Transcript Lecture 4 - Queen's University

MGT 821/ECON 873
Volatility Smiles
&
Extension of Models
1
What is a Volatility Smile?



It is the relationship between implied volatility
and strike price for options with a certain
maturity
The volatility smile for European call options
should be exactly the same as that for
European put options
The same is at least approximately true for
American options
2
Why the Volatility Smile is the Same
for Calls and Put




Put-call parity p +S0e-qT = c +Ke–r T holds for market
prices (pmkt and cmkt) and for Black-Scholes prices (pbs
and cbs)
It follows that pmkt−pbs=cmkt−cbs
When pbs=pmkt, it must be true that cbs=cmkt
It follows that the implied volatility calculated from a
European call option should be the same as that
calculated from a European put option when both
have the same strike price and maturity
3
The Volatility Smile for Foreign Currency
Options
Implied
Volatility
Strike
Price
4
Implied Distribution for Foreign Currency
Options


Both tails are heavier than the lognormal
distribution
It is also “more peaked” than the lognormal
distribution
5
The Volatility Smile for Equity
Options
Implied
Volatility
Strike
Price
6
Implied Distribution for Equity Options

The left tail is heavier and the right tail
is less heavy than the lognormal
distribution
7
Other Volatility Smiles?


What is the volatility smile if
True distribution has a less heavy left tail and
heavier right tail
True distribution has both a less heavy left
tail and a less heavy right tail
8
Ways of Characterizing the Volatility
Smiles



Plot implied volatility against K/S0 (The volatility
smile is then more stable)
Plot implied volatility against K/F0 (Traders
usually define an option as at-the-money when K
equals the forward price, F0, not when it equals
the spot price S0)
Plot implied volatility against delta of the option
(This approach allows the volatility smile to be
applied to some non-standard options. At-the
money is defined as a call with a delta of 0.5 or a
put with a delta of −0.5. These are referred to as
50-delta options)
9
Possible Causes of Volatility Smile


Asset price exhibits jumps rather than
continuous changes
Volatility for asset price is stochastic


In the case of an exchange rate volatility is not
heavily correlated with the exchange rate. The
effect of a stochastic volatility is to create a
symmetrical smile
In the case of equities volatility is negatively
related to stock prices because of the impact of
leverage. This is consistent with the skew that is
observed in practice
10
Volatility Term Structure


In addition to calculating a volatility smile,
traders also calculate a volatility term
structure
This shows the variation of implied volatility
with the time to maturity of the option
11
Volatility Term Structure
The volatility term structure tends to be
downward sloping when volatility is high and
upward sloping when it is low
12
Example of a Volatility Surface
K/S0
0.90
0.95
1.00
1.05
1.10
1 mnth 14.2
13.0
12.0
13.1
14.5
3 mnth 14.0
13.0
12.0
13.1
14.2
6 mnth 14.1
13.3
12.5
13.4
14.3
1 year 14.7
14.0
13.5
14.0
14.8
2 year 15.0
14.4
14.0
14.5
15.1
5 year 14.8
14.6
14.4
14.7
15.0
13
Greek Letters

If the Black-Scholes price, cBS is expressed
as a function of the stock price, S, and the
implied volatility, simp, the delta of a call is
cBS cBS simp

S simp S

Is the delta higher or lower than
cBS
S
14
Three Alternatives to Geometric Brownian
Motion

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Constant elasticity of variance (CEV)
Mixed Jump diffusion
Variance Gamma
15
CEV Model
a
dS  (r  q)Sdt  sS dz



When a = 1 the model is Black-Scholes
When a > 1 volatility rises as stock price
rises
When a < 1 volatility falls as stock price
rises
European option can be value analytically
in terms of the cumulative non-central chi
square distribution
16
CEV Models Implied Volatilities
simp
a<1
a>1
K
17
Mixed Jump Diffusion
Merton produced a pricing formula when the asset
price follows a diffusion process overlaid with
random jumps
dS / S  (r  q  lk )dt  sdz  dp



dp is the random jump
k is the expected size of the jump
l dt is the probability that a jump occurs in the
next interval of length dt
18
Jumps and the Smile
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
Jumps have a big effect on the
implied volatility of short term options
They have a much smaller effect on
the implied volatility of long term
options
19
The Variance-Gamma Model


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Define g as change over time T in a variable
that follows a gamma process. This is a
process where small jumps occur frequently
and there are occasional large jumps
Conditional on g, ln ST is normal. Its variance
proportional to g
There are 3 parameters



v, the variance rate of the gamma process
s2, the average variance rate of ln S per unit time
q, a parameter defining skewness
20
Understanding the Variance-Gamma
Model



g defines the rate at which information arrives
during time T (g is sometimes referred to as
measuring economic time)
If g is large the change in ln S has a relatively
large mean and variance
If g is small relatively little information arrives
and the change in ln S has a relatively small
mean and variance
21
Time Varying Volatility

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Suppose the volatility is s1 for the first year
and s2 for the second and third
Total accumulated variance at the end of
three years is s12 + 2s22
The 3-year average volatility is
2
2
s

2
s
1
2
3s2  s12  2s22 ; s 
3
22
Stochastic Volatility Models
dS
 (r  q)dt  V dzS
S
dV  a(VL  V )dt  V a dzV
When V and S are uncorrelated a
European option price is the BlackScholes price integrated over the
distribution of the average variance
23
Stochastic Volatility Models continued


When V and S are negatively correlated we
obtain a downward sloping volatility skew
similar to that observed in the market for
equities
When V and S are positively correlated the
skew is upward sloping. (This pattern is
sometimes observed for commodities)
24
The IVF Model
T heimplied volat ility funct ionmodelis
designed t o creat ea processfor t heasset
price t hatexact lymat chesobservedopt ion
prices.T heusual geomericBrownian mot ion
model
dS  ( r  q ) Sdt  sSdz
is replacedby
dS  [r (t )  q (t )]Sdt  s ( S , t ) Sdz
25
The Volatility Function
The volatility function that leads to the model
matching all European option prices is
[s ( K , t )]2 
cmkt t  q(t )cmkt  K [r (t )  q(t )]cmkt K
2
K 2 ( 2 cmkt K 2 )
26
Strengths and Weaknesses of the IVF
Model
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
The model matches the probability distribution
of asset prices assumed by the market at
each future time
The models does not necessarily get the joint
probability distribution of asset prices at two
or more times correct
27