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 S_t ⩾ 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 S_t ⩽ 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 S_T > K.

Example 7.1.1 Payoffs 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 S_0.25 = 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:

1. an up-and-in call, i.e. an option with payoff ; and
2. 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 < S_T < 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 (S_T − 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 7.1.2 Financial 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 < … < t_n ⩽ 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: (m_T − K)₊ or (M_T − K)₊;
• puts: (K − m_T)₊ or (K − M_T)₊.

Example 7.1.3 Payoffs 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 S_T − m_T = 16 − 10.25 = 5.75;
• floating-strike put option: the payoff is M_T − S_T = 22.84 − 16 = 6.84;
• fixed-strike call option on the maximum: the payoff is (M_T − K)₊ = (22.84 − 20)₊ = 2.84;
• fixed-strike call option on the minimum: the payoff is (m_T − K)₊ = (10.25 − 12)₊ = 0;
• fixed-strike put option on the maximum: the payoff is (K − M_T)₊ = (20 − 22.84)₊ = 0;
• fixed-strike put option on the minimum: the payoff is (K − m_T)₊ = (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:

1. monitoring, i.e. the frequency at which prices are computed in the average;
2. type of average, i.e. arithmetic or geometric.

Using the fixed set of dates 0 ⩽ t₁ < t₂ < … < t_n ⩽ T for discrete monitoring, we have the following four possible ways of computing the average stock price :

1. Discrete monitoring, arithmetic average: .
2. Discrete monitoring, geometric average: .
3. Continuous monitoring, arithmetic average: .
4. 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 < … < t_n ⩽ 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 7.1.4 Payoff 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 S_0.5 = 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 7.1.5 Payoff 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 C_t the value at time t of the option to buy S⁽¹⁾ for S⁽²⁾ and P_t 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 = S_t 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 C_t + Ke^{− r(T − t)} = P_t + S_t, which is the put-call parity of equation (6.2.4).

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 7.2.1 Weather 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 7.2.2 Weather 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 7.2.3 Earthquake 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 7.2.4 Hurricane 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 7.2.5 Longevity 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 7.2.6 Longevity 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).

 ◼