2024-E-8
Bank of Japan Review
Changes in Risk Perceptions on Yen Interest Rates and Exchange Rates
Observed in Options Markets:
Developments in Implied Probability Distributions amid Rate Hikes
in the United States and Europe from 2022 to 2023
Financial Markets Department
YONEYAMA Akihito*, TSUCHIYA Akitaka**, FUKUMA Noritaka

August 2024
In this paper, we extract the implied probability distributions from daily option prices for the future levelsof yen swap rates and the U.S. dollar/yen exchange rate, which came under upward pressure amid rate
hikes by central banks in the United States and Europe from 2022 to 2023, using data through the end of2023. The extracted implied probability distributions allow us to understand the overall picture of market
participants' risk perceptions, which cannot be captured by other simpler indicators such as risk reversal.
The results suggest that caution about a higher interest rate increased significantly in the yen interest
rate swaps market toward the middle of 2022, as overseas interest rates rose. Such caution further
heightened after the Bank of Japan decided to expand the range of 10-year Japanese government bond
(JGB) yield fluctuations from the target level at the Monetary Policy Meeting (MPM) held in December
2022, but subsequently eased. In the U.S. dollar/yen market, caution about future yen depreciation
heightened toward fall 2022 as the yen weakened. However, when the yen weakened against U.S.
dollar again to around the same level in fall 2023, market participants did not become as cautious as infall 2022 about the future yen depreciation.

Introduction

In this paper, we extract implied probability
distributions of the future levels of yen swap rates andthe U.S. dollar/yen exchange rate from daily option
prices using data through the end of 2023, andsummarize the characteristics of changes in market
participants' risk perceptions.

Long-term interest rates in the United States andEurope significantly increased from 2022 as central
banks, especially those in the United States andEurope, rapidly raised interest rates in response toelevated global inflation (upper panel of Chart 1).
With overseas interest rate rising, Japanese long-term
interest rates, particularly in the swaps market, cameunder upward pressure, although fluctuations in those
rates were contained under the Bank's conduct of yield
curve control (YCC). In the foreign exchange market,
the yen depreciated against the U.S. dollar, as the
U.S.-Japan yield differential widened amid a
difference in the direction of monetary policy between
the two economies (lower panel of Chart 1).
As such, Japanese interest rates and the U.S.
dollar/yen exchange rate were brought under upward
pressure from 2022. Under these circumstances,market participants' risk perceptions on the futurelevels of Japanese interest rates and the U.S.
dollar/yen exchange rate likely changed, partly
reflecting their views on the future course of monetary
policy in the United States and Europe, as well as their
outlooks on Japan's economic and price developments
and speculation about the Bank's future conduct ofmonetary policy.

Overview of Analysis Methodology
Options are a form of financial derivatives instruments(derivatives) that give the holder the right to buy (call)or sell (put) a specific asset (underlying asset) at a
specific price (strike price) at some point in the future(expiration date). Various types of underlying assets
are involved in option trading. For example, in foreignexchange markets, options are based on a currency
pair (e.g., the U.S. dollar/yen exchange rate). In
interest rate markets, options on an interest rate swap
(a transaction where two parties exchange different
types of interest payments, such as fixed and floating
rates, called a swaption) are traded.1
The price of an option is determined based onmarket participants' expectations about the future
price of the underlying asset. As such, information
about market participants' expectations about the
future performance of the underlying asset can bederived from option prices.
1

Bank of Japan August 2024

exchange rate from daily option prices (see BOX 1 foran overview of the estimation procedure). 3 The
distributions thus extracted are called implied
probability distributions and have several advantages
over other simpler indicators such as IV and RR. For
instance, they can capture market participants'
perceptions of tail risk (see BOX 2).

[Chart 1] Long-Term Interest Rates and U.S.
Dollar/Yen Exchange Rate<Long-Term Interest Rates (10-year)>
6(%)
Japan (JGB yield)
Japan (swap rate)
United States
Germany

5
4

Yen Interest Rate Swaps Market

3
2

We extract the implied probability distributions of
2-year and 10-year yen swap rates from transaction
prices of swaptions expiring three months after the
trade date.4,5 Please note that the implied probability
distributions calculated in this paper do notnecessarily correspond to the probability distributionof cash JGBs yields, which are another kind of yen
interest rates.6 This is because yen swap rates andcash JGB yields have diverged rate levels and move in
different manners (upper panel of Chart 1), and also
there is a difference in the composition of trading
entities between the two markets; for instance, the yenswap rate market has a relatively high proportion ofoverseas participants.7
Looking at the extracted implied probability
distributions from a long-term perspective, thedispersion of the distributions (i.e., the dispersion ofmarket participants' views) increased, particularly forthe 10-year rate, after the Bank introduced
Quantitative and Qualitative Monetary Easing (QQE)in April 2013 (Chart 2). The distributions became
further dispersed with a longer tail in the direction oflower interest rates following the Bank's decision to
introduce negative interest rates in January 2016. Thiswas particularly notable for the 2-year rate, which ishighly correlated with the outlook for policy rates,
indicating that market participants' forecasts became
more dispersed and skewed toward a cautious stancetoward a decline in interest rates. After the
introduction of YCC in September 2016, however, thedistributions became less dispersed for both the 2-year
and 10-year rates. Thereafter, while the dispersion
temporarily increased around March 2020 in responseto the instability in financial markets at the onset of
the COVID-19 pandemic, the distributions generally
stayed within a limited range compared to the
pre-YCC period until the beginning of 2022.

1
0
-1

Jan
2022

Apr

Jul

Oct

Jan
23

Apr

Jul

Oct

<U.S. Dollar/Yen Exchange Rate>
160(%pts)

(yen)

6
5

150

4

140

3
130

2

120

1

110

0
Jan
2022

May

Sep

Jan
23

May

Sep

U.S. dollar/yen exchange rate
U.S.-Japan 2-year yield differential (rhs)
U.S.-Japan 10-year yield differential (rhs)Note: "Japan (swap rate)" is the OIS rate based on theuncollateralized overnight call rate (TONA). Latest data as ofDecember 22, 2023.
Sources: Bloomberg; LSEG Eikon.

For example, implied volatility (IV), an estimateof the volatility (standard deviation) of the underlying
asset price at a future time extracted from an option
price, is widely used in financial markets. Also, risk
reversal (RR), the difference between the IV of a call
option and that of a put option, is used in foreignexchange markets particularly, as an indicator of thebias in market participants' risk perceptions.2
However, while IV and RR are relatively easy to
calculate, they represent only two of the numerous
parameters (moments) that characterize the probability
distribution of future underlying asset prices andtherefore cannot capture the overall risk perception ofmarket participants, especially with regard to tail risk.
In this paper, we extract the entire probability
distributions of the market's expectation about futurelevels of yen swap rates and the U.S. dollar/yen
2

Bank of Japan August 2024

[Chart 2] Implied Probability Distributions<2-Year Rate>
2.0[Chart 3] Dispersion and Skewness of
Probability Distributions<Dispersion><Skewness>(%)
1st-99th percentile
5th-95th percentile
25th-75th percentile
Median

1.5

1.0

1.8

(%)

(%)
1.0

1.6
1.4

0.8

1.2

0.5

0.6

1.0

0.0

-0.5

0.4

0.6

0.2

15

16

17

18

19

20

21

22

23

0.0
Jan Jul
2022(%)
1st-99th percentile
5th-95th percentile
25th-75th percentile
Median

2.5

2.0

-0.2
Jan
23

Jul

Jan Jul
2022
24

Jan
23

Jul

10-year rate

10-year rate

2-year rate

2-year rate

Note: "Dispersion" indicates the range between the 95th percentile(tail in the direction of higher interest rates) and the 5th
percentile (tail in the direction of lower interest rates).
"Skewness" is calculated by subtracting the range between
the 5th percentile and the median (caution about lower interest
rates) from the range between the 95th percentile and the
median (caution about higher interest rates). Latest data as ofDecember 22, 2023.

1.5
1.0
0.5

0.0

Source: Authors' estimates based on Bloomberg, TraditionData, andLSEG Eikon

-0.5
-1.0
CY13

0.0

0.2

14

<10-Year Rate>
3.0

0.8
0.4

-1.0
CY 13

↑ Cautious about
higher interest rates

14

15

16

17

18

19

20

21

22

yield fluctuations from the target level: from between
around plus and minus 0.25 percentage points to
between around plus and minus 0.5 percentage points,
caution about higher interest rates further heightenedfor both the 2-year and 10-year rates. (5) Subsequently,
caution about higher interest rates eased for both the
2-year and 10-year rates, as the Bank decided tomaintain its monetary policy and enhance theFunds-Supplying Operations against Pooled Collateralat the MPM held in January 2023 and overseasinterest rates declined in response to the failure of anddeterioration in the business conditions of somefinancial institutions in the United States and Europe
in March 2023. (6) From September 2023 onward,
although there were some occasions when thedispersion of the distributions, particularly of the
2-year rate, increased and caution about higher
interest rates heightened somewhat with speculation
over the Bank's monetary policy, in none of them did
the dispersion reach the levels seen after the
December 2022 MPM.

23

Note: Estimated from swaptions whose underlying asset is swap rates
based on 6-month yen LIBOR for figures through May 26, 2022
(synthetic yen LIBOR for figures from the beginning of 2022
through May 26, 2022) and those based on uncollateralized
overnight call rate (TONA) for figures during the subsequent
period. Please note that there is a discontinuity between thedata before and after May 26, 2022 (the same applies
hereafter). Latest data as of December 22, 2023.
Source: Authors' estimates based on Bloomberg, TraditionData, andLSEG Eikon.

From 2022 onward, the probability distributions ofboth the 2-year and 10-year rates became fat-tailed in
the direction of higher interest rates, and the
distributions became more dispersed and skewed
toward a cautious stance toward higher interest rates(Chart 2 and 3).8 Specifically, (1) the distributions
began to disperse after March 2022, when the Federal
Reserve started raising interest rates, and (2) the
distributions became further dispersed andsignificantly skewed in the direction of a cautious
stance toward higher interest rates (i.e., market
participants became cautious about an increase ininterest rates) around June 2022, when overseasinterest rates were rising further (upper panel of Chart 1).
(3) Thereafter, caution about higher interest rates,
particularly for the 10-year rate, kept increasing
toward December 2022, except for isolated occasions,as U.S. and European interest rates continued to rise.
(4) After the MPM held in December 2022, where the
Bank decided to expand the range of 10-year JGB
3

Bank of Japan August 2024

24

[Chart 4] Implied Probability Distributions of Yen Swap Rates (2-Year)<December 2022 MPM>
<July 2023 MPM>
<December 2023 MPM>
(Probability density)
First day of
meetingFirst day of
meeting
Second day of
meetingFirst day of
meeting
Second day of
meeting

Second day of
meeting-0.3 0 0.3 0.6 0.9 1.2
3-month-ahead 2-year rate (%)(Probability density)(Probability density)-0.3 0 0.3 0.6 0.9 1.2
3-month-ahead 2-year rate (%)-0.3 0 0.3 0.6 0.9 1.2
3-month-ahead 2-year rate (%)

Note: The horizontal axis indicates the level of the yen swap rate (2-year) on the expiration date. "Second day of the meeting" indicates the implied
probability distribution after the policy decision (the same applies hereafter).
Source: Authors' estimates based on Bloomberg and TraditionData.

[Chart 5] Implied Probability Distributions of Yen Swap Rates (10-Year)<December 2022 MPM>
<July 2023 MPM>
<December 2023 MPM>
(Probability density)
First day of
meeting
Second day of
meeting

0 0.5 1 1.5 2 2.5
3-month-ahead 10-year rate (%)(Probability density)(Probability density)

First day of
meeting
Second day of
meetingFirst day of
meeting
Second day of
meeting

0 0.5 1 1.5 2 2.5
3-month-ahead 10-year rate (%)

0 0.5 1 1.5 2 2.5
3-month-ahead 10-year rate (%)

Note: The horizontal axis indicates the level of the yen swap rate (10-year) on the expiration date.
Source: Authors' estimates based on Bloomberg and TraditionData.

Meanwhile, the Bank decided at the MPM held in
July 2023 to conduct YCC with greater flexibility
regarding the 10-year JGB yield fluctuation range ofaround ±0.5 percentage points as a reference.
Furthermore, at the MPM held in October 2023, it
decided to further increase the flexibility in the
conduct of YCC regarding the upper bound for
10-year JGB yields of 1.0 percent as a reference. In
the following, we compare the implied probability
distributions of the yen swap rates before and after the
MPMs in December 2022 and July and October 2023.
With regard to the 2-year rate, the probability
distribution became fat-tailed in the direction ofhigher interest rates after the December 2022 MPM(Chart 4). Given the high correlation of the 2-year ratewith the outlook for the policy rate, there may havebeen heightened caution about possible policychanges, including a policy rate hike, ahead of thenext MPM in January 2023. In contrast, the shape ofthe distribution remained largely unchanged before
and after the MPMs in July and October 2023,
suggesting that caution about a near-term change inthe policy rate did not increase much.
As for the 10-year rate, the probability distribution
became fat-tailed in the direction of higher interest
rates after the December 2022 and July 2023 MPMs
(Chart 5). In contrast, the shape of the distribution waslargely unchanged before and after the October 2023
MPM, indicating that risk perceptions regarding future
long-term interest rates did not change significantly.
4

Bank of Japan August 2024

U.S. Dollar/Yen Market[Chart 6] Implied Probability Distribution<U.S. Dollar/Yen Exchange Rate>

Next, we extract the implied probability distributionof the U.S. dollar/yen exchange rate from transaction
prices of options expiring three months after the tradedate.9Looking at the extracted implied probability
distribution of the U.S. dollar/yen exchange rate froma long-term perspective, the dispersion of the
distribution increased in 2013, when the U.S.
dollar/yen spot rate rose in response to mainly the
introduction of QQE in April of that year (Chart 6).
The distribution subsequently became less dispersed,
with the spot rate remaining more or less unchanged.
However, it dispersed again, mainly due to the
expansion of QQE, in October 2014 and remained so
until around the end of 2016. The distribution became
particularly dispersed and skewed in the direction of a
cautious stance toward yen appreciation toward thefirst half of 2016. This was against the backdrop ofrisk-off yen buying due to concerns about a slowdown
in the global economy, particularly China, as well asheightened uncertainty over overseas politicaldevelopments such as the referendum on the United
Kingdom's withdrawal from the European Union. The
dispersion declined moderately as a trend from 2017
onward, and the distribution generally stayed within alimited range until the beginning of 2022, compared toaround the mid-2010s, although the dispersion
temporarily increased around March 2020 in responseto the instability in financial markets at the onset of
the COVID-19 pandemic.
From 2022 onward, the distribution significantly
dispersed as the yen weakened against the U.S. dollar
with the widening of the yield differential between
Japan and the United States (Charts 6 and 7).
Specifically, (1) the distribution became moredispersed and skewed in the direction of a cautious
stance toward yen depreciation, as the U.S. dollar/yen
spot rate rose to around 150 in fall 2022 (the lower
panel of Chart 1). However, (2) caution about yen
appreciation heightened toward March 2023, as the
U.S. dollar/yen spot rate declined. This was mainly
against the background of a decline in U.S. interest
rates due to weaker-than-expected U.S. CPI data
released in November and December 2022, the Bank's
decision to expand the range of 10-year JGB yieldfluctuations from the target level at the December
2022 MPM, and the failure of and deterioration in thebusiness conditions of some financial institutions in
the United States and Europe in March 2023. (3)Thereafter, there was no significant increase in the

180

(yen)
1st-99th percentile
5th-95th percentile
25th-75th percentile
Median

160
140
120
100
80

60
CY13

14

15

16

17

18

19

20

21

22

23

Note: Latest data as of December 22, 2023
Source: Authors' estimates based on LSEG Eikon.

[Chart 7] Dispersion and Skewness of
Probability Distribution<Dispersion><Skewness>
45

(yen)

40

6
Dispersion

35

4
2

(yen)
Skewness↑ Cautious about
yen depreciation

30
0

25

-2

20

-4

15

-6

10
Jan Jul
2022

Jan
23

Jul

Jan Jul
202224

Jan
23

Jul

Note: "Dispersion" indicates the range between the 95th percentile(tail in the direction of yen depreciation) and the 5th percentile(tail in the direction of yen appreciation). "Skewness" iscalculated by subtracting the range between the 5th percentile
and the median (caution about yen appreciation) from therange between the 95th percentile and the median (caution
about yen depreciation). Latest data as of December 22,
2023.
Source: Authors' estimates based on LSEG Eikon.

dispersion, nor was there a significant increase incaution about yen depreciation, unlike the second halfof 2022. However, the dispersion of the distribution
remained larger than that in and before 2021.

5Bank of Japan August 202424

probability distributions of the 2-year US dollar swap
rate at the same days of the year, it can be inferred thatthe distribution was less dispersed in fall 2023 than infall 2022, with a particularly thin tail in the directionof higher interest rates (Chart 9). This is because,
while the dispersion in market participants' views on
the final destination of the U.S. policy rate (theterminal rate) could reasonably be called considerable
in fall 2022, when the policy rate was being increasedin the United States, the variation in views on the
future U.S. policy rate was not as significant as in fall2022 and there was limited caution about a further risein interest rates in fall 2023, as more market
participants expected a near-term end of the tightening
cycle. This difference in risk perception regarding U.S.
interest rates between the two periods may have been
one of the reasons why caution about future yen
depreciation was not as strong in fall 2023 when the
yen weakened as in fall 2022.

[Chart 8] Implied Probability Distributions of
U.S. Dollar/Yen Exchange Rate
(Probability density)
Oct. 20, 2022
Oct. 25, 2023

-30

-20

-10
0
10
20
Difference in 3-month-ahead
U.S. dollar/yen exchange rate from spot rate (yen)30

Note: The horizontal axis indicates the difference in the U.S. dollar/yen
exchange rate on the expiration date from the spot rate. The
data are on the days when the U.S. dollar/yen closed above 150(at 5 p.m. in New York) for the first time that year.
Source: Authors' estimates based on LSEG Eikon.

[Chart 9] Implied Probability Distributions of
USD Swap Rates (2-year)

Concluding Remarks(Probability density)
Oct. 20, 2022

In this paper, we extracted the implied probability
distributions from daily option prices for the futurelevels of yen swap rates and the U.S. dollar/yen
exchange rate, which came under upward pressureamid rate hikes by central banks in the United Statesand Europe from 2022 to 2023. The results showed
that, in the yen interest rate swaps market, caution
about higher interest rates increased significantly
toward the middle of 2022, as overseas interest rates
rose. Such caution further increased after the Bank's
decision at the December 2022 MPM to expand thefluctuation range for 10-year JGB yields from thetarget level, but subsequently eased. In the U.S.
dollar/yen market, caution about future yen
depreciation heightened toward fall 2022 with theweakening of the yen. However, even as the yen
weakened to the same level as the previous year in fall2023, caution about future yen depreciation did notincrease as much as in fall 2022.
Implied probability distributions represent the risk
perceptions of option market participants and are notnecessarily strong predictors of future pricemovements. 10 However, as observed in this paper,
estimating implied probability distributions enables us
to analyze market participants' risk perceptions,
including with regard to tail risk, in a moreinterpretable form. Thus, implied probability
distributions are expected to be further utilized infinancial market monitoring, along with existingquestionnaire surveys and indicators.

Oct. 25, 2023

-3

-2
-1
0
1
2
Difference in the 3-month-ahead 2-year ratefrom spot rate (%pts)

3Note: The horizontal axis indicates the difference in the USD swap
rate on the expiration date (2-year, based on SOFR) from the
spot rate. The data are on the days when the dollar/yen closed
above 150 (at 5 p.m. in New York) for the first time that year.
Source: Authors' estimates based on LSEG Eikon.

The U.S. dollar/yen spot rate rose to around 150 infall 2022, but slid back below 130 thereafter. However,
the rate rose again to about 150 toward fall 2023
(lower panel of Chart 1). Nevertheless, a comparisonof the implied probability distributions when the yen
depreciated in fall 2022 and fall 2023 (on the days
when the closing price of the U.S. dollar/yen
exchange rate in the New York market exceeded 150
for the first time that year) shows that the distribution
was less dispersed in fall 2023 than in fall 2022, witha particularly thin tail in the direction of yen
depreciation (Chart 8). In other words, in fall 2023,
the dispersion of expectations about the future level of
the U.S. dollar/yen exchange rate did not increase as
much as in fall 2022, and caution about yen
depreciation did not heighten.
In regard to this, when comparing the implied
6

Bank of Japan August 2024

BOX 1: Extracting an Implied Probability Distribution from Option Prices
Implied probability distributions can be computed by converting the transaction prices of options at various strikeprices into IVs, interpolating these IVs to plot volatility smiles, and then converting these into probabilities.
Volatility smiles are converted into probabilities by calculating the cost of creating a butterfly spread position (anoption trading strategy for betting that the price of the underlying asset will be near a certain strike price on the
expiration date). Below is the specific procedure of computing an implied probability distribution.11
1.
2.
3.

4.

Create a function V(K) that gives the volatility at strike price K by converting market transaction prices of
options at various strike prices into IVs and then interpolating them.12
Convert V(K) into a call option price C(K) using an option price valuation formula (in this paper, theBachelier formula for yen swap rates and the Black-Scholes formula for the U.S. dollar/yen exchange rate).13
Take an appropriate tick size Δ (in this paper, 0.01 percentage points for yen swap rates and 0.1 yen for the
U.S. dollar/yen exchange rate), and for each K, calculate the cost BF(K) of establishing a butterfly spread
position at a strike price K as C(K+Δ) + C(K-Δ) - 2C(K) (BOX 1 Chart).14
Plot an implied probability distribution by putting strike price K on the horizontal axis and the cost ofbuilding a butterfly spread position BF(K) on the vertical axis and rescaling the vertical axis so that it can beinterpreted as a probability at a given time point.
[BOX 1 Chart] Payoff of a butterfly spread
payoff

Δ

0

K-Δ

K

7

K+Δ

underlying
asset priceBank of Japan August 2024

BOX 2: Usefulness of Implied Probability Distributions
There are a variety of indicators that represent the risk perceptions of option market participants. Typical examples
include IVs of at-the-money options in each market and delta 25 RRs in foreign exchange markets. However,these examples are merely single values of the volatility smile at a particular observation point (IV) and reflect
only certain of the numerous parameters (moments) that characterize the probability distribution. Compared tothese indicators, implied probability distributions are useful in the following ways.
1.
2.
3.

Implied probability distributions represent a wide range of market participants' assumptions, because theyreflect information from the entire volatility smile (comprehensiveness).
Implied probability distributions can capture risk perceptions of large price movements, such as those above
25 delta (tail information).
Risk perceptions can be assessed in terms of both the level of the underlying asset price and its probability ofrealization (unit).
[BOX 2 Chart] Volatility Smile and Implied Probability Distribution<Volatility Smile>

11.5<Implied Probability Distribution>(Probability density)

(%)

ATM

11.0

←spot rate

10.5

25 delta call25 delta put

10.0

tail information

RR (25 delta)

9.5

9.0

comprehensivenessIV (ATM)

8.5
10P

25P

ATM

25C

10C delta

130

135

140

145
unit

150

155

160
yen

Note: Based on the volatility smile of the U.S. dollar/yen exchange rate on August 25, 2023, as an example. "C" and "P" in the left chart indicate "call" and"put," respectively. "Tail information" in the right chart indicates the area above the 95th percentile.
Source: Author's estimates based on LSEG Eikon.

8Bank of Japan August 2024

ones. However, due to data constraints, the data in this paper
are based on synthetic Yen LIBOR-based interest rate swaps for
figures from the beginning of 2022 through May 26, 2022, as
described in note 4. Therefore, the results for the above period
should be interpreted with some latitude.
9 Data on currency options are usually provided by vendors infive series: IV at ATM; RRs at delta 10 and delta 25; and thestrangles at delta 10 and delta 25. By adding and subtracting
these series, the IVs of a call option at deltas 10, 25, 50, 75, and90 can be obtained. To convert delta into a strike price, an
explicit formula for delta obtained from the Black-Scholes
formula is solved for the strike price.
10 For example, the following paper points out that the
time-difference correlation coefficient between the 25 delta RR
and spot rate for the U.S. dollar/yen exchange rate is low andthe predictive power of RRs for future spot rates is low.
Kato, H., Fukunaga, I., and Yamada, K., "Risuku ribāsaru kara
mita kawase hendō eno risuku ninshiki" [Risk perceptions on
exchange rate fluctuations observed from risk reversals]<available only in Japanese>, Bank of Japan Review Series,
2012-J-14.
11 For more details on the method of extracting implied
probability distributions from option prices, see, for example,
the following.
Malz, A. M., "A Simple and Reliable Way to Compute
Option-Based Risk-Neutral Distributions," New York Fed Staff
Reports, No. 677 (2014).
12 Interpolation methods for volatility smiles can be classified
into nonparametric methods, such as spline interpolation, andparametric methods based on modeling volatilities. When
spline interpolation is applied, the corresponding obtained
distribution tends to be non-intuitive, with multimodality. For
this reason, this paper uses a parametric interpolation formula(the Hagan approximation) based on the SABR model. For
more information on this interpolation formula, see thefollowing.
Hagan, P. S., Kumar, D., Lesniewski, A. S. and Woodward, D.
E., "Managing Smile Risk," The Best of Wilmott, 1 249-296(2002).
13 With respect to the yen swap rate, the Black-Scholes model
does not assume a situation where the value of the underlying
asset is negative. Given the recent market developments whereinterest rates can be negative, either an appropriate shift range
should be introduced or a Bachelier model that assumes that theprice of the underlying asset follows a normal distribution(rather than a log-normal distribution) should be applied.
14 A butterfly spread is specifically a portfolio consisting ofselling two units of call options at strike K and buying one unitof call options at each strike price K ± Δ (Δ is an appropriate
tick size).

*Currently, Fukuoka Branch**Currently, Personnel and Corporate Affairs Department
1

The term "swaption" originally referred to an option on aswap transaction in general. However, in today's financial
markets, a "swaption" usually refers to an option on an interest
rate swap, which is the definition we use in this paper. Amongswaptions, options on an interest rate swap that give the right toreceive a fixed rate and pay a floating rate are called receiver
swaptions. Conversely, options on an interest rate swap thatgive the right to pay a fixed rate and receive a floating rate arepayer swaptions. For instance, when the holder of a receiver
swaption exercises their right on the expiration date, they canreceive a pre-contracted fixed rate while paying a floating rate
for a pre-contracted period of time.
2 In the Japanese foreign exchange market, RRs are often used
for hedging purposes, mainly by domestic non-financial
corporations, and thus have high representativeness compared
to RRs of other assets.
3 It should be noted that the implied probability distributions
calculated in this paper are risk-neutral probability distributions
and not natural probability distributions. See the following for astudy on the conversion of a risk-neutral probability
distribution into a natural probability distribution.
Ross, S., "The Recovery Theorem," J. Finance, 70 615-648(2015).
4 Besides swaptions, another type of yen interest rate options
are options on JGB futures. However, in this paper, the analysis
is focused on swaptions, because they are easier to handle forthe following reasons.
(i) Swap periods can be freely set, so interest rates with various
maturities can be analyzed.
(ii) Since the expiration is not fixed on specific dates, data witha fixed tenor can be obtained.
(iii) Swaptions are usually European options, which are easy tohandle theoretically.
The traded swaption price data used in the analysis are IVs ateach of the following 13 strike prices: At the Money (ATM);±25 bps; ±50 bps; ±75 bps; ±100 bps; ±150 bps; and ±200 bps.
The underlying asset of the swaptions is the swap rate based on
6-month yen LIBOR through May 26, 2022 (synthetic yen
LIBOR from the beginning of 2022 through May 26, 2022) andthe overnight uncollateralized call rate (TONA) for thesubsequent period.
5 For example, the following is an analysis based on implied
probability distributions in Japan.
Suganuma, K., and Yamada, T., "Mainasu kinri o kōryo shita
fōwādo rēto moderu to sijō no kinri mitōshi" [Forward rate
models considering negative interest rates and market outlook
for interest rates] <available only in Japanese>, IMES
Discussion Paper Series, 2017-J-18.
6 For example, see the following for a study that uses highly
granular data to organize facts about the yen interest rate swaps
market.
Inoue, S., Miki, S., and Gemma, Y., "The Japanese Yen Interest
Rate Swap Market Observed from OTC Derivative Transaction
Data: the Impact of COVID-19," Bank of Japan Review Series,2021-E-3.
7 Swaptions at strike prices whose interest rate levels aresubstantially distant from the ATM are considered less liquidthan those near the ATM. For this reason, some latitude should
be given to the interpretation of the extreme tails of the
distributions.
8 With the discontinuation of the publication of Yen LIBOR at
the end of 2021, swap transactions have shifted to TONA-based

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9Bank of Japan August 2024