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Part 3 - Derivatives with exotic embedded features Knock-out and knock-in features Averaging feature Lookback feature Reset and shout feature

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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

asset price

time

t0

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.

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.

- 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).

Knock-in and Knock-out

up-barrier

barrier level

time

knock-out

Extinguished or activated upon achievement of relevant asset price level.

* 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.

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:

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.”

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.

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

- 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.
- Barrier options often have regions of high gamma, which
- greatly increase the hedging error associated with dynamic
- hedging.

- 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.

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-

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).

•

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.

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.

•

•

- 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.

- 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.

- 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.

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).

- 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.

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.

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.

- 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.

Explain why the price of this callable option lies within

the prices of the 1-year and 3-year non-callable counterparts.

1.

2.

What is the impact of dividend yield on the optimal calling

policy of this callable option?

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.

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).

- 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.

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.

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.

- 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.

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.

risk

interest

rate risk

FX

risk

volatility

risk

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.

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 underlying’s credit risk is challenging.

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.

• 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’.)

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)+

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.

•

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.

“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

- 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.

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.

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.

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.

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).

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

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.

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.

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.

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

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

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.

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.

- 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.

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.

- 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.

- 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.

Gamma trading and vega trading

Time decay profit:

Gamma trading Net profit from realized volatility

Vega trading Net profit from changes in implied volatility

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.

•

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.

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.

- 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.

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)

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

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

- 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.

- 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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