Transcript No Slide Title

Part 3 - Derivatives with exotic embedded features
• Knock-out and knock-in features
• Averaging feature
• Lookback feature
• Reset and shout feature
• Chooser feature
• Credit derivatives
• Volatility trading and products
1
Path dependent feature
asset price
t0
time
T
The payoff of the option contract depends on
the realization of the asset price within the
whole life or part of the life of the option.
2
Most common types of path dependent options
 Option is knocked out or activated when the asset price
breaches some threshold value  Barrier Options.
 Average value of the asset prices over a certain period is
used as the strike  Asian Options.
 The strike price is determined by the realized maximum
value of the asset price over a certain period
 Lookback Options.
3
The market for exotic options
Development of exotic products
• increased flexibility for risk transfer and hedging
• highly structured expression of expectation of asset
price movements
• facilitation of trading in new risk dimension such as the
correlation between key financial variables
Modest volumes of trading and a relative lack of liquidity.
These are associated with the difficulty in pricing,
hedging / replicating (due to complex risk profiles).
4
asset price
Knock-in and Knock-out
up-barrier
barrier level
knock-out
Extinguished or activated upon achievement of
relevant asset price level.
5
time
Features
*
*
*
*
*
*
barrier periods may cover only part of the option’s life
discretely monitored
can be in both European and American exercise format
barrier variable other than the underlying asset price
two-sided barriers (up-down) and sequential breaching
rebate may be paid upon knock out
Advantage
To achieve savings in premium; no need to pay for states
believed to be unlikely to occur.
6
delta:
it is typically positive (for a call) but it becomes
negative as it approaches the barrier
 gamma: demonstrate very high gamma when the asset price
is close to barrier
vega:
theta:
usually higher than the non-barrier counterpart
pattern of time decay is not smooth, with sharp
discontinuity when close to barrier
7


Hedging difficulties  circuit breaker effect upon knock out
Market manipulation near barrier to trigger
knock-out.
“Soros (1995)  knock-out options relate to ordinary
options the way crack relates to cocaine.”
8
More complex versions of barrier options
• The option could have two barrier levels (double
barriers), one above the and below the current level of
the index. The knockout condition then be (i) touching
either one, or (ii) sequential breaching.
• The barrier level could be based on another market
(external barrier), say, the knock out of FTSE-100
option could be subject to the S&P 500 trading below
a given level.
• The barrier condition could exist for only part of the
life time of the option (partial barriers).
• Variable rather than a fixed barrier.
9
Down-and-out call option
The call option is nullified when the asset price hits a down barrier B
during the life of the option. The price formula for the continuously
monitored down-and-out barrier call option is given by
1 2 2r
2

S  B
c( S ,  )  cE ( S ,  )    cE 
,  
B
 S 
where cE(S, ) is the price of the vanilla counterpart.
cvanilla  cdown and out  cdown and in .
1 2 2r
2

S  B

The second term   cE  ,   then gives the price of the
B
 S 
corresponding down-and-in call option.
10
Difficulties with dynamic hedging of barrier securities
1. The underlying asset as the dynamic hedging instrument is
insensitive to changes in volatility. Option’s vega for
barrier securities is usually high. Vega risk is unhedgeable
except with other option-like securities.
2. Barrier options often have regions of high gamma, which
greatly increase the hedging error associated with dynamic
hedging.
11
Digital options (binary)
• A pre-determined fixed payout if the option is at- or
in-the-money (also called all-or-nothing, bet or lottery
options). Primarily European in style.
• Suited to markets where support and resistance levels
are found, say, in the currency and bond markets. If an
investor believes that a currency will not fall below a
certain level, he can write a digital option to earn
premium.
• Writer faced with greater hedging challenges due to
large gamma.
12
Note with embedded options
Customer pays notional of 100 today. We pay a coupon of
x% (p.a.) in 3 months. If spot price is above 100 at the end of
the 3-month period, then the deal is terminated and we pay
back 100 to him on that date.
If the spot price is below 100, then a further coupon of 2%
(p.a.) is paid in 6 months. The final redemption amount that
the customer would obtain is given by
Customer gets notional  S/100 if S < 90 or S > 110,
otherwise he would get back the notional.
13
The problem is to work out x%.
The interesting thing is the barrier condition at the end of 3
months. The final payout for the customer can be decomposed
into a combination of call option, put option and binary options.
14
Asian options
Asian options are averaging options whose terminal payoff depends on
some form of average.
1 n
 Si
Arithmetic averaging =
n i 1 1
n
n


Geometric averaging =   Si 
 i 1 
• Used by investors who are interested to hedge against the average
price of a commodity over a period, rather than the end-of-theperiod price
e.g. Japanese exporters to the US, who are receiving stream of US
dollar receipts over certain period, may use the Asian currency
option to hedge the currency exposure.
• To minimize the impact of abnormal price fluctuation near expiration
(avoid the price manipulation near expiration, in particular for 15thinlytraded commodities).
Asian Averaging Options
Average rate call: maxSAVE  X ,0
Average strike call: maxST  SAVE ,0
S AVE
1

M
M
 S (t )
i 1
i
or
S AVE  S (t1 ) S (t2 )  S (t M )
1
M
Uses
 Exposure as a future series of asset prices e.g. cost of
production is sensitive to the prices of raw material.
 To prevent abnormal price manipulation on expiration
date, arising perhaps from a lack of depth in the market.
16
Fixed strike Asian call: max(A  X ,0)
•
•
The option premium is expected to be lower than that of the vanilla
options since the volatility of the average asset value should be
lower than that of the terminal asset value;
The delta and gamma tend to zero as time is approaching expiration.
Floating strike Asian call: max(ST  A,0)
•
•
Set the strike to the average of prices over a period so as to avoid
the exposure of market.
The delta and gamma tend to that of the vanilla option with
identical expiration data and strike equal to the average.
17
Shout options
•
The payoff upon shouting is another derivative with
contractual specifications different from the original
derivative.
•
The embedded shout feature in a call option allows its
holder to lock in the profit via shouting while retaining the
right to benefit from any future upside move in the payoff.
The terminal payoff of a shout call option is the form:
C = max(ST – K, L – K, 0),
where K is the strike price, ST is the terminal stock price and L
is some ladder value installed at shouting.
The ladder value L is set to be the prevailing stock price St at
the shouting instant t.
18
Shout feature
•
The terminal payoff is guaranteed to be at least St – K.
•
Obviously, the holder should shout only when St > K.
•
The number of shouting rights throughout the life of the
contract may be more than one.
•
Some other restrictions may apply, say, the shouting
instants are limited to some predetermined times.
19
Reset feature
This is the right given to the derivative holder to reset
certain contract specifications in the original derivative.
Strike reset – strike reset to a lower strike for a call or to a
higher strike for a put.
Maturity reset – extension of the maturity of a bond.
Constraints on reset
• A limit to the magnitude of the strike adjustment.
• Triggered by underlying price reaching certain level.
• Reset allowed only on specific dates or limited
period.
20
Example - Reset strike put option
• The strike price is reset to the prevailing stock
price upon shouting.
The shouting payoff is given by
max(St – ST, 0) = max(ST – K, St – K, 0) – (ST – K).
• The shout call option can be replicated by the reset
strike put and a forward contract
(put-call parity relation).
21
Example – Extendible bonds
• Gives the holder the option of extending the term
of the instrument, on or before a fixed date at a
pre-determined coupon rate.
The 5.5 percent Government of Canada extendible
bond was issued on October 1, 1959. It was
exchangeable on or before June 1, 1962 into 5.5
percent bonds maturing October 1, 1975.
The three year initial bond was extendible into a 16
year bond at the holder’s option.
22
Example - S&P 500 index bear warrants with a
three-month reset
• Launched in the Chicago Board Options Exchange
and the New York Stock Exchange (late 1996).
• These warrants are index puts, where the strike
price is automatically reset to the prevailing index
value if the index value is higher than the original
strike price on the reset date three months after the
original issuance.
23
Lookback options
Reset the strike to the realized lowest or highest market price
during the lookback period. Payoff of the following forms:





[ 0,T ]
[ 0,T ]
[ 0,T ]
[ 0,T ]
max Smax
 X ,0 , max Smax
 ST ,0 , Smax
 Smin
 X ,0 , etc.
Partial lookbacks: selects a subset of the period from
commencement to expiry as the lookback period. The
premium increases with the length of the lookback period.
Strike bonus rollover hedging strategy
For the floating strike put, whenever a new maximum asset
price is realized, replace the old put with a new put that has
strike equal to the new maximum.
24
Uses of lookback options
Offshore debt or equity investments where the investor
wishes to achieve the best currency over the relevant time
period and wishes to uncouple the timing of the
investment from the currency rate setting.
Perspectives of holder
• Most advantageous if the realized volatility of the
underlying asset price is higher that the implied
volatility.
• There will be a sharp move in the underlying asset
price but is unsure when and for how long the price
will move.
25
Callable Options
Consider a 3-year call option with a fixed strike. After the first
year and at every 6-month interval thereafter, the issuer has
the right to call back the option. Upon calling, the holder is
forced to exercise at the intrinsic value, or if the option is
out-of-the-money, then the call option is terminated without
any payment.
Questions:1. Explain why the price of this callable option lies within
the prices of the 1-year and 3-year non-callable counterparts.
2. What is the impact of dividend yield on the optimal calling
policy of this callable option?
26
Range notes
Provide investors with an above market coupon, but they must agree
to forego coupon payments when LIBOR falls outside prescribed
bounds.
Example
Suppose the market coupon for a conventional note is 6.5%.
A range note pays 8.8% coupon semi-annually conditional on
the 6-month LIBOR remains within 4.5-7.5%. The true coupon is
computed on a daily accrual basis (coupons are counted on those dates
when the LIBOR falls within the range).
27
Corridor risk
•
The investor loses coupon of rate 8.8% when LIBOR either
exceeds 7.5% or below 4.5%. This is like the payoff of a digital
cap and digital floor, respectively. This is called the “corridor
risk”.
•
In essence, the investor shorts these two options in return for a
higher coupon rate – selling volatility.
•
Investors have a strong view that rates will stay within a range
and often they are structured to reflect an investor’s view that is
contrary to a particular forward rate curve.
28
Example
The Kingdom of Sweden issued dollar-denominated corridor
Eurobonds in January 1994. The 200 million 2-year Sweden deal,
for example, paid out Libor + 75 bp when the 3-month Libor fell
between the following rates:
07/02/94 – 07/08/94; 3% to 4%
07/08/94 – 07/02/95; 3% to 4.75%
07/02/95 – 07/08/95; 3% to 5.50%
07/08/95 – 07/02/96; 3% to 6%
The principal is fully protected, and the coupon is sacrificed only
on days in which the 3-month Libor is outside the range.
29
Zero coupon accrual notes
A hybrid version of a zero-coupon bond and an accrual note.
•
In a plain vanilla accrual note, an investor receives a
coupon based on the number of days that a fixed income
benchmark rate stays within a pre-specified range.
•
In a zero coupon bond, the investor knows at the time of
purchase the bond’s maturity and effective yield.
The zero coupon accrual note investor buys the note at a
discount. Instead of a set maturity, there is a maximum
maturity date. The note’s payout is capped at par. When the
total return of the principal and the accrued coupon reaches
par, the zero coupon accrual note matures.
30
Uses of zero coupon accrual notes
In a rising interest rate environment, the maturity of the notes
accelerates. Fixed income investors are thus able to reinvest
their capital at the prevailing higher rates.
•
The inherent high convexity built into the zero coupon
accrual notes benefits the buyer greatly by reducing the
duration of the note as rates rise while lengthening
duration as rates fall.
•
Unlike range notes where ranges are specified, this
product allows investors to bet on a general move up in
rates rather than the actual move in basis points.
31
Example of zero coupon accrual note
A 3-year zero coupon accrual note linked to 6-month LIBOR
sold at a price of 90 and a minimum annualized coupon of
2.5% (minimum coupon feature).
•
If the 6-month LIBOR does not rise substantially during
the 3-year life of the note, the note will mature in 3 years.
32
Callable Range Accrual Note
• The call options enable the investor to enhance his yield,
compared to a standard Range Accrual Note. Even if the Note is
called on the first call date, he would have benefited from a high
coupon compared to the market conditions.
• The Range Accrual structures are very popular with investors,
especially when the implied volatility is high compared to the
historical movements of the underlying index.
• The Note will pay a higher coupon if, based on the forward
curve, there is a high probability that the reference index will
fix outside the range.
The range can be tailored to match investor’s view on interest rates.
33
• The graph below shows the forward distribution of the 6m Euribor as
well as the upper barriers of the structure, and thus the probability for
the index to fix within the range according to market conditions at the
time of pricing.
34
Risk de-aggregation
Credit derivatives are over-the-counter contracts which allow the
isolation and management of credit risk from all other components of
risk.
interest
rate risk
credit
risk
volatility
risk
FX
risk
Off-balance sheet financial instruments that allow end users to buy
and sell credit risk.
35
Product nature of credit derivatives
Payoff depends on the occurrence of a credit event:
•
•
•
default: any non-compliance with the exact specification of a contract
price or yield change of a bond
credit rating downgrade
In the case of the default of a bond, any loss in value from the default date until the
pricing date (a specified time period after the default date) becomes the value of
the underlying.
Credit derivatives can take the form of swaps or options.
1.
2.
In a credit swap, one party pays a fixed cashflow stream and the other party
pays only if a credit event occurs (or payment based on yield spread).
A credit option would require the upfront premium and would pay off based
on the occurrence of a credit event (or on a yield spread).
Pricing a credit derivative is not straightforward since modeling the
stochastic process driving the underlying’s credit risk is challenging.
36
Uses of credit derivatives
To hedge against an increase in risk, or to gain exposure to a market
with higher risk.
•
Creating customized exposure; e.g. gain exposure to Russian debts
(rated below the manager’s criteria per her investment mandate).
•
Leveraging credit views - restructuring the risk/return profiles of
credits.
•
Allow investors to eliminate credit risk from other risks in the
investment instruments.
Credit derivatives allow investors to take advantage of relative value
opportunities by exploiting inefficiencies in the credit markets.
37
Credit spread derivatives
• Options, forwards and swaps that are linked to credit spread.
Credit spread = yield of debt – risk-free or reference yield
• Investors gain protection from any degree of credit deterioration
resulting from ratings downgrade, poor earnings etc.
(This is unlike default swaps which provide protection against
defaults and other clearly defined ‘credit events’.)
38
Credit spread option
Use credit spread option to
• hedge against rising credit spreads;
• target the future purchase of assets at favorable prices.
Example
An investor wishing to buy a bond at a price below market can sell
a credit spread option to target the purchase of that bond if the credit
spread increases (earn the premium if spread narrows).
at trade date, option premium
counterparty
investor
if spread > strike spread at maturity
Payout = notional  (final spread – strike spread)+
39
Example
The holder of the put has the right to sell the bond at the strike spread
(say, spread = 330 bps) when the spread moves above the strike spread
(corresponding to drop of bond price).
May be used to target the future purchase of an asset
at a favorable price.
The investor intends to purchase the bond below current market price
(300 bps above US Treasury) in the next year and has targeted a
forward purchase price corresponding to a spread of 350 bps. She
sells for 20 bps a one-year credit spread put struck at 330 bps to a
counterparty (currently holding the bond and would like to protect
the market price against spread above 330 bps).
• spread < 330; investor earns the premium
• spread > 330; investor acquires the bond at 350 bps
40
Implied volatilities
The only unobservable parameter in the Black-Scholes formulas is the
volatility value, . By inputting an estimated volatility value, we obtain
the option price. Conversely, given the market price of an option, we
can back out the corresponding Black-Scholes implied volatility.
• Several implied volatility values obtained simultaneously from
different options (varying strikes and maturities) on the same
underlying asset provide the market view about the volatility of
the stochastic movement of the asset price.
41
Black wrote
“It is rare that the value of an option comes out exactly equal to the price at which
it trades on the exchange.
There are several reasons for a difference between the value and price:
(i) we may have the correct value;
(ii) the option price may be out of line;
(iii) we may have used the wrong inputs to the Black-Scholes formula;
(iv) the Black-Scholes may be wrong.
Normally, all reasons play a part in explaining a difference between value and price.”
The market prices are correct (in the presence of sufficient
liquidity) and one should build a model around the prices.
42
Different volatilities for different strike prices
Stock options – higher volatilities at lower strike and lower
volatilities at higher strikes
• In a falling market, everyone needs out-of the-money puts
for insurance and will pay a higher price for the lower strike
options.
• Equity fund managers are long billions of dollars worth of
stock and writing out-of-the-money call options against their
holdings as a way of generating extra income.
43
Commodity options – higher volatilities at higher strike and
lower volatilities at lower strikes
• Government intervention – no worry about a large price fall.
Speculators are tempted to sell puts aggressively.
• Risk of shortages – no upper limit on the price. Demand for
higher strike price options.
44
Volatility smiles
Interest rate options – at-the-money option has a low volatility
and either side the volatility is higher
Propensity to sell at-the-money options and buy out-ofthe-money options.
For example, in the butterfly strategy, two at-the-money
options are sold and one-out-of the-money option and one
in-the-money option are bought.
45
Different volatilities across time
Supply and demand
When markets are very quiet, the implied volatilities of the near month
options are generally lower than those of the far month. When markets
are very volatile, the reverse is generally true.
• In very volatile markets, everyone wants or needs to load with gamma.
Near-dated options provide the most gamma and the resultant buying
pressure will have the effect of pushing prices up.
• In quiet markets no one wants a portfolio long of near dated options.
Use of a two-dimensional implied volatility matrix.
46
Floating volatilities
As the stock price moves, the entire skewed profile also moves.
This is because what was out-of-the-money option now
becomes in-the-money option.
Example
If an investor is long a given option and believes that the
market will price it at a lower volatility at a higher stock price
then he may adjust the delta downwards (since the price
appreciation is lower with a lower volatility).
47
Terminal asset price distribution as implied by
market data
probability
In real markets, it is common that
when the asset price is high,
volatility tends to decrease, making
it less probable for high asset price
to be realized. When the asset
price is low, volatility tends to
increase, so it is more probable
that the asset price plummets
further down.
S
solid curve: distribution as implied by
market data
dotted curve: theoretical lognormal
distribution
48
Extreme events in stock price movements
Probability distributions of stock market returns have typically been
estimated from historical time series. Unfortunately, common
hypotheses may not capture the probability of extreme events, and the
events of interest are rare and may not be present in the historical record.
Examples
1. On October 19, 1987, the two-month S & P 500 futures price fell 29%.
Under the lognormal hypothesis of annualized volatility of 20%, this
is a 27 standard deviation event with probability 10160 (virtually
impossible).
2. On October 13, 1989, the S & P 500 index fell about 6%, a 5 standard
deviation event. Under the maintained hypothesis, this should occur
only once in 14,756 years.
49
The market behavior of higher probability of large decline in stock
index is better known to practitioners after Oct., 87 market crash.
• The market price of
out-of-the-money call (puts)
has become cheaper (more
expensive) than the BlackScholes theoretical price after
the 1987 crash because of
the thickening (thinning)
of the left-end (right-end)
tail of the terminal asset
price distribution.
Implied
volatility
X/S
1.0
A typical pattern of post-crash smile.
The implied volatility drops against X/S.
50
Theoretical and implied volatilities
Theoretical volatility
• When valuing an option, a trader’s theoretical volatility will be a
critical input in a pricing model.
• The strategy of trading on theoretical volatilities involves holding
the option until expiry – common strategy of option users.
Market implied volatility
• Volatility extrapolated from, or implied by, an option price.
• Trading on implied volatility involves implementing and reversing
positions over short time periods.
51
It is always necessary to provide prices of European options of strikes
and expirations that may not appear in the market. These prices are
supplied by means of interpolation (within data range) or extrapolation
(outside data range).
Implied
• A smooth curve is plotted
through the data points
(shown as “crosses”). The
estimated implied volatility
at a given strike can be
read off from the dotted
point on the curve.
volatility






X/S
52
Time dependent volatility
• Given the market prices of European call options with different
maturities (all have the strike prices of 105, current asset price is
106.25 and short-term interest rate over the period is flat at 5.6%).
maturity
Value
Implied volatility
•
1-month
3.50
21.2%
3-month
5.76
30.5%
7-month
7.97
19.4%
Extend the assumption of constant volatility to allow for time
dependent deterministic volatility (t).
53
The Black-Scholes formulas remain valid for time dependent volatility
except that
1 T
2

(

)
d
t
T t
is used to replace .
How to obtain (t) given the implied volatility measured at time t  of
a European option expiring at time t. Now
 imp (t , t ) 

1
t
2

(

)
d

 t
t t
54
so that
  ( ) d   (t , t )(t  t ).
t
t
2
2
imp


Differentiate with respect to t, we obtain
 (t )   imp (t  , t ) 2  2(t  t  ) imp (t  , t )
2 imp (t  , t )
t
.
55
Practically, we do not have a continuous differentiable implied volatility
function  imp (t  , t ) , but rather implied volatilities are available at
discrete instants ti. Suppose we assume (t) to be piecewise constant
over (ti1, ti), then
2
2
(ti  t  ) imp
(t  , ti )  (ti 1  t  ) imp
(t  , ti 1 )
 tii1  2 ( )d   2 (t )(ti  ti 1 ),
t
 (t ) 
ti 1  t  ti ,
2
2
(ti  t  ) imp
(t  , ti 1 )  (ti 1  t  ) imp
(t  , ti 1 )
ti  ti 1
,
ti 1  t  ti .
56
Implied volatility tree
An implied volatility tree is a binomial tree that prices a given set of
input options correctly.
The implied volatility trees are used:
1. To compute hedge parameters that make sense for the given option
market.
2. To price non-standard and exotic options.
The implied volatility tree model uses all of the implied volatilities of
options on the underlying - it deduces the best flexible binomial tree
(or trinomial tree) based on all the implied volatilities.
57
Volatility trading
Trading based on taking a view on market volatility different from
that contained in the current set of market prices. This is different
from position trading where the trades are based on the expectation
of where prices are going.
Example
A certain stock is trading at $100. Two one-year calls with strikes of
$100 and $110 priced at $5.98 and $5.04, respectively.
These prices imply volatilities of 15% and 22%, respectively.
Strategy Long the cheap $100 strike option and short of the expensive
$110 strike option.
58
Trading volatilities
Short term players
• Sensitive to the market prices of the options.
• This is more of a speculative trading strategy, applicable only to
liquid options markets, where the cost of trading positions is small
relative to spreads captured in implied volatility moves.
Long term players
• If a trader’s theoretical value is higher than the implied volatility,
he would buy options since he believes they are undervalued.
59
60
Market data: Stock price = $99, call price = $5.46, delta = 0.5
portfolio A: 50 shares of stock;  A
t 0
 $4,550
B
t 0
 $546
portfolio B: 100 call options;
profit
solid line: option portfolio
dotted line: stock portfolio
87

stock price
99
600
Both portfolios are delta equivalent.
Since the option price curve is concave upward, the call option portfolio
always outperforms the delta equivalent stock portfolio.
61
Long volatility trade
Whichever way the stock price moves, the holder always make a profit.
This is the essence of the long volatility trade.
• By rehedging, one is forced to sell in rising markets and buy in
falling market – trade in the opposite direction of the market trend.
Where is the catch
• The option loses time value throughout the life of the option.
Long volatility strategy
Competition between the original price paid and the subsequent
volatility experienced. If the price paid is low and the volatility is high,
the long volatility player will win overall.
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Vega risk
Vega is defined as the change in option price caused by a
change in volatility of 1%.
• Shorter dated options are less sensitive of volatility inputs.
That is, vega decreases with time.
• Near-the-money options are most sensitive and deep
out-of-the-money options are less sensitive.
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Gamma trading and vega trading
(asset price)2
 (implied volatility)2
Time decay profit: positiongamma
2
Gamma trading
Net profit from realized volatility
(asset price)2
positiongamma
 [(realized volatility)2  (implied volatility)2 ]
2
Vega trading
Net profit from changes in implied volatility
vega  (currentimplied volatility  originalimplied volatility)
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Maturity and moneyness
The ability of individual derivative positions to realize profits from
gamma and vega trading is crucially dependent on the average
maturity and degree of moneyness of the derivatives book.
• For at-the-money options, long maturity options display high vega
and low gamma; short maturity options display low vega and high
gamma.
• For out-of-the-money options, long maturity options display lower
vega and high gamma, and short maturity options higher vega and
lower gamma.
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Balance between gamma-based and
vega-based volatility trading
1. If a trader desires high gamma but zero vega exposure, then
a suitable position would be a large quantity of short
maturity at-the-money options hedged with a small quantity
of long maturity at-the-money options.
2. If a trader desires high vega but zero gamma exposure, then
a suitable position would be a large quantity of long
maturity at-the-money options hedged with a small quantity
of short maturity at-the-money options.
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Long gamma – holding a straddle
A trader believes that the current implied volatility of
at-the-money options is lower than he expects to be realized.
He may buy a straddle: a combination of an at-the-money call
and an at-the-money put to acquire a delta neutral, gamma
position.
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Trading mispriced options
If options are offered at an implied volatility of 15% and a
manager believes that the real volatility is going to be higher
in the future, say, 25%. How to profit?
He should set up a delta neutral portfolio.
If his prediction is correct, he can profit in two ways:
1. The rest of the market begin to agree with him, then the
option price will mark up. He gains by unwinding his
option position.
2. The market continues to price options at 15%,. He keeps the
portfolio delta neutral (delta calculated based on market
volatility). His rehedging profit will exceed the time decay
losses.
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Variance swap contract
The terminal payoff of a variance swap contract is
notional  (v  strike)
where v is the realized annualized variance of the logarithm of the daily
return of the stock.
2

 
N n1 Si 1
   ln
v
   
n  1  i  0  Si
 

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Variance swap contract (cont’d)
where n = number of trading days to maturity
N = number of trading days in one year (252)
 = realized average of the logarithm of daily return of the stock
1 n 1 Si 1 1 S n
  ln
 ln .
n i 0 Si n S 0
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• The payoff could be positive or negative.
• The objective is to find the fair price of the strike, as indicated by the
prices of various instruments on the trade date, such that the initial
value of the swap is zero.
Observe that


  ln Si1   ln Si1  2 
N 
Si
Si 
.
v

 n  
n 1 
n

 

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Volatility Note
• A Volatility Note is an interest rate investment product, which pays a
coupon linked to the absolute variation of an Index over a period of
time.
• The coupon is equal to Cn = G  Abs (Indexn – Indexn1).
• The Volatility Note represents a natural hedging solution to longterm bond investors, such as insurance companies, whose portfolios
bear natural negative volatility:
(a) if rates rise, the value of their existing portfolio of fixed rate
vanilla and callable bonds will fall,
(b) if rates fall they will be unable to reinvest any income at a
reasonable level.
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Opportunity
• In a volatile market, the volatility bond investor takes advantage
of any movements of the Index, without having to take a view on
the direction of the market.
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