Part 3 Derivatives with exotic embedded features

1

• 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

2
Path dependent feature
asset price
time
T
t0
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.
3
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.

4

• 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).

5
asset price
Knock-in and Knock-out
up-barrier
barrier level
time
knock-out
Extinguished or activated upon achievement of
relevant asset price level.
6
Features
barrier periods may cover only part of the
options 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.
7
it is typically positive (for a call) but it
becomes negative as it approaches the barrier
delta
?
gamma
demonstrate very high gamma when the asset
price is close to barrier
?
usually higher than the non-barrier counterpart
vega
?
pattern of time decay is not smooth, with sharp
discontinuity when close to barrier
?
theta
8
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.
9

• 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 SP 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.

10
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
where cE(S, t) is the price of the vanilla
counterpart.
11
Difficulties with dynamic hedging of barrier
securities

• 1. The underlying asset as the dynamic hedging
  instrument is
• insensitive to changes in volatility. Options
  vega for
• barrier securities is usually high. Vega risk
  is unhedgeable
• except with other option-like securities.
• Barrier options often have regions of high gamma,
  which
• greatly increase the hedging error associated
  with dynamic
• hedging.

12

• 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.

13
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 lt 90 or S gt
110, otherwise he would get back the notional.
14
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.
15
Asian options
Asian options are averaging options whose
terminal payoff depends on some form of average.
Arithmetic averaging Geometric averaging

Used by investors who are interested to hedge
against the average price of a commodity over a
period, rather than the end-of-the- period
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 thinly- traded commodities).

16
Asian Averaging Options
Average rate call
Average strike call
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.
?
17
Fixed strike Asian call
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.


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.


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• 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.

19

• Shout feature
• The terminal payoff is guaranteed to be at
  least St K.
• Obviously, the holder should shout only when St
  gt 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.
•  

20

• 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.

21

• 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).

22

• 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 holders option.

23

• Example - SP 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.

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Lookback options
Reset the strike to the realized lowest or
highest market price during the lookback period.
Payoff of the following forms
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.
25

• 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.

26
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.
27
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).
28
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 investors view that is contrary to a
  particular forward rate curve.

29
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.
30
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
bonds 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 notes payout is
capped at par. When the total return of the
principal and the accrued coupon reaches par, the
zero coupon accrual note matures.  
31
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.  
32
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.
33
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 investors
view on interest rates.
34

• 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.

35
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.
Off-balance sheet financial instruments that
allow end users to buy and sell credit risk.
36
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.
• 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 underlyings
  credit risk is challenging.

37
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
managers 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.
38
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.)
39
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 gt strike spread at maturity
Payout notional ? (final spread strike
spread)
40
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 lt 330 investor earns the
premium spread gt 330 investor acquires the
bond at 350 bps
41
Implied volatilities
The only unobservable parameter in the
Black-Scholes formulas is the volatility value,
s. 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.
42
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.
43
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.

44

• 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.

45
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-of- the-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.
46
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.

47
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).
48
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
49
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
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).
1.
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.
2.
50
The market behavior of higher probability of
large decline in stock index is better known to
practitioners after Oct., 87 market crash.
Implied volatility
The market price of out-of-the-money call
(puts) has become cheaper (more expensive) than
the Black- Scholes theoretical price after the
1987 crash because of the thickening
(thinning) of the left-end (right-end) tail of
the terminal asset price distribution.

1.0
A typical pattern of post-crash smile. The
implied volatility drops against X/S.
51
Theoretical and implied volatilities

• Theoretical volatility
• When valuing an option, a traders 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.

52
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 volatility
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.

?
?
?
?
?
?
X/S
53
Time dependent volatility
54
The Black-Scholes formulas remain valid for time
dependent volatility except that
is used to replace s.
How to obtain s(t) given the implied volatility
measured at time t ? of a European option
expiring at time t. Now
55
so that
Differentiate with respect to t, we obtain
56
Practically, we do not have a continuous
differentiable implied volatility function
, but rather implied volatilities are
available at discrete instants ti. Suppose we
assume s(t) to be piecewise constant over (ti-1,
ti), then
57
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.
58
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.
59
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 traders theoretical value is higher than
  the implied volatility,
• he would buy options since he believes they are
  undervalued.

60
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61
Market data
Stock price 99, call price 5.46, delta 0.5
portfolio A 50 shares of stock portfolio B
100 call options
profit
solid line option portfolio dotted line stock
portfolio
?
stock price
87
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.
62
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.

63
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.

64
Gamma trading and vega trading
Time decay profit
Gamma trading Net profit from realized
volatility
Vega trading Net profit from changes in
implied volatility
65
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.

66
Balance between gamma-based and vega-based
volatility trading

• 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.
• 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.

67
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.
68

• 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
• The rest of the market begin to agree with him,
  then the
• option price will mark up. He gains by unwinding
  his
• option position.
• 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.

69
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.
70
Variance swap contract (contd)
where n number of trading days to maturity N
number of trading days in one year (252) m
realized average of the logarithm of daily return
of the stock
71
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
72
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 long-term 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.

73
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.