Yield Curve Volatility: An Insightful Study
P
r
e
s
e
n
t
a
t
i
o
n
o
f
t
h
e
p
a
p
e
r
:
Y
i
e
l
d
C
u
r
v
e
s
a
n
d
V
o
l
a
t
i
l
i
t
y
S
m
i
l
e
s
,
b
y
P
e
t
e
r
C
a
r
r
J
ingsheng Zhang
October
9, 2018
A
b
s
t
r
a
c
t
Yield Curve
Volatility Smile
In this paper, the author found particular measures of implied volatility and moneyness so
that the resulting volatility smile is analogous to the yield curve for zero coupon bonds. The
relation was used to develop arbitrage-free curves in both cases.
C
o
n
c
l
u
s
i
o
n
L
a
y
o
u
t
1.
Introduction
2.
Benchmark models
3.
Market models
4.
Summary
C
o
n
c
l
u
s
i
o
n
I
n
t
r
o
d
u
c
t
i
o
n
C
o
n
c
l
u
s
i
o
n
B
e
n
c
h
m
a
r
k
M
o
d
e
l
s
C
o
n
c
l
u
s
i
o
n
B
e
n
c
h
m
a
r
k
M
o
d
e
l
s
C
o
n
c
l
u
s
i
o
n
B
e
n
c
h
m
a
r
k
M
o
d
e
l
s
C
o
n
c
l
u
s
i
o
n
B
e
n
c
h
m
a
r
k
M
o
d
e
l
s
C
o
n
c
l
u
s
i
o
n
B
e
n
c
h
m
a
r
k
M
o
d
e
l
s
C
o
n
c
l
u
s
i
o
n
B
e
n
c
h
m
a
r
k
M
o
d
e
l
s
C
o
n
c
l
u
s
i
o
n
M
a
r
k
e
t
M
o
d
e
l
s
In this section, the author described two additional models. A market model of stochastic yields is used
to build an arbitrage-free yield curve. Analogously, a market model of stochastic implied volatilities
is used to build an arbitrage-free volatility smile. He also drew an analogy between these two models.
1. Market model for yields
Suppose that under Q, the risk-neutral yield dynamics are given by the solution to the following
stochastic differential equation (SDE):
The absence of arbitrage implies that at each time t, each bond's price grows in expectation at
the short rate r
t
. As a result, we obtain the following no arbitrage constraint on yields:
C
o
n
c
l
u
s
i
o
n
M
a
r
k
e
t
M
o
d
e
l
s
We have the following three Greeks:
Then a specification of the risk-neutral drift and difusion processes governing yields determines an
arbitrage-free yield curve.
C
o
n
c
l
u
s
i
o
n
M
a
r
k
e
t
M
o
d
e
l
s
M
a
r
k
e
t
M
o
d
e
l
s
M
a
r
k
e
t
M
o
d
e
l
s
3
. Comparing Market Models
2 Benchmark models.
Connection of yield and implied volatility under these models.
2 Market models.
Connection of yield and implied volatility under these models.
1
2
1
2
C
o
n
c
l
u
s
i
o
n
O
v
e
r
v
i
e
w
S
u
m
m
a
r
y
In this paper, the author found particular measures of implied volatility and moneyness of a European
swaption, so that the resulting volatility smile is analogous to the yield curve for zero coupon bonds.
The relation was used to develop arbitrage-free curves in both cases.
Extension
s
1.
Notation
2.
Market
support e.g. yields and implied vol not parallel shift, coupon
bonds
3.
Bachelier model vs Bla
ck model
T
h
a
n
k
Y
o
u
!
This paper delves into the relationship between implied volatility and moneyness, drawing parallels between volatility smiles and yield curves. It explores benchmark and market models, providing a detailed analysis and conclusions on arbitrage-free curves.
Download Presentation
Please find below an Image/Link to download the presentation.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author.If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.
You are allowed to download the files provided on this website for personal or commercial use, subject to the condition that they are used lawfully. All files are the property of their respective owners.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author.
E N D
Presentation Transcript
Presentation of the paper: Yield Curves and Volatility Smiles, by Peter Carr Jingsheng Zhang October 9, 2018 0
Abstract Yield Curve Volatility Smile In this paper, the author found particular measures of implied volatility and moneyness so that the resulting volatility smile is analogous to the yield curve for zero coupon bonds. The relation was used to develop arbitrage-free curves in both cases. 1
Conclusion Layout 1. Introduction 2. Benchmark models 3. Market models 4. Summary 2
Conclusion Introduction 1. Fruitful interplay between models developed to price bonds and models developed to price options, e.g. Binomial models, similar results on the counterparty. 2. This paper explored the connection between the continuously compounded yield on a zero coupon bond and the implied volatility of a European swaption and draws a precise mathematical connection between a volatility smile and a yield curve. 3. This paper used the Bachelier Model, and considered moneyness in this model, e.g. ?? ? , ?? which can be interpreted as the number of annualized standard deviations that the forward swap rate exceeds the strike rate. 3
Conclusion Benchmark Models In this section, the author described two different models. A constant short interest rate model is used to dene the yield to maturity of a bond. Similarly, a constant short normal volatility model is used to dene the normal implied volatility of a swaption. And he drew analogies between these two models. 1. Constant Interest Rate Bond Model The value b of the T-bond at any time ??[0,?] must be: The yield-to-maturity of this T-bond is defined as the solution to the equation: 4
Conclusion Benchmark Models The only yield curve which is free of our model-based arbitrage is: 2. Constant Normal Volatility Swaption Model Under the forward swap measure ??corresponding to that term and tenor, the forward swap rate dynamics are assumed to be: ? is the instantaneous volatility of the forward swap rate. 5
Conclusion Benchmark Models Let c(F; t;K; T; ) be the the arbitrage-free call swaption valuation function in our constant normal volatility model. This function is the unique solution to the following terminal value problem: Very similar to Black s equation. Let ? = ? ?,? = ? ?, (13) is equivalent to: The solution is: 6
Conclusion Benchmark Models Plug in the moneyness ? =?? ? , and the normal implied variance rate?? 2(?), ? In the Bachelier model, the usual normal implied volatility smile is flat and constant over time: 3. Comparing Benchmark Models (Draw the analogy) 7
Conclusion Benchmark Models 8
Conclusion Benchmark Models Besides being fundamental solutions, the two functions and each arise in the Rodrigues formula and inner product for Laguerre polynomials and for the probabilists' Hermite polynomials. Not quite understood. 9
Conclusion Market Models In this section, the author described two additional models. A market model of stochastic yields is used to build an arbitrage-free yield curve. Analogously, a market model of stochastic implied volatilities is used to build an arbitrage-free volatility smile. He also drew an analogy between these two models. 1. Market model for yields Suppose that under Q, the risk-neutral yield dynamics are given by the solution to the following stochastic differential equation (SDE): The absence of arbitrage implies that at each time t, each bond's price grows in expectation at the short rate rt. As a result, we obtain the following no arbitrage constraint on yields: 10
Conclusion Market Models We have the following three Greeks: Then a specification of the risk-neutral drift and difusion processes governing yields determines an arbitrage-free yield curve. 11
Conclusion Market Models 2. Market Model for Implied Volatility Suppose that under ??, the forward swap rate process F solves the following stochastic differential equation (SDE): To compensate for the absence of a speficication of the instantaneous normal variance rate V , we suppose that under ??, the implied volatility process ?? stochastic differential equation (SDE): 2(?) is the solution to the following Let ???[ 1,1] be the bounded stochastic process governing the correlation between the two standard Brownian motions W and Z at time t. The absence of arbitrage implies that at each time t, each swaption's price is a local martingale. 12
Market Models 3. Comparing Market Models 14
Conclusion Overview 2 Benchmark models. 1 Connection of yield and implied volatility under these models. 2 2 Market models. 1 2 Connection of yield and implied volatility under these models. 15
Summary In this paper, the author found particular measures of implied volatility and moneyness of a European swaption, so that the resulting volatility smile is analogous to the yield curve for zero coupon bonds. The relation was used to develop arbitrage-free curves in both cases. Extensions 1. Notation 2. Market support e.g. yields and implied vol not parallel shift, coupon bonds 3. Bachelier model vs Black model 16
Thank You! 17
