In Chapter 6, we engineered basic derivatives, including an investment guarantee, binary options and gap options. European call and put options, in addition to the abovementioned derivatives, fall into the category of simple options because their payoffs depend only on the underlying asset price ST at the maturity of the derivative.
However, on financial markets, institutional investors such as banks, insurance companies and pension plans design and trade more sophisticated contracts tailored for their specific needs. Those derivatives/options are said to be exotic or path-dependent, while others are called event-triggered derivatives.
A derivative is said to be exotic or path-dependent if its payoff depends on the underlying asset price at more than one date during the life of the derivative. Event-triggered derivatives are contracts whose payoff depends on the occurrence and/or severity of events such as natural catastrophes. Just like forwards, futures and simple options, exotic options and event-triggered derivatives are used by investors for speculation or hedging purposes.
The main objective of this chapter is to familiarize the reader with exotic/path-dependent options and event-triggered derivatives. The specific objectives are to:
understand and compute the payoffs of barrier, Asian, lookback and exchange options;
understand and compute the payoffs of weather, catastrophe and longevity derivatives;
understand why complex derivatives exist and how they can be used;
use no-arbitrage arguments to identify relationships between the prices of some of these derivatives.
7.1 Exotic options
Introduced originally as fully-customized agreements between two investors, most exotic options are now commonly traded over the counter. We will now look at the following four categories of exotic options:
barrier options;
Asian options;
lookback options;
exchange options.
7.1.1 Barrier options
A barrier option is an option to buy (call) or an option to sell (put) an asset S for a strike price K at maturity time T if the underlying asset price crosses (or not) a predetermined barrier level H during the life of the option.
There are two main types of barrier options: knock-in and knock-out. A knock-in barrier option is said to be activated (or triggered) if the asset price reaches the barrier level H before maturity. A knock-out option is activated if the asset price never reaches the barrier level H before maturity. Said differently, a knock-out option is deactivated if the barrier level is crossed.
Depending on whether S0 is below or above H, knock-in and knock-out barrier options can be further divided into two sub-groups:
a knock-in option can be either an up-and-in (barrier) option or a down-and-in (barrier) option;
a knock-out option can be either an up-and-out (barrier) option or a down-and-out (barrier) option.
To be more precise, first assume S0 < H and then denote by τ+H the first (random) time the stock price crosses the barrier level H (coming from below). Mathematically,
(7.1.1)
Similarly, if we assume S0 > H, then we can define τ−H, the first (random) time the stock price attains the barrier level H (coming from above), by
(7.1.2)
Figure 7.1 gives possible realizations of the random variables τ+H and τ−H, i.e. when barrier options are activated/deactivated.
Therefore, the event {τ+H ⩽ T} is realized whenever the barrier is crossed during the life of the option, i.e. there is a time 0 ⩽ t ⩽ T such that St ⩾ H. Similarly, the event {τ−H ⩽ T} occurs when the stock price attains the barrier during the life of the option, i.e. there is a time 0 ⩽ t ⩽ T such that St ⩽ H.
These events can be used to define indicator random variables determining whether a barrier has been attained or not. For example, is equal to 1 if the barrier is crossed (coming from below) during the life of the option and it is equal to 0 otherwise. This notation allows us to summarize the eight different barrier options whose payoffs are described below:
up-and-in options: in these cases, we have S0 < H and the payoffs are
– up-and-in call (UIC):
– up-and-in put (UIP):
up-and-out options: in these cases, we have S0 < H and the payoffs are
– up-and-out call (UOC):
– up-and-out put (UOP):
down-and-in options: in these cases, we have S0 > H and the payoffs are
– down-and-in call (DIC):
– down-and-in put (DIP):
down-and-out options: in these cases, we have S0 > H and the payoffs are
– down-and-out call (DOC):
– down-and-out put (DOP):
For example, an up-and-in call option provides the opportunity to buy a share of stock for K only if the stock price has attained H (and S0 < H) before time T. When a barrier option has been activated, it does not mean the option will necessarily be exercised. In the case of the up-and-in call option, the option to buy is exercised only if ST > K.
Example 7.1.1Payoffs of barrier options
Consider down-and-in and down-and-out call and put options with maturity in 3 months, initial stock price $50, strike price of $50 and barrier level at $47. Assume the following scenario has occurred on the market: the stock price (after 3 months) is $45 and, in between the option’s inception and maturity, the minimum stock price reached $41 (after 20 days). Let us compute the payoffs in this scenario.
First of all, the stock price started at S0 = 50 > 47 = H, reached 41 after 20 days and ended up at S0.25 = 45 at maturity time T = 0.25. Therefore, the barrier H = 47 has been crossed (coming from above).
Therefore, we have
DIC: the barrier is crossed, the option is activated but the call option is out of the money at maturity and consequently the payoff is
DIP: the barrier is crossed, the option is activated and the put option is in the money at maturity and consequently the payoff is
DOC or DOP: the barrier is crossed so both knock-out options are deactivated and consequently their payoffs are 0.
◼
It should be emphasized, despite the similarities from a notational point of view, that barrier options are significantly different from gap options studied in Chapter 6. Even if both types of options have final payments depending on the crossing of a barrier, those events differ a lot. To illustrate the situation, let us consider:
an up-and-in call, i.e. an option with payoff ; and
a gap call option, i.e. an option with payoff .
For the up-and-in call, the payoff is strictly positive if the underlying stock price crosses level H before or at maturity and if the final stock price is greater than K. Its payoff is always non-negative. For the gap call, the payoff is strictly positive if the final stock price is greater than both H and K, it is equal to zero if the final stock price is less than H, and it can be negative in some situations (if H < ST < K).
Financial engineering with barrier options
Barrier options can be combined to create vanilla options. Indeed, as the barrier is either crossed or not crossed, we have
The financial interpretation of the latter is that
Consequently, the payoff of a vanilla call option (ST − K)+ can be further decomposed as the sum of two barrier options. First, we can write
In other words, a vanilla call option can be replicated by a long up-and-in call and a long up-and-out call. Second, we can write
In other words, a vanilla call option can be replicated by a long down-and-in call and a long down-and-out call. Of course, similar relationships hold for barrier put options.
Therefore, by the no-arbitrage assumption, we must also observe, at any time 0 ⩽ t ⩽ T:
and
where for example is the time-t value of a down-and-in call option.
Example 7.1.2Financial engineering with barrier options
You are given the following:
a share of stock trades for $75;
a 72-strike call option trades for $7.50;
an up-an-out put option with strike $72 and barrier $80 sells for $3;
a 3-month Treasury zero-coupon bond trades for $98.50 (face value of $100);
all options mature in 3 months.
Let us find the price of an up-and-in put option.
Using the put-call parity of equation (6.2.4), we have
Moreover,
and therefore, to avoid arbitrage opportunities, the up-and-in put should sell for $0.42.
◼
Alternative representation with max and min
There are important relationships between the first-passage-time random variables τ+H and τ−H and the minimum/maximum value of the underlying asset price. Let us define the continuously monitored maximum and minimum values taken by S over the time interval [0, T] by
where
We deduce the following equalities of events:
and
If S0 < H, then the stock price has to increase to reach H at the random time τ+H. This also means the maximum attained by the stock price between time 0 and time T has to be larger than H for the stock price to cross H. A similar reasoning applies if S0 > H. Therefore, we can rewrite each of the above eight payoffs using the maximum or the minimum value of the underlying asset during the life of the option.
In practice, the minimum/maximum is monitored daily, weekly or monthly, i.e. at the beginning or end of each day, week or month. First, let us set the monitoring dates 0 ⩽ t1 < t2 < … < tn ⩽ T. Then, we can define the corresponding discretely monitored maximum and minimum values taken by S over the time interval [0, T] by
Of course, the monitoring frequency affects the value of the maximum/minimum: the continuously monitored maximum is always greater than or equal to the discretely monitored maximum, while the continuously monitored minimum is always less than or equal to the discretely monitored minimum.
Figure 7.2 shows a sample path (or times series) for the price of a stock. In this example, the continuously monitored maximum is observed between time t2 and time t3. If the maximum is discretely monitored at times t1, t2, t3, t4 and T, then the maximum is observed at time t2. As illustrated by this figure, the monitoring frequency affects the value of the maximum/minimum.
Consequently, we could have a second look at the eight different barrier options and define discretely monitored barrier options.
In general, discretely monitored exotic options are difficult to price so continuously monitored exotic options are often used as an approximation.
7.1.2 Lookback options
Lookback options are options whose payoff is based upon the minimum or maximum asset price observed during the life of the option. The contract usually specifies the frequency at which the minimum or maximum is monitored: continuously or discretely.
There are two types of lookback options: floating-strike and fixed-strike. In floating-strike lookback call and put options, we replace the strike price of a vanilla option by the minimum and maximum price of the underlying, respectively. The payoffs are then given by
for a floating-strike lookback call and a floating-strike lookback put, respectively. Because these payoffs are already positive, there is no need to use the positive part function.
In a fixed-strike lookback call or put option, we replace the asset price in a vanilla option by the minimum or maximum price of the underlying. There are four possibilities for the payoff:
calls: (mT − K)+ or (MT − K)+;
puts: (K − mT)+ or (K − MT)+.
Example 7.1.3Payoffs of lookback options
One-month lookback options are issued on the stock of ABC inc. The fixed strike on the minimum is $12 and the fixed strike on the maximum is $20. Assume you observe the following scenario over the next month:
stock price after a month: $16;
minimum stock price, during that month, monitored at the end of each business day: $10.25;
maximum stock price, during that month, monitored at the end of each business day: $22.84.
Let us compute the payoffs of lookback call and put options of either floating-strike or fixed-strike types.
We get:
floating-strike call option: the payoff is ST − mT = 16 − 10.25 = 5.75;
floating-strike put option: the payoff is MT − ST = 22.84 − 16 = 6.84;
fixed-strike call option on the maximum: the payoff is (MT − K)+ = (22.84 − 20)+ = 2.84;
fixed-strike call option on the minimum: the payoff is (mT − K)+ = (10.25 − 12)+ = 0;
fixed-strike put option on the maximum: the payoff is (K − MT)+ = (20 − 22.84)+ = 0;
fixed-strike put option on the minimum: the payoff is (K − mT)+ = (12 − 10.25)+ = 1.75.
◼
Equity-linked insurance and annuities (see Chapter 8) offer benefits that are similar to discretely monitored lookback options. For example, the high watermark indexing scheme offered in equity-indexed annuities is based on the maximum value of a reference index observed on anniversary or tri-anniversary dates. Variable annuities may also offer automatic resets, a feature similar to a protective put whose strike price is updated periodically (once every year or two years) to reflect the best market conditions.
7.1.3 Asian options
An Asian option is an option whose payoff is based upon the average price of the underlying asset during the life of the option. Asian options limit the effects, on the value of the option, of speculative transactions on the underlying stock. They are generally cheaper than vanilla call and put options.
Let us denote by the average asset price between inception and maturity. In general, the calculation of the average is specified in the contract with respect to:
monitoring, i.e. the frequency at which prices are computed in the average;
type of average, i.e. arithmetic or geometric.
Using the fixed set of dates 0 ⩽ t1 < t2 < … < tn ⩽ T for discrete monitoring, we have the following four possible ways of computing the average stock price :
Discrete monitoring, arithmetic average: .
Discrete monitoring, geometric average: .
Continuous monitoring, arithmetic average: .
Continuous monitoring, geometric average: .
Note that the continuous versions are obtained as limits of the discrete versions when the fixed set of dates 0 ⩽ t1 < t2 < … < tn ⩽ T gets larger in order to cover the whole time-interval [0, T].
Asian options can be of two types: average price or average strike. For an average price Asian option, the payoff is based on the difference between the average price and a fixed strike price K. For an average strike Asian option, the price at which we can buy/sell the underlying asset is random and given by the average stock price. More precisely, the payoff of an average price Asian call option (resp. average price Asian put option) with strike price K is defined as
and the payoff of an average strike Asian call option (resp. average strike Asian put option) by
In practice, the most popular type of average is the arithmetic average computed over a discrete set of dates (discrete monitoring). For example, the average stock price could be calculated as the arithmetic mean of each end-of-day price. However, in many financial models, calculating the exact no-arbitrage price of Asian options in that popular case is very difficult and the geometric average or the continuous monitoring are used as a way to approximate the price of such options.
Example 7.1.4Payoff of Asian options
In an Asian option contract having a 6-month maturity, the average is said to be computed on the observed stock prices at the end of each week (every Friday). Average price Asian call/put options both have a strike of $70.
Here is a possible scenario for the next 6 months: the average stock price, during those 6 months, is $69.41 and the final stock price is $72. Let us calculate the payoff of average strike/price Asian call/put options in this scenario.
We have T = 0.5 and K = 70. In this scenario, we also have and S0.5 = 72. Consequently,
average strike Asian call: ;
average strike Asian put: ;
average price Asian call: ;
average price Asian put: .
◼
7.1.4 Exchange options
An exchange option gives its holder the right to exchange an asset for another asset. It is a generalization of vanilla call and put options, which are options to exchange the underlying asset for a cash amount K.
More generally, let us denote by S(1)t and S(2)t the time-t prices of two risky assets. The payoff of an exchange option to obtain a share of S(1) in exchange for a share of S(2) at maturity time T is given by
while the payoff to obtain a share of S(2) in exchange for a share of S(1) is given by
Example 7.1.5Payoff of an exchange option
You enter into an exchange option to buy one share of ABC inc. in exchange for one share of XYZ inc. Both shares currently trade at 100. Find the payoff of that exchange option if, at maturity, the stock of ABC inc. is worth 105 whereas the stock of XYZ is worth 112.
The option gives you the right, not the obligation, to buy a share of ABC inc. in exchange for a share of XYZ. Therefore, you can buy for 112 an asset worth 105 and you decide not to exercise this option. The payoff is 0.
◼
Of course, exchange options make sense when the two assets being exchanged have comparable prices. In general, an exchange option gives its holder the right to exchange a given quantity of S(1) for another given quantity of S(2).
Using the translation property of the maximum function, we can write
Denoting by Ct the value at time t of the option to buy S(1) for S(2) and Pt the price at time t of the option to buy S(2) for S(1), we can deduce the following parity relationship between exchange options:
for all 0 ⩽ t ⩽ T.
This is a generalization of the classical put-call parity of Chapter 6. Indeed, suppose that S(1) is a stock such that S(1)t = St whereas S(2) is a zero-coupon bond with face value K and time-t value S(2)t = Ke− r(T − t). As previously discussed, an option to buy S(1) (S(2)) for S(2) (S(1)) is a regular call (put) option. Therefore, and as expected, the above parity relationship becomes Ct + Ke− r(T − t) = Pt + St, which is the put-call parity of equation (6.2.4).
Other exotic options
It is generally recognized that Mark Rubinstein1 coined the term “exotic options” in the early 1990s to name a set of derivatives whose payoff structure is unconventional. The expression has remained in contrast to vanilla options that comprise European and American call/put options. Vanilla options are usually exchange-traded whereas most exotic options are traded OTC.
In addition to barrier, Asian, lookback and exchange options, the following designs have appeared regularly in the markets (shown alphabetically):
Chooser option: option that allows the holder to choose whether the contract is a European call or put prior to maturity.
Cliquet (or ratchet) option: portfolio of forward start options (see below) such that the k-th option applies only during the k-th period.
Compound option: option written on another option (e.g. call on a call, call on a put, etc.), i.e. in which the underlying asset is another option.
Forward start option: option issued today, starting at time T1 > 0 and expiring at time T2 > T1. More precisely, the strike price is given by , i.e. the option is started at-the-money and so it is unknown at inception.
Parisian option and step option: option whose payoff depends on the time spent by the underlying asset below, above or between barrier(s).
Package: portfolio of standard European/American call/put options. These typically involve the strategies of Section 5.4.
Reset option: option allowing the strike price to be reset before maturity.
7.2 Event-triggered derivatives
An event-triggered derivative is a financial contract whose payoff is tied to the occurrence and/or severity of one or several events or variables, such as temperature, rainfall, natural catastrophes, mortality or longevity of a pool of individuals. These derivatives are useful to transfer the risk tied to these events to other investors in exchange for a premium. For example, pension plan sponsors use longevity derivatives as a risk transfer mechanism to offset the adverse effects of longevity risk (see Chapter 1).
Why would someone else accept to bear weather, natural catastrophe or longevity risks? Simply because investors can use these instruments to diversify their typical investment portfolios (stock, bonds, commodities, retail, etc.). Indeed, it is widely recognized that weather, natural catastrophe or longevity risks are relatively independent from common financial variables such as stock prices and interest rates. Those derivatives can also be used for speculation.
Catastrophe and longevity derivatives are the most common examples of insurance-linked securities (ILS). It is a class of instruments whose payoff is linked to insurance losses. They are used by insurance companies for risk transfer.
In this section, we will look at weather, catastrophe and longevity derivatives and we will see how pension plans, life and property and casualty insurers can use these derivatives to risk manage their liabilities.
7.2.1 Weather derivatives
A weather derivative is a financial contract whose payoff is contingent on the temperature or amount of precipitation (rain, snow) observed at a given location during a predetermined period. For industries whose profitability is largely determined by the weather, it is easy to imagine how weather derivatives can be used for risk management:
An industrial farmer loses whenever there is too much rain (flooded lands) or not enough (droughts). He could enter into a weather derivative on rainfall to receive a payment if one of these adverse events occurs.
A ski station suffers a lot if snowfall is low during winter. It could enter into a weather derivative on snowfall or temperature so that it receives a payment if snowfall is well below average or if temperature during winter is too high.
A city pays for snow plowing of its streets and if snowfall is well above average, it will incur major costs. It could enter into a weather derivative to receive a payment if snowfall is well above average.
The first weather derivative was created between two energy companies in 1996. The contract offered a rebate to one party if temperature was cooler than expected. Weather derivatives have been trading OTC since 1997 and on the Chicago Mercentile Exchange (CME) since 1999. For example, the CME offers weather derivatives on the temperature for major cities in the U.S. and in Canada, and rain/snowfall derivatives for some cities in the U.S.
Example 7.2.1Weather derivative for a P&C insurer
A property and casualty insurance company covers flood risk in its homeowner’s insurance policies. The company would like to offset losses resulting from a flood occurring on a specific river. The actuary knows that whenever the accumulated rainfall during a week is more than 10 inches near that river, a flood is very likely to occur.
A rainfall derivative is available: it pays $1 per inch of accumulated rain in excess of 10 inches, measured between May 1st and May 7th, at a specific weather station. If the cumulative rainfall is below 10 inches, the derivative pays 0. If the cumulative rainfall is 13 inches, then the derivative will pay $3 per unit.
Hopefully, this will be enough to offset the losses the company will incur. Remember that, in all scenarios, the insurance company is required to pay an initial premium on these derivatives.
◼
The most common measures to determine the payoff of temperature-based weather derivatives are heating degree days (HDD) and cooling degree days (CDD). The idea is that for a temperature below 65°F (18°C), people start heating buildings whereas for temperatures above 65°F, people need cooling (air conditioning). The HDD (resp. CDD) is a measure of the extent by which heating (resp. cooling) is required. On a given day, the HDD and the CDD are computed as follows:
and
where X is the average between the minimum and maximum temperature on that day. For a day whose average temperature X is 80°F, the corresponding HDD is 0 (no heating required) and the CDD is 15 (cooling required). Finally, we define the cumulative HDD (resp. CDD) as the sum of the HDD (resp. CDD) over a given number of days.
Example 7.2.2Weather derivative for a life insurer
A life insurance company wants to protect itself against excess mortality due to heat waves for the first week of July. It decides to enter into a temperature-based weather derivative. This derivative will pay $1 per cumulative CDD in excess of 105. Let us calculate the payoff of the weather derivative in the scenario where temperatures are as follows during that week:
Temp/Day
S
M
T
W
T
F
S
Min
55
57
59
65
64
63
59
Max
75
80
85
90
88
89
78
We calculate the daily mean between the minimum and maximum temperature, in addition to the daily CDD. The results for this scenario are shown in the next table.
Temp/Day
S
M
T
W
T
F
S
Mean
65
68.5
72
77.5
76
76
68.5
CDD
0
3.5
7
12.5
11
11
3.5
For example, for Monday, 68.5 is calculated as (80 + 57)/2 and the CDD on Monday is 68.5 − 65. Summing the CDD over the seven days of the week, we get that the cumulative CDD is 48.5. Hence, in this scenario, the weather derivative will not pay since the cumulative CDD is well below 105.
◼
7.2.2 Catastrophe derivatives
For an insurance company covering losses resulting from natural catastrophes, a common risk-transfer mechanism is reinsurance: in exchange for a premium, the reinsurer covers losses in excess of some predetermined amount. Catastrophe derivatives are an alternative to reinsurance: instead of transferring the risk to one of many reinsurers, other investors can assume a part of the natural catastrophe risk in exchange for a premium.
A catastrophe (CAT) derivative is a financial contract whose payment is affected by the occurrence and/or severity of natural catastrophes in a given region during a given period of time. According to the OECD, they have “[...] appeared in the aftermath of Hurricane Andrew in 1992 in the belief that the capacity offered by the traditional reinsurance market and the retrocession market would shrink.” The most important CAT derivatives are CAT bonds, options and futures.
The first CAT derivative that appeared on the market was the CAT bond. A CAT bond is like a regular bond but principal (and maybe coupons as well) is reduced whenever a catastrophic event occurs. The triggering events have to be well defined and they are usually based on either the actual losses of the issuer or based upon a catastrophe loss index. It was created in 1994 as a customized agreement between insurers and reinsurers.
CAT options and CAT futures are issued on catastrophe loss indices and provide the opportunity to buy or sell the index for a predetermined price. Such derivatives were introduced by the Chicago Board of Trade in 1995. Nowadays, the main providers of catastrophe loss indices are the Property Claims Services (PCS) and the CME. They are meant as proxies for industry losses due to hurricanes, earthquakes or other natural catastrophes.
We illustrate two popular CAT derivatives in the examples below.
Example 7.2.3Earthquake bond
In January, a 5-year earthquake bond with annual coupons of 8% and principal of $100 is issued by an insurance company. Principal is reduced by 50% if the Earthquake Loss Index (computed in a given region) is over 500 at maturity. Let us describe the cash flows of this bond if it is issued at par, i.e. for a price of $100.
At inception, the insurance company receives $100 and pays annual coupons of $8. In the meantime, if earthquakes are rare and the index ends up below 500 after 5 years, then the insurance company will repay the full principal, i.e. $100. But if earthquake losses are high during this 5-year period, it may have to repay only 50% of the principal (as specified in the contract), i.e. $50. Investors in the market lose $50 in this scenario but they are usually compensated by a high coupon rate that provides a large return whenever earthquake losses are small.
◼
Example 7.2.4Hurricane options
A call option on the Hurricane Loss Index is issued on May 31st just before hurricane season in the North Atlantic. The index is at 100. The option provides the right to buy the index for 130 on November 30th, at the end of hurricane season. The call option currently sells for $15. Let us describe the cash flows of the call option if we assume there are only two possible scenarios: the Hurricane Loss Index is either 160 or 110, at maturity.
For a company exposed to hurricane losses, this call option acts as an insurance. The company pays $15 at inception. If there are many severe hurricanes, the index will increase a lot and the option will mature in the money. If the index is 160 at maturity, the payment will be 160 − 130 = 30 per unit of option, which should help offset losses an insurer might suffer for a severe hurricane season. Otherwise, for an index at 110 on November 30th, the call option will be out of the money (payoff equal to zero) at maturity.
◼
7.2.3 Longevity derivatives
As defined in Chapter 1, longevity risk is the uncertainty related to the overall improvement of life expectancy. It is difficult to predict how much longer people will live in the future. Longevity risk is an important systematic risk faced by pension plans, putting a significant pressure on social security systems.
Life insurance companies are also exposed to longevity risk but to a lesser extent. Higher longevity means their annuity business is more expensive (additional annuity payments) whereas their insurance business is cheaper (lower present value). As a result, life insurers have a natural hedge against large variations in mortality patterns. Pension plan sponsors, as annuity providers/sellers, traditionally manage longevity risk by buying annuities from a life insurer (buy-in) or by selling the pension liability (along with its related assets) to a life insurer (buy-out).
Longevity derivatives were introduced in the early 2000s as an alternative scheme to transfer longevity risk. A longevity derivative is a financial contract whose payoff is contingent on how many individuals, from a given group, survive during a given period of time. For example, the group could be the retirees of a pension plan or it could be the 65-year old males in a given country. Survivorship indices are also common and are provided by Credit Suisse and the Life & Longevity Markets Association (LLMA).
Three types of longevity derivatives have appeared on the market: longevity bonds, forwards and swaps. A longevity bond is a bond whose coupons decrease with the number of survivors in the pool of individuals. As in Chapter 4, in a typical longevity swap, a fixed amount is exchanged periodically for a variable (floating) amount computed with fixed and floating mortality rates. A longevity forward is a single exchange of cash flows based upon a fixed and a floating mortality rate. These derivatives are illustrated in the following examples.
Example 7.2.5Longevity bond
A bank issues a longevity bond with a principal of $100. The annual coupons decrease according to the number of survivors in a population of 100,000 65-year-old individuals. The first coupon is $5 while the subsequent coupons are equal to the proportion of survivors times $5. Let us describe the first few cash flows of the bond in the following scenario: after 1 year, the number of survivors is 98,500, and after 2 years it is 97,000.
First of all, the principal is received at inception and it is repaid at maturity. The first coupon paid, at the end of the first year, is 5. In this scenario, the second coupon, paid after 2 years, will be 0.985 × 5 = 4.925, whereas the third coupon, paid after 3 years, will be 0.97 × 5 = 4.85.
If the mortality experience of a pension plan is similar to this population (65-year-old individuals) and if each retiree had identical benefits, then the pension plan sponsor could invest in this longevity bond to (partially) hedge out the longevity risk, as the decreasing coupon pattern could (partially) offset the annuity payments made by the plan. Indeed, in this case, it would not matter (for the sponsor) how much longer retirees might live.
◼
Example 7.2.6Longevity forward
You need to manage longevity risk tied to 70-year-old females. You enter into a longevity forward known as a q-forward whose payoff design is simple: for a zero initial cost, you pay 100 times the mortality rate experienced by 70-year-old females next year and you receive a fixed amount of 1.56 in exchange. Let us describe the cash flows in the following two scenarios: the mortality experience of 70-year-old females is either 1500 or 1600 deaths per 100,000.
In a year, you will receive 1.56 for a notional of 100, which translates into a fixed mortality rate of 1560 per 100,000. In the case where 1500 people die, you would pay 1.50 and receive 1.56 at maturity, for a gain of 6 cents per 100 of notional. Hence, to protect against improvements in mortality, you should be the fixed-rate receiver. If 1600 people die, you would pay 1.60 and still receive 1.56 for a loss of 4 cents per 100 of notional.
Depending on whether you are the fixed-rate receiver or payer, a longevity forward can protect against decreases/increases in mortality (longevity).
◼
7.3 Summary
Barrier option
Option to buy/sell an asset only if a barrier has been crossed (or never been crossed) prior to or at maturity.
Knock-in: activated if the barrier has been crossed.
Knock-out: deactivated if the barrier has been crossed.
Types:
– up-and-in call (UIC):
– up-and-in put (UIP):
– up-and-out call (UOC):
– up-and-out put (UOP):
– down-and-in call (DIC):
– down-and-in put (DIP):
– down-and-out call (DOC):
– down-and-out put (DOP):
Parity relationships:
Lookback options
Options based on the minimum/maximum price of the underlying asset.
Option based on the average price of the underlying asset.
Average price of the underlying asset: .
Types:
– Average price Asian call: .
– Average price Asian put: .
– Average strike Asian call: .
– Average strike Asian put: .
Types of monitoring and averages:
– Discrete monitoring, arithmetic average: .
– Discrete monitoring, geometric average: .
– Continuous monitoring, arithmetic average: .
– Continuous monitoring, geometric average: .
Exchange options
Option to exchange one risky asset for another.
Call option: receive S(1)T in exchange for S(2)T.
Put option: receive S(2)T in exchange for S(1)T.
Regular call and put options: exchange cash (strike price) for a stock (or vice versa).
Similar parity relationship:
Weather derivative
Derivative whose payoff is contingent on the temperature or amount of precipitation (rain, snow) observed at a given location during a predetermined period.
Temperature-based weather derivatives are generally built with cumulative HDD and CDD:
where X is the average between the minimum and maximum temperature observed on a given day.
Catastrophe derivative
Derivative whose payoff is based upon the occurrence and/or severity of specified natural catastrophes during a given period of time and in a given region.
Types: bonds, futures, options.
Payoff based on a catastrophe loss index.
Longevity derivative
Derivative whose payoff is contingent on how many individuals, from a given group, survive during a given period of time.
Types: bonds, swaps and forwards.
Payoff based on a survivorship index.
7.4 Exercises
In Figure 7.3 and in Table 7.1, you will find the (fictional) evolution of the stock price of ABC inc. Calculate the payoff of the following exotic options (or investment strategies) given that they are purchased (or initiated) at the beginning of the year and mature at the end.
Table 7.1Fictional evolution of the stock price over a year. Note that k labels the days within the year, i.e. k = 0, 1, 2, …, 240
k
Sk
k
Sk
k
Sk
k
Sk
k
Sk
k
Sk
0
52.00
1
52.90
41
54.57
81
47.82
121
50.63
161
50.20
201
48.81
2
52.45
42
55.05
82
48.24
122
51.61
162
51.08
202
48.18
3
52.86
43
56.21
83
49.16
123
52.51
163
51.51
203
47.09
4
53.14
44
56.61
84
48.83
124
52.18
164
50.70
204
47.03
5
53.88
45
56.48
85
49.02
125
52.39
165
50.30
205
47.47
6
53.57
46
56.29
86
48.91
126
53.62
166
52.35
206
47.89
7
54.01
47
55.90
87
49.44
127
53.60
167
50.36
207
47.57
8
53.37
48
56.30
88
49.27
128
53.35
168
50.92
208
48.81
9
52.32
49
56.18
89
49.03
129
52.99
169
50.74
209
49.63
10
51.90
50
56.30
90
49.79
130
52.44
170
49.92
210
49.57
11
52.01
51
57.16
91
50.06
131
53.60
171
48.79
211
50.01
12
52.40
52
57.26
92
50.47
132
53.37
172
48.33
212
50.58
13
53.10
53
57.78
93
51.15
133
54.35
173
47.43
213
51.77
14
53.78
54
58.79
94
51.41
134
54.77
174
47.54
214
51.15
15
53.39
55
59.71
95
51.04
135
54.75
175
48.17
215
50.54
16
54.31
56
58.15
96
51.41
136
54.04
176
48.73
216
50.70
17
53.77
57
56.98
97
50.55
137
53.93
177
48.33
217
50.78
18
53.80
58
57.82
98
50.96
138
53.72
178
47.80
218
52.20
19
54.13
59
57.49
99
50.54
139
54.12
179
47.41
219
51.52
20
54.68
60
56.23
100
50.54
140
54.10
180
47.23
220
52.45
21
54.82
61
55.66
101
48.82
141
53.80
181
47.25
221
52.57
22
54.78
62
54.30
102
46.94
142
53.43
182
47.03
222
51.90
23
53.63
63
54.75
103
46.81
143
52.16
183
46.89
223
52.74
24
54.71
64
53.54
104
46.52
144
52.09
184
47.65
224
52.93
25
54.35
65
52.79
105
46.63
145
53.02
185
46.36
225
52.88
26
54.60
66
52.00
106
46.70
146
53.28
186
47.49
226
53.43
27
54.77
67
51.73
107
47.23
147
52.91
187
48.72
227
53.14
28
54.03
68
51.52
108
47.09
148
53.60
188
49.34
228
52.90
29
54.53
69
51.47
109
47.44
149
53.66
189
48.71
229
52.79
30
54.77
70
50.58
110
46.60
150
53.79
190
48.87
230
53.17
31
53.22
71
49.98
111
46.64
151
53.60
191
48.86
231
53.48
32
52.31
72
50.16
112
47.81
152
54.19
192
48.33
232
52.68
33
53.79
73
50.19
113
47.92
153
53.57
193
48.39
233
54.03
34
54.35
74
49.59
114
48.63
154
54.72
194
49.35
234
54.96
35
54.21
75
48.81
115
48.24
155
54.40
195
49.20
235
55.38
36
54.78
76
48.47
116
48.43
156
52.97
196
48.98
236
55.87
37
53.56
77
48.03
117
49.12
157
53.71
197
48.95
237
55.77
38
52.84
78
46.93
118
49.19
158
51.57
198
48.44
238
54.84
39
52.77
79
47.02
119
49.67
159
50.42
199
48.51
239
53.29
40
53.69
80
47.73
120
51.29
160
50.23
200
48.35
240
53.68
Note that in this example a year is made up of 240 business days or 48 business weeks of 5 days or 12 months of 20 business days.
Call option with strike price of $50.
Covered call with strike price of $49.
Straddle at $50.
Gap put option with strike price of $52 and trigger price of $50.
Up-and-in call option with barrier of $56 and strike price of $51.
Average strike Asian call option whose arithmetic average is computed at the end of each month (on S20, S40, …).
Floating-strike lookback put option where the maximum/minimum is monitored daily.
Cash-or-nothing put option with strike price of $50.
Fixed-strike lookback call option on the minimum where the maximum/minimum is monitored every week on Fridays (i.e. k = 5, 10, 15, …) with a strike of $40.
Bear spread with strike prices of $47 and $52.
Investment guarantee with K = 1.01S0 (minimum return of 1% on the initial investment).
You are given the following prices for barrier options with strike price $50 all maturing at the same time:
H = 40
H = 60
Call
Put
Call
Put
DO
5.18
0.81
0
0
DI
0.05
1.98
5.23
2.79
UO
0
0
0.59
2.68
UI
5.23
2.79
4.64
0.11
The current stock price is $50.
What is the price of regular call and put options?
Calculate the current price of an at-the-money call option that is deactivated if the stock price ever attains $40 or $60 during the life of the option. Repeat for a similar put option.
An average strike Asian call option was issued 5 months ago and matures in a month. The average stock price over the last 5 months is $14 and the payoff of the option is $2. If the average is computed arithmetically on prices observed at the end of each month, determine the stock price at maturity.
Suppose that you can buy today for $5.75 a contract that pays off at maturity. Given that a stock currently trades for $6.14 and an average strike Asian call option sells for $1.78, find the current no-arbitrage price of an average strike Asian put option.
Four floating-strike lookback put options maturing in 6 months are available in the market. They differ only in the monitoring frequency of the maximum:
Lookback # 1: maximum is monitored continuously.
Lookback # 2: maximum is monitored monthly.
Lookback # 3: maximum is monitored weekly.
Lookback # 4: maximum is monitored daily.
To avoid arbitrage opportunities, order the price of these lookback options, starting from the lowest to the highest price.
Suppose the temperature (°F) in Montpelier, VT, in the first week of February is as follows: