MCP detector was used to register the images of the phases.
(b) An illustration of the radial-motion ("magnetron", "cyclotron", and
"center") projection of the 75As+ ions onto the position-sensitive MCP
detector. The magnetron phase spot is displayed on the left side and the
cyclotron phase spot on the right. The angle difference between the two spots
relative to the center spot is utilized to deduce the cyclotron frequency of
the measured ion. The number of ions in each pixel is illustrated by color
bars.

##  II Experimental method

The atomic mass of 75As was measured with the JYFLTRAP double PTMS Eronen and
Hardy (2012) at the Ion Guide Isotope Separator On-Line facility (IGISOL)
Moore et al. (2013) of the University of Jyväskylä. The stable 75As+ ions were
produced with an offline glow-discharge ion source. To measure the mass of
75As precisely, 76Ge+ ions with well-known mass value were used as reference
ion, co-produced from the same ion source. As shown in Fig. 1 (a), the gas
cell of the glow-discharge ion source contains two sharp electrodes, of which
one is made of the mixture of natural abundance arsenic and germanium,
allowing simultaneous production of ions of these elements. The produced ions
were stopped in the gas cell and extracted out with gas flow and electric
fields. They were subsequently accelerated with a voltage of $\sim$ 30 kV and
transported further to a 90∘ electrostatic bender to the main horizontal
beamline before the dipole magnet. With a mass resolving power of
$M/\Delta{M}$ $\sim$ 500, the dipole magnet is used to selectively transport
the 75As+ or 76Ge+ ions to a radiofrequency quadrupole cooler-buncher (RFQ)
Nieminen et al. (2001). After cooling and bunching the ions with the RFQ, they
were transported to the JYFLTRAP double PTMS as schematically shown in Fig. 1
(a).

After the dipole magnet, in principle pure ion beams of 75As+ and 76Ge+ with
unambiguous identification from the offline ion source are separated and
selected. To ensure that any possible isobaric or molecular contaminations are
removed, the first (preparation) trap is used via the sideband buffer gas
cooling technique Savard et al. (1991) with a typical resolving power of
around $10^{5}$. Finally, the purified sample of 75As+ or 76Ge+ ions are
injected into the second (measurement) trap. The ion’s cyclotron frequency
$\nu_{c}=\frac{1}{2\pi}\frac{qB}{m}$, where $B$ is the magnetic field
strength, $q$ is the charge state and $m$ the mass of the ion, is determined
in the second Penning trap.

The phase-imaging ion-cyclotron-resonance (PI-ICR) technique Nesterenko et al.
(2018) at JYFLTRAP is used to measure the cyclotron frequency in this work.
This technique depends on projecting the ion motion in the Penning trap onto a
position-sensitive multichannel-plate ion detector (schematic shown in Fig. 1
(a)) and provides around 40 times better resolving power and is 25 times
faster than the conventional TOF-ICR method Nesterenko et al. (2018); Eliseev
et al. (2014, 2013). Measurement scheme 2 described in Eliseev et al. (2014)
was applied to measure the cyclotron frequency $\nu_{c}$ of the corresponding
nuclide.

The PI-ICR measurement is initiated by exciting a cooled ion bunch with a
dipolar radio-frequency electric field at the trap-modified cyclotron
frequency $\nu_{+}$. This is a short excitation lasting approximately 1 ms.
Afterwards, the trap-modified cyclotron motion is converted to magnetron
motion using a quadrupole excitation with the cyclotron frequency $\nu_{c}$
and a duration of approximately 2 ms. Two different delays for the application
of this excitation are alternatively used. In one, the conversion pulse is
applied right after the dipolar excitation and in the other after a longer
time. The time difference of the excitation pulses is called the phase
accumulation time, denoted $t$. During this time, in a cycle with a very short
conversion pulse delay, the ions perform magnetron motion and in the one with
a long delay modified cyclotron motion. After, the ions are extracted from the
trap (the time difference between the initial dipolar excitation and the
extraction is always constant irrespective of the quadrupole conversion delay)
and their positions are recorded with a position-sensitive MCP detector PS-
(2021). The spot obtained with the small conversion time delay is called the
magnetron spot and the one with the long delay is the cyclotron spot. The
center spot of the ions of interest is collected by extracting the ions
directly to project them onto the MCP detector without applying any
excitation. The angle between the spots of the cyclotron and magnetron phases
with respect to the center spot is $\alpha_{c}$ = $\alpha_{+}$-$\alpha_{-}$,
where $\alpha_{+}$ and $\alpha_{-}$ are the polar angles of the magnetron
motion phase and cyclotron motion phases, respectively. The cyclotron
frequency $\nu_{c}$ is deduced from:

$\nu_{c}=\frac{\alpha_{c}+2\pi n_{c}}{2\pi{t}},$ (1)

where $n_{c}$ are the number of complete revolutions of the measured ion
during the phase accumulation time $t$. The accumulation time $t$ for 75As+
and 76Ge+ was 400 ms, which ensures the spot of interest was resolved from any
possible isobaric, and molecular contamination. The positions of the
magnetron-motion and cyclotron-motion phase spots were chosen such that the
angle $\alpha_{c}$ did not exceed a few degrees. By this way, the shift in the
frequency ratio measurements, due to the conversion of the cyclotron motion to
magnetron motion and the possible distortion of the ion-motion projection, can
be reduced to a level well below 10-10 Eliseev et al. (2014) . In addition,
the start of the initial dipolar excitation with frequency $\nu_{+}$ was
repeatedly scanned over one magnetron period (6 points) and the extraction was
scanned over one cyclotron period (6 points) to average out any residual
magnetron and cyclotron motion that could shift the different spots. One
collected data of phase imaging of on the MCP detector on Fig. 1 (b).

The atomic mass of the ion of interest was derived from the measured cyclotron
frequency ratio:

$M_{ioi}=R(M_{ref}-m_{e})+m_{e}+(R\cdot B_{ref}-B_{ioi})/c^{2},$ (2)

where $M_{ioi}$ and $M_{ref}$ are the masses of the ion of interest (75As+)
and reference (76Ge+) atoms, respectively.
$R=\frac{\nu_{c,{ref}}}{\nu_{c,ioi}}$ is their cyclotron frequency ratio for
singly charged ions. $m_{e}$ is the mass of electron and $c$ is the speed of
light in vacuum. $B_{ref}$ and $B_{ioi}$ are the electron binding energy of
76Ge+ and 75As+, which are 7.899440(10) eV and 9.78855(25) eV from Kramida et
al. (2020), respectively. The main contributions of the final mass uncertainty
of 75As are from the statistical uncertainty of the measurements of $R$ and
the reference mass uncertainty (0.018 keV from Wang et al. (2021)).

Table 1: Final results from the analysis of mean cyclotron frequency ratio between the ion of interest and reference nuclei. The first column gives lists of ion of interest and the reference (IOI-Ref) nuclei. The measured frequency ratio $\overline{R}$ is in the second column. The mass excess (ME) in keV/c2 of ion of interest determined in this work in comparison to the AME2020 values Wang et al. (2021) are listed in the third column and splited in two rows. The last column demonstrates the difference of the ME value from this work and that adopted from AME2020. IOI-Ref | $\overline{R}$ |  ME (this work) ME (AME2020) | Diff.
---|---|---|---
75As-76Ge | 0.986 830 896 52(53) |  -73035.521(42) -73034.20(90) | -1.32(90)
77Se-76Se | 1.013 181 218 27(79) |  -74599.443(58) -74599.490(60) | 0.047(84)
94Mo-95Mo | 0.989 455 235 86(72) |  -88414.101(136) -88414.06(14) | -0.041(195)
Table 2: Transitions from the ground states of parent nuclei 75Ge and 75Se to the excited states of the daughter 75As. The first and second columns illustrate the experimental spin-parities of the initial ground states and the half-lives of the parent nuclei. The third column gives the measured spin-parities of the excited final states for the transitions. The fourth column gives the decay type. The fifth column gives the gs-to-gs decay $Q$ values ($Q_{\beta^{-}}^{0}$ for $\beta^{-}$-decay and $Q_{EC}^{0}$ for EC) from literature (AME2020) Wang et al. (2021) and the sixth column from this work. The seventh and eighth columns give the gs-to-es decay $Q$ values ($Q^{*}$) from literature and this work, respectively. The last column gives the excitation energies E∗ from NND (2021). All the energy values are in units of keV. Spin-parity assignments enclosed in braces indicate that these are uncertain, which results in an uncertainty in the decay type, indicated by a {?}. FNU denotes forbidden non-unique.  Initial state | Half-life | Final state | Decay type |  $Q^{0}$ (AME2020) |  $Q^{0}$ (This work) |  $Q^{*}$ (AME2020) |  $Q^{*}$ (This work) | $E^{*}$
---|---|---|---|---|---|---|---|---
75Se (5/2+) | 119.78(5) d |  75As ({3/2-, 5/2-}) | EC: 1st FNU | 864.72(90) | 866.041(81) | -0.7(10) | 0.64(51) | 865.40(50)
75Se (5/2+) | 119.78(5) d |  75As (1/2+) | EC: 2nd FNU | 864.72(90) | 866.041(81) | 4.7(10) | 6.04(41) | 860.00(40)
75Ge (1/2-) | 82.78(4) m |  75As ({1/2-:7/2-}) |  $\beta^{-}$: Allowed{?} | 1177.24(90) | 1178.561(65) | 5.2(11) | 6.56(60) | 1172.00(60)
Figure 2: (Color online). The measured cyclotron frequency ratios $R$ (
$\nu_{c}$(76Ge+)/$\nu_{c}$(75As+) (left axis) and mass excess (right axis) in
this work compared to values adopted from AME2020. The red dots with
uncertainties are the measured PI-ICR single ratios in five time slots, which
are separated with vertical brown dashed lines. The weighted average value in
this work $\overline{R}$ = 0.986 830 896 52(53) represents by the solid red
line and its 1$\sigma$ uncertainty band is shaded in red. The dashed blue line
indicates the value adopted from AME2020 with its 1$\sigma$ uncertainty area
shaded in blue.

##  III Results and discussion

###  III.1 $Q$ value determination

Five data sets for PI-ICR measurements were performed in different time slots.
Every data set was accumulated by switching between the ion species 75As+ and
76Ge+ every four cycles. Each cycle took less than 2 minutes to complete a
full scanning measurements of the magnetron phase, cyclotron phase and center
spot in sequence. In the analysis, typically 8 to 16 cycles were summed before
determining the position of each spot by fitting with the maximum likelihood
method. The phase angles were calculated accordingly to deduce the cyclotron
frequencies of each ion species. Closest measured cyclotron frequencies
$\nu_{c}$ of the reference was linearly interpolated to the time for
measurement of the ion of interest to deduce the cyclotron frequency ratio
$R$. Only events less than five ions/bunch were considered in the analysis in
order to reduce the possible cyclotron frequency shifts due to ion-ion
interaction Kellerbauer et al. (2003); Roux et al. (2013). A countrate-class
analysis Roux et al. (2013) was performed and the count rate related frequency
shifts were not observed in the analysis. A temporal fluctuation of the
magnetic field $\delta_{B}(\nu_{ref})/\nu_{ref}=\Delta t\times 2.01(25)\times
10^{-12}$/min Nesterenko et al. (2021), where $\Delta t$ is the time interval
between two consecutive reference measurements, is considered in the analysis.
Furthermore, a mass-dependent uncertainty of $\delta_{m}(r)/r=\Delta m\times
2.35(81)\times 10^{-10}/u$ Nesterenko et al. (2021), was taken into account as
there is one mass unit (1 $u$) difference of the two measured ion species. To
check whether some other systematic error should be added due to this
technique, a crosscheck Nesterenko et al. (2020) has been carried out using
well-known mass pairs 77Se-76Se and 94Mo-95Mo Huang et al. (2021); Wang et al.
(2021), which both have one mass unit difference. The measured results are
shown in Table 1. The measured mass excesses of both cases are consistent with
AME2020 values within 1$\sigma$ uncertainty. Therefore, in our final results,
we did not include any further systematic error.

The weighted mean ratio $\overline{R}$ of all single ratios was calculated
along with the inner and outer errors to deduce the Birge rato Birge (1932).
The maximum of the inner and outer errors was taken as the weight to calculate
$\overline{R}$. Results of the analysis including all PI-ICR data sets in
comparison to literature values are demonstrated in Fig. 2. The final
frequency ratio $\overline{R}$ as well as the derived mass-excess value are
respectively 0.986 830 896 52(53) and -73035.521(42) keV/c2.

Figure 3: (Color online). Partial decay scheme of 75Se and 75Ge to 75As using
$Q$ values from AME2020 Huang et al. (2021); Wang et al. (2021) in comparison
to this work. The energies of the excited states and spin-parites indicated in
the figure are from NND (2021). The solid lines demonstrate the levels of the
excited states with the $Q$ values from AME2020 and dashed lines from the
refined $Q$ values in this work. The shaded or hatched areas (in blue for
$\beta^{-}$ decay or in red for EC) illustrate the corresponding 1 $\sigma$
uncertainties of the $Q$ values. See Table. 2 for more details on the $Q$
values.

The mass-excess value of 75As in AME2020 is evaluated from two indirect
reaction measurements, 75As(p,n)75Se and 78Se(p,a)75As, which have the
influences of 85.3% and 14.7% on the primary nuclide, respectively. The
refined mass-excess value of 75As from our direct measurements is 21 times
more precise. The value is 1.32(90) keV/c2 lower than that adopted in AME2020
and indicates that 75As is more bound. The high-precision mass-excess value of
75As was used to deduce the gs-to-gs $Q$ values ($Q^{0}$ deduced from the mass
difference of the decay-parent and decay-daughter nuclei) of the 75Se
$\rightarrow$ 75As and 75Ge $\rightarrow$ 75As transitions. Combined with the
excitation energies $E^{*}$ from NND (2021) for the excited states of 75As,
the gs-to-es $Q$ values ($Q^{*}$ = $Q^{0}$\- $E^{*}$) were calculated. The
comparison of the $Q$ values from this work to the corresponding $Q$ values
from literature is performed in Table. 2. The presently measured gs-to-es $Q$
values of the transitions tabulated in Table 2 are determined with
uncertainties less than 100 eV, while the uncertainties of the gs-to-es $Q$
values are much larger than 100 eV, the main contributions coming from the
large uncertainties of the excitation energies of the final states. Our
results confirm that three low $Q$-value transitions from the ground-states of
75Se and 75Ge to the corresponding excited states of 75Se, as tabulated in
Table. 2, are all energetically allowed. Furthermore, our measurements confirm
that the $Q^{*}$ value corresponding to the EC transition 75Se $\rightarrow$
75As* (865.40(50) keV) is ultra-low and positive at a level of $\sim$
1.3$\sigma$. This transition and the transition to the $1/2^{+}$ state at
860.00(40) keV are non-unique EC transitions and thus their characteristics
depend on nuclear-structure details through the nuclear matrix elements Ejiri
et al. (2019); Behrens and Bühring (1982). This excludes their use as reliable
sources of information about the electron-neutrino mass. On the other hand,
the $\beta^{-}$ decay to the excited state at 1172.00(60) keV might be of
interest for the electron-antineutrino mass measurements if the spin of the
state is either $1/2^{-}$ or $3/2^{-}$, corresponding to an allowed
$\beta^{-}$ transition. Hence, accurate determination of the excitation energy
and spin of the state is highly important and called for.

###  III.2 Beta-decay study via nuclear shell-model

The low-$Q$ EC transitions of interest go to a $1/2^{+}$ state at 860.00(40)
keV and a negative-parity state at 865.40(50) keV, experimentally either a
$3/2^{-}$ or a $5/2^{-}$ state. The $\beta^{-}$ transition goes to a negative-
parity state at 1172.00(60) keV, the experimental spin assignment being
between a $1/2^{-}$ and a $7/2^{-}$ state. One may try to shed light on these
spin-assignment ambiguities by performing nuclear shell-model (NSM)
calculations. States of the isotope 75As were computed using the software
NuShellX Brown and Rae (2014) in a single-particle model space consisting of
the orbitals $1f_{5/2}$, $2p_{3/2}$, $2p_{1/2}$, and $1g_{9/2}$ for both
neutrons and protons. Effective single-particle energies have been fitted to
be used within the region of interest $A$ = 63 - 96 in Honma et al. (2009). In
the present study both jun45pn and jj44bpn interactions were applied. The
interaction jun45pn was originally applied to a neighbouring isotope (76Ge)
Mukhopadhyay et al. (2017) while the jj44bpn was fitted to work within the
presently studied region Honma et al. (2009).

The sequence of the spin-parities of the levels in 75As are well reproduced by
both interactions up to the experimental negative-parity state at 865.40(50)
keV. The level energies predicted by the NSM agree with the experimental ones
within a range of 50 to 200 keV. The two interactions suggest consistently the
spin assignment $5/2^{-}$ for the experimental 865.40(50) keV state. The
corresponding EC transition is thus predicted to be a first-forbidden non-
unique $\Delta J=0$ transition and thus its properties depend sensitively on
nuclear-structure details.

Concerning the experimental state at $1172.00$ keV, jun45pn predicts a
$3/2^{-}$ state at $910.0$ keV and a $5/2^{-}$ state at $1296.0$ keV. This
suggests that the most likely spin-assignments for the $1172.0$ keV state are
either $3/2^{-}$ or $5/2^{-}$. The jj44bpn interaction predicts a $7/2^{-}$
state at $1129.0$ keV and a $3/2^{-}$ state at $1269.0$ keV. Considering the
predictions of both interactions together leaves the $3/2^{-}$ as the most
likely spin-parity for the $1172.00$ keV state. This, in turn, makes the
$\beta^{-}$ transition from the $1/2^{-}$ state of 75Ge to this state an
allowed one, with an available $Q_{\beta^{-}}$ of 6.56(60) keV. This would
make the transition interesting from the point of view of electron-
antineutrino mass measurements. Eventually, a re-measurement of the spin-
parities of the states in question is called for.

##  IV Conclusion

The atomic mass of 75As is precisely measured directly with the PI-ICR
technique at the JYFLTRAP double PTMS. The precision of the 75As mass has been
improved by a factor of 21 and was found to be 1.32(90) keV/c2 more bound than
the evaluated value in the AME2020 mass evaluation. This allows a high-
precision determination of the gs-to-gs $Q$ values of the decays of 75Ge and
75Se to 75As with an uncertainty of better than 100 eV. Exploiting the energy-
level data of the excites states of 75As, the gs-to-es $Q$ values of three low
$Q$-value transitions have been determined to a sub-keV precision. The derived
positive gs-to-es $Q$ values have all been confirmed to be associated with
energetically-allowed low $Q$-value EC and $\beta^{-}$ transitions. Among
them, a first ultra-low (< 1 keV) $Q$-value EC transition has been found. One
of the transitions, namely the low-$Q$ $\beta^{-}$ transition, has the
potential to be used in determination of the value of the electron-
antineutrino mass. For this to be realized, a more precise determination of
the excitation energy and spin of the corresponding final state is of
paramount importance.

###### Acknowledgements.

We acknowledge the support by the Academy of Finland under the Finnish Centre
of Excellence Programme 2012-2017 (Nuclear and Accelerator Based Physics
Research at JYFL) and projects No. 306980, 312544, 275389, 284516, 295207,
314733, 318043, 327629 and 320062. The support by the EU Horizon 2020 research
and innovation program under grant No. 771036 (ERC CoG MAIDEN) is
acknowledged.

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