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The convenience yield implied by quadratic volatility smiles presentation [compatibility mode]
1. The convenience yield implied
by quadratic volatility smiles
(2001, 2002)
2001, 2002)
By Prof. Haim Reisman
Presented by
Yigal Ben Tal & David Feldman
2. Introduction
The Implied Convenience Yield (ICY)
The Moneyness, what is it?
Moneyness,
The Implied Volatility and its Smile ☺
Black & Scholes Model representation
3. Basic definitions - ICY
The Implied Convenience Yield of an illiquid option
is the rate of locally risk-less profit obtained from
risk-
hedging this option using other liquid at-the-money
at-the-
(ATM) options as hedging instruments.
instruments.
(Prof. Haim Reisman, 06/2002)
4. Basic definitions - Moneyness
Let the function m(t,T,r,K,St) be a function of time,
maturity date, interest rate, strike price and underlying
price. Then the moneyness X=Xt at time t (0 ≤ t<T) is
price. X=X t<T)
generally defined as
X= m(t,T,r,K,St).
m(
The function m(·) is referred to as the moneyness
function.
function.
It is required that the moneyness to be increasing in K.
(Reinhold Hafner, 2004)
5. Basic definitions - IV
The Implied Volatility is the value of the expected
volatility imputed from an option pricing model (such
as B&S), given the option price, the asset’s price,
the exercise price, the time to maturity, and the risk-
risk-
free interest rate.
rate.
(OECD Economic Outlook Glossary)
6. Basic definitions - IV
(the math definition)
If C = f(σ,·) is a theoretical value of an option, and
option,
f(·) is a pricing model that depends on volatility σ
f(·)
plus other inputs, and f(·)
f(·) is monotonically
increasing in σ, than if exists some inverse function
f -1(·),
·), such that σC* = f -1(C*,·),
(C*,·), where C* is the
market price of an option, than the value σC* is the
implied volatility by the market price C*.
C*.
(Reinhold Hafner, 2004)
7. Basic definitions - Smile
For any fixed maturity date t (t ≤ T), the function
σ(K,·) of implied volatility against strike price K
K,·)
(K>0) is called the volatility smile or just smile (for
K>0
maturity T) at date t (0 ≤ t<T).
t<T)
IV
X = Ke − r ∆t S t
ATM
Call ITM Call OTM
Put OTM Put ITM
1 X
8. The economic assumptions
• The ICY of the liquid options is zero and
one of the non-liquid options isn’t zero.
non- zero.
• The stock index and European options are
traded continuously.
continuously.
• The interest rate is constant over the time.
time.
• The ATM options are traded at no
transaction costs, but those that are away
from the money are traded with it.
it.
• The paper analyzes a set of fix expiration
options.
options.
9. The mathematic assumptions
The Volatility Smile is quadratic with coefficients as
Ito's processes.
processes.
This is the IV of ATM option.
option.
This is the slope of the volatility smile.
smile.
This is curvature measure of the smile.
smile.
This is moneyness.
moneyness.
10. Black & Scholes Model changes
The standard formula for European Call option is:
C ( t ,T , S t , K ) = S t N ( d 1 ) − Ke − r ∆t N ( d 2 )
ln ( S t K ) + ( r + σ 2 2 ) ∆ t
d1 = , d 2 = d1 − σ ∆t
σ ∆t
∆t
The changed B&S formula, that used in the paper is:
C ( t ,T , St , K ) = St N ( d1 ) − XN ( d2 ) , X = Ke−r∆t St
− ln X V ( t ,T , X ) ∆t
d1 = + , d2 = d1 −V ( t ,T , X ) ∆t
V ( t ,T , X ) ∆t 2
11. The target of the paper
The ICY for non-liquid options may be explained as
non-
a stream of the cash (received/paid) for the
discomfort of the option holding.
holding.
The non-liquid options may have non-zero ICY.
non- non- ICY.
The target of the article is a creation of an exact
formula for the ICY and for its hedging coefficients.
coefficients.
12. The getting formula
dC − rC dt = ∆ * ⋅ ( dS t − rS t dt ) +
A
B
2
+ ∑ V ega * ⋅ ( d z k − α k d t ) +
k
k =0
C
+ ε ( t ,T , K ) dt
D
Where µ is very complicated expression of the B&S model’s
partial derivatives.
∆* = ∆ − X ⋅ (Vega S ) z1 + 2 ( X − 1) z2 , Vega* = ( X − 1) ⋅ Vega
k
k
∂ 1 ∂2
α0 ( X = 1) = µ Vega , α1 ( X = 1) = ( µ Vega ) , α2 ( X = 1) = 2 (
µ Vega )
∂X 2 ∂X
ε = µ − ∑αk ⋅ Vega* , ε ( t,T ,K ) = o ( ( X − 1) )
2
3
k
k =0
13. The advantage remarks
• The received formulas are simple
computation and depend just on currently
observable parameters.
parameters.
• There is no need for any historical data or
some arbitrary assumption on the
behavior of processes in the future.
future.
14. There are some question points
• The model has many different initial
parameters (zk, cov(dw,dwk), ets.).
cov( ets.
• There are many undefined expressions
used by the author (cov(dw,dwk), coefficients of zk).
cov(
• Various economic and mathematical
assumptions, that are not clear (ICY, the
formula of the hedging portfolio options).
