An approximation formula for normal implied volatility under general local stochastic volatility models

• Author: Yasaman Karami,Kenichiro Shiraya
• Published date: 01 September 2018
• Date: 01 September 2018
• DOI: http://doi.org/10.1002/fut.21931

Received: 1 February 2018

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Revised: 16 April 2018

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Accepted: 17 April 2018

DOI: 10.1002/fut.21931

RESEARCH ARTICLE

An approximation formula for normal implied volatility

under general local stochastic volatility models

Yasaman Karami

|

Kenichiro Shiraya

Quantitative Finance, Graduate School of

Economics, The University of Tokyo,

Tokyo, Japan

Correspondence

Kenichiro Shiraya, Quantitative Finance,

Graduate School of Economics,

The University of Tokyo,

7‐3‐1, Hongo, Bunkyo‐ku,

Tokyo 113‐0033, Japan.

Email: kenichiro.shiraya@gmail.com

Funding information

Center for Advanced Research in

Finance (CARF)

We approximate normal implied volatilities by means of an asymptotic

expansion method. The contribution of this paper is twofold: to our knowledge,

this paper is the first to provide a unified approximation method for the normal

implied volatility under general local stochastic volatility models. Second, we

applied our framework to polynomial local stochastic volatility models with

various degrees and could replicate the swaptions market data accurately. In

addition we examined the accuracy of the results by comparison with the

Monte‐Carlo simulations.

KEYWORDS

approximation formula, local stochastic volatility, normal implied volatility

1

|

INTRODUCTION

The construction of implied volatility (IV) surfaces is a critical step in pricing exotic instruments and options with

illiquid strike‐expiry pairs.

To obtain the smooth IV surface, we need to choose an interpolation method or a model that fits accurately with the

market IV data. However, the interpolation method cannot be applied to evaluate path‐dependent exotic derivatives.

Local volatility (LV), stochastic volatility (SV), and the hybrid local stochastic volatility (LSV) models are commonly

used to construct the surface. Especially, LSV models have become increasingly popular among practitioners in foreign

exchange (Dadachanji, 2015), equity (Homescu, 2014), and interest rate (Mercurio, 2006) derivatives markets.

Analytical approximations of IV are useful for exploring the relationships between IV and fair prices of volatilities

derivatives while numerical inversion cannot be used to analyze such relationships. There is a substantial body of

research on the approximation of Black’s IV. The asymptotic expansion method is one of the approximation techniques,

and has been widely used in this field of research. For example, Forde and Jacquier used the method to study the short‐

term behavior of IVs (Forde and Jacquier, 2009). Hagan, Kumar, Lesniewski, and Woodward (2002) used the

asymptotic expansion in stochastic alpha beta rho (SABR) LSV model. Gatheral, Hsu, Laurence, Ouyang, and Wang

(2012), Takahashi and Yamada (2012), and Lorig, Pagliarani, and Pascucci (2017) derived an approximation method of

IVs under LV, SV, and LSV models, respectively. Homescu (2011) provided an extensive survey of the works on the IV

surface.

Previously, Black model was used as the market standard for quoting the IVs. However, in the interest rate

derivatives market, the emergence of negative rates led practitioners to seek for other models that are more general and

compatible with negative rates since the Black’s model requires the underlying value to be strictly nonnegative. Market

practitioners, especially in markets with low and negative interest rates, have been using normal volatility instead,

which is defined by Bachelier’s option pricing formula and allows for both negative and positive prices. This paper

provides a unified approximation method of the normal implied volatilities from general LSV model parameters.

J Futures Markets. 2018;38:1043–1061. wileyonlinelibrary.com/journal/fut © 2018 Wiley Periodicals, Inc.

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1043

For dealing with negative rates, some market practitioners use the shifted SABR model for describing the properties

(such as the smile) and dynamics of the volatility term. This is done by adding a shift parameter to the underlying rate

term in SABR model, and it is artistically decided by market practitioners.

Another type of LSV models, Heston‐type SV with LV models are also widely used in several markets, and also

studied for the interest rate market. The Heston SV model involves less approximation in the measure change

from the forward measure to the swaption measure (see, e.g., Wu & Zhang, 2006; Shira ya, Takahashi, and

Yamazaki, 2012). In our numerical examples, we apply our formula to the polynomial LV functions with Heston

SV model.

This paper is organized as follows: In Section 2, we derive an approximation formula for normal IV for interest rate

swaptions under a general LSV model. In Section 3, we show the numerical examples by using Heston‐type SV with

some LV functions to replicate the actual market IVs. Then, we test the accuracy of the results by comparison with the

ones obtained from Monte‐Carlo simulations and numerical inversion method.

2

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NORMAL VOLATILITY APPROXIMATION

In this section, we derive an approximation formula for normal IV.

2.1

|

Basic setup

This subsection defines basic concepts such as tenor structures, and the swap rates. First, a tenor structure is given by a

finite set of dates:

=<<⋯<TT T

0

,

N01

where

T

i

=…iN

(

0, 1, ,

)

are prespecified dates, and ≔−

−

δTT

jjj

1

.

Pt(

)

j

denotes the price of the discount bond with

maturity

Tj

at time

t

, where =PT()

1

jj and =Pt() 0

j

for

>

t

T

j

.

An interest rate swap is a financial contract between two counterparties to exchange cash flows with a fixed

predetermined interest rate and floating interest rates on a certain notional amount (also called a swap principal). We

denote the swap rate at time

t

as

S

t()

. This is the fixed predetermined rate at which one is willing to enter a swap that

starts at

=

t

T

α

and expires at =

t

T.

β

At the money (ATM) swap rate can be shown as:

=−

∑

=+

S

tPt Pt

δP t

() () ()

()

.

αβ

jα

βjj

1

(1)

A swaption is a contingent claim that gives the holder the right to enter a specific swap contract.

Let FQ

(

Ω, ,

)

denote a complete probability space satisfying the usual conditions where

Q

is the swap measure for

which

∑

=+δP t(

)

jα

βjj

1is the numeraire. In this setting, the corresponding swap rate

S

t()

is a martingale under the swap

measure

Q

.

Under no arbitrage conditions, we can express the swaption forward premium

V

TK(, )

under swap measure as:

=−

+

V

TK ST K NE(, ) [(() )]

,

αβ,

(2)

∑

=

=+

δP(0)

.

αβ

jα

β

jj,

1

(3)

E

is the expectation taken under the swap measure, and

K

denotes the strike rate of the swaption. Hereafter, we set

=

1

αβ,for simplicity.

Suppose we can express the forward swap rate

S

ϵt(, )and its volatility

σ

ϵt(, )as the solutions to the following

stochastic integral equations, for

∈ϵ(0, 1]

.

1044

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KARAMI AND SHIRAYA