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. Path dependent feature. asset price. time. t 0. T.
Path dependent feature
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.
Most common types of path dependent options
breaches some threshold value Barrier Options.
value of the asset price over a certain period
Knock-in and Knock-out
Extinguished or activated upon achievement of relevant asset price level.
*barrier periods may cover only part of the option’s life
*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
To achieve savings in premium; no need to pay for states
believed to be unlikely to occur.
it is typically positive (for a call) but it becomes
negative as it approaches the barrier
demonstrate very high gamma when the asset price
is close to barrier
usually higher than the non-barrier counterpart
pattern of time decay is not smooth, with sharp
discontinuity when close to barrier
Hedging difficulties circuit breaker effect upon knock out
Market manipulation near barrier to trigger
“Soros (1995) knock-out options relate to ordinary
options the way crack relates to cocaine.”
The second term then gives the price of the
corresponding down-and-in call option.
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.
Difficulties with dynamic hedging of barrier securities
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.
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.
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-
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-
Asian Averaging Options
Average rate call:
Average strike call:
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.
Floating strike Asian call:
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.
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.
Explain why the price of this callable option lies within
the prices of the 1-year and 3-year non-callable counterparts.
What is the impact of dividend yield on the optimal calling
policy of this callable option?
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
Provide investors with an above market coupon, but they must agree
to forego coupon payments when LIBOR falls outside prescribed
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).
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.
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.
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.
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.
Callable Range Accrual Note
The range can be tailored to match investor’s view on interest rates.
Credit derivatives are over-the-counter contracts which allow the
isolation and management of credit risk from all other components of
Off-balance sheet financial instruments that allow end users to buy
and sell credit risk.
Product nature of credit derivatives
Payoff depends on the occurrence of a credit event:
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
•Allow investors to eliminate credit risk from other risks in the
Credit derivatives allow investors to take advantage of relative value
opportunities by exploiting inefficiencies in the credit markets.
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’.)
Credit spread option
Use credit spread option to
•hedge against rising credit spreads;
•target the future purchase of assets at favorable prices.
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
if spread > strike spread at maturity
Payout = notional (final spread – strike spread)+
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
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.
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.
“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.
Different volatilities for different strike prices
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-
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.
Different volatilities across time
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.
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).
Terminal asset price distribution as implied by
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
solid curve:distribution as implied by
dotted curve:theoretical lognormal
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.
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
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.
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 Black-
Scholes theoretical price after
the 1987 crash because of
the thickening (thinning)
of the left-end (right-end)
tail of the terminal asset
A typical pattern of post-crash smile.
The implied volatility drops against X/S.
Theoretical and implied volatilities
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).
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.
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%).
Extend the assumption of constant volatility to allow for time
dependent deterministic volatilitys(t).
Time dependent volatility
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
Differentiate with respect to t, we obtain
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
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
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.
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.
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.
Stock price = $99, call price = $5.46, delta = 0.5
portfolio A: 50 shares of stock;
portfolio B: 100 call options;
solid line: option portfolio
dotted line: stock portfolio
Both portfolios are delta equivalent.
Since the option price curve is concave upward, the call option portfolio
always outperforms the delta equivalent stock portfolio.
Long volatility trade
Gamma trading and vega trading
Time decay profit:
Gamma trading Net profit from realized volatility
Vega trading Net profit from changes in implied volatility
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
For out-of-the-money options, long maturity options display lower
vega and high gamma, and short maturity options higher vega and
Balance between gamma-based and
vega-based volatility trading
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
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.
Variance swap contract (cont’d)
wheren = 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
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.