Embed Size (px)
Citation preview
CBOE Risk Management Conference Europe
September 2013
Understanding Volatility
Sheldon Natenberg & Tim Weithers
Chicago Trading Co.
440 South LaSalle St.
Chicago, IL 60605
(312) 863-8000
What is the value of a call option at expiration?
intrinsic value
0
exercise
price
+1
+1
: maximum [S - X, 0]
security price
pro
ba
bil
ity
Suppose we create a probability distribution
and then overlay the option’s intrinsic value
on the probability distribution.
n
i=1
For each underlying price we have an intrinsic
value and a probability, p.
The expected value for the option at expiration
is the sum of all these individual values.
p * intrinsic value = p * maximum[S - X, 0]
Σ pi * maximum[Si - X, 0]
The theoretical value is the present value of
this amount.
What probability
distribution
should we
use?
With slight modification most traditional pricing
models (Black-Scholes, binomial) assume a
normal distribution
of underlying
prices.
All normal distributions
are defined by their mean (μ)
and their standard deviation (σ).
Mean – where the
peak of the curve
is locatedStandard deviation –
how fast the curve
spreads out.
low standard deviation
high peak
narrow body
high standard deviation
low peak
wide body
underlying price
120 call
option value
80 put
+1 S.D.
+1 S.D. ≈ 34%
-1 S.D.
-1 S.D. ≈ 34%
+2 S.D.-2 S.D.
+2 S.D. ≈ 47.5%
-2 S.D. ≈ 47.5%
±1 S.D. ≈
68% (2/3)
±2 S.D. ≈
95% (19/20)
mean
exercise price
time to expiration
underlying price
interest rate
volatility
(dividends)
mean?
standard
deviation?
Mean –
Standard deviation –
(underlying price, time to expiration,
interest rates, dividends)
Volatility: one standard deviation, in percent,
over a one year period.
stock: S * (1+r*t) - D
foreign currency: S *1+rd*t
1+rf*t
futures contract: F
forward price
volatility
1-year forward price = 100.00
volatility = 20%
One year from now:
• 2/3 chance the contract will be
between 80 and 120 (100 ± 20%)
• 19/20 chance the contract will be
between 60 to 140 (100 ± 2*20%)
• 1/20 chance the contract will be
less than 60 or more than 140
1-year later
underlying price = 180
Was 20% an accurate volatility?
If 20% was correct, how many standard deviations
did the market move? (180-100) / 20 = 4
What is the likelihood of a 4 standard
deviation occurrence? ≈ 1 / 16,000
Is one chance in 16,000 impossible?
What does an annual volatility tell us about
movement over some other time period?
monthly price movement?
weekly price movement?
daily price movement?
Volatilityt = Volatilityannual * t√
Daily volatility (standard deviation)
Trading days in a year? 250 – 260
Assume 256 trading days
Volatilitydaily ≈ Volatilityannual / 16
t = 1/256 =t√ √ 1/256 = 1/16
current price = 100.00
volatilitydaily ≈ 20% / 16 = 1¼%
One trading day from now:
• 2/3 chance the contract will be trading
between 98.75 and 101.25 (100 ± 1¼%)
• 19/20 chance the contract will be trading
between 97.50 and 102.50 (100 ± 2*1¼%)
Weekly volatility:
Volatilityweekly ≈ Volatilityannual / 7.2
t = 1/52 =t√ √ 1/52 ≈ 1/7.2
Volatilitymonthly ≈ Volatilityannual / 3.5
t = 1/12 ≈ 1/3.5
Monthly volatility:
=t√ √ 1/12
daily standard deviation?
stock = 68.50; volatility = 42.0%
≈ 68.50 * 42% / 16
= 68.50 * 2.625% ≈ 1.80
weekly standard deviation?
≈ 68.50 * 42% / 7.2
= 68.50 * 5.83% ≈ 4.00
daily standard deviation?
stock = 68.50; volatility = 42.0%
+1.25 -.95 +.35+.70 -1.60
Is 42% a reasonable volatility estimate?
How often do you expect to see an occurrence
greater than one standard deviation?
≈ 1.80
∞+∞–
normal
distribution
lognormal
distribution
0
normal
distribution
110 call
lognormal
distribution
forward price = 100
3.00
90 put 3.00
3.20
2.80
2.90
3.10
price
Are the options mispriced?
Maybe the marketplace is right.
Maybe the marketplace thinks the model is wrong.
The volatility of the
underlying contract over some period
of time (historical, future)
realized volatility:
derived from the prices of options in
the marketplace
implied volatility: The marketplace’s
consensus forecast of future volatility;
pricing
model
theoretical
value
5.50
6.75
exercise price
time to expiration
underlying price
interest rate
volatility 27%volatility
???31%
implied volatility
today
realized
volatility
backward
looking
(what has occurred)
implied
volatility
forward
looking
(what the marketplace
thinks will occur)
implied volatility = price
realized volatility = value
30 August 2013
S&P 500 = 1632.97
Time to October expiration = 7 weeks
1525 call
1625 call
1725 call
1525 put
1625 put
1725 put
price 12% 16% 20% implied
115.95 21.67%
40.35 17.32%
3.70 13.29%
21.63%
17.32%
13.29%
105.74
28.66
2.50
108.94
37.45
6.87
113.66
46.24
12.71
11.55
35.95
99.25
1.40
24.27
98.04
4.60
33.05
102.42
41.84
108.26
October
9.32
Interest rate = .50%
30 August 2013
S&P 500 = 1632.97
Time to October expiration = 7 weeks
1525 call
1625 call
1725 call
12% 16%
105.74
28.66
2.50
108.94
37.45
6.87
October
Interest rate = .50%
ITM
ATM
OTM
increase %
3%
31%
175%
3.20
8.79
4.37
1525 put
1625 put
1725 put
1.40
24.27
98.04
4.60
33.05
102.42
OTM
ATM
ITM
229%
36%
4%
8.78
4.38
3.20
30 August 2013 S&P 500 = 1632.97
Time to December Expiration = 16 weeks
1525 call
1625 call
1725 call
increase implied
21.67%
17.32%
13.29%
3.20
8.79
4.37
October
Time to October Expiration = 7 weeks
price 12% 16%
115.95
40.35
3.70
105.74
28.66
2.50
108.94
37.45
6.87
1525 call
1625 call
1725 call
increase implied
19.23%
16.28%
13.57%
9.90
14.33
10.90
December price 12% 16%
127.95
58.15
15.15
108.98
42.81
11.11
118.88
57.14
22.01
1. In total points an at-the-money option is
always more sensitive to a change in volatility
than an equivalent in- or out-of-the-money
option.
2. In percent terms an out-of-the-money option
is always more sensitive to a change in volatility
than an equivalent in- or at-the-money option.
3. A long-term option is always more sensitive
to a change in volatility than an equivalent
short-term option.
Volatility Characteristics
serial correlation – in the absence of other
information, the best estimate of volatility over
the next time period is the volatility which
occurred over the previous time period.
mean reversion – over long periods of time
volatility tends to revert to its average.
momentum – when volatility begins to rise or
fall, it tends to continue in the same direction.
GARCH model (generalized auto-regressive
conditional heteroscedasticity)
Changes in implied volatility
March implied
Mean volatility = 20%
June implied
September implied
20%
20%
20%
25%
23%
21%
15%
17%
19%
Term Structure of Volatility
time to expiration
Imp
lie
d v
ola
tility
CBOE Risk Management Conference Europe
September 2013
Option Risk Measures
Sheldon Natenberg & Tim Weithers
Chicago Trading Co.
440 South LaSalle St.
Chicago, IL 60605
(312) 863-8000
option
pricing
model
theoretical
value
42.17
exercise price
time to expiration
underlying price
interest rate
volatilityvolatility
theta: θθθθ
delta: ∆: ∆: ∆: ∆
gamma: Γ: Γ: Γ: Γ
rho: ρ
vega: ν: ν: ν: ν
Option Valuation or Option Pricing
Bloomberg SPX Option Valuation Page (OV)
Delta: ∆∆∆∆A measure of how long or short an option makes you.
An option’s underlying price sensitivity:
∆ Hedge ratio
Equivalent position in the underlying (∆ = 742).
If a Call is deep in-the-money (X = 100, S = 1685), ∆ = If a Call is at-the-money (X = 1685, S = 1685), ∆ =
If a Call is out-of-the-money (X = 2000, S = 1685), ∆ =
The probability the option will end up in-the-money.
A measure of directional market (spot price) risk.
S
C
∂
∂=
Delta: ∆∆∆∆
Before, C = 42.17 with S = 1685.
Now, with S = 1686 . . .
since ∆ = dC/dS and ∆ = .4845
dC = +.48, so . . .
new C should = 42.65
(Pretty good, huh?)
Graphically, Delta = Slope
What is ∆ at expiration?
1685-Strike Call Option
0
Value
Underlying Price
X = 1685
+ C
∆∆∆∆ =dC
dS
{dC
{dS
Vega: ν (ν (ν (ν (Tau: τ): τ): τ): τ)
An option’s underlying volatility sensitivity:
ν
The higher the volatility (of the underlying),
the higher option values will be.
Longer-dated options are more volatility sensitive.
A measure of (implied) volatility risk.
σ∂
∂=
C
Vega: νννν (Tau: ττττ)
Before, C = 42.17 with σ = 13.784.
Now, with σ = 14.784 . . .
since ν = dC/dσ and ν = 3.32
dC should = + 3.32, so . . .
new C should = 45.49
(That worked well!)
1685-Strike Call Option
0
Value
Underlying PriceX = 1685
+ C
νννν =dC
dσ
{dC
C(σ=13%) C(σ=14%)
Graphically, Vega
Theta: θθθθ
An option’s sensitivity to the passage of time:
θ
Typically, option values fall over time.
Some refer to theta as “decay”.
If you have a one-year option and a day goes by, . . .
Option values depend on time (in multiple ways).
t
C
∂
∂=
If you have a one-week option and a day goes by, . . .
Theta: θθθθ
Before, C = 42.17 with t = 90 days.
Now, with t = 89 days . . .
since θ = dC/dt and θ = -.21
dC = -.21, so . . .
new C should = 41.96
(Not too bad.)
Graphically, Theta
1685-Strike Call Option
0
Value
Underlying PriceX = 1685
+ C
θθθθ =dC
dt
{dC
Rho: ρ ρ ρ ρ
An option’s underlying volatility sensitivity:
ρ
All these option risk measures assume you hold
all the other “variables” constant.
All these option risk measures are additive
A measure of interest rate risk.
r
C
∂
∂=
Rho: ρρρρ
Gamma: Γ Γ Γ Γ
An option’s Delta sensitivity with respect to price:
Γ
If Delta is the hedge ratio, Gamma is a measure
of how quickly you’re becoming unhedged.
When/where is Gamma large?
A second-order risk measure.
2
2
S
C
S
∂
∂=
∂
∆∂=
Option traders live and die for gamma/volatility.
Gamma: ΓΓΓΓ
Before, ∆ = 48.45 with S = 1685.
Now, with S = 1686 . . .
since Γ = d∆/dS and Γ = 5.79
d∆ should = +5.79 per 1% (16.85),
so . . . d∆ should = +.34
so new ∆ should = 48.79
1685-Strike Call Option
0
Value
Spot Price
X =1685
+ C
∆∆∆∆ = .10
{
dS
{ {
∆∆∆∆ = .50
∆∆∆∆ = .90
Graphically, Gamma = Curvature
Implied Probability Distributions
If you have enough market option prices,
you can back out or “infer” a probability density.
There’s a large academic literature on this:D. Breeden and R. Litzenberger (1978) “Prices of State-Contingent Claims in Option Prices”
J.-C. Jackwerth and M. Rubinstein (1996) “Recovering Probability Distributions from Contemporary Security Prices”
W. Melick and C. Thomas (1997) “Recovering an Asset’s PDF from Option Prices: An Application to Crude Oil”
Y. Ait-Sahalia (2001) “Do Option Markets Correctly Price the Probabilities of Movement of the Underlying Asset?”
The logic behind these sorts of approaches:
99 100 101 S
1.00
If the 99-100-101 butterfly is trading at 0.05, . . .
CBOE Risk Management Conference Europe
September 2013
Capturing Volatility Value
Sheldon Natenberg & Tim Weithers
Chicago Trading Co.
440 South LaSalle St.
Chicago, IL 60605
(312) 863-8000
Suppose an option is trading at an implied
volatility of 20%, but we “know” that the true
realized volatility of the underlying contract
over the life of the option will be 25%.
option price = 4.50 (implied volatility = 20%)
option value = 5.00 (future realized volatility = 25%)
If 25% does in fact turn out to be the actual
volatility, how can we turn the difference
between the price of the option (4.50) and its
value (5.00) into a profit of .50?
buy a call option
the
ore
tica
l va
lue
underlying price
exercise
price
value at expiration?
value prior to
expiration?
buy a call option
the
ore
tica
l va
lue
determine the
option’s delta
(ΔC)
ΔC-ΔC
take an opposing
delta position in the
underlying contract
(-ΔC)
delta neutral or flat
underlying price
current
underlying
price
Due to the option’s curvature, as market conditions
change the position will become unhedged.
the
ore
tica
l va
lue
{
ΔC-ΔC
unhedged
amount
underlying price
current
underlying
price
the
ore
tica
l va
lue
new ΔC
Determine the new delta of the option.
Rehedge the position to return to delta neutral
new -ΔC
underlying price
current
underlying
price
the
ore
tica
l va
lue
new ΔC
new -ΔC
Continue the rehedging process throughout
the life of the option.
underlying price
}
current
underlying
price
the
ore
tica
l va
lue
Dynamic Hedging
underlying price
ΔC
-ΔC
current
underlying
price
Continue the rehedging process throughout
the life of the option.
Suppose we add up all the profit opportunities
over the life of the option which result from the
rehedging process.
the option’s theoretical value
The rehedging process is a type of
statistical arbitrage.
What should they add up to?
Each time the position becomes unhedged
there is a potential profit opportunity. We can
capture this profit by rehedging the position.
