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P RICING LOOKBACK O PTIONS AND D YNAMICG UARANTEES

Hans U. Gerber* and Elias S.W. Shiu

A BSTRACT

Pricing exotic options or guarantees in equity-indexed annuities can be problematic. The authorspresent closed-form formulas for pricing lookback options and dynamic guarantees that facilitatethe hedging and reserving for such products. The principal tool used is a closed-form expression for B (u , T ), the Laplace-Stieltjes transform of the expected excess of the running maximum of a Wiener process above a positive constant u in a nite time interval of length T . If the aggregate netincome of a company is modeled with a Wiener process, then the excess of the running maximumabove u can be interpreted as aggregate dividend payments, and the quantity B (u , T ) is theexpectation of the discounted value of the dividend payments up to time T . The formula for B (u , T )is used to price European lookback options (call and put, xed and oating strike). It is also usedto price dynamic fund protection, which is a guarantee on an investment fund: The number of unitsof the investment fund is increased whenever necessary, so that their total value does not fall belowa guaranteed level. The guaranteed level can be stochastic, such as that given by a stock index.Some well-known results for the rst passage time of the Wiener process are explained in theappendix.

1. I NTRODUCTION A main motivation for this paper is the pricing of exotic options or guarantees in certain equity-indexedannuities (EIAs). EIAs, introduced in the U. S. marketplace in the last decade of the last century, have

been considered among the most innovative products sold by U.S. life insurance companies. Expositionson EIAs can be found in a report by a task force of the American Academy of Actuaries (AAA 1997), Lee(2002), Mitchell and Slater (1996), Streiff and DiBiase (1999), and Tiong (2000a, b).

The principal tool described in this paper is a three-term formula for the Laplace-Stieltjes transformof the expected excess of the running maximum of a Wiener process above a positive constant in a nitetime interval. Let { X (t); t 0}, with X (0) 0, be a Wiener process or Brownian motion and

M t max X s , 0 s t (1)

be the maximal value of the process until time t. For u 0, let D(t) be the excess of M (t) over u ,

D t M t u . (2)

The basic building block in this paper is the function

B u , T E0

T

e t dD t , (3)

* Hans U. Gerber, A.S.A., Ph.D., is Professor of Actuarial Science, Ecole des H.E.C., Universite de Lausanne, CH-1015 Lausanne, Switzerland,e-mail: [email protected]. Elias S.W. Shiu, A.S.A., Ph.D., is Principal Financial Group Foundation Professor of Actuarial Science, Department of Statistics and ActuarialScience, University of Iowa, Iowa City, Iowa 52242-1409, and Visiting Chair Professor of Actuarial Science, Department of Applied Mathematics,Hong Kong Polytechnic University, Hung Hom, Hong Kong, e-mail: [email protected].

48

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where is a constant. Because D(t) is a nondecreasing function of t along each sample path, the integralcan be interpreted as a Stieltjes integral. Thus, in treating the integral, we do not need sophisticatedmathematics such as stochastic calculus, even though D(t) is a function of a Wiener process. We shallderive closed-form expressions for the function B( u , T). The main formula is (13).

A positive can be viewed as a force of interest. Then the expectation (3) can be interpreted as theexpected discounted dividend payments, if D(t) models the aggregate dividends paid by a company upto time t. That is, we model the company s aggregate net income with { X (t)}, and consider that thecompany has a policy to pay dividends whenever the aggregate net income, or more precisely, themodi ed aggregate net income, X (t) D(t), reaches the level u . The word modi ed signi es that theaggregate net income is reduced by the total of all previous dividend payments. The company s policyis that, whenever the modi ed aggregate net income reaches the level u , the over ow is immediatelypaid out as dividends.

Formula (13) remains valid for evaluating (3) even for negative values of . In Section 4, it is used toprice European lookback options (call and put, xed and oating strike). In Section 5, it is used to pricedynamic fund protection, which is a guarantee on an investment fund; the number of units of theinvestment fund is increased whenever necessary, so that their total value does not fall below aguaranteed level. In Section 6, we show that the guaranteed level can be stochastic, such as that given

by (a fraction of) a stock index.For the convenience of the reader, there is an appendix in the end of the paper. Using the optional

sampling theorem, it derives some well-known results for the rst passage time of the Wiener process.

2. EVALUATING B ( U , T )The purpose of this section is to evaluate the function B( u , T). The main result is given by (13) and (14).The proof is somewhat lengthy. The impatient reader may proceed to the next section after reviewingthese two formulas. Note that, in this section, the drift and diffusion parameters of the Wiener process{ X (t)} are denoted by and , respectively.

Because the expectation of a positive random variable is the integral of its tail probabilities, we have

E D t E M t u0

Pr M t u y dy

0

Pr M t u y dy u

Pr M t x dx . (4)

There is a close connection between M ( t ) and the rst passage time of the process { X ( t )}. Theappendix is an exposition on the rst passage time. For x 0, let T x denote the rst time theprocess { X ( t )} attains the level x . Observe that, for any x 0, t 0, the event { M ( t ) x } isidentical to the event { T x t }. Hence, the integrand in (4) can be replaced by Pr[ T x t ], whichwe denote as G x ( t ), and (4) becomes

E D t u

G x t dx . (5)

By (A15) in the appendix,

G x t x t

t e2 x / 2

x t

t, t 0, (6)

where is the cumulative distribution function of a standard normal random variable.

49PRICING LOOKBACK OPTIONS AND DYNAMIC GUARANTEES

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To evaluate (5), rst consider the case 0. Then,

G x t 2 x

t.

Integration by parts yields

E D t 2 u u

tt

u

tfor 0, (7)

where is the probability density function of a standard normal random variable. For 0, theintegration of (6) seems formidable. Nevertheless, the integral can be evaluated, and we claim:

E D t t2

2 u u t

t

2

2 e2 u / 2

u t

tt

u t

tfor 0. (8)

To verify (8), observe that its right-hand side tends toward 0 as u tends toward . It remains to check that its derivative, with respect to u , is G u (t). Note that the variable u appears ve times in (8). Thus,the derivative is a sum of ve terms. The rst and third terms already yield G u (t). Hence, we need toverify that the second, fourth, and fth terms cancel. This step is facilitated by the identity

e2 u / 2 u t

t

u t

t, (9)

which follows from the identity

e2 ab a b a b . (10)

Formula (9) can also be derived in the context of changing probability measure; see (D29) of Huang andShiu (2001).

As a check for the validity of (8), we take the limit 3 0 to see if we can retrieve (7). Grouping thetwo terms with in the denominator and applying the rule of Bernoulli-L Ho pital, we obtain (7).

We are now ready to evaluate (3). For 0,

B u , T E D T ,

for which the closed-form expressions (7) and (8) are available. To obtain a formula for B( u , T) for 0, we start by rewriting (3) as

B u , T0

T

e td

dt E D t dt . (11)

It follows from (7) and (8) in combination with (9) that

d dt E D t

u t

t t

u t

t. (12)

We claim that, for 0, equation (11) is

B u , T e T u T

T

21

2 e2 u

uT

2 1 T2

22

2 e1 u

uT

1 2 T2 ,

(13)

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where 1 and 2 are the solutions of the quadratic equation2

22 0. (14)

To prove formula (13), note that its right-hand side tends toward 0 as T tends toward 0. To show thatits derivative, with respect to T, is e T times the right-hand side of (12) (with t replaced by T), we rstobserve that the roots of (14) satisfy

1 22

2 (15)

and

1 22

2 . (16)

Also note that

e 2 u uT

2 1 T2 e

1 2 2T / 2 uT

1 2 T2 e

T u TT

(17)

by (10), (16) and (15). Similarly,

e 1 u u

T1 2 T

2 eT

u T

T. (18)

Using the chain rule, noting

d dT

u T

T

u2 T

3/ 2

2 T1/ 2 ,

and applying (17) and (18), we nd that the derivative of the right-hand side of (13) is

e T u T

T e T

u T

T, (19)

where

u2 T

3/ 2

2 T1/ 2 1 2

u4 T

3/ 23

1 22

8 T1/ 2

2

2

31 2

2

8 T1/ 2 (20)

by (15). It follows from (15) and (16) that

1 22 4

2

4

82 . (21)

Hence, equation (20) simpli es as / T. This completes the proof of (13). We remark that equation (14) is equivalent to the condition that the stochastic process { e t X ( t ) , t

0} is a martingale. An approach to pricing exotic options by means of such martingales can be found inLin (1998). Note that (14) is the same as (A6) in the appendix, except that z in (A6) is assumed to bepositive, whereas in (14) may be negative. Even though the two roots of (14) may be complex, formula(13) remains valid, extending Theorem 1 of Gerber and Pafumi (1998) to negative values of .

In the context of actuarial risk theory, equation (14) is sometimes called Lundberg s equation. Thisis to honor the Swedish actuary F. Lundberg, who pointed out that an equation analogous to (14) isfundamental to the whole of collective risk theory (Lundberg 1932, p. 144).


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As a check for the validity of formula (13), we assume 0 and take the limit 3 0 to obtain (8). With 0, the roots of equation (14) are 0 and 2 / 2 . The limit of formula (13) cannot be determinedby directly substituting these three values. However, because each term on the right-hand side of formula (13) has a in its denominator, the rule of Bernoulli-L Ho pital is readily applicable. Writing ( ) in (14) and differentiating it implicitly, we obtain the derivative formula

12 . (22)

Then we nd that (8), with t replaced by T, is indeed a limiting case of (13).

3. T WO EXPECTATIONSTo price various lookback options, we evaluate in this section the expectations

C u , E e M T e u (23)

and

P u , E e u e M T , (24)

where u is a positive constant, is the drift parameter of the Wiener process { X (t)}, and the randomvariable M (T) is de ned by (1). (The diffusion parameter, , is not exhibited explicitly, because it is notvaried in most of this paper.) Here, C and P stand for call and put, respectively. In the remainder of thepaper,

B u , T B u , , . (25)

That is, we eliminate T from the notation (because T is a xed time) and exhibit the dependence on thedrift parameter and the force of interest . Both expectations (23) and (24) can be expressed interms of the function B( u , , ). See (38) and (39) below. The impatient reader may proceed to the nextsection after reviewing these two formulas.

Because

Pr M T x Pr T x T 1 G x T , (26)

we know the cumulative distribution function of M (T), with which we could determine the expectations(23) and (24) by integration. Luckily, these tedious integrations can be avoided by means of thefunction B( u , , ). We remark that the cumulative distribution function of M (T) can also be derived byusing the re ection principle and the method of Esscher transforms; see Gerber and Shiu (2000) orHuang and Shiu (2001).

Two cases of formula (13) are of particular interest here. The rst case is 2 /2. Then thesolutions of (14) are 1 2 /

2 and 2 1. With the de nitions

d1 u , u T

T

u 2 /2 TT , (27)

d 2 u , u

T1 2 / 2 T

2 d1 u , T, (28)

and

d 3 u , uT

2 / 2 1 T2 d1 u ,

2 T, (29)


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formula (13) becomes

B u ,2

2 , 12

2 eT d1 u , e u d2 u ,

2

2 e2 u / 2 d3 u ,

e T d1 u , e u d2 u ,2

2 eT d1 u , e 2 u /

2 d3 u , . (30)

The other case is 2 /2. Noting that , 1 and 2 are the negative of those in the case above, weobtain from formula (13) that

B u ,2

2 , e u d2 u , e T d1 u ,

2

2 [e2 u/ 2 ( d3 u , ) e T d1 u , ]. (31)

For the evaluation of (23) and (24), we need one more formula for the function B( u , , ), which isobtained by differentiating (5) with respect to t and substituting the derivative in (11),

B u , , u 0

T

e t g x t ; dt dx . (32)

Here, by (A10) of the appendix,

g x t ; d dt G x t

x

2 t 3/ 2 exp

t x 2

2 2t (33)

is the rst passage time density function. We also note that, for each number , we have

e x g x t ; e2 2 / 2 t g x t ; 2 , (34)

which can be veri ed algebraically.Because the expectation of a positive random variable is the integral of its tail probabilities,

C u ,0

Pr e M T e u y dy0

Pr e M T e u y dy . (35)

With the change of variable, y e x e u , equation (35) becomes

C u , u

Pr M T x e x dx u

G x T e x dx . (36)

Substituting G x (T) 0T g x (t; ) dt in (36) yields

C u , u 0

T

e x g x t ; dt dx . (37)


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Applying the identity (34), with 1, to the integrand in (37) and comparing the resulting integral with(32), we see that

C u , B u , 2,2

2 , (38)

which can be evaluated using (30). Analogous to the derivation in the last paragraph, we have

P u ,0

Pr e u e M T y dy u

Pr M T x e x dx

u 0

T

e x g x t ; dt dx B u , 2 ,2

2 . (39)

In the last step, we used (34) with 1. Formula (39) can be evaluated with (31).

4. EUROPEAN LOOKBACK OPTIONSThe purpose of this section is to derive closed-form formulas for the prices of various European lookback options. Many EIAs contain lookback, high water mark or low water mark options (see AAA 1997, Streiff and DiBiase 1999, and Tiong 2000a, b).

4.1. European Fixed Strike Lookback Call OptionLet S(t) denote the price of a stock or asset at time t, t 0. Assume

S t S 0 e X t , (40)

where the process { X (t)} is a Wiener process with parameters and as in previous sections. A European lookback call option with exercise date T, T 0, and strike price (exercise price) K , K 0,is a security whose payoff, payable at time T, is

W max L, max0 t T

S t K . (41)

Here, L is a positive constant with L S(0); it can be interpreted as the maximum level of the stock spast ( t 0) prices. A main goal in this section is to determine E[ W ], and hence the price of the lookback call option. To this end, we need to distinguish two cases: (1) K L (strike price higher than pastmaximum) and (2) K L (strike price lower than past maximum).

For K L, equation (41) simpli es as

W max0 t T

S t K S 0 e M T K

S 0 , (42)

where the random variable M (T) is de ned by (1). It follows from (23) and (38) that

E W S 0 C ln K / S 0 , S 0 B ln K / S 0 , 2,2

2 , (43)

with B evaluated by (30).For a given force of interest , the discounted expected value of the payoff of the European lookback

call option is e T E[W ]. Before the appearance of the celebrated paper by Black and Scholes (1973), one


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would attempt to price an option by estimating the parameters and with historical data and usinga force of interest which is the sum of the risk-free (Treasury) rate and a risk premium. However, therisk premium depends on the investor s attitude toward risk. Whereas this attitude can be strictlyde ned in theory, it is dif cult or impossible to observe in practice. A crucial contribution made byBlack and Scholes is the insight that it is, in fact, not necessary to use any risk premium when valuingan option or a derivative security. (This does not mean that the risk premium disappears; instead, it isalready included in the stock price.) Black and Scholes would specify the drift parameter used in theoption-pricing formula to be

r2

2 . (44)

Here, r is the risk-free force of interest, which is assumed to be constant. It is also assumed that thestock is to pay dividends continuously at a nonnegative, constant dividend-yield rate . That is, the stock pays dividends of amount S(t) dt between time t and time t dt . If we consider all dividends beingimmediately reinvested in the stock, then each share of the stock at time 0 will become e t shares of thestock at time t. Equation (44) is the condition that the discounted value process, { e rt e t S(t)}, is amartingale. (It follows from equation 44 that there is no need to estimate the actual drift parameter inorder to price an option.)

Substituting the risk-neutral drift given by (44) in (43), we have that, in the Black-Scholes framework,the price of the European lookback call option with a xed strike price K and past maximum less than K is

e rT S 0 B ln K / S 0 , r2

2 , r . (45)

We remark that the quantity r is normally negative. It turns out that in all of Section 4, isalways equal to r .

To apply (30) to evaluate (45), we introduce the notation

d j, K d j ln K / S 0 , r , j 1, 2, 3, (46)

where d j( , ) on the right-hand side is de ned by (27), (28) or (29). Then (45) is

S 0 e T d 1, K Ke rT d 2, K S 0 2

2 r eT d 1, K

K S 0

2 r / 2

e rT d 3, K . (47)

Note that the rst two terms in (47) constitute the Black-Scholes formula for a plain-vanilla Europeancall option on a dividend-paying stock and, hence, the last term is the extra price for the privilege of exercising the option at the maximum stock price.

Now let us evaluate E[ W ] for the case K L. Here,

W max L, S 0 e M T K L K S 0 e M T L . (48)

The second half of the right-hand side of (48) is the same as the right-hand side of (42), with K replacedby L. Hence,

E W L K S 0 B ln L / S 0 , 2, 2

2 . (49)


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Replacing by the right-hand side of (44), we obtain the time-0 Black-Scholes price of a Europeanlookback call option with xed strike price K and past maximum level L, L K ,

e rT L K S 0 B ln L / S 0 , r2

2 , r

S 0 e T d1, L Le rT d2, L Ke rT S 0 2

2 r eT d1, L

L S 0

2 r / 2

e rT d3, L . (50)

For j 1, 2, 3, d j, L is de ned by (46), with K replaced by L. With 0, formulas (47), (50), (60) and (63) were rst derived by Conze and Viswanathan (1991).

4.2. European Floating Strike Lookback Put OptionHere, the payoff of the option at exercise date T is

max L, max0 t T

S t S T , (51)

where L S(0); L can be interpreted as the maximum level of past ( t 0) stock prices. Comparing (51)with (48), we see that the time-0 Black-Scholes price of this option is (50), with the discounted valueof K , Ke rT replaced by the time-0 price of the time- T payoff S(T), e T S(0), namely,

Le rT d2, L S 0 e T d1, L S 0 2

2 r eT d1, L

L S 0

2 r / 2

e rT d3, L . (52)

Note that rst two terms in (52) constitute the Black-Scholes formula for a plain-vanilla European putoption on a dividend-paying stock with strike price L. With 0, formulas (52) and (65) were rstderived by Goldman, Sosin, and Gatto (1979).

4.3. European Fixed Strike Lookback Put Option A European lookback put option with exercise date T, T 0, and strike price K , K 0, is a securitywhose payoff, payable at time T, is

Y K min L, min0 t T

S t . (53)

Here, L is a positive constant, with L S(0); it can be interpreted as the minimum level of the stock spast ( t 0) prices. We now de ne

M T max X t , 0 t T (54)

and note that { X (t)} is a Wiener process with parameters and . Then

min0 t T

S t S 0 e M T . (55)

Thus,

Y K min L, S 0 e M T . (56)

To determine E[ Y ], we distinguish two cases: (1) K L (strike price lower than past minimum) and (2) K L (strike price higher than past minimum).

For K L, equation (56) simpli es as

Y K S 0 e M T S 0K

S 0 e M T . (57)


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It follows from (24) and (39) that

E Y S 0 P ln S 0 / K , S 0 B ln S 0 / K , 2,2

2 . (58)

Replacing by the right-hand side of (44), we see that the time-0 Black-Scholes price of the Europeanlookback put option with a xed strike price K and past minimum higher than K is

e rT S 0 B ln S 0 / K , r2

2 , r . (59)

It follows from (31) that (59) is

Ke rT d2, K S 0 e T d1, K S 0 2

2 rK

S 0

2 r / 2

e rT d3, K e T d1, K . (60)

The rst two terms in (60) constitute the Black-Scholes formula for a plain-vanilla European put optionon a dividend-paying stock and, hence, the last term is the cost for the privilege of being able to exercisethe option at the minimum stock price.

Now let us evaluate E[ Y ] for the case K L. Here,

Y K min L, S 0 e M T K L L S 0 e M T . (61)


E Y K L S 0 P ln S 0 / L , K L S 0 B ln S 0 / L , 2,2

2 . (62)

Hence, the time-0 Black-Scholes price of a European lookback put option with xed strike price K andpast minimum level lower than K is

e rT K L S 0 B ln S 0 / L , r2

2, r

Ke rT Le rT d2, L S 0 e T d1, L S 0 2

2 rL

S 0

2 r / 2

e rT d3, L e T d1, L . (63)

4.4. European Floating Strike Lookback Call OptionHere, the payoff of the option at exercise date T is

S T min L, min0 t T

S t , (64)

where 0 L S(0); L can be interpreted as the minimum level of past ( t 0) stock prices. Comparing(64) with (61), we see that the time-0 Black-Scholes price of this option is (63) with the discounted

value of K , Ke rT , replaced by the time-0 price for the time- T payoff S(T), e T S(0), namely,

S 0 e T d 1, L Le rT d 2, L S 0 2

2 rL

S 0

2 r / 2

e rT d3, L e T d1, L . (65)

4.5. High-Low Option At maturity date T, the payoff of the high-low option or length-of-range option is

max Lmax , max0 t T

S t min Lmin , min0 t T

S t , (66)


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where 0 Lmin S(0) Lmax . The parameters Lmin and Lmax can be interpreted as the past ( t 0)stock price minimum and maximum, respectively. Comparing (66) with (51) and (64) or with (48) and(61), we see that the pricing formula is given by the sum of (52) and (65) or by the sum of (50) and (63),with the Ls appropriately modi ed.

In the special case that Lmin S(0) Lmax , formula (66) simpli es as

max0 t T

S t min0 t T

S t (67)

Furthermore, the d s also simplify. With the subscript L or K dropped from the notation of d s, we have

d 1 r 2 / 2 T

and

d 2 d 1 T d3.

5. D YNAMIC FUND PROTECTIONThis section examines the pricing of an investment fund with dynamic fund protection. Such a productdesign has been discussed by Gerber and Pafumi (2000), but their approach is different (also see Imaiand Boyle 2001).

For t 0, let I (t) denote the value of an index (for example, the Standard & Poor s 500) at time t. If the index does not include the reinvestment of dividends from its underlying assets (which is the casewith the S&P 500), assume that the dividends are a continuous stream of cash ow at a constantdividend-yield rate , 0. That is, assume that, between time t and time t dt , dividends of amount I (t) dt are paid. Also, assume that the index follows a geometric Brownian motion,

I t I 0 eY t , (68)

where the process { Y (t)} is a Wiener process with diffusion parameter . Let F(t) denote the time- t valueof a unit of a mutual fund, with

F 0 f . (69)

The fund has a participation rate p with respect to the index, according to the formula:

F t fe pY t , t 0. (70)

Furthermore, the fund has a dynamic fund protection that is de ned by a boundary or barrier function K (t), t 0, with K (0) 1. An example is

K t 0.9 1.03 t , t 0, (71)

which arises from a standard nonforfeiture requirement for insurers. Consider a customer purchasingone unit of the fund at time 0 for a payment of f . The maturity date of the contract is T, T 0. Thenthe company guarantees that the total value of the customer s account at time t, 0 t T, will not beless than fK (t). As soon as the customer s account value drops below the guaranteed level fK (t), thecompany will credit the customer s account with a suf cient number of fund units to restore the accountvalue to the guaranteed level. Then, the total number of fund units in the customer s account at timet must be

n t max 1, max0 s t

fK s F s , 0 t T. (72)


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In other words, for an initial payment of f , the customer has the amount n (t) F(t) in his account at timet , 0 t T.

Now let us determine the cost of dynamic fund protection. The differential dn (t) gives the additionalnumber of fund units to be credited to the customer s account at time t. Because the customer is to holdhis or her funds to maturity, the maturity value of the additional fund units is F(T) dn (t). Hence, thetime-0 value of dynamic fund protection is

E0

T

e rT F T dn t , (73)

where the expectation is taken with respect to the risk-neutral measure. That is, the expectation (73)is evaluated with the drift of { Y (t)}, , given by (44). To determine (73), we seek a constant such that{ e t F(t)} is a martingale (under the risk-neutral measure). Then (73) is

e r T E0

T

e T F T dn t e r T E0

T

e t F t dn t . (74)

With the de nition

D t ln n t max0 s t

ln fK s F s , (75)

we have

dn t n t dD t . (76)

(Both n (t) and D(t) are nite-variation functions; hence, we do not need to invoke Ito s lemma.) Forthose values of t for which the differentials are not zero,

n t fK t F t . (77)

Thus, it follows from (77) that (76) can be rewritten as

F t dn t fK t dD t , (78)

the substitution of which in the right-hand side of (74) yields

E0

T

e rT F T dn t fe r T E0

T

e t K t dD t . (79)

Motivated by (71), we now assume that

K t ke t, (80)

where 0 k 1 and is a guaranteed rate of return. Such a boundary function has been consideredin Gerber and Pafumi (2000, Sect. 4). Then the right-hand side of (79) simpli es as

fke r T E0

T

e t dD t . (81)


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ln fK s F s s pY s ln k . (82)

With the identi cations

X s s pY s (83)

and

u ln k , (84)

we see that the function D(t) de ned by (75) is of the same form as (2); hence, the expectation in (81)is our basic function B, with and with the drift and diffusion parameters of { X (t)} being p and p , respectively. Finally, for the process { e t F(t)} to be a martingale, we must have

ln E e pY 1 p p2 2

2 . (85)

Then it follows from (81) that the time-0 value of the dynamic fund protection is

fk exp r p p2 2

2 T B ln k ; p , p p2 2

2 , (86)

with given by (44). The function B in (86) can be evaluated using (30). However, for the diffusionparameter, we need to use p in place of . With p 1, 0 and 0, formula (86) reduces to (2.11)of Gerber and Pafumi (2000).

6. S TOCHASTIC GUARANTEED LEVELThe analysis in the previous section can be extended to the case of a stochastic guaranteed level. Inplace of fK (t) and F(t), consider S1 (t) and S2 (t) with 0 S1 (0) S2 (0). In the context of dynamic fundprotection, S2 (t) is the time- t value of one unit of an investment fund and S1 (t) is the guaranteed levelat time t. For t 0,

n t max 1, max0 s t

S1 s S2 s

(87)

is the number of units of the investment fund in the investor s account at time t , because it isthe minimal nondecreasing function satisfying n (0) 1 and

n t S2 t S1 t for all t 0.

We can also interpret S1 (t) and S2 (t) as the time- t prices of two stocks. Consider an investor whosepreference is to invest in stocks of type 2, but at any time wants to be able to buy one unit of stock 1.Hence, whenever the value of his portfolio (consisting of n (t) units of stock 2 at time t) threatens to fallbelow S1 (t), an adequate number of additional units of stock 2 must be bought. What is the time-0 priceof these additional units (purchased between time 0 and time T)? An alternative application of themodel is asset and liability management, with S2 (t) and S1 (t) being interpreted as the time- t values of assets and liabilities, respectively; however, we shall save such a discussion for a future paper.


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Consider S1 (t) S1 (0) e X 1 ( t ) and S2 (t) S2 (0) e

X 2 ( t ) , t 0, and assume that {( X 1 (t), X 2 (t)); t 0} is atwo-dimensional Wiener process. For j 1, 2, the drift and diffusion parameters of { X j(t)} are j and j,respectively. Equation (44) takes the form

j r

j

j2

2. (88)

For a contract with maturity date T, let V denote the time-0 value of dynamic fund protection,

V E0

T

e rT S2 T dn t . (89)

Because { e ( r 2 ) t S2 (t)} is a martingale,

V e 2T E

0

T

e r 2 t S2 t dn t , (90)

which generalizes (74).Now, let

X t X 1 t X 2 t . (91)

Dene

2 Var X 1 12 2 1 2 22, (92)

where is the correlation coef cient between X 1 (1) and X 2 (1). With

u ln S2 0 / S1 0 , (93)

and D(t) de ned by (2) and (1), formula (78) is generalized as

S2 t dn t S1 t dD t ; (94)

hence, equation (90) becomes

V e 2T E0

T

e r 2 t S1 t dD t S1 0 e 2T E0

T

e r 2 t e X 1 t dD t . (95)

Using the random variable e X 1 ( t ) /E[ e X 1 ( t ) ] e X 1 ( t ) e ( r 1 ) t as the Radon-Nikodym derivative for changingthe probability measure, we have

E e X 1 t dD t e r 1 t E dD t , (96)

where the tilde signi es that the expectation is to be taken with respect to the changed measure. Underthe changed measure, { X ( s)} is still a Wiener process, with drift parameter

E X 1 1 E X 2 1 1 12 2 1 2 2 12

2 . (97)

The last equality follows from (88) and (92). For more detail on the second equality in (97), see Gerberand Shiu (1994, 1996). The diffusion parameter of { X ( s)} remains under the changed measure.


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Hence, equation (95) is

V S1 0 e 2T E 0

T

e 1 2 t dD t S1 0 e 2T B u , , 1 2

S1 0 e 2T B ln S2 0 / S1 0 , 2 12

2 , 1 2 . (98)

We can use (30) to evaluate (98) if we replace u with ln[ S2 (0)/ S1 (0)], with 1 2 , and with . Withthe de nitions

d 1ln S1 0 / S2 0 T

T, (99)

d 2 d 1 T , (100)

d 3 d 12 1 2 T

, (101)

M S1 0 e 1T d 1 S2 0 e 2T d 2 , (102)

and

R2 1 2

2 , (103)

applying (30) to (98) yields

V M S1 0

R e2T

S1 0 S2 0

R

d 3 e 1T d 1 , (104)

which generalizes (3.1) of Gerber and Pafumi (2000). The quantity M , dened by (102), is the time-0price of a European option with exercise date T and payoff ( S1 (T) S2 (T)) (see Gerber and Shiu 1996,Eq. 10.18 or Panjer 1998, Eq. 10.8.8). Formula (102), with 1 2 0, was rst given by Margrabe(1978).

Finally, let us consider the special case where S1 (t) S(t) and S2 (t) L. Then, if we multiply (87)by L, we obtain W in (48) with K 0. Hence, formula (98) can be used to verify (49).

7. C ONCLUDING REMARKSThis paper has derived a three-term formula for B( u , T), the Laplace-Stieltjes transform of the expectedexcess of the running maximum of a Wiener process above a positive constant u in a nite time intervalof length T. The formula can be used to price various European lookback options and dynamic fundprotection. Exotic options can now be found in many nancial and insurance products. Closed-form

pricing formulas facilitate the hedging and reserving for such products. For some recent discussions onexotic options embedded in EIAs, see AAA (1997), Lee (2002), Mitchell and Slater (1996), Streiff andDiBiase (1999), and Tiong (2000a, b).

Treatments on exotic options have appeared in books by Bjo rk (1998), Briys et al. (1998), Clewlow and Strickland (1997), Haug (1998), Kwok (1998), Nelken (1996), Wilmott (1998), and Zhang (1998). Also note a recent paper on lookback options by Bermin (2000).

Our formula (13) is an extension of Theorem 1 of Gerber and Pafumi (1998); we are hopeful that itwill have more applications. To extend some of the results in this paper, one may consider Americanoptions instead of European options; for the case of perpetual American options, we have establishedsome closed-form formulas in Gerber and Shiu (2003). Also, the evaluation of (3), with 0 X (0) u


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and T being the time of ruin random variable, rather than a xed time, is a much easier problem (see,for example, Gerber and Shiu 2002).

ACKNOWLEDGMENTElias Shiu gratefully acknowledges the generous support from the Principal Financial Group Foundationand Robert J. Myers, F.C.A., F.C.A.S., F.S.A.

APPENDIXTHE FIRST PASSAGE TIME OF THE W IENER PROCESS

This appendix explains some well-known results in probability theory. The rst passage time of the Wiener process at a given level is closely related to the inverse Gaussian distribution (Chhikara andFolks 1989 and Seshadri 1993). Recall that an inverse Gaussian random variable Y , with shape parameter 0 and scale parameter 0, has a probability density function

f y2

y 3/ 2 exp y 2

2 y, y 0, (A1)

and a cumulative distribution function

F y y

y e2

y y

, y 0. (A2)

(It is straightforward to verify that F(0) 0, F( ) 1, and F ( y) f ( y), y 0.) We can nd an explicitexpression for the Laplace transform of (A1),

f z0

e zt f t dt E e zY , z 0,

by writing the integrand, e zt f (t), as a factor times an inverse Gaussian probability density function with

shape parameter 12 z

and scale parameter 12 z

. The factor is

f z exp 1 12 z

, z 0. (A3)

Now, let { X (t), t 0} be a Wiener process with X (0) 0, drift parameter and diffusion parameter. For a 0, let T a denote the rst time the process attains the level a ,

T a inf t 0 : X t a . (A4)

In the literature, the random variable T a is called a rst passage time. A goal of this appendix is to

determine its probability density function, g a (t), by means of the Laplace transform

g a z0

e zt g a t dt E e zT a , z 0. (A5)

For z 0, consider the stochastic process { e zt rX ( t ) , t 0}. This is a martingale if the number rsatis es the condition

2

2 r2 r z 0. (A6)


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The martingale is bounded for t T a if r 0, that is, if

r2 2 z 22 . (A7)

With this r , it follows from the optional sampling theorem thatE e zT a ra 1.

Hence,

g a z E e zT a e ra , z 0. (A8)

Substituting (A7) into the right-hand side of (A8) yields

g a z exp a

22 2 z 2 . (A9)

Using (A9), (A3), and (A1) we shall show that the probability density function of T a is

g a t a

2 t 3/ 2 exp

t a 2

2 2t , t 0. (A10)

To verify (A10), we need to take into account the sign of the drift parameter . For 0, the process{ X (t)} will attain the level a with certainty, so g a (t) is a proper probability density function. In fact,comparing (A9) with (A3), it follows immediately that T a has an inverse Gaussian distribution withshape parameter a / 2 and scale parameter 2 / 2 . Substituting these two values in theright-hand side of (A1) yields (A10). For 0, we take the limit ( 3 0 ) in (A10) and obtain

g a t a

2 t 3/ 2 exp

a 2

2 2t , t 0, (A11)

which is the probability density function of a one-sided stable distribution with index 1/2 (Feller 1971).For 0, note that T a with positive probability. The probability that T a assumes a nite valuecan be determined as follows:

Pr T a lim z3 0

E e zT a lim z3 0

e ra

by (A8). Because the value of r is given by (A7), we obtain

Pr T a e2 a / 2. (A12)

It follows that the distribution of T a is improper (defective) in this case. Thus, let us consider theconditional distribution of T a , given that T a . For z 0,

E e zT a

T a E e zT a

/Pr T a g a z / e2 a / 2

. (A13) Applying (A9) to (A13), we obtain

E e zT a T a exp a

22 2 z 2 . (A14)

Comparing (A14) with (A3), we see that the conditional distribution of T a is inverse Gaussian withshape parameter a / 2 and scale parameter 2 / 2 . The (defective) probability densityfunction of T a is the product of (A12) and the conditional probability density function of T a . Thus wend that (A10) also holds in this case of negative drift.


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Finally, we can apply (A2) to nd the cumulative distribution function of T a :

G a t a t

t e2 a / 2

a t

t, t 0. (A15)

Formula (A15) holds independently of the sign of . It is easy to check that

limt3

G a t 1

if 0, and that

limt3

G a t e2 a / 2

if 0, con rming (A12).

References

A MERICAN A CADEMY OF A CTUARIES (AAA). 1997. Final Report of theEquity Indexed Products Task Force, American Academyof Actuaries, Washington, D.C. Reprinted in part as Soci-ety of Actuaries Study Note 8V-312-01, Schaumburg, IL:Society of Acturaries.

BERMIN, H.-P. 2000. Hedging Lookback and Partial Lookback Options Using Malliavin Calculus, Applied Mathematical Finance 7: 75 100.

BJO RK , T. 1998. Arbitrage Theory in Continuous Time. Oxford:Oxford University Press.

BLACK , F., AND M. SCHOLES . 1973. The Pricing of Options andCorporate Liabilities, Journal of Political Economy 81:637 54.

BRIYS, E., M. BELLALAH, H. M. M AI, AND F. DE V ARENNE. 1998. Op-

tions, Futures and Exotic Derivatives. New York: Wiley.CHHIKARA , R. S., AND J. L. F OLKS. 1989. The Inverse Gaussian

Distribution: Theory, Methodology, and Applications. New York: Marcel Dekker.

CLEWLOW , L., AND C. STRICKLAND, EDS. 1997. Exotic Options: The State of the Art. London: International Thomson.

CONZE, A., AND V ISWANATHAN . 1991. Path Dependent Options: TheCase of Lookback Options, Journal of Finance 46: 1893 1907.

FELLER , W. 1971. An Introduction to Probability Theory and Its Applications. Vol 2. 2nd ed. New York: Wiley.

GERBER , H. U., AND G. P AFUMI. 1998. Stop-loss a Tempo Continuoe Protezione Dinamica di un Fondo d Investimento,

Rivista di Matematica per le Scienze Economiche e Sociali21: 125 46.. 2000. Pricing Dynamic Investment Fund Protection,

North American Actuarial Journal 4 (2): 28 37; also seeDiscussion in 5 (1): 153 7.

GERBER , H. U., AND E. S. W. S HIU . 1994. Option Pricing byEsscher Transforms, Transactions of the Society of Actu- aries 36: 99 140; Discussions 141 91. Reprinted, 1999, Investment Section Monograph, Society of Actuaries, 69 88.

. 1996. Actuarial Bridges to Dynamic Hedging and Op-

tion Pricing, Insurance: Mathematics & Economics 18:183 218.

. 2000. Discussion of Valuing Equity-Indexed Annu-ities, North American Actuarial Journal 4 (4): 164 9.

. 2002. Economic Ideas of Bruno De Finetti in the Wie-ner Process Model, in Metodi Statistici per la Finanza e le Assicurazioni, edited by B. V. Frosini. Milan, Italy: Vita ePensiero.

. 2003. Pricing Perpetual Fund Protection with With-drawal Option North American Actuarial Journal 7 (2),Forthcoming.

GOLDMAN, M. B., H. B. SOSIN , AND M. A. G ATTO . 1979. Path De-pendent Options: Buy at the Low, Sell at the High , Jour- nal of Finance 34: 1111 27.

H AUG, E. P. 1998. The Complete Guide to Option Pricing For- mulas. New York: McGraw-Hill.

HUANG, Y.-C., AND E. S. W. S HIU . 2001. Discussion of PricingDynamic Investment Fund Protection, North American Actuarial Journal 5 (1): 153 57.

I MAI, J., AND P. P. B OYLE. 2001. Dynamic Fund Protection, North American Actuarial Journal 5 (3):31 49; Discussion 49 51.

K WOK , Y.-K. 1998. Mathematical Models of Financial Deriva-tives. Singapore: Springer-Verlag.

LEE, H. 2002. Pricing Exotic Options with Applications toEquity-Indexed Annuities. Ph.D. thesis, University of Iowa, Iowa City.

LIN, X. S. 1998. Double Barrier Hitting Time Distribution with

Applications to Exotic Options, Insurance: Mathematics & Economics 23: 45 58.LUNDBERG, F. 1932. Some Supplementary Researches on the

Collective Risk Theory, Skandinavisk Aktuarietidskrift15: 137 58.

M ARGRABE, W. 1978. The Value of an Option to Exchange One Asset for Another, Journal of Finance 33: 177 86.

MITCHELL , G. T., AND J. SLATER JR . 1996. Equity-Indexed Annu-ities: New Territory on the Ef cient Frontier, Society of Actuaries Study Note 441-99-96, Schaumburg, IL: Societyof Actuaries.


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NELKEN, I., ED. 1996. The Handbook of Exotic Options: Instru- ments, Analysis, and Applications. New York: Irwin.

P ANJER , H. H., ED. 1998. Financial Economics: With Applicationsto Investments, Insurance, and Pensions. Schaumburg,IL: The Actuarial Foundation.

SESHADRI, V. 1993. The Inverse Gaussian Distribution: A Case Study in Exponential Family. Oxford: Oxford University Press.

STREIFF , T. F., AND C. A. DIBIASE. 1999. Equity-Indexed Annuities.Chicago: Dearborn Financial.

TIONG , S. 2000a. Equity-Indexed Annuities in the Black-ScholesEnvironment. Ph.D. thesis, University of Iowa, Iowa City.

. 2000b. Valuing Equity-Indexed Annuities, North American Actuarial Journal 4 (4): 149 70; also see Dis-cussions in 5 (3): 128 36.

W ILMOTT , P. 1998. Derivatives: The Theory and Practice of Fi- nancial Engineering. New York: Wiley.

ZHANG, P. G. 1998. Exotic Options: A Guide to Second Genera-tion Options. 2nd ed. Singapore: World Scienti c.

DISCUSSION

GRISELDA DEELSTRA*Hans U. Gerber and Elias S. W. Shiu have written a very interesting paper in which they study theproblem of pricing exotic options and guarantees in equity-indexed annuities. They present closed-formformulas for pricing lookback options and dynamic guarantees in a Black-Scholes framework with

constant parameters, in particular with constant interest rates. Their principal tool was a closed-formexpression of the Laplace-Stieltjes transform of the expected excess of the running maximum of a Wiener process (with constant drift) above a positive constant in a nite time interval of length T.

In this discussion note, I want to consider the idea of generalizing the interest rates to stochasticinterest rates, and I will present some dif culties encountered in doing so.

It is well-known (see, e.g., Lamberton and Lapeyre 1991, Musiela and Rutkowski 1997) that in astochastic interest rate framework, the price at time t of a zero-coupon bond P(t, T) with expiration dateT is given by

P t , T E Q expt

T

r u du t ,

where Q denotes the risk-neutral measure. For each maturity date T, there exists an adapted process(t, T) such that, under Q ,

dP t , T P t , T r t dt t , T dB t,

where ( Bt , t 0) is a Brownian motion under Q . For simplicity, assume that this Brownian motion isthe same as the Brownian motion generating the asset prices in the Black-Scholes model. Thishypothesis could be relaxed later on.

It is common practice in derivative pricing to use a change-of-numeraire to a forward-neutralprobability in order to get rid of the discounting term in the pricing formula. For example, let usconsider a European xed strike lookback call option with payoff (41)

C t E Q expt

T

r u du max L, max0 t T

S t K t P t, T E QT max L, max

0 t T S t K t ,

where QT is the forward-neutral probability related with maturity date T. This probability is de ned insuch a way that, for each price process ( X (t), t 0), ( X (t)/ P(t, T), t 0) is a martingale under QT . ByGirsanov s theorem, one easily veri es that ( Bt 0

t ( u , T) du , t 0) is a Brownian motion under QT ,

* Griselda Deelstra is a Professor at the Universite Libre de Bruxelles, I.S.R.O. and ECARES, Boulevard du Triomphe CP 210, 1050 Brussels,Belgium, e-mail: [email protected].


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which we will denote by ( BtT , t 0). As a consequence, the dynamics of the risky asset under QT are

given by

dS t S t

r t , T dt dB tT.

We see that, under QT , ( St , t 0) is no longer a geometric Brownian motion because ( (t, T), t 0)might even be stochastic, as in the Cox-Ingersoll-Ross model (Cox, Ingersoll, and Ross 1985). However,a common hypothesis in option pricing is to suppose that ( (t, T), t 0) is deterministic. This can bemotivated by the exponential volatilities in the Vasicek (1977) model and in the frequently appliedGaussian cases of the Heath-Jarrow-Morton model (Heath, Jarrow, and Morton 1992).

But, even then, in order to generalize the results in the Gerber and Shiu paper, one should be able tocalculate the Laplace-Stieltjes transform of the expected excess of the running maximum of a Brownianmotion with a time-dependent drift term. In the literature, for example, Durbin (1985, 1992) andSacerdote and Tomassetti (1996) have studied crossing probabilities in this more general setting, but byusing numerical approximating algorithms. The elegance in the Gerber and Shiu paper leans, however,in large part, on the explicit form of the Laplace-Stieltjes transform. Can this be generalized to the casewith time-dependent drift term?

Concerning the pricing of guarantees in equity-indexed annuities, the same problem holds whengeneralizing to stochastic interest rates. For example (81) would turn out to be equal to

fke TE exp0

T

r u du0

T

e t dD t fke T P 0, T E QT

0

T

e t dD t .

Now, X in (83) has, under the forward-neutral probability QT , a time-dependent drift term so that oneshould need the Laplace-Stieltjes transform for that case.

In conclusion, Hans Gerber and Elias Shiu have written an excellent paper, which will be dif cult togeneralize further.

REFERENCES

COX , J. C., J. E. I NGERSOLL , AND S. A. R OSS . 1985. A Theory of theTerm Structure of Interest Rates, Econometrica 53(2):385 407.

DURBIN, J. 1985. The First-Passage Density of a ContinuousGaussian Process to a General Boundary, Journal of Ap- plied Probability 22: 99 122.

. 1992. The First-Passage Density of the Brownian Mo-tion to a Curved Boundary, Journal of Applied Probabil- ity 29: 291 304.

HEATH , D., R. A. J ARROW , AND A. MORTON . 1992. Bond Pricing andthe Term Structure of Interest Rates: A New Methodologyfor Contingent Claim Valuations, Econometrica 60(1):77 105.

L AMBERTON, D., AND B. L APEYRE. 1991. Introduction au Calcul Stochastique Applique la Finance. Paris: Editions S.M.A.I.

MUSIELA , M., AND M. R UTKOWSKI. 1997. Mathematical Methods in Financial Modelling. New York: Springer-Verlag.

S ACERDOTE , L., AND F. T OMASSETTI . 1996. On Evaluation and As-ymptotic Approximations of First-Passage-Time Probabili-ties, Advances in Applied Probability 28: 270 84.

V ASICEK , O. A. 1977. An Equilibrium Characterisation of the TermStructure, Journal of Financial Economics 5(2): 177 88.

Additional discussions on this paper can be sub- mitted until July 1, 2003. The authors reserve the right to reply to any discussion. Please see the Sub- mission Guidelines for Authors on the inside back cover for instructions on the submission of discussions.


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Similarly, we have

h x e 1 x

x

e 2 1 y

y

e 2 t t d t dy . (16)

Both can be veried by identifying their Laplacetransforms. Furthermore, 1 and 2 are exchange-able in the above formulas. Second, for w ( x1 , x2 ) 1,

h i u0

g i x dx , and h u u

g x dx , (17)

and, similarly, for w ( x1 , x 2 ) I ( x 2 u ) ,

i h i 00

g i x dx ,

and

h 00

g x dx . (18)

We can now convert the Laplace transforms inEquation (10), using Equations (13) and (14), andobtain the following renewal equation:

u

12

0

u

u x g x dx

12 h u . (19)

It can be shown that, given that the rst dropbelow the initial surplus occurs, the density g( x)/ g (0) is the discounted rst drop density as fol-lows. For any y, let w ( x1 , x 2 ) I ( x 2 y ) and u 0;then, (18) and (19) imply that the survival func-tion of the discounted rst drop is (0), and it isgiven by 1/ 2 y g( x) dx . Furthermore, (17) and(19) together imply that the Laplace transform of the time of ruin is the tail of a compound geomet-ric distribution with geometric parameter g (0)/ 2

and the secondary distribution having the density g( x)/ g (0).

REFERENCESBOWERS JR ., N., H. G ERBER , J. H ICKMAN, D. JONES AND C. NESBITT .

1997. Actuarial Mathematics, 2nd ed., Schaumburg, IL:Society of Actuaries.

GERBER , H. U. AND E. S. W. S HIU . 1998. On the Time Value of Ruin, North American Actuarial Journal 2(1): 4878.

GERBER , H. U. AND E. S. W. S HIU . 2003. Discussion on Moments of the Surplus before Ruin and the Decit at Ruin in theErlang(2) Risk Process, North American Actuarial Jour- nal 7(3): 11719.

Pricing Lookback Options andDynamic Guarantees, Hans U.Gerber and Elias S. W. Shiu, January 2003

CARISA K. W. Y U*Professors Gerber and Shiu have derived a three-term formula for the Laplace-Stieltjes transformof the expected excess of the running maximumof a Wiener process above a positive constant in anite time interval. This closed-form formula canbe used to price European lookback options (calland put, xed and oating strike) and dynamicfund protection. I have several comments. First, Iwant to present an alternative derivation for for-mula (12), which gives an expression for the de-rivative ( d / dt )E[ D(t)]. Second, I shall evaluate theexpectation (89) in another way. Finally, I shallshow how to price lookback options with partialor fractional payoff by means of the formulas inthe paper.

Alternative Derivation of Formula (12) A key step for evaluating B( u , T) is an expressionfor ( d / dt )E[ D(t)], given by formula (12) in thepaper. The authors derived formula (12) usingequations (7) and (8). I now present an alterna-tive derivation of formula (12).

Differentiating formula (5) in the paper, wehave

d dt E D t

d dt

u

G x t dx u

d dt G x t dx ,

which, by formula (A10) or (33) in the paper, is

u

x

2t 3/ 2 exp

t x 2

2 2t dx . (D1)

* Carisa K. W. Yu is an M. Phil. student in the Department of AppliedMathematics, Hong Kong Polytechnic University, Hung Hom, HongKong, e-mail: [email protected].


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Replace the rst x in the integrand by t ( t x).Then the integral (D1) becomes the sum of twointegrals,

u

t 1/ 2

2 exp

t x 2

2 2t dx

u

t x

2t 3/ 2 exp

t x 2

2 2t dx , (D2)

the rst of which is

t u

t. (D3)

expression (D3) is the rst term on the right-hand

side of formula (12) in the paper. To evaluate thesecond integral in (D2), we introduce a new vari-able

y expt x 2

2 2t ,

whose differential is

d y d expt x 2

2 2t

t x2t exp

t x 2

2 2t dx .

Thus, the second integral in expression (D2) is

2t 1/ 2 0 exp

t u

t

2

/2

t

t u

t, (D4)

which is the same as the second term on theright-hand side of formula (12).

Alternative Evaluation of Expectation(89)In Section 6, the authors extend the analysis of dynamic fund protection to the case of a stochas-tic guaranteed level. The time-0 value of thedynamic fund protection with a xed maturitydate T is given by the expectation (89) in thepaper. Because the integrand in expectation (89)

does not depend on t, the integral in expectation(89) is

e rT S2 T n T n 0 e rT S2 T n T 1 .

Thus, expectation (89) is equivalent to

V E e rT S2 T n T 1 . (D5)

To evaluate this expectation, let us rst consider n(T). As in equation (91) in the paper, we let

X s X 1 s X 2 s . (D6)

Then it follows from formula (87) in the paperthat

n T max 1, max0 s T

S1 s S2 s

max 1, S1 0 S2 0

e M T

1 S1 0 S2 0

e M T S2 0 S1 0

, (D7)

where the random variable M (T) is dened byequation (1) in the paper. Note that { X ( s)} is a Wiener process with drift parameter 1 2and diffusion parameter , where 1 and 2 aregiven by equation (88) and is dened by equa-tion (92).

By using the factorization formula (Gerber andShiu 1994, p. 177, eq. R.13; 1996, p. 188, eq. 5.7),

we can factorize the right-hand side of equation(D5) as the product of two expectations,

V E e rT S2 T E* n T 1 S2 0 e 2TE* n T 1

S1 0 e 2TE* e M T S2 0 S1 0

(D8)

by equation (D7). Here, the asterisk signies thatthe expectation is taken with respect to a changedprobability measure in which { X ( s)} is a Wienerprocess with drift parameter

* E* X 1 1 E* X 2 1

1 1 2 2 22

2 1

2

2 .

(D9)

The diffusion parameter of { X ( s)} remains underthe changed measure. It follows from formulas(23) and (38) in the paper that

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E* e M T S2 0 S1 0

C ln S2 0 / S1 0 , *

B ln S2 0 / S1 0 , * 2, *2

2

B ln S2 0 / S1 0 , 2 12

2 , 1 2 , (D10)

with B evaluated by formula (30). Substitutingequation (D10) in (D8) yields formula (98) of thepaper.

Floating-Strike Lookback Put OptionLookback options are rather expensive. Partial

or fractional lookback options are cheaper al-ternatives. We now investigate the pricing of these options using the formulas in the paper. Forsimplicity, assume that no dividends are paid.That is, assume 1 2 0.

Consider a European lookback option with thefollowing time- T payoff

P max0 t T

S t S T , (D11)

where 0 1. If 1, then

P max0 t T

S t S T ,

which is a special case of the payoff formula (51)in the paper. With M (T) dened by formula (1) of the paper, equation (D11) can be rewritten as

P S T e M T X T1

. (D12)

Thus, the time-0 price of the option is

E e rT P E e rT S T e M T X T1

E e rT S T E* e M T X T1

S 0 E* e M T X T1

(D13)

by applying the factorization formula. Here, theasterisk signies that the expectation is takenwith respect to a changed probability measure in

which { X (t)} is a Wiener process with drift param-eter

* 2 r2

2 (D14)

and diffusion parameter .To evaluate the expectation on the right-hand

side of equation (D13), note that

M T X T max0 t T

X t X T , (D15)

and that the increment X (T) X (t) has the samedistribution as X (T t). Also,

max0 t T

X T t max0 s T

X s , (D16)

which is the running maximum of a Brownian

motion with drift the negative of that of { X ( s)}.Thus

E* e M T X T1

E e M T 1 , (D17)

where M (T) is the running maximum of a Brown-ian motion with drift

* r2

2

by equation (D14). Consequently, by formulas (23)

and (38) in the paper, equation (D13) becomes

E e rT P S 0 C ln , r2

2

S 0 B ln , r2

2 , r . (D18)

To apply formula (30) in the paper to evaluate thefunction B in equation (D18), we dene

d j d j ln , r , j 1, 2, 3,

where d j( , ) on the right-hand side is dened by

formulas (27), (28), and (29) in the paper. Thenformula (D18) is

E e rT P S 0 e rT d1 S 0 d22

2 r S 0 e rT d1 2 r /

2 d3 . (D19)

Let us compare formula (D19) with formula(25) in Section 19.2.4 of Briys et al. (1998). Theformulas are the same if M T 0

0 in Briys et al. (1998)


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is S(0). (There is a typographical error in Briys etal. (1998). The minus sign in the second term onthe right-hand side of their formula (25) shouldbe a plus sign.) Also, formula (D19) is the same asformula (5.7) in Heynen and Kat (1997) if their E0is S(0).

Floating-Strike Lookback Call OptionConsider a European lookback option with time- Tpayoff

C S T min0 t T

S t , (D20)

where 1. If 1, then equation (D20) sim-plies as

C S T min0 t T

S t ,

which is a special case of the payoff formula (64)in the paper. To nd the time-0 price of theoption, we dene

m T max X t , 0 t T . (D21)

Then

min0 t T

S t S 0 e m T . (D22)

Thus,

C S T S 0 e m T

S T1

e m T X T . (D23)

Hence, the time-0 price of the option is

E e rTC E e rT S T1

e m T X T

S 0 E*1

e m T X T (D24)

by applying the factorization formula. As in the

last section, the last expectation above is takenwith respect to a changed probability measure inwhich { X (t)} is a Wiener process with drift param-

eter * given by equation (D14) and diffusionparameter . Now,

m T X T max0 t T

X T X t , (D25)

which has the same distribution as M (T). Thus,

E*1

e m T X T E*1

e M T

P ln , r2

2 B ln , r2

2 , r (D26)

by formulas (24) and (39) in the paper. To applyformula (31) in the paper to evaluate the functionB in equation (D26), we dene

d j d j ln , r , j 1, 2, 3,

where d j( , ) on the right-hand side is dened byequations (27), (28), and (29). Then it followsfrom equations (D24) and (D26) that

E e rTC S 0 d2 S 0 e rT d12

2 r S 02 r / 2 d3 e rT d1 . (D27)

Let us compare formula (D27) with formula (23)in Section 19.2.4 of Briys et al. (1998). They arethe same if their m T 0

0 is S(0) and their is .Formula (D27) is also the same as the formula

(5.6) in Heynen and Kat (1997) if their E0 is S(0)and their is .

REFERENCESBRIYS, E., M. BELLALAH, H. M. M AI, AND F. DE V ARENNE. 1998. Op-

tions, Futures and Exotic Derivatives. New York: Wiley.GERBER , H. U., AND E. S. W. S HIU . 1994. Option Pricing by

Esscher Transforms, Transactions of the Society of Actu- aries 36: 99-140; Discussions 14191.

. 1996. Actuarial Bridges to Dynamic Hedging and Op-tion Pricing, Insurance: Mathematics & Economics 18:183218.

HEYNEN, R. C., AND H. M. K AT . 1997. Lookback OptionsPricingand Applications, in Exotic Options: The State of the Art ,pp. 99123, edited by L. Clewlow and C. Strickland. Lon-don: International Thomson.


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Pricing Lookback Options andDynamic Guarantees, Hans U.

Gerber and Elias S. W. Shiu, January 2003

MARC DECAMPS * AND MARC J.GOOVAERTS

Actuarial mathematics has been applied in vari-ous elds of economics over the years. Amongmany others, we can cite Hans Buhlmann (1980)who proved that the Esscher transform can beused to describe the Walrasian equilibrium in apure exchange economy and Hans U. Gerber andElias S. W. Shiu (1994) who wrote the seminalactuarial paper on option pricing in incompletemarkets. In this excellent paper, Gerber and Shiudemonstrate once again that actuarial conceptsare suitable in nance. They derive closed-formexpressions for the price of lookback options anddynamic guarantees with arguments arising inrisk theory. Lookback options have also been in-vestigated in nance and some methods rely onthe notion of local time. In this discussion, weaim to relate both approaches and show that thenotion of local time can be applicable to actuarial

purposes.The Wiener process { X (t) x 0 t B(t),t 0} models the log-return of the underlyingasset, and the authors are interested in the stop-loss premiums E[ D(t)] E[( M (t) u ) ] of therunning maximum M (t) max 0 t X ( ). Asmentioned in the introduction, we can interpret D(t) as the aggregate dividends paid by a com-pany up to time t. The notion of local time wasdevised by Paul Levy to measure the time spentby a diffusion in the vicinity of a point. We canbravely dene the local time of the diffusion X atthe point a as

Lt a X

0

t

X s a ds . (D1)

where ( x) is the Dirac delta function. In thecases 0 and 1, according to the cele-brated result of P. Levy, the processes

M t B t , 0 t

and

B t 12 Lt0 B , 0 t

have the same law. In particular, { M t } and{12 Lt

0 ( B ) Lt0 ( B)} are equal in law. Thus, the

moment generating function of D(t) ( M (t) u )

is m , t , x 0 , u E x0 e D t

E x0 u e / 2 Lt

0 B

E x0 u e Lt

0 B . (D2)

The Laplace transform of the last expectation canbe expressed by means of a Green function , seeSection 2.8 in Ito and McKean (1974),

0

e st m , t , x 0 , u dt

G x 0 u , x , s dx

where

G x 0 u , x , s G x 0 u , x , s

G 0, x , s G x 0 u , 0, s1 G 0, 0, s

and G( a , b , s) e 2 s b a / 2 s. By integratingwith respect to x , we obtain the Laplace trans-form of E[ D(t)] that simplies as follows for x 00:

0

e st E D t dt0

e st m 0, t, x 0, u dt

2 e u 2 s

2 s 3/ 2 . (D3)

* Marc Decamps is an Assistant at the K.U. Leuven, FETEW, Naam-sestraat 69, 3000 Leuven, Belgium; e-mail: [email protected]. Marc J. Goovaerts is Professor at the K.U. Leuven and at the U.v. Amsterdam, FETEW, Naamsestraat 69, 3000 Leuven, Belgium;e-mail: [email protected].

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The inversion of equation (D3) leads to expres-sion (6) in the paper for 0 and 1. Thequantity of interest is the expectation of the dis-counted aggregate dividends denoted B( u , T) andits Laplace transform is obtained by elementaryoperations on equation (D3)

0

e std

dt E D t dt e u 2 s

2 sE D 0

0

e s td

dt E D t dt e u 2 s

2 s

0

dTe sT

0

T

dte td

dt E D t e u 2 s

s 2 s.

The inversion of the Laplace operator yields theformula for B( u , T). The stochastic representation(D2) of the moment generating function is thesolution of the heat ow problem (see Sect. 2.3 inIto and McKean 1974),

12 x0

2 m t m

x0 m , t , u , u m , t , u , u

m , 0, x 0, u 1. (D4)

The previous partial differential equation stoppedat the rst hitting time H inf{t 0 : B u }reduces to equation (4.3) in Gerber and Shiu(2003) for 0. Figure 2 from that paper illus-trates clearly the connection between Lt0 ( B )and dividends. In the cases 0 and 1, wecan adapt formula (D2) by means of a Girsanovtransformation and a time scaling, see for in-stance Decamps et al. (2003). As pointed out onpage 165 of Tiong (2000), the Girsanov change of measure can be implemented with a Esscher

transform.In this contribution, Gerber and Shiu proposean actuarial alternative to the Brownian localtime for pricing options whose payoff involves therunning maximum of a Wiener process. As thesimulation of local time is a hard task, formulaesuch as those derived by the authors are morethan valuable. Moreover, they permit us to relateproblems in risk theory such as optimal dividendsstrategy to the concept of local time.

ACKNOWLEDGMENTThe authors wish to thank professors Gerber andShiu for discussions on this topic while they werevisiting the Actuarial Workshop held in Leuvenon March 6 7, 2003.

REFERENCESBU HLMANN, H ANS. 1980. An Economic Premium Principle, Astin

Bulletin 11: 52 60.DECAMPS, M ARC, M ARC J. G OOVAERTS , AND A NN DE SCHEPPER . 2003.

The Solution of the Fokker-Planck Equation with Bound-ary Conditions by Feynman-Kac Integration. Researchreport, Department of Applied Economics, K.U. Leuven,Belgium.

GERBER , H ANS U., AND ELIAS S. W. S HIU . 1994. Option Pricing byEsscher Transforms, Transactions of the Society of Actu-

aries 46: 99 140.. 2003. Optimal Dividends: Analysis with Brownian Mo-

tion. Technical report, University of Iowa.ITO , K IYOSHI, AND HENRY P. MCK EAN, JR . 1974. Diffusion Processes

and their Sample Paths. New York: Springer-Verlag.TIONG , S ERENA . 2000. Valuing Equity-Indexed Annuities, North

American Actuarial Journal 4(4): 149 70.

AUTHORS REPLY We are grateful to have received three excellentdiscussions of the paper. Deelstra (2003) has exam-ined the dif culties in generalizing the results to thecase of stochastic interest rates. A recent paper of related interest is Benninga, Bjo rk and Wiener(2002). Yu (2003) has presented an elegant alterna-tive derivation for two results in the paper. Ms. Yualso shows how the method in the paper can beadapted to price partial or fractional lookback options at time 0. Mr. Decamps and Professor Goo-vaerts have pointed out the applicability of the con-cept of local time. Their discussion has reminded usof the book by Harrison (1985).

REFERENCESBENNINGA , S IMON, TOMAS BJO RK , AND Z VI W IENER . 2002. On the Use

of Numeraires in Option Pricing, Journal of Derivatives10(2): 43 58.

DEELSTRA , GRISELDA . 2003. Discussion of Pricing Lookback Op-tions and Dynamic Guarantees, North American Actuar- ial Journal 7(1): 66 67.

H ARRISON, J . MICHAEL . 1985. Brownian Motion and Stochastic Flow Systems. New York: Wiley. Reprinted by Krieger, Malabar, FL, 1990.


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Y U , C ARISA K. W. 2003. Discussion of Pricing Lookback Optionsand Dynamic Guarantees, North American Actuarial Journal 7(3): 124 27.

Moments of the Surplus beforeRuin and the Decit at Ruin inthe Erlang(2) Risk Process,Yebin Cheng and Qihe Tang, January 2003

HANS U. GERBER* AND ELIAS S.W. S HIU

This discussion is inspired by the discussions of the Cheng and Tang paper by Li (2003) and Lin(2003). A main goal here is to present two alter-native derivations for Li s Theorem 2.

Let us begin with the integro-differential equa-tion (D10) in Gerber and Shiu (2003):

j 0

n 1

1 j 1

I c

j 1 D 0 u

0

u

0 u x p x dx

u

w u , x u p x dx .

For simplicity, we write

u 0 u ,

s j 0

n 1

1 j 1

c

j 1 s , (D1)

which is an n -th degree polynomial in s , and

u u

w u , x u p x dx . (D2)

Then the integro-differential equation can bewritten more concisely as

D p . (D3)

LAPLACE TRANSFORM OFIntegro-differential equations similar to (D3) canbe found in the literature, and they are usuallysolved by means of Laplace transforms ; for ex-ample, see Section 5.3 of Cox and Smith (1961)or Willmot (1999). Let f ( u) be a function withLaplace transform f ( ). The Laplace transform of the m -th derivative f ( m )( u) is

m f m 1 f 0 m 2 f 0 f m 1 0

(Spiegel 1965, p. 10). Thus, the Laplace trans-form of (D3) is

q p , Re 0,

(D4)

where q( ) is a polynomial of degree n 1 or less,with coef cients in terms of , c, 1 , 2 , . . . , n ,and the values of ( u) and its rst n 1 deriva-tives at u 0. It follows from (D4) that

q

p , Re 0. (D5)

Because ( ) is nite for Re 0, the numeratoron the right-hand side of (D5) must be zero when-ever the denominator is zero. The equation

p 0 (D6)

is a generalization of Lundberg s fundamental equation . As pointed out by Li (2003), Rouche sTheorem implies that, in the right half of thecomplex plane, equation (D6) has exactly n roots.

We denote them as 1 , 2 , . . . , n , and, for sim-plicity, assume that they are distinct. It followsthat q( ) is the collocation polynomial of thefunction ( ) with respect to these n points.Then, by the Lagrange interpolation formula weobtain

qj 1

n

jk 1k j

nk

j k. (D7)

* Hans U. Gerber, A.S.A., Ph.D., is Professor of ActuarialScience, Ecoledes hautes e tudes commerciales, Universite de Lausanne, CH-1015Lausanne, Switzerland, e-mail: [email protected]. Elias S. W. Shiu, A.S.A., Ph.D., is Principal Financial Group Founda-tion Professor of Actuarial Science, Department of Statistics and Actuarial Science, University of Iowa, Iowa City, Iowa 52242-1409,and visiting Chair Professor of Actuarial Science, Department of Ap-plied Mathematics, Hong Kong Polytechnic University, Hung Hom,Hong Kong e mail: eshiu@stat uiowa edu


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