On October 23rd, 2014, we updated our
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increased flexibility for risk transfer and hedging
highly structured expression of expectation of asset
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 Extinguished or activated upon achievement of relevant asset price level. asset price up-barrier barrier level time knock-out
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.
delta: gamma: 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 vega: usually higher than the non-barrier counterpart theta: 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 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.
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 c E ( S , ) is the price of the vanilla counterpart. The second term then gives the price of the corresponding down-and-in call option.
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
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.
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 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). •
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.
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. • Floating strike Asian call:
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( S T – K, L – K , 0),
where K is the strike price, S T 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 S t at the shouting instant t .
The terminal payoff is guaranteed to be at least S t – K .
Obviously, the holder should shout only when S t > 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( S t – S T , 0) = max( S T – K, S t – K , 0) – ( S T – K ).
The shout call option can be replicated by the reset strike put and a forward contract
(put-call parity relation).
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.
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.
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.
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.
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?
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).
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.
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.
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 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 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. interest rate risk volatility risk FX risk 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
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.
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.
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’.)
if spread > strike spread at maturity 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). investor counterparty at trade date, option premium Payout = notional (final spread – strike spread) +
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
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.
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.
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
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.
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.
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.
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).
Terminal asset price distribution as implied by market data 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. probability 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 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.
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 price distribution. 1.0 Implied volatility A typical pattern of post-crash smile. The implied volatility drops against X / S . X / S
Theoretical and implied volatilities
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). • 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. Implied volatility X / S
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%). • Extend the assumption of constant volatility to allow for time dependent deterministic volatility ( t ).
The Black-Scholes formulas remain valid for time dependent volatility except that 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
so that 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 t i . Suppose we assume ( t ) to be piecewise constant over ( t i 1 , t i ), 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 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.
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.
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.
Market data: 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 87 99 600 stock price profit 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
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
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. •
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.
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
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
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) 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
• 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 C n = G Abs (Index n – Index n 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
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.