autocallable structured products · they have to sell longer-dated zero coupon bonds and buy more...

1

AUTOCALLABLE STRUCTURED PRODUCTS

Author : Tristan Guillaume

Professional title : Lecturer and research fellow

Mailing address : Université de Cergy-Pontoise, Laboratoire Thema, 33 boulevard du port, F-95011

Cergy-Pontoise Cedex, France

E-mail address : [email protected]

Telephone number : + 33 6 12 22 45 88

Fax number : + 33 1 34 25 62 33

Abstract

In this article, a general, flexible form of autocallable note is analytically valued, and its payoff profile

and risk management properties are discussed. Two models are considered : a two-asset Black-Scholes

one (Black and Scholes, 1973), and a Merton jump-diffusion framework (Merton, 1976) combined

with a Ho-Lee two-factor stochastic yield curve (Ho and Lee, 1986). In the former setting, the general

autocallable structure under consideration includes many popular features : regular coupons, reverse

convertible provision, down-and-out American barrier, best-of mechanism and snowball effect. These

features are more or less fully addressed according to the entailed valuation difficulties. Simpler notes

are easily designed and priced on the basis of this general structure. In the second analytical

framework, the autocallable payoff structure is restricted to the following features : regular coupons, a

reverse convertible provision and possible participation in the growth of a single underlying equity

asset.

The formulae provided in this paper can be expected to be a valuable tool for both buyers and issuers

in terms of risk management. Indeed, they enable investors to assess their chances of early redemption

as well as their expected return on investment as a function of the contract’s specifications, while they

allow issuers to analyze and compute their various risk exposures in an accurate and efficient manner.

1. Introduction

Autocallables, also known as auto-trigger structures or kick-out plans, are very popular in the world of

structured products. They have captured a large part of the market share in recent years. Product

providers use them to offer higher payoffs than those on structured products that automatically run to a

full term. In its standard form, an autocallable is a note that is linked to an underlying risky asset

2

(usually a single stock, a basket of stocks or an equity index) and that has no fixed maturity. What is

referred to as the maturity of the autocallable is actually the maximum duration this product can stay

alive, usually ranging from 2 to 5 years. Several observation dates within the product’s life are

prespecified in the contract, typically on an annual or semi-annual basis. At each observation date, if

the value of the underlying is at or above a prespecified level, usually called the autocall trigger level

or autocall barrier, then the principal amount is paid back by the issuer to the holder of the note, along

with a coupon rate. It is said then that the note autocalls. The prespecified autocall trigger level is often

defined as the level of the underlying asset at the contract’s inception, but it does not have to be. It

may also vary in time. If there is no early redemption, the note proceeds to the next observation date,

where there is again the possibility of early redemption. A lot of plans kick out in year one or two,

leaving investors with the choice to reinvest in rollover substitutes from the same provider, switch to

another offer or opt back into the markets.

Another level can be prespecified for each observation date, below the autocall trigger level, such that

if the note does not autocall but the underlying is above that lower level, usually called the coupon

barrier, then the note pays a coupon rate. Some autocallable notes have a memory function embedded

– also called a snowball effect. With a memory function, the product will pay any coupons that have

not been paid on previous observation dates, if on a subsequent observation date all prerequisites are

met.

Several variants of the standard structure are traded on the markets. One way to allow for yield

enhancement is to include a reverse convertible mechanism : if the note has not been autocalled and if

the underlying asset has fallen below a prespecified conversion level at maturity, which is a kind of

safety threshold from the point of view of the investor, then the principal amount is not redeemed and

shares (or their equivalent cash amount) are paid back instead, which entails a capital loss for the

investor. This amounts to the sale of an out-of-the-money put by the investor to the issuer. This is a

suitable feature in the current market environment. Indeed, low interest rates leave little room for yield

enhancement if investors want full capital protection, as providers must use more of the initial capital

to guarantee its complete return. Moreover, in high volatility markets, volatility has to be sold in order

to generate a higher income. The answer is then to introduce capital-at-risk structures, typically losing

money if the underlying index has fallen 50% or further from its initial level.

Another way to allow for yield enhancement is to make the coupon payments at a given observation

date contingent on the underlying asset not having crossed a prespecified lower level since the latest

observation date. This amounts to the introduction of a down-and-out American barrier (i.e., a

continuous barrier) in a time interval between two observation dates.

Investors may also want to be given the opportunity to participate in potential increases in the value of

the underlying instead of receiving fixed coupons. More specifically, some contracts define a

prespecified upper level such that, if the underlying is above that level at a given observation date,

then the note autocalls and the coupon paid to the holder of the note is a percentage of the spot value

3

of the underlying. Finally, some autocallable notes include best-of (worst-of) features, which usually

consist in providing investors with a percentage of the maximum (minimum) between two or more

assets in case the contract allows for a participation in the market upward potential.

Despite their widespread use in the markets, there are not many academic studies on

autocallable products. While Bouzoubaa and Osseiran [2010] describe a variety of payoffs and analyse

risk management issues, all the other contributions focus on numerical pricing schemes. Fries and

Joshi [2008] study a product specific variance reduction scheme for Monte Carlo simulation purposes.

Deng et al. [2011] discuss finite difference methods in a numerical partial differential equation

framework. Kamtchueng C. [2011] discusses smoothing algorithms for the Monte Carlo simulation of

the Greeks. Alm et al. [2013] develop a Monte Carlo algorithm that allows stability with respect to

differentiation. No research article has yet come up with an analytical solution to the valuation

problem raised by the autocallable structured products traded in the markets, i.e. autocallables with

discrete observation dates. The main purpose of this paper is to fill this gap by providing a closed form

formula for a flexible autocallable structure with discrete observation dates. In a Black-Scholes

framework (Black and Scholes, 1973), our formula can capture the following features : regular

coupons, reverse convertible provision, down-and-out American barrier, best-of mechanism and

snowball effect. The way in which all these features can be embedded into an analytical formula is

more or less restrictive according to the entailed valuation difficulties : while regular coupons, the

reverse convertible provision and the snowball effect can be fully tackled, the American barrier and

best-of features are only partially covered, in the sense that the American barrier is not alive during the

entire product life and the best-of mechanism is limited to two assets. The advantage of using a Black-

Scholes model is that it enables to value sophisticated payoffs and it provides a simple hedging

method. However, it is well known that the model assumptions are flawed. In particular, it has long

been known that equity prices are not purely continuous and exhibit jumps. The latter are a way to

account for the skew observed in the options market, especially the steeper skew for short expirations

(Gatheral, 2006). Secondly, the assumption that interest rates are constant is particularly spurious for

the valuation of autocallable notes. Indeed, the latter are a combination of fixed income and equity

components which are usually long-dated. Moreover, the correlation between equity and interest rate

sources of randomness has a significant impact. When the stock market goes up, the duration of the

autocallable structure goes down. If there is a positive correlation between equity and interest rate,

sellers of the note make losses while hedging their interest rate exposure, whether equity increases – as

they have to sell longer-dated zero coupon bonds and buy more short term zero coupon bonds under

higher interest rates, or decreases – as they need to sell short term bonds and buy more longer-dated

bonds under lower interest rates. Conversely, if there is a negative correlation between equity and

interest rate, sellers of the notes make a net profit while hedging their interest rate exposure, whether

equity goes up and down because of the opposite directions of equity and interest rate. As a

consequence, pricing models that do not take the correlation between equity and interest rate into

4

account will underprice the autocallable structure when this correlation is positive and overprice it

when that correlation is negative.

That is why a second model is considered, which consists of a combination of a Merton-type jump-

diffusion (Merton, 1976) and a Ho-Lee two-factor stochastic yield curve (Ho and Lee, 1986). As a

result of the increased sophistication of the model assumptions, fewer payoff specifications can be

tackled, namely : regular coupons, a reverse convertible provision and possible participation in the

growth of a single underlying equity asset.

In both models, the way to achieve an analytical solution is to price the structure as a whole in the

form of an option valuation formula that is a function of the contract’s specifications and of fixed

levels of volatility. The obtained formulae can be expected to be a valuable tool both for investors and

issuers in terms of risk management, as they allow to analyze the influence of each variable as well as

to study the global properties of the structure in an accurate and efficient manner. They also provide a

sensible, fast approximation of the structure’s fair price before more general numerical schemes may

be tested, allowing for more stochastic factors.

The article is organized as follows : in Section 2, the autocallable structure priced under the Black-

Scholes model is described in detail; in Section 3, risk management issues are analysed; in Section 4,

the formula under the Black-Scholes model is stated; mathematical proof of this formula is not

reported because it is cumbersome; in Section 5, the formula under the jump-diffusion model with a

stochastic yield curve is stated and the proof of this formula is provided. An appendix discusses the

numerical implementation of the formulae.

2. A flexible autocallable structure

Let us begin by describing in detail the payoff structure that will be valued in a Black-Scholes setting

in Section 4. Several observation dates1 2, ,...,

nt t t T are set within the product life

00,t T

,

where T is the maximum duration of the product (its “maturity”). Two risky assets,1

S and2

S , are

picked. At each observation datemt , 1 m n , prior to expiry, autocall may occur in two ways :

(i) If the value of1

S at timemt , denoted by 1 m

S t , lies within the range ,m m

D U

,m m

D U ,

then the investor’s initial capital or notional N is redeemed at timemt , along with a coupon rate

my .

Both the range ,m m

D U

and the coupon ratem

y are prespecified in the contract. The autocall trigger

levelm

D is typically the value of the underlying at inception, i.e. 1 0S t .

5

(ii) If 1 mS t is greater than the level

mU , the note autocalls too, but, instead of yielding a

prespecified coupon rate, it provides the investor with a percentage1 of the return ratio

1 1 0/

mS t S t if this ratio is greater than the return ratio 2 2 0

/m

S t S t or with a percentage2

of 2 2 0/

mS t S t if the latter is greater than 1 1 0

/m

S t S t . Thus, the asset2

S was introduced

in the first place to allow for a best-of mechanism in the event of early redemption with a participation

rate.

Besides, if 1 mS t lies within a range ,

m mC D

,m m

C D , then the note does not autocall but a

prespecified coupon ratem

z is paid out to the investor; furthermore, all prior coupons that may have

been lost because the coupon barriersk

C , 1 k m , had not been crossed, are also paid out to the

investor at timemt . This amounts to embedding a memory function or snowball effect into the note.

Thus, there are three different barriers at each observation datemt : the autocall trigger level without

participation denoted bym

D , the autocall trigger level with participation denoted bym

U , and the

coupon barrier without autocall denoted bym

C . The relative positions of these barriers are as follows:

m m mC D U , with

mD being defined as 1 0

S t in most traded contracts. The following table

(Table 1) recapitulates all possible payoffs at any observation date prior to expiry.

Table 1 : Possible payoffs at each observation datemt , 1 m n prior to expiry, under a Black-

Scholes model

Event Outcome

1 m mS t C the note proceeds to the next observation date

1m m mD S t C the note pays out a prespecified coupon

mN z

, foregoing missed coupons are recovered and the noteproceeds to the next observation date

1m m mU S t D early exit with coupon : the investor’s initial

capital is fully redeemed, foregoing missed couponsare recovered and the note pays out a final

prespecified coupon 1m

N y

1 m mS t U and 1 2m m

S t S t early exit with participation in1

S : the investor’s

initial capital is fully redeemed, foregoing missedcoupons are recovered and the note pays out a final

coupon equal to a prespecified percentage1 of the

ratio 1 1 0/

mS t S t

1 m mS t U and 1 2m m

S t S t early exit with participation in2

S : the investor’s

initial capital is fully redeemed, foregoing missed

6

coupons are recovered and the note pays out a final

coupon equal to a prespecified percentage2

of the

ratio 2 2 0/

mS t S t

Notes : This table shows the possible payoffs provided by the autocallable structure under consideration at an observationdate prior to expiry

Let us now turn to the profile of the note at expiry or maximum durationnt T . The safety barrier,

denoted by B , is the European-type barrier under which the notional N will not be redeemed at par

and the reverse convertible mechanism will be activated.For the investor to receive a final couponn

Y ,

two conditions must be met : 1 nS t must lie in the range ,

n nD U

,n n

D U , and the minimum

value reached by1

S at any time between1n

t

andnt must not be smaller than a prespecified

American-type down-and-out barrier H . The way in which the investor may participate in the growth

of assets1

S and2

S also differs from previous observation dates, as potential participation in the

growth of2

S is not contingent on 1 nS t being above level

nU . Rather, a European-type up-and-in

barrier W is specifically attached to2

S and may trigger participation in the growth of2

S even if

1 nS t lies below

nU . The best-of mechanism is thus less restrictive at expiry than at the previous

observation dates. Finally, a European-type barriern

C determines whether the memory function or

snowball effect can be activated.

Thus, there are six different barriers at expiry or maximum durationnt T :

- the safety barrier, denoted by B , under which the reverse convertible mechanism is activated

- an exit barrier denoted byn

D triggering a final coupon payment

- an American-type down-and-out barrier denoted by H and monitored during time interval1,

n nt t

,

that determines the possibility of receiving a final coupon payment but not the possibility of receiving

a final payment through equity growth participation

- an exit barrier denoted byn

U triggering a final payment contingent on the value of 1 nS t

- an exit barrier denoted by W triggering a final payment contingent on the value of 2 nS t

- a coupon barriern

C triggering the memory function of the note

The following order holds :n n n

B C D U . The barrier W has to stand above B , but its

position can be freely chosen relative to barriers ,n n

C D andn

U , depending on how large the

7

influence of the best-of mechanism should be. The barrier H has to lie belown

D , but its position

can be freely chosen relative to barriers B andn

C .

The following table (Table 2) recapitulates all possible payoffs at expiry.

Table 2 : Possible payoffs at expiry or maximum durationnt T , under a Black-Scholes model

Event Outcome

1 nS t B the reverse convertible mechanism is triggered :

only a fraction 1 1 0/

nS t S t of the investor’s

initial capital is redeemed, foregoing missed couponsare definitely lost and no final coupon payment isreceived

1,

n nS t B C

and 2 nS t W the investor’s initial capital is fully redeemed, but

foregoing missed coupons are definitely lost and nofinal coupon is received

1,

n nS t B C

and 2 nS t W foregoing missed coupons are definitely lost, but

the investor’s initial capital is fully redeemed and thenote pays out a final coupon equal to a prespecified

percentage2

of the ratio 2 2 0/

nS t S t

1,

n n nS t C D

and 2 nS t W the investor’s initial capital is fully redeemed,

foregoing missed coupons are recovered but no finalcoupon payment is received

1,

n n nS t C D

and 2 nS t W the investor’s initial capital is fully redeemed,

foregoing missed coupons are recovered, and the notepays out a final coupon equal to a prespecified

percentage2

of the ratio 2 2 0/

nS t S t

1,

n n nS t D U

and 1

1inf

n nt t t

S t H

the investor’s initial capital is fully redeemed,

foregoing missed coupons are recovered, and the note

pays out a prespecified final couponn

N y

1,

n n nS t D U

and 1

1inf

n nt t t

S t H

the investor’s initial capital is fully redeemed,

foregoing missed coupons are recovered, but no finalcoupon is received as the American down-and-outbarrier has been breached

1 n nS t U and 1 2n n

S t S t the investor’s initial capital is fully redeemed,foregoing missed coupons are recovered and the notepays out a final coupon equal to a prespecified

percentage1 of the ratio 1 1 0

/n

S t S t

1 n nS t U and 1 2n n

S t S t the investor’s initial capital is fully redeemed,foregoing missed coupons are recovered and the notepays out a final coupon equal to a prespecified

percentage2

of the ratio 2 2 0/

nS t S t

Notes : This table shows the possible payoffs provided by the autocallable structure under consideration at expiry ormaximum duration

These multiple features are introduced in order to span a large variety of possible contracts. Despite its

rather complex payoff function, the fair value of the general autocallable structure under consideration

can be analytically obtained. Forthcoming Proposition 1 provides a formula for any number of

8

observation dates without snowball effect, while Proposition 2 provides a formula for four observation

dates. The reason why a formula is written down in the specific case of four observation dates is three-

fold : first, it is useful to provide a fully explicit example of how Proposition 1 is expanded, in case the

latter might look somewhat terse to the reader at first sight; second, an autocallable note with four

observation dates will serve as the basis for subsequent numerical examples; last but not least,

Proposition 2 includes the pricing of the snowball effect. The reason why the latter is not included in

Proposition 1 is because it seems impossible to come up with a compact formula for it in general, i.e.

without specifying the number of observation dates.

3. Risk management issues

Simpler notes can be designed on the basis of the above general autocallable structure.

Proposition 1 and Proposition 2 nest the valuation of all kinds of simpler structures by putting the

suitable inputs into the formulae :

- notes that do not allow exit through equity participation are valued by setting the parameter1

A

equal to one and the parameters2

A and3

A equal to zero

- notes that allow exit through participation in asset1

S only are valued by setting the parameter2

A

equal to one and the parameters1

A and3

A equal to zero

- the best-of provision is activated by setting the parameter3

A equal to one and the parameters1

A

and the parameters2

A equal to zero

- the snowball effect is activated by setting the parameter4

A equal to one in Proposition 2

All other adjustments are fairly obvious. For example, if you do not want an American barrier to

condition exit with a coupon rate at expiry, just let H tend to zero; if you do not want coupon

payments prior to expiry, just set all thei

z parameters equal to zero.

Investors need to be aware of the cost of benefitting from additional opportunities, compared to a

standard contract. Introducing intermediary coupon payments before expiry increases the note’s value

in a relatively straightforward manner, as long as they are fixed coupon payments. It is less easy to

anticipate precisely the effect of exit through equity participation, or the effect of a memory function,

or that of a best-of provision. In Table 3, the prices of four different types of contracts with a growing

number of options are compared. The first one is a plain single-asset note, with no autocall through

participation in the growth of1

S , and no memory function. The second one adds to the first one the

possibility of autocall through participation in the growth of1

S . The third one adds to the second one

a memory function. The fourth one adds to the third one a best-of provision.

9

In all three volatility settings of Table 3, the sharpest increase in value comes from adding the

possibility of autocall through participation in the growth of asset1

S . This effect is more and more

pronounced as volatility rises. The smallest increase in value results from adding a memory function,

but it is not negligible and it would be greater if the coupon barriers were higher. In the low volatility

setting, it is only slightly smaller than the increases in value resulting from adding other options. This

overall behavior is essentially a reflection of the probability of early autocall, as will be seen shortly.

The impact of the best-of provision is substantial, especially in the high volatility setting. Overall, the

introduction of non-standard features thus increases the value of the autocallable note to a large extent

: averaging across all three volatility settings, the price of contract 4 is 23.25% higher than that of

contract 1. There is a major difference between contracts 1, 2 and 3 on the one hand, and contract 4 on

the other hand, as regards the effect of volatility : the value of the former is a decreasing function of

volatility, whereas the value of the latter grows with volatility. We will come back in more detail to

the issue of price sensitivity later in this Section.

Table 3 : Prices of various types of 4-year maturity autocallable notes

Low volatility

1 212%, 15%

Medium volatility

1 228%, 35%

High volatility

1 245%, 60%

Value of contract 1 102.138322 90.7335053 80.5106173




Notes : This table compares the values of four types of autocallable notes with maximum expiry 4 years and annual

observation dates. Contract 1 is a single-asset note linked to1

S , with no early exit through participation, and no memory

function. Contract 2 includes the possibility of early exit through participation in the growth of1

S . Contract 3 includes a

memory function. Contract 4 includes a best-of provision. All reported prices are obtained using forthcoming Proposition 2

with the following inputs :

1 2 1 2 1 2 3 4 1 2 3 40 0 100, 2.5%, 0, 35%, 100, 115S S r D D D D U U U U

1 2 3 4 1 2 3 4 1 2 3 4 1 290, 75, 100, 80, 8%, 5%, 1C C C C B W H y y y y z z z z

Contract 1 is valued by setting1

1 and2 3 4

0 . Contract 2 is valued by setting2

1 and

1 3 40 . Contract 3 is valued by setting

2 4 1 31, 0 . Contract 4 is valued by setting

1 2 3 40, 1 .

Whatever the number of options available in an autocallable contract, the issue of the expected

autocall date is fundamental. Indeed, potential automatic early redemption is the specific property of

these products, relative to other index-linked notes, and most investors who choose to put their money

in autocallable structures hope to get their capital back at an early stage. The invested notional can be

10

redeemed as soon as on the first observation date,1t . In this respect, moderately bullish investors

would be well advised to set a high value for1

y , the coupon rate offered upon exit without

participation in either1

S or2

S at time1t . They will thus be in a position to benefit from higher

returns than if they invest in standard fixed-income. On another hand, strongly bullish investors had

better trade off a high value of1

y for high levels of participation in the growth of assets1

S and2

S .

They will thus be able to benefit from an amplification of upside market movements, relative to a long

position in the stocks themselves, while being protected from downside risk. Overall, in terms of

contract specifications, the chances to benefit from autocall as soon as the first or maybe the second

observation date depend mostly on the choice of the autocall trigger level. Investors who contemplate

lowering that level must be aware that this will cause the cost of the note to rise. For a given price, this

means that the coupon rates and the participation rates will be lower. The autocall trigger levels may

increase or decrease during the note’s life. The former case is often referred to as a step-up autocall in

the markets, while the latter is called a step-down autocall. When autocall levels vary, they usually do

so in a step-down pattern, so that the investor is more and more likely to kick out as time passes. When

dealing with notes that provide possible early exit with participation in the growth of risky assets, it is

obvious that lowering thei

U ’s will increase the likelihood of early redemption with participation,

which may be called the probability effect. But investors should be aware that this positive probability

effect may be offset by the lower expected return received when the note autocalls, which may be

called the return effect, so that the overall effect on the value of the note is ambiguous. Roughly

speaking, the higher the volatility, the more the return effect tends to prevail over the probability

effect, as the likelihood of reaching relatively highi

U ’s increases. Table 4 reports the maturity

breakdown on a four-year autocallable note including equity participation and a best-of provision, as a

function of volatility; for simplicity, volatility is supposed to be flat, i.e. the skew and the term

structure are not taken into account. Table 4 shows a couple of noticeable results. First, the most likely

outcome is to autocall on the first observation date, whether volatility is low or high. The probability

of autocall occurring as early as the first or the second observation date increases with volatility. This

increase is not linear, it accelerates when volatility is shifted from the “medium volatility” category to

the “high volatility” one. When volatility is low, the chances of autocall occurring at the fourth and

final observation date are quite substantial, but they go down as volatility gets higher. Again, this

decrease is not linear, it considerably accelerates as volatility moves to the “high volatility” regime, at

a faster pace than the observed increase in the probability of autocall at the first observation date.

Interestingly, autocall at the third observation date is always the least likely outcome, whatever the

volatility input, and its probability of occurrence changes more or less linearly, unlike the probability

of autocall at the first or at the last observation date.

11

Table 4 : Maturity breakdown of a 4-year autocallable note as a function of volatility under risk

neutrality

Low volatility

1 212%, 15%

Medium volatility

1 228%, 35%

High volatility

1 245%, 60%

Maturity = 1 year 0.59292711 0.60704343 0.66483994

Maturity = 2 years 0.14079933 0.14795149 0.16333123

Maturity = 3 years 0.06899081 0.0738965 0.08193993

Maturity = 4 years 0.19728274 0.17110857 0.0898889

Notes : This table reports the risk-neutral probabilities that the actual maturity on a 4-year autocallable note providing the

payoff described in Table 1 and Table 2 will be either 1, 2, 3 or 4 years. For simplicity, the volatilities of1

S and2

S are

assumed to be flat. The reported probabilities are obtained using forthcoming Proposition 2 with the following inputs :

1 2 3 1 2 1 20, 1, 0 0 100, 2.5%, 0, 35%,S S r

1 2 3 4 1 2 3 4100, 115D D D D U U U U

Other inputs to Proposition 2 have no relevance here. The probability that the actual maturity is 1 year is obtained by adding

up3 5 6 . The probability that the actual maturity is 2 years is obtained by adding up

9 11 12 . The

probability that the actual maturity is 3 years is obtained by adding up16 18 19 . The probability that the actual

maturity is 4 years is equal to one minus the probability that the actual maturity is 1, 2 or 3 years.

For risk management purposes, the probabilities of autocall need to be computed under the physical

measure. In Table 5, the risky assets1

S and2

S include risk premia. Overall, the structure of the

maturity breakdown remains the same as under the risk-neutral measure. There is little difference

between the probabilities of early exit in year 2 and in year 3 in Table 4 and Table 5. However, under

the risk-averse, physical measure, the likelihood that the note will extend to its maximum maturity of

four years is even lower, while the probability that autocall will occur as early as in year 1 is even

higher. Under the high volatility regime, exit at maximum expiry becomes the least likely outcome in

Table 5. Interestingly, the probability of autocall in year 1 does not increase monotonically with

volatility, as it is lower in the medium volatility regime than in the low volatility one. This observation

points out the need to accurately compute the sensitivity of the probability of early exit at any

observation date with respect to volatility. This is easily done by mere differentiation thanks to the

formulae provided in this paper.

Table 5 : Maturity breakdown of a 4-year autocallable note as a function of volatility under risk

aversion

Low volatility

1 212%, 15%

Medium volatility

1 228%, 35%

High volatility

1 245%, 60%

Maturity = 1 year 0.663790051 0.641161764 0.687729344

12

Maturity = 2 years 0.147111224 0.153577508 0.167175316

Maturity = 3 years 0.068736885 0.076020303 0.083416916

Maturity = 4 years 0.120361840 0.129240425 0.061678424

Notes : This table reports the probabilities that the actual maturity on a 4-year autocallable note providing the payoff

described in Table 1 and Table 2 will be either 1, 2, 3 or 4 years. The difference with Table 4 is that the returns on assets1

S

and2

S now include risk premia of 2% and 3% respectively.

Finally, two other features should be borne in mind by investors. First, it must be noticed that the best-

of provision plays a more important role at expirynt than at previous observation dates, as it is not

contingent on1

S being above then

U barrier. Indeed, as long as 2 nS t trades above the independent

barrier W , the best-of provision can be activated even if 1 nS t trades at levels below

nD , unless

the reverse convertible mechanism is activated (i.e., the reverse convertible provision dominates the

best-of one). One should also pay attention to the fact that the main role of the American down-and-

out barrier is to determine whether a final coupon payment is received in case the note does not

autocall before expiry, so that this barrier will not have a significant impact on the value of the note

unless a sufficiently high value forn

y is agreed on in the contract, which makes sense as a

compensation to the investor when early redemption has not taken place.

Now, from the perspective of traders who sell these products, the risks associated with the various

components of the autocallable structure are very similar to those associated with digitals. These

components have positive Delta, and sellers of the option will have to buy Delta of the underlying,

meaning that they will be long dividends, short interest rates and long borrow costs of the underlying.

To tackle the discontinuity risk around the coupon barriers, a barrier shift will usually be applied so

that the resulting shifted payoff over-replicates the initial payoff by the least amount, while making the

Greeks of the new payoff manageable near the barrier. The position in volatility depends on the

coupon barrier levels and on the forward price of the underlying asset1

S . The Vega of the

autocallable’s components is positive if the underlying’s forward price is lower than the trigger level;

otherwise, Vega is negative. Hence, the Vega of the general autocallable structure under consideration

in this paper will depend on the respective weights of the downside and upside components within the

overall structure. Typically, with the 'i

D s equal to the starting spot price 10S and the '

iC s being

smaller than the 'i

D s , the components of the autocallable structure providing fixed coupon payments

will have a negative Vega. The reverse holds for the components that provide participating payments.

In view of the potentially offsetting effects from the various components, one should always check

13

whether the Vega of the overall structure is positive or negative. Once endowed with the formulae

provided in this paper, this becomes an easy task using analytical or numerical differentiation.

One of the difficulties associated with hedging the sale of an autocallable note is that this product is

sensitive to the skew. Traders can hedge the components providing fixed payments by implementing a

simple roll-over strategy as follows : at each observation dateit , if there is no autocall and the note

proceeds to the next observation date1i

t

, they will take a long position in call spreads expiring at

time1i

t

. These call spreads will be more or less tight or wide depending on how conservative the

trader is, bearing in mind that the tighter the call spread the larger Gamma can become near the

coupon barrier. The catch is that the moneyness of the involved future call options cannot be known at

inception. As a result, the forward implied volatility inputs that might be used to price the structure at

time0

t will generally be wrong. Moreover, all components with expiry greater than1t are conditional

on not having autocalled at a previous observation date, which introduces a path-dependent element in

the valuation problem. In this respect, the usefulness of the formulae provided in this article is two-

fold : first, they are a function of the skew at time0

t , which, by definition, is observable, i.e. they do

not involve pricing a combination of forward start digitals that would inevitably require the input of

the forward implied volatility surface; second, by giving the risk-neutral conditional probabilities of

autocalling at each observation date, these formulae provide traders with a simple way to accurately

weight the call spreads put in place for each maturity. Table 6 compares the prices of four-year

maturity autocallable notes with and without the possibility of exit through participation under three

stylized regimes of skew. For the sake of simplicity, the at-the-money volatility remains flat up until

maximum duration. In line with most observed data on equity, the slope of the upper part of the skew

curve (i.e. the part to the left of the at-the-money level) is greater than the slope of the lower part of the

skew curve (i.e. the part to the right of the at-the-money level); besides, the “small” skew decreases

more or less linearly with time, while the decrease in the “large” skew with respect to time displays

more curvature. As the downside components of the autocallable structure have negative Vega and the

upside components have positive Vega, the effect of the skew is to lower the value of the autocallable

structure, as the skew increases the volatility of downside components and decreases the volatility of

the upside components with respect to the at-the-money line. The effect of the skew on non-

participating autocallable notes is moderate but non-negligible, and would obviously be bigger under

more extreme skew curves than those chosen here. The effect of the skew on participating autocallable

notes is of great magnitude, so that pricing the autocallable note using flat at-the-money volatility will

largely overestimate the product’s value. It must be noticed that, for some components of the

autocallable structure, it is not obvious to know which point of the implied volatility surface to use,

due to the path-dependent nature of the product and to the many different coupon and early

redemption barriers. The rule followed in this paper is, for every component involved, to pick the

14

point in the implied volatility surface that matches the last barrier and the observation date in the

digital; for example, the digital providing a coupon equal to3

y upon early exit at time3

t when

1 3 3 3,S t D U

, conditional on autocall not having occurred prior to3

t , is priced using the point

with coordinates 3 3,D t in the

1S volatility surface.

For implied volatility curves that “smile”, i.e. that display higher implied volatilities to the left but also

to the right of the at-the-money level, the previous analysis does not hold anymore. Indeed, the higher

volatility on the out-of-the-money components of the autocallable structure raises their prices since

they are Vega positive, so that the net effect of the smile is ambiguous and depends on the weight of

the upside digitals with respect to the downside digitals, “weight” meaning their respective

contribution to the total note’s value. Clearly, notes without the possibility to autocall through upward

market participation are skew negative, whether the implied volatility curve is skew- or smile-shaped.

Table 6 : Prices of two different kinds of autocallable notes under different regimes of skew

No skew Small skew Large skewContract 1 : no possibility

of exit throughparticipation

87.4125025 86.76425608 85.425556

Contract 2 : possibility ofexit through participation,

including a best-ofprovision

111.102677 108.869329 105.180181

Notes : This table compares the prices of four-year maturity autocallable notes with and without the possibility of early exitthrough participation under three stylized regimes of skew. The reported prices are computed using Proposition 2. The“small skew” and “large skew” regimes are defined by the implied volatility surface displayed in Table 7. The “no skew”

regime assumes an identically flat volatility at1

36% and2

42% . The term-structure of the risk-free rate is given

by : 1-year at 2.5%, 2-year at 2.75%, 3-year at 3% and 4-year at 3.25%. The contract specifications are as follows:

1 2

0 0 100S S , 35% ,1

1t ,2

2t ,3

3t ,4

4t ,1 2 3 4

90C C C C

1 2 3 4100D D D D ,

1 2 3 4115U U U U , 75B , 100W , 80H

1 2 3 48%y y y y

1 2 35%z z z ,

1 4 2 3A A 1, A A 0 for contract 1,

1 2 3 4A A 0, A A 1 for contract 2

Table 7 : Implied volatility surface used to compute the prices in Table 6

Small skew

1S volatility

surface

Small skew

2S volatility

surface

Large skew

1S volatility

surface

Large skew

2S volatility

surfaceCoordinates

1 1,C t

0.38 0.41

2 2,C t

0.375 0.40

3 3,C t

0.37 0.39

4 4,C t

0.365 0.38

15

1 1,D t

0.36 0.36

2 2,D t

0.36 0.36

3 3,D t

0.36 0.36

4 4,D t

0.36 0.36

1 1,U t

0.35 0.39 0.32 0.38

2 2,U t

0.35 0.39 0.33 0.39

3 3,U t

0.355 0.395 0.34 0.40

4 4,U t

0.355 0.395 0.35 0.41

4

,B t0.40 0.48

4

,W t0.42 0.42

Notes : This table reports the implied volatility inputs used to define the small skew and the large skew in Table 6. Only thepoints that are needed to compute the autocallable note’s value are reported.

4. Valuation formulae under a Black-Scholes model

This section includes the two formulae denoted by Proposition 1 and Proposition 2 in Section 1 and 2.

Proposition 1

Let 1S t and 2

S t be two geometric Brownian motions driven by standard Brownian motions

1B t and 2

B t with correlation coefficient . Under the risk-neutral measure denoted by RN ,

the dynamics of 1S t and 2

S t are given by :

1 1 1 1 1 1dS t r S t S t dB t (1)

2 2 2 2 2 2dS t r S t S t dB t (2)

where r is the risk-free rate,1 and

2 are two constant continuous dividend rates, and

1 20, 0 are two volatility inputs assumed to be extracted from an implied volatility surface

across a continuous range of strikes and maturities for both assets1

S and2

S .

Let the function 1 2 1 2 1, ,..., ; , ,...,

n n n

, , 1, 1 ,

i ii

, be defined as

follows : for 1n and 2n ,n

is the standard gaussian cumulative distribution function ; for

2n ,n

is given by the following special case of the multivariate standard gaussian cumulative

distribution function :

16

1 2

22 1

11

21

1 21/2

2

1

1 2

exp2 2 1

... ...

2 1n

ni i i

i i

n nnn

ii

n

x x x

x xx

dx dx dx

(3)

The numerical computation of the functionn

is explained in the Appendix.

Let the following notations hold :

- 1

ln0i

i

Dd

S

, 1

ln0i

i

Uu

S

, 1

ln0i

i

Cc

S

, 1,...,i n (4)

- 1

ln0

Bb

S

, 2

ln ,0

Hh

S

2

ln0

Ww

S

,

1

12

2

0ln

0

Sx

S

(5)

- 2

1 1 1/ 2r , 2

1 1 1/ 2r , 2 2

12 1 2 1 2 1 2/ 2 (6)

- 2 2

12 1 1 2 22 , 1 2

12

(7)

- 2 2

12 2 1 1 2 1 2ˆ / 2 , 2

1 1 1 2 1ˆ / 2,r 2

2 2 2ˆ / 2r (8)

-. is the indicator function, taking value 1 if the argument inside the braces is true and value zero

otherwise

-1

is a parameter that takes value one if the autocall note does not allow exit through participation

in equity growth, and zero otherwise

-2

is a parameter that takes value one if the autocall note allows exit through participation in asset

1S and

1S only, and zero otherwise

-3

is a parameter that takes value one if the autocall note allows exit through participation in asset

1S or in asset

2S , i.e. if it includes a best-of provision, and zero otherwise

- the symbol ; ...i

a i j k denotes the sequence of real numbersi

a , where i ranges from j to k

Then, the no-arbitrage value,AUTOCALL

V , of an autocallable note with n observation dates until

maximum expirynt , n , including all the features defined in Table 1 and Table 2 except for the

snowball effect, that is, except for the possibility of recovering foregoing missed coupons at any

observation date, is given by the following formula :

1

11

expn

AUTOCALL i ii

V r t N z

(9)

1 2 2 3 3exp 1

i ir t N y

17

2 1 1 4 3 1 1 5exp exp

i it N t N

3 2 2 6exp

it N

7 1 2 8exp

nN r t N

3 9 3 2 2 10exp exp

n nr t N t N

1 11 2 3 12exp 1

n nr t N y

1 13 2 3 14exp

nr t N

2 1 1 15 3 1 1 16exp exp

n nt N t N

3 2 2 17exp

nt N

where :

1 is the probability, under the money market numeraire, that the coupon

iN z is paid out to the

investor at timeit . It is given by : (10)

1 11 1

1 1

1 1 1 1

1

1 1

2 1 2 1

1 1

; 1,..., 1 , ; ; 1,..., 1 , ;

; 1,..., 2 , ; 1,..., 2 ,

j j j ji i i i

i i

j i j i

i i

j ji i

i i i i

j i j i

d t d tc t d tj i j i

t t t t

t tt tj i j i

t t t t

2 is the probability, under the money market numeraire, of autocall at time

it when the note does not

allow exit through equity participation. It is given by :

1 1

1

1 1

2

1

2 1

1

; 1,..., 1 , ;

; 1,..., 2 ,

j j i i

i

j i

i

j i

i i

j i

d t d tj i

t t

t tj i

t t

(11)

3 is the probability, under the money market numeraire, of autocall at time

it without equity

participation when the note allows equity participation. It is given by : (12)

18

1 11 1

1 1

1 1 1 1

3

1 1

2 1 2 1

1 1

; 1,..., 1 , ; ; 1,..., 1 , ;

; 1,..., 2 , ; 1,..., 2 ,

j j j ji i i i

i i

j i j i

i i

j ji i

i i i i

j i j i

d t d td t u tj i j i

t t t t

t tt tj i j i

t t t t

4 is the probability, under the

1S numeraire, of autocall at time

it with participation in the growth

of1

S in the absence of a best-of provision. It is given by :

1 1

1

1 1

4

1

2 1

1

; 1,..., 1 , ;

; 1,..., 2 ,

j j i i

i

j i

i

j i

i i

j i

d t u tj i

t t

t tj i

t t

(13)




of1

S when the contract includes a best-of provision. It is given by :

1 1 12 12

1

1 1 12

5 1

1

2 1

1

; 1,..., 1 , , ;

; 1,..., 2 , ,

j j i i i

i

j i i

i

j i

i i

j i

d t u t x tj i

t t t

t tj i

t t

(14)




of2

S when the contract includes a best-of provision. It is given by :

1 1 12 12

1

1 1 12

6 1

1

2 1

1

ˆ ˆ ˆ; 1,..., 1 , , ;

; 1,..., 2 , ,

j j i i i

i

j i i

i

j i

i i

j i

d t u t x tj i

t t t

t tj i

t t

(15)


1S numeraire, of exit at maximum expiry

nt under the reverse

convertible provision. It is given by :

1 17

11 1

; 1,..., 1 , ; ; 1,..., 1i i n in

ii n

d t b t ti n i n

tt t

(16)

19

8 is the probability, under the money market numeraire, of exit at maximum expiry

nt without a

final coupon because the autocall barrier has not been reached, in the absence of a best-of provision. It

is given by :

1 1 1 1 1

2

1 1 1 18

1

2

1

; 1,..., 2 , , ;

; 1,..., 2 ,

i i n n n

n

i n nn

i n

n

i n

d t d t b ti n

t t t

t ti n

t t

(17)

1 1 1 1 1

2

1 1 1 1

1

2

1

; 1,..., 2 , , ;

; 1,..., 2 ,

i i n n n n

n

i n nn

i n

n

i n

d t d t d ti n

t t t

t ti n

t t


nt without a

final coupon because the autocall barrier has not been reached, when the contract includes a best-of

provision. It is given by :

1 1 1 1 1 2

2

1 1 1 1 29 1

1

2

1

; 1,..., 2 , , , ;

; 1,..., 2 , ,

i i n n n n

n

i n n nn

i n

n

i n

d t d t b t w ti n

t t t t

t ti n

t t

(18)

1 1 1 1 1 2

2

1 1 1 1 21

1

2

1

; 1,..., 2 , , , ;

; 1,..., 2 , ,

i i n n n n n

n

i n n nn

i n

n

i n

d t d t d t w ti n

t t t t

t ti n

t t



nt with participation in

the growth of2

S , when the contract includes a best-of provision and when1

S is below the

participation barriern

U at timent . It is given by :

20

1 1 1 1 1 2

2

1 1 1 1 210 1

1

2

1

ˆ ˆ ˆ ˆ; 1,..., 2 , , , ;

; 1,..., 2 , ,

i i n n n n

n

i n n nn

i n

n

i n

d t d t b t w ti n

t t t t

t ti n

t t

(19)

1 1 1 1 1 2

2

1 1 1 1 21

1

2

1

ˆ ˆ ˆ ˆ; 1,..., 2 , , , ;

; 1,..., 2 , ,

i i n n n n n

n

i n n nn

i n

n

i n

d t d t d t w ti n

t t t t

t ti n

t t


nt with a final

couponn

N y when the note does not allow exit through equity participation. It is given by :

1 1 1 1

2

1 1 1 111

2 1

3 2

1 1

; 1,..., 2 , , ;

; 1,..., 3 , ,

i i n n n

n

i n nn

i n n

n n

i n n

d t h t d ti n

t t t

t t ti n

t t t

(20)

1

21

1 1 1 12 2

1 1 1 1

2 1

3 2

1 1

2; 1,..., 2 , , ;

; 1,..., 3 , ,

i i n n n

nh

i n nn

i n n

n n

i n n

d t h t d h ti n

t t te

t t ti n

t t t

1 1 1 1 1

2

1 1 1 1

2 1

3 2

1 1

; 1,..., 2 , , ;

; 1,..., 3 , ,

i i n n n n

n

i n nn

i n n

n n

i n n

d t d t d ti n

t t t

t t ti n

t t t

1

21

1 1 1 1 12 2

1 1 1 1

2 1

3 2

1 1

2; 1,..., 2 , , ;

; 1,..., 3 , ,

i i n n n n

nh

i n nn

i n n

n n

i n n

d t d t d h ti n

t t te

t t ti n

t t t


nt with a final

couponn

N y when the note allows equity participation. It is given by :

21

1 1 1 1

2

1 1 1 112 11

2 1

3 2

1 1

; 1,..., 2 , , ;

; 1,..., 3 , ,

i i n n n

n

i n nn

i n n

n n

i n n

d t h t u ti n

t t t

t t ti n

t t t

(21)

1

21

1 1 1 12 2

1 1 1 1

2 1

3 2

1 1

2; 1,..., 2 , , ;

; 1,..., 3 , ,

i i n n n

nh

i n nn

i n n

n n

i n n

d t h t u h ti n

t t te

t t ti n

t t t

1 1 1 1 1

2

1 1 1 1

2 1

3 2

1 1

; 1,..., 2 , , ;

; 1,..., 3 , ,

i i n n n n

n

i n nn

i n n

n n

i n n

d t d t u ti n

t t t

t t ti n

t t t

1

21

1 1 1 1 12 2

1 1 1 1

2 1

3 2

1 1

2; 1,..., 2 , , ;

; 1,..., 3 , ,

i i n n n n

nh

i n nn

i n n

n n

i n n

d t d t u h ti n

t t te

t t ti n

t t t


nt without

receiving a final couponn

N y because the down-and-out barrier has been crossed in the time

interval1,

n nt t

, when the note does not allow exit through equity participation. It is given by :

1 1 1 1 1

2

1 1 1 113 11

1

2

1

; 1,..., 2 , , ;

; 1,..., 2 ,

i i n n n n

n

i n nn

i n

n

i n

d t d t d ti n

t t t

t ti n

t t

(22)


nt without

receiving a final couponn

N y because the down-and-out barrier has been crossed in the time

interval1,

n nt t

, when the note allows exit through equity participation. It is given by :

22

1 1 1 1 1

2

1 1 1 114

1

2

1

; 1,..., 2 , , ;

; 1,..., 2 ,

i i n n n n

n

i n nn

i n

n

i n

d t d t d ti n

t t t

t ti n

t t

(23)

1 1 1 1 1

2

1 1 1 112

1

2

1

; 1,..., 2 , , ;

; 1,..., 2 ,

i i n n n n

n

i n nn

i n

n

i n

d t d t u ti n

t t t

t ti n

t t




the growth of1

S in the absence of a best-of provision. It is given by :

1 1 1 1 1

2

1 1 1 115

1

2

1

; 1,..., 2 , , ;

; 1,..., 2 ,

i i n n n n

n

i n nn

i n

n

i n

d t d t u ti n

t t t

t ti n

t t

(24)




the growth of1

S , when the contract includes a best-of provision. It is given by :

1 1 1 1 1 12 12

2

1 1 1 1 1216 1

1

2

1

; 1,..., 2 , , , ;

; 1,..., 2 , ,

i i n n n n n

n

i n n nn

i n

n

i n

d t d t u t x ti n

t t t t

t ti n

t t

(25)




the growth of2

S , when the contract includes a best-of provision and when1

S is above the

participation barriern

U at timent . It is given by :

1 1 1 1 1 12 12

2

1 1 1 1 1217 1

1

2

1

ˆ ˆ ˆ ˆ; 1,..., 2 , , , ;

; 1,..., 2 , ,

i i n n n n n

n

i n n nn

i n

n

i n

d t d t u t x ti n

t t t t

t ti n

t t

(26)

23

End of Proposition 1

Although Proposition 1 looks bulky, it may actually be regarded as quite compact, given that it

accomodates any number n of observation dates and that it nests a large variety of possible payoffs.

However, as mentioned earlier, Proposition 1 does not tackle the memory function embedded into

some autocallable notes. So let us now introduce Proposition 2, which expands Proposition 1 in the

special case 4n , thus providing an easy way for readers to make sure they fully understand how to

expand Proposition 1 for any integer n ; furthermore, Proposition 2 prices the snowball effect.

Proposition 2

Let the assumptions of Proposition 1 hold. Then, the no-arbitrage value,AUTOCALL

V , of an

autocallable note with 4 observation dates until expiry4

t , endowed with all the features described in

Table 1 and Table 2, including the snowball effect, is given by the following formula :

1 1 1exp

AUTOCALLV r t N z (27)

1 1 1 2 2 3 3exp 1r t N y

2 1 1 1 4exp t N

3 1 1 1 5 3 2 1 2 6exp expt N t N

2 2 7exp r t N z

2 2 1 8 2 3 9exp 1r t N y

2 1 2 1 10 3 1 2 1 11exp expt N t N

3 2 2 2 12 4 2 1 13exp expt N r t N z

3 3 14exp r t N z

3 3 1 15 2 3 16exp 1r t N y

2 1 3 1 17 3 1 3 1 18exp expt N t N

3 2 3 2 19exp t N

4 3 1 2 20 4 3 2 21exp expr t N z z r t N z

22 1 2 4 23 3 4 24exp expN r t N r t N

3 2 4 2 25exp t N

4 4 1 26 2 3 27exp 1r t N y

4 1 28 2 3 29exp r t N

24

2 1 4 1 30 3 1 4 1 31exp expt N t N

3 2 4 2 32exp t N

4 4 1 2 3 33exp r t N z z z

4 4 2 3 34 4 4 3 35exp expr t N z z r t N z

The parameter4

takes value one if the autocall note includes a memory function and value zero

otherwise. The full expansion of allm terms, m 1,2,..., 35 , is given in Appendix 2. There are

6 out of the 35m terms in this formula that are not a direct expansion of Proposition 1 for 4n ,

the role of which is to value the coupon-related memory function; they are 13 , 20 , 21 , 33 , 34

and 35 .

The meaning of these 6 terms can be intuitively defined as follows :

(i) 2 1 13exp r t N z is the risk-neutral value of the digital allowing to recover at time 2t

the coupon that was not received at time 1t ;

(ii) 3 1 2 20exp r t N z z is the risk-neutral value of the digital allowing to recover at

time 3t the coupon that was not received at time 1t and the coupon that was not received at time 2t ;

(iii) 3 2 21exp r t N z is the risk-neutral value of the digital allowing to recover at time

3t the coupon that was not received at time 2t ;

(iv) 4 1 2 3 33exp r t N z z z is the risk-neutral value of the digital allowing to

recover at time 4t the coupon that was not received at time 1t and the coupon that was not received at

time 2t and the coupon that was not received at time 3t ;

(v) 4 2 3 34exp r t N z z is the risk-neutral value of the digital allowing to recover at

time 4t the coupon that was not received at time 2t and the coupon that was not received at time 3t ;

(vi) 4 3 35exp r t N z is the risk-neutral value of the digital allowing to recover at time

4t the coupon that was not received at time 3t

The full expansion of the 35m terms in Proposition 2 is as follows :

25

1 1 1 1 1 1 1 1 11 1 1 2 1

1 1 1 1 1 1

c t d t d t

t t t

1 1 1 1 1 1 1 1 13 1 1 4 1

1 1 1 1 1 1

d t u t u t

t t t

1 1 1 12 12 1

5 2

1 1 12 1

, ;u t x t

t t

1 1 1 12 12 1

6 2

1 1 12 1

ˆ ˆ, ;

u t x t

t t

1 1 1 2 1 2 1 1 1 1 2 1 2 1

7 2 2

2 21 1 1 2 1 1 1 2

, ; , ;d t c t t d t d t t

t tt t t t

1 1 1 2 1 2 1

8 2

21 1 1 2

, ;d t d t t

tt t

1 1 1 2 1 2 1 1 1 1 2 1 2 1

9 2 2

2 21 1 1 2 1 1 1 2

, ; , ;d t d t t d t u t t

t tt t t t

1 1 1 2 1 2 1

10 2

21 1 1 2

, ;d t u t t

tt t

1 1 1 2 1 2 12 12 2 1

11 3

21 1 1 2 12 2

, , ; ,d t u t x t t

tt t t

1 1 1 2 1 2 12 12 2 1

12 3

21 1 1 2 12 2

ˆ ˆ ˆ, , ; ,

d t u t x t t

tt t t

1 1 1 2 1 2 1

13 2

21 1 1 2

, ;c t c t t

tt t

1 1 1 2 1 2 3 1 3 1 214 3

2 31 1 1 2 1 3

, , ; ,d t d t c t t t

t tt t t

1 1 1 2 1 2 3 1 3 1 23

2 31 1 1 2 1 3

, , ; ,d t d t d t t t

t tt t t

1 1 1 2 1 2 3 1 3 1 2

15 3

2 31 1 1 2 1 3

, , ; ,d t d t d t t t

t tt t t

1 1 1 2 1 2 3 1 3 1 2

16 3

2 31 1 1 2 1 3

, , ; ,d t d t d t t t

t tt t t

1 1 1 2 1 2 3 1 3 1 23

2 31 1 1 2 1 3

, , ; ,d t d t u t t t

t tt t t

26

1 1 1 2 1 2 3 1 3 1 2

17 3

2 31 1 1 2 1 3

, , ; ,d t d t u t t t

t tt t t

1 1 1 2 1 2 3 1 3 12 12 3 1 2

18 4

2 31 1 1 2 1 3 12 3

, , , ; , ,d t d t u t x t t t

t tt t t t

1 1 1 2 1 2 3 1 3 12 12 3 1 2

19 4

2 31 1 1 2 1 3 12 3

ˆ ˆ ˆ ˆ, , , ; , ,

d t d t u t x t t t

t tt t t t

1 1 1 2 1 2 3 1 3 1 2

20 3

2 31 1 1 2 1 3

, , ; ,c t c t c t t t

t tt t t

1 1 1 2 1 2 3 1 3 1 2

21 3

2 31 1 1 2 1 3

, , ; ,c t c t c t t t

t tt t t

1 1 1 2 1 2 3 1 3 1 2

3

2 31 1 1 2 1 3

, , ; ,d t c t c t t t

t tt t t

1 1 1 2 1 2 3 1 3 1 4 1 2 3

22 4

2 3 41 1 1 2 1 3 1 4

, , , ; , ,d t d t d t b t t t t

t t tt t t t

1 1 1 2 1 2 3 1 3 1 4 1 2 323 4

2 3 41 1 1 2 1 3 1 4

, , , ; , ,d t d t d t b t t t t

t t tt t t t

1 1 1 2 1 2 3 1 3 4 1 4 1 2 34

2 3 41 1 1 2 1 3 1 4

, , , ; , ,d t d t d t d t t t t

t t tt t t t

1 1 1 2 1 2 3 1 3 1 4 2 4 1 2 324 5

2 3 41 1 1 2 1 3 1 4 2 4

, , , , ; , , ,d t d t d t b t w t t t t

t t tt t t t t

1 1 1 2 1 2 3 1 3 4 1 4 2 4 1 2 35

2 3 41 1 1 2 1 3 1 4 2 4

, , , , ; , , ,d t d t d t d t w t t t t

t t tt t t t t

1 1 1 2 1 2 3 1 3 1 4 2 4 1 2 325 5

2 3 41 1 1 2 1 3 1 4 2 4

ˆ ˆ ˆ ˆ ˆ, , , , ; , , ,

d t d t d t b t w t t t t

t t tt t t t t

1 1 1 2 1 2 3 1 3 4 1 4 2 4 1 2 3

5

2 3 41 1 1 2 1 3 1 4 2 4

ˆ ˆ ˆ ˆ ˆ, , , , ; , , ,

d t d t d t d t w t t t t

t t tt t t t t

1 1 1 2 1 2 1 3 4 1 4 1 2 3

26 4

2 3 41 1 1 2 1 3 1 4

, , , ; , ,d t d t h t d t t t t

t t tt t t t

1

21

2

1 1 1 2 1 2 1 3 4 1 4 1 2 3

4

2 3 41 1 1 2 1 3 1 4

2, , , ; , ,

h d t d t h t d h t t t te

t t tt t t t

27

1 1 1 2 1 2 3 1 3 4 1 4 1 2 3

4

2 3 41 1 1 2 1 3 1 4

, , , ; , ,d t d t d t d t t t t

t t tt t t t

1

21

2

1 1 1 2 1 2 3 1 3 4 1 4 1 2 34

2 3 41 1 1 2 1 3 1 4

2, , , ; , ,

h d t d t d t d h t t t te

t t tt t t t

1 1 1 2 1 2 1 3 4 1 4 1 2 3

27 26 4

2 3 41 1 1 2 1 3 1 4

, , , ; , ,d t d t h t u t t t t

t t tt t t t

1

21

2

1 1 1 2 1 2 1 3 4 1 4 1 2 3

4

2 3 41 1 1 2 1 3 1 4

2, , , ; , ,

h d t d t h t u h t t t te

t t tt t t t

1 1 1 2 1 2 3 1 3 4 1 4 1 2 3

4

2 3 41 1 1 2 1 3 1 4

, , , ; , ,d t d t d t u t t t t

t t tt t t t

1

21

2

1 1 1 2 1 2 3 1 3 4 1 4 1 2 3

4

2 3 41 1 1 2 1 3 1 4

2, , , ; , ,

h d t d t d t u h t t t te

t t tt t t t

1 1 1 2 1 2 3 1 3 4 1 4 1 2 328 4 26

2 3 41 1 1 2 1 3 1 4

, , , ; , ,d t d t d t d t t t t

t t tt t t t

1 1 1 2 1 2 3 1 3 4 1 4 1 2 3

29 4

2 3 41 1 1 2 1 3 1 4

, , , ; , ,d t d t d t d t t t t

t t tt t t t

1 1 1 2 1 2 3 1 3 4 1 4 1 2 34 27

2 3 41 1 1 2 1 3 1 4

, , , ; , ,d t d t d t u t t t t

t t tt t t t

1 1 1 2 1 2 3 1 3 4 1 4 1 2 3

30 4

2 3 41 1 1 2 1 3 1 4

, , , ; , ,d t d t d t u t t t t

t t tt t t t

1 1 1 2 1 2 3 1 3 4 1 4 12 12 4 1 2 3

31 5

2 3 41 1 1 2 1 3 1 4 12 4

, , , , ; , , ,d t d t d t u t x t t t t

t t tt t t t t

1 1 1 2 1 2 3 1 3 4 1 4 12 12 4 1 2 3

32 5

2 3 41 1 1 2 1 3 1 4 12 4

ˆ ˆ ˆ ˆ ˆ, , , , ; , , ,

d t d t d t u t x t t t t

t t tt t t t t

1 1 1 2 1 2 3 1 3 4 1 4 1 2 3

33 4

2 3 41 1 1 2 1 3 1 4

, , , ; , ,c t c t c t c t t t t

t t tt t t t

1 1 1 2 1 2 3 1 3 4 1 4 1 2 334 4

2 3 41 1 1 2 1 3 1 4

, , , ; , ,c t c t c t c t t t t

t t tt t t t

1 1 1 2 1 2 3 1 3 4 1 4 1 2 3

4

2 3 41 1 1 2 1 3 1 4

, , , ; , ,d t c t c t c t t t t

t t tt t t t

28

1 1 1 1 1 1 2 1 2 2 1 2 3 1 3 4 1 4 1 2 335 4

2 3 41 1 1 1 1 2 1 2 1 3 1 4

, , , , , ; , ,c t d t c t d t c t c t t t t

t t tt t t t t t

where the function 4 11 12 21 22 3 4 1 2 3, , , , , ; , ,

is the same as in eq. (3) except for the

bounds of the1

x integral which are changed from1

x (lower bound) and1 1

x (upper

bound) to1 11

x and1 12

x respectively, and for the bounds of the2

x integral which are

changed from2

x (lower bound) and2 2

x (upper bound) to2 21

x and2 22

x

respectively.

5. Analytical formula for the value of a general autocallable note under a jump-diffusion equity

model correlated with a two-factor stochastic interest rate process

The autocallable payoff structure considered in this section is described in the following Table 8 and

Table 9.

Table 8 : Possible payoffs at each observation date mt , 1 m n prior to expiry under the

jump-diffusion stochastic interest rate model

Event Consequence m mS t C the note proceeds to the next observation date

m m mD S t C the note pays out a prespecified coupon

mM z and the note proceeds to the next

observation date m m mU S t D early exit with coupon : the investor’s initial

capital N is fully redeemed and the note paysout a final prespecified coupon mM y

m mS t U early exit with participation in S : theinvestor’s initial capital M is fully redeemedand the note pays out a final coupon equal to aprespecified percentage 1 of the ratio

0/mS t S t

29

Table 9 : Possible payoffs at expiry or maximum duration nt T under the jump-diffusion

stochastic interest rate model

Event Consequence nS t H the reverse convertible mechanism is

triggered : a fraction 0/nS t S t of the

investor’s initial capital M is redeemed n nH S t D the investor’s initial capital M is fully

redeemed n n nU S t D the investor’s initial capital M is fully

redeemed and the note pays out a prespecifiedcoupon nM y

n nS t U the investor’s initial capital M is fullyredeemed and the note pays out a final couponequal to a prespecified percentage 1 of the

ratio 0/nS t S t

The new modeling framework is now introduced.

Let 1 , 0W t t , 2 , 0W t t and 3 , 0W t t be three correlated Brownian motions,

whose constant pairwise correlation coefficients are denoted by 1.2 , 1.3 and 2.3 .

The default-free interest rate process , 0r t t is driven by :

1 1 2 2dr t t dt dW t dW t (28)

where 1 and 2 are two positive constants and t is a non-random, piecewise continuous function

satisfying a linear growth condition. Eq. (28) is a two-factor Ho-Lee model (1986). The reason that it

was chosen is two-fold : (i) it allows calibration to the observed data by fitting a suitable t ; (ii) the

finite-dimensional distributions of , 0r t t are multivariate normal, which enables to preserve

the analytical tractability of the full model, as will be seen later in the proof of forthcoming

Proposition 3. The latter point should be emphasized as there are other classical interest rate models

that are univariate normal and that can be appropriately calibrated but whose finite-dimensional

distributions are not multivariate normal, such as the two-factor Hull and White model (1994) and its

generalization known as the G2++ model. The reader wishing details about interest rate models and

how they can be fitted to market data may refer to Brigo and Mercurio (2006).

The underlying equity asset process , 0S t t is driven by :

3

dS tr t dt t dW t I t dN t

S t

(29)

30

where :

(i) t is a positive, piecewise continuous, non-random function such that 2

0

t

s ds ,

0t

(ii) , 0N t t is a Poisson process of constant intensity 0 .

(iii) 1,n n

nn

I t U t

, where inf 0,n t N t n and nU is a sequence of

independent, identically distributed random variables taking values in 1,

Let ln 1n nJ U be normally distributed with mean and variance 2 ; then, the parameter

is defined by : 2exp / 2 1 .

It is assumed that all random processes are defined on a suitable probability space , , ,t , in

which , 0t t is the smallest algebra generated by the random variables iW s , N s and

nU n N t , : 0s s t , n , 1,2, 3i . It is recalled that a Brownian motion

and a Poisson process relative to the same filtration must be independent (Shreve, 2004).

Thus, the stochastic differential equation for the equity asset price is a jump-diffusion process

extending the seminal Merton model (Merton, 1976) by integrating a stochastic interest rate process

driven by eq. (28). The reason for that extension is that it is essential to factor in the correlation

between sources of randomness in the equity world and in the fixed income world, as explained in the

introduction of this article.

The valuation formula can now be stated under the heading “Proposition 3”.

Proposition 3

Under the model assumptions of Section 5, the fair value, V , at time 0 0t , of the autocallable

contingent claim whose payoff is defined by Table 8 and Table 9, with four observation dates

1 2 3 4t t t t , is given by :

2 3 41 2 3 4 1

1 2 3 4

1 2 1 3 2 4 34

0 0 0 0 1 2 3 4

exp! ! ! !

n n nn n n n n

n n n n

t t t t t t tV M t

n n n n

(30)

1 1 1 2 1 4 5 1 3

2 2 6 7 2 9 10 2 8

3 3 11 12 3 14 15 3 13

4 4 16 17 19 20 4 18 21

1 1

1 1

1 1

1 1

t y z t

t y z t

t y z t

t y t

31

1 0 1 1 1 1 11

1

ln / ,U S t t t t t

t

1 0 1 1 1 1 12

1

ln / ,D S t t t t t

t

0 1 1 1 1 1 13

1

ln / ,S U t t t t t

t

1 0 1 1 1 1 14

1

ln / ,D S t t t t t

t

1 0 1 1 1 1 15

1

ln / ,C S t t t t t

t

1 0 1 1 2 1 2 2 0 2 2 2 2 2 16 2

1 2 2

ln / , ln / ,, ;

D S t t t t t U S t t t t t t

t t t

1 0 1 1 2 1 2 2 0 2 2 2 2 2 17 2

1 2 2

ln / , ln / ,, ;

D S t t t t t D S t t t t t t

t t t

1 0 1 1 2 1 2 0 2 2 2 2 2 2 18 2

1 2 2

ln / , ln / ,, ;

D S t t t t t S U t t t t t t

t t t

1 0 1 1 2 1 2 2 0 2 2 2 2 2 19 2

1 2 2

ln / , ln / ,, ;

D S t t t t t D S t t t t t t

t t t

1 0 1 1 2 1 2 2 0 2 2 2 2 2 110 2

1 2 2

ln / , ln / ,, ;

D S t t t t t C S t t t t t t

t t t

1 0 1 1 3 1 3 2 0 2 2 3 2 3

1 211 3

3 0 3 3 3 3 3 1 2

3 2 3

ln / , ln / ,, ,

ln / ,; ,

D S t t t t t D S t t t t t

t t

U S t t t t t t t

t t t

1 0 1 1 3 1 3 2 0 2 2 3 2 3

1 212 3

3 0 3 3 3 3 3 1 2

3 2 3

ln / , ln / ,, ,

ln / ,; ,


t t

D S t t t t t t t

t t t

1 0 1 1 3 1 3 2 0 2 2 3 2 3

1 213 3

0 3 3 3 3 3 3 1 2

3 2 3

ln / , ln / ,, ,

ln / ,; ,


t t

S U t t t t t t t

t t t

32

1 0 1 1 3 1 3 2 0 2 2 3 2 3

1 214 3

3 0 3 3 3 3 3 1 2

3 2 3

ln / , ln / ,, ,

ln / ,; ,


t t

D S t t t t t t t

t t t

1 0 1 1 3 1 3 2 0 2 2 3 2 3

1 215 3

3 0 3 3 3 3 3 1 2

3 2 3

ln / , ln / ,, ,

ln / ,; ,


t t

C S t t t t t t t

t t t

1 0 1 1 4 1 4 2 0 2 2 4 2 4

1 2

3 0 3 3 4 3 4 4 0 4 4 4 4 416 4

3 4

1 2 3

2 3 4

ln / , ln / ,, ,

ln / , ln / ,, ;

, ,


t t

D S t t t t t U S t t t t t

t t

t t t

t t t

1 0 1 1 4 1 4 2 0 2 2 4 2 4

1 2

3 0 3 3 4 3 4 4 0 4 4 4 4 417 4

3 4

1 2 3

2 3 4

ln / , ln / ,, ,

ln / , ln / ,, ;

, ,


t t


t t

t t t

t t t

1 0 1 1 4 1 4 2 0 2 2 4 2 4

1 2

3 0 3 3 4 3 4 0 4 4 4 4 4 418 4

3 4

1 2 3

2 3 4

ln / , ln / ,, ,

ln / , ln / ,, ;

, ,


t t

D S t t t t t S U t t t t t

t t

t t t

t t t

1 0 1 1 4 1 4 2 0 2 2 4 2 4

1 2

3 0 3 3 4 3 4 4 0 4 4 4 4 419 4

3 4

1 2 3

2 3 4

ln / , ln / ,, ,

ln / , ln / ,, ;

, ,


t t


t t

t t t

t t t

1 0 1 1 4 1 4 2 0 2 2 4 2 4

1 2

3 0 3 3 4 3 4 0 4 4 4 4 420 4

3 4

1 2 3

2 3 4

ln / , ln / ,, ,

ln / , ln / ,, ;

, ,


t t

D S t t t t t H S t t t t t

t t

t t t

t t t

33

1 0 1 1 4 1 4 2 0 2 2 4 2 4

1 2

3 0 3 3 4 3 4 0 4 4 4 4 421 4

3 4

1 2 3

2 3 4

ln / , ln / ,, ,

ln / , ln / ,, ;

, ,


t t

D S t t t t t H S t t t t t

t t

t t t

t t t

where the following definitions hold :

23 2 2

1 2 1.2 2 2 1

0 0

exp 06

it ui

i i

tt r t s dsdu

(31)

2 2 2

1 10 0

1 1exp

2 2

i it ti i

i i k kk k

t t s ds n s ds n

(32)

2

10 0 0

10

2

i it tu i

i i i kk

t r t s dsdu t s ds n

(33)

it

1/22 2

3/23/22 2 11 2 1.22 2 2

1.3 2.3 11 0 0

2 23 1.2

0

3 3

i i

i

t tiii

kk

t

ttn s ds s ds

s ds

(34)

1/223 2 2

1 2 1.2 2 2 1

3

i

i

tt

(35)

1/2

2 2

10

it i

i kk

t s ds n

(36)

3/2 3/21 2 1.2 1 2 1.2 2

1.3

0

3/2 3/22 2 1 2 2 12

2.3 1

0

3 3, /

3 3

i

i

t

i i

i j iti i

t ts ds

t t tt t

s ds

jt

(37)

34

3/21 2 1.22 2

1.3 1.3

0 0

3/22 2 12 2 2 2 2

2.3 1 2.3 1 3 1.210 0 0

,

3/

3

i i

i i i

i j

t t

i

i jt t t ii

kk

t t

ts ds s ds

t tt

s ds s ds s ds n

(38)

2 2 22.3 1.2 1.32 1 1.2 3 1.2 1.32.3 1 2.3 1

2 1

1 , , 1

(39)

The functions 1b and 2 1 2, ;b b are equal to the univariate and bivariate standard normal

cumulative distribution functions, respectively. For 2n , the function

1 1 1 2 1, ..., , ; ,..., ,n n n n nb b b is defined by eq. (3).

End of Proposition 3.

We now proceed to the proof of Proposition 3.

The proof relies on three lemmas called Lemma 1, Lemma 2 and Lemma 3. Unless stated otherwise,

the notations used here have already been defined in (28) – (39).

Lemma 1

Let , 0r t t and , 0S t t be driven by eq. (28) and eq. (29), respectively. Let b be a

positive number. Then, given N t , we have :

0ln /b S t

S t bt

(40)

Proof of Lemma 1 :

Given N t , let 0 0ln / ln /P S t b S t S b S

For 0t , the solution of eq. (29) yields, conditional on N t :

23

10 0 0

1ln

0 2

t t t N t

nn

S tN t r s ds t s ds s dW s J

S

(41)

where each nJ is a normal random variable with expectation and variance 2 , which will be

denoted as follows :

2;nJ

Integrating eq. (28) yields :

35

1 1 2 2

0

0

t

r t r s ds W t W t (42)

Let us denote by 2L the Hilbert space of random variables with finite variance on , ,F . Then,

given 1W t , it is well known that 2W t admits the following orthogonal decomposition in 2L (see,

e.g., Shreve, 2004) :

2 1.2 1 2 1 2W t W t B t (43)

where 2B is a standard Brownian motion defined on the same probability space as 1W and 2W and

independent of 1W . Furthermore, for any given Brownian motion , 0W s s defined on a

suitable probability space and any fixed 0t , we have:

3

0 0

0;3

t tt

W s ds t s dW s

(44)

which implies that, for any given x and fixed 0t , we have :

0 3

tt

W s ds x W t x (45)

Thus, as far as the computation of P is concerned, the integral 0

t

r s ds can be expanded as

follows :

2 2 11 2 1.2

1 2

0 0

03 3

t u ttr t s dsdu W t B t

(46)

Next, given 1W t and 2W t , the random variable 3W t admits an orthogonal decomposition in

2L as follows :

3 1.3 1 2 2 3 3W t W t a B t a B t (47)

where 2a and 3a are real-valued scalars and 3B is a standard Brownian motion defined on the same

probability space as the processes 1W , 2W and 3W and independent of the processes 1W , 2W and 2B .

Note that 3a must be positive since, by definition of the multivariate normal random vector

1 2 3, ,W t W t W t , 3a t is the standard deviation of 3W t conditional on 1W t and 2W t

From the definition of linear correlation and the bilinearity of covariance, we obtain :

2 3

2.3 1.2 1 1.3 1 2 1 2 2 2

cov , 1cov , cov ,

W t W tW t W t B t a B t

t t

(48)

36

2.3 1.2 1.32 2.3 1

2 1

a

The real number 2.3 1 is the partial correlation between 2W t and 3W t conditional on 1W t .

Next, from the standard deviation of 3W t and the stability under addition of a set of uncorrelated

normal random variables, we obtain :

2 2 2 2 21.3 3 3 1.3 3 1.22.3 1 2.3 1

1t a t a (49)

Now, as the function t is non-random, the isometry property of the Ito integral implies that the

random variable 3

0

t

s dW s is normally distributed with mean zero and variance 2

0

t

s ds ,

0t .

Hence, with regard to the computation of P , the integral 3

0

t

s dW s can be expanded as :

21 1.3 2 2.3 1 3 3 1.2

0

1t

s ds W t B t B tt

(50)

Since the random variables nJ are pairwise independent and are also independent of 1W t , 2W t

and 3W t , the following equality thus holds in distribution for any given 0t :

ln0

S tN t

S

3/2

1 2 1.22 21 1.3

0 0 0 0

10

2 3

t u t tt

r t s dsdu t s ds N t Z s ds

3/2

2 2 1 2 22 2.3 1 3 3 1.2 4

0 03

t ttZ s ds Z s ds N t Z

(51)

where 1 2 3, ,Z Z Z and 4Z are four uncorrelated standard normal random variables

Consequently, given N t , the random variable

ln0

S t

S

is normally distributed with mean given

by t and standard deviation given by t .

37

Lemma 2

Let , 0r t t and , 0S t t be driven by eq. (28) and eq. (29), respectively. Let 1t , 2t , 3t ,

4t be four non-random times such that : 1 2 3 40 t t t t . Let 1b , 2b , 3b and 4b be four

positive numbers. Then,

1 1 2 2 3 3 4 4, , ,S t b S t b S t b S t b

2 3 41 2 3 4 1

1 2 3 4

1 2 1 3 2 4 34

0 0 0 0 1 2 3 4

exp! ! ! !

n n nn n n n n

n n n n

t t t t t t tt

n n n n

1 0 1 2 0 2 3 0 3 4 0 4

1 2 3 44

1 2 3

2 3 4

ln / ln / ln / ln /, , , ;

, ,

b S t b S t b S t b S t

t t t t

t t t

t t t

(52)

Proof of Lemma 2 :

Given 4N t , let 1 1 2 2 3 3 4 4, , ,P S t b S t b S t b S t b . The first step is to shift

to log coordinates as in the proof of Lemma 1. Each random variable 0ln / , 1,2, 3, 4iS t S i ,

is continuous. Hence, one can express P as the following quadruple integral :

1 0 2 0 3 0 4 0ln / ln / ln / ln /

4 3 2 1

1 2 3 41 2 3 4

0 0 0 0

ln , ln , ln , ln

b S b S b S b S

P dx dx dx dx

S t S t S t S tdx dx dx dx

S S S S

(53)

Then, by conditioning and using the weak Markov property of the process , 0S t t , we have :

1 0 2 0 3 0 4 0ln / ln / ln / ln /

4 3 2 1

1 2 11 2 1

0 0 0

3 23

0 0

ln ln ln

ln ln

b S b S b S b S

P dx dx dx dx

S t S t S tdx dx dx

S S S

S t S tdx

S S

4 3

2 4 30 0

ln lnS t S t

dx dx dxS S

(54)

Using (51), it is straightforward to show that the correlation between 0ln /iS t S and

0ln /jS t S , i j , is equal to the standard deviation of 0ln /iS t S divided by the standard

deviation of 0ln /jS t S . Thus, from the definition of the bivariate normal distribution, we have :

38

0 0

ln lnj i

j i

S t S tdx dx

S S

(55)

2

2 22 2

1 1exp

2 1 /2 1 /

j j i i i

ij ji jj i j

x t t x t

tt tt tt t t

Hence, (53) becomes :

1 0 1 2 0 2 3 0 3 4 0 4

1 2 3 4

ln / ln / ln / ln /

4 3 2 1

232

1 11

3 2 212 2 2 1

11

/1 1exp

2 2 1 /4 1 /

b S t b S t b S t b S t

t t t t

i i i i

i i ii i

i

P dx dx dx dx

x t t xx

t tt t

(56)

Summing over all possible values of the process N t in each time interval 1,i it t and multiplying

by their respective probabilities of occurrence yields Lemma 2.

The smaller-dimensional case of 1 1 2 2 3 3, ,S t b S t b S t b is nested by Lemma 2 in an

obvious manner.

Lemma 3

Let Y be a normal random variable with mean Y and variance 2Y . Let 1 2 3 4, , ,X X X X

be a

quadrivariate normal random vector such that :

1 1 2 2 3 3 4 4, , ,X x X x X x X x

1 2 3 4

1 2 2 3 3 4

1 2 3 4

1 2 3 4

4 . . ., , , ; , ,X X X X

X X X X X XX X X X

x x x x

(57)

whereiX and

iX are the mean and the standard deviation of iX , respectively, and .i jX X is the

correlation coefficient between iX and jX , 2,i j . Then, if the random vector

1 2 3 4, , , ,Y X X X X is multivariate normal, we have :

1 1 2 2 3 3 4 4, , ,exp X x X x X x X xE Y

39

1 1 1 2 2 2

1 2

3 3 3 4 4 4

3 4

1 2 2 3 3 4

1 . 2 .

23 . 4 .

4

. . .

, ,

exp , ;2

, ,

X X Y X Y X X Y X Y

X X

X X Y X Y X X Y X YYY

X X

X X X X X X

x x

x x

(58)

where .iX Y is the correlation coefficient between iX and Y

Proof of Lemma 3 :

Let 1 2 3 4, , , , 1 2 3 4, , , ,Y X X X Xf y x x x x denote the joint density of the standardized random vector

1 2 3 4, , , ,Y X X X X , where Y

Y

YY

and i

i

i X

iX

XX

.

Then,

1 1 2 2 3 3 4 4, , ,exp X x X x X x X xE Y

(59)

1 2 3 41 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

, , , , 1 2 3 4 4 3 2 1exp , , , ,

X X X X

X X X X

x x x x

Y X X X X

y z z z z

y f y z z z z dydz dz dz dz

Following the same approach as in the proof of Lemma 1, a little algebra shows that the normal

random variables iX , 1,2, 3, 4i , admit the following orthogonal decompositions :

1 11 . 1X Y X YX Y Z (60)

2 1 2 2 12 . . 1 . 2X Y X X Y X Y XX Y Z Z (61)

3 1 3 2 3 1 3 1 23 . . 1 . . 2 . . 3X Y X X Y X X Y X X Y X XX Y Z Z Z (62)

4 1 4 2 4 1 3 4 1 2 4 1 2 34 . . 1 . . 2 . . . 3 . . . 4X Y X X Y X X Y X X X Y X X X Y X X XX Y Z Z Z Z (63)

where the 'iZ s are pairwise independent standard normal random variables that are all independent of

Y and where the following definitions hold :

. . .2 2 2. .. . .

1 , , 1a b a b

a a b b aa b a b

a

X X X Y X Y

X Y X X Y X Y XX Y X Y X X YX Y

(64)

. . . . .

. ..

b c b c a b a c

b c a

b a

X X X Y X Y X X Y X X Y

X X Y XX Y X

(65)

2 2 2. . . . . .

1c a b b a b b c a

X Y X X X Y X X Y X X Y X (66)

40

. . . . . . . . . . . .. .

1c d a b c d c d a d a c b d a b c a

c a b

X X Y X X X X X Y X Y X X Y X X Y X X Y X X X Y XX Y X X

(67)

2 2 2 2. . . . . . . . . .

1d a b c b a b b c a c d a b

X Y X X X X Y X X Y X X Y X X X Y X X (68)

Thus, we have :

1 11 . ;X Y X YX Y Y (69)

1

2 2 1 2 1

1

1 .

2 1 . . ., ;X Y

X Y X X Y X Y XX Y

X YX Y X Y

(70)

3 1 2, ,X Y X X (71)

3 2 11 1

3 3 1 2 2 1

1 2 1 1

3 1 2

. .1 . 1 .

. . 2 . ..

. .

;X X Y XX Y X Y

X Y X X Y X Y X X YX Y X Y X X Y

X Y X X

X Y X YY X Y

4 1 2 3, , ,X Y X X X (72)

4 2 11 1

4 4 1 2 2 1

1 2 1 1

4 3 1 2 3 2 11 1

3 3 1 2 2 1

3 1 2 1 2 1 1

. .1 . 1 .

. . 2 . ..

. . . . .1 . 1 .

3 . . 2 . .. . .

X X Y XX Y X Y

X Y X X Y X Y X X YX Y X Y X X Y

X X Y X X X X Y XX Y X Y

X Y X X Y X Y X X YX Y X X X Y X Y X X

X Y X YY X Y

X Y X YX Y X Y

4 1 2 3. . .

;Y

X Y X X X

From (69) - (72), one can derive 1 2 3 4, , , , 1 2 3 4, , , ,Y X X X Xf y z z z z as the following product of conditional

densities :

1 2 3 4, , , , 1 2 3 4, , , ,Y X X X Xf y z z z z (73)

1 2 1 3 1 2 4 1 2 31 , 2 1 , , 3 1 2 , , , 4 1 2 3, , , , , ,Y X Y X Y X X Y X X X Y X X Xf y f z y f z y z f z y z z f z y z z z

Substituting (73) into (59) and performing the necessary calculations then yields Lemma 3.

The proof of Proposition 3 can now ensue. According to the no-arbitrage theory of valuation in a

complete market (Harrison and Pliska [1981] ), the price of the contingent claim under consideration is

equal to the expectation of its discounted payoff under the equivalent martingale measure. But the

market here is incomplete, due to the introduction of jumps. This raises the question of the choice of a

relevant equivalent martingale measure. A necessary condition for the discounted price process

0

exp , 0

t

r s ds S t t

to be a martingale is to set the drift coefficient in the dynamics of

41

, 0S t t equal to r t . This can be shown in exactly the same way as when the riskless

rate process , 0r t t is constant (see, e.g., Lamberton and Lapeyre, 1997), so the details are

omitted. According to the classical argument by Merton (1976) that jump risk is diversifiable and

therefore not rewardable with excess return, this condition should be both necessary and sufficient. If

we consider this argument to be true, then the dynamics of , 0S t t can only be given by eq.

(29). However, one must bear in mind that this argument is debatable, as empirical evidence suggests

that there are industry wide shocks and even country wide shocks that are not easily diversifiable, so

that the possibility of perfect hedging remains theoretical. The literature on mean-variance hedging

(Schweizer, 1992) and quantile hedging (Föllmer and Leukert, 1999) can be consulted for alternative

approaches.

Denoting by the probability measure under which eq. (29) holds, and assuming that this is the

correct pricing measure, then the autocallable payoff defined in Table 8 and Table 9 implies that, at

each fixing date jt , 1,2, 3j , prior to maximum expiry 4t , the following expectations must be

computed :

1 1 1 1,..., ,

0

exp 1j

j j j j j

t

j jS t D S t D D S t Ur s ds M y N t

(74)

1 1 1 1,..., ,

0

exp 10

j

j j j j

t

jjS t D S t D S t U

S tr s ds M N t

S

(75)

1 1 1 1,..., ,

0

expj

j j j j j

t

j jS t D S t D C S t Dr s ds M z N t

(76)

At maximum expiry 4t , the following expectation is also required :

4

1 1 2 2 3 3 4

4, , , 4

0

exp0

t

S t D S t D S t D S t H

S tr s ds M N t

S

(77)

It is clear that the valuation problem comes down to computing two kinds of expectations, denoted by

1E and 2E , , 1,2, 3, 4 ,i j i j :

1

0

expj

i i

t

S t c jE r s ds N t

(78)

2

0

exp0

j

i i

t

jS t c j

S tE r s ds N t

S

(79)

42

where the symbol . inside the indicator functions is a vector notation and where the components of

ic , 0 i j , are positive constants

Thus, the value of the contingent claim under consideration can be obtained by applying Lemma 3,

provided the right parameters are entered into the formula.

Let 0

jt

Y r s ds and let

ln0

ii i

S tX t N t

S

. From eq. (46), the expectation of Y is

given by :

0 0

0jt u

jr t s dsdu (80)

while the standard deviation of Y is given by jt .

From eq. (51), the expectation and standard deviation of iX t are given by it and by it ,

respectively. Hence, the covariance between Y and iX t is given by the numerator in ,i jt t .

Now, let

0

ln0

jt

jj

S tY r s ds N t

S

. Then, the expectation and the variance of Y are

given, respectively, by :

2

10

1

2

jt j

j kk

E Y t s ds n

(81)

2 2

10

varjt j

kk

Y s ds n

(82)

Hence, the covariance between Y and iX t is given by the numerator in ,i jt t .

Adding all required 1E type and 2E type expectations together and then summing over all

possible values of the process N t in each time interval 1,i it t weighted by their respective

probabilities of occurrence, one can obtain Proposition 3.

It must be emphasized that the validity of the proof depends on the multivariate normality of the

random vector 1 2 3 4, , , ,Y X X X X in Lemma 3. It is well known that the univariate normality of each

marginal does not imply the multivariate normality of the joint density (see, e.g., Tong [1989]).

43

References

Alm, T., Harrach, B., Harrach, D., Keller, M., A Monte Carlo pricing algorithm for autocallables thatallows for stable differentiation, to appear in Journal of Computational Finance

Black, F., Scholes, M., The Pricing of Options and Corporate Liabilities, The Journal of PoliticalEconomy, Vol. 81, No. 3 (May - Jun., 1973), pp. 637-654

Bouzoubaa, M., Osseiran, A. (2010) Exotic Options and Hybrids, Wiley Finance

Broadie, M., Yamamoto, Y. (2005), A double-exponential fast Gauss transform algorithm for pricingdiscrete path-dependent options, Operations Research, vol. 53 (5), 764-779

Brigo, D., Mercurio, F. (2006), Interest Rate Models – Theory and Practice, Springer Finance

Davis, P.J., Rabinowitz, P. (2007), Methods of Numerical Integration, Dover Publications

De Weert, F. (2008), Exotic Options Trading, Wiley Finance

Deng G., Mallett, J., McCann, C. (2011), Modeling autocallable structured products, Journal ofDerivatives and Hedge Funds, vol.17, 326-340

Föllmer, H., Leukert, P. (1999), Quantile hedging, Finance and Stochastics, 3, 251-273

Fries C.P., Joshi, M.S. (2011), Perturbation stable conditional analytic Monte-Carlo pricing schemefor autocallable products, International Journal of Theoretical and Applied Finance, vol. 14 (02), 197-219

Gatheral, J. (2006), The Volatility Surface, Wiley Finance

Harrison, J.M., Pliska, S. (1981) Martingales and stochastic integrals in the theory of continuoustrading, Stochastic Processes and their Applications, 11, 215-260

Hull, J., White, A. (1994), Numerical procedures for implementing term structure models II : two-factor models, The Journal of Derivatives, 2, 37-47

Ho S.Y., Lee S.B. (1986), Term structure movements and pricing interest rate contingent claims,Journal of Finance, vol. 41 (5), 1011-1029

Kamtchueng, C. (2011), Autocallable more American than European ?, Working Paper, BarclaysCapital

Lamberton, D., Lapeyre, B. (1997), Introduction to Stochastic Calculus Applied to Finance, Chapman& Hall, 1997

Merton, R.C. (1976), Option pricing when underlying stock returns are discontinuous, Journal ofFinancial Economics, 3, 125-144

Schweizer, M. (1992), Mean variance hedging for general claims, Annals of Applied Probability, 2 (1),171-179

Shreve, S.E. (2004), Stochastic Calculus For Finance II Continuous-time Models, Springer Finance

Tong, Y.L. (1990), The multivariate normal distribution, Springer

44

Appendix

From a numerical point of view, the valuation formulae presented in this paper raise the question of

the computation of the functionn

. As the number n of possible exit dates increases, so does the

dimension of numerical integration. The latter can be reduced by using the following identities :

(i) when there are three observation dates, the actual numerical dimension of the function3

can be

brought down from 3 to 1 by using :

2

2

2 1 1 2 3 2 2

3 1 2 3 1 2 1 1 22 2

1 2

2 exp / 2, , ; ,

2 1 1x

x x xdx

(83)

(ii) when there are four observation dates, the actual numerical dimension of the function4

can be

reduced by a factor of two by using :

2 3

4 1 2 3 4 1 2 3

2

22

3 2

2 1

, , , ; , ,

x

x x

(84)

2 2

2 3

2

4 3 2 3 3 2 21 1 2

1 1 2 32 2

1 3

exp2 1

2 1 1

x x

x xxdx dx

(iii) more generally, when there are over four observation dates, the actual numerical dimension of the

functionn

can always be reduced by a factor of 2 by using :

1 2 1 1 2 1, , ..., , ; , ..., ,

n n n n n

(85)

2 13

2 13

...

n

n

xx x

22 2

2

3 2 2 1 2 22 2

2 2

222 2

2

1 1exp ...

2 2 1 2 1

1 2

n n n

n

nn

ii

xx x x x

1 1 2 1 1

1 1 2 3 12 2

1 1

...1 1

n n n

n

n

x xdx dx dx

45

The functions to be integrated in (83)-(84) are so smooth that a mere 16-point Gauss-Legendre

quadrature with a lower bound cut off at 5 is enough to achieve a high level of accuracy for pricing

purposes. When 2 is close to 1, the upper bound of the inner integral in (84) becomes very

« large ». To avoid numerical errors, it is safe to prespecify a maximum value for all possible multiple

integral endpoints. Given the standard normal nature of the integrands, bounding at an absolute value

of 5 will entail negligible loss of accuracy. The fact that the arguments in the1

x functions in (83)-

(85) may rise sharply when thei

coefficients, 1,..., 1i n , are close to 1 , is not a cause for

concern since these functions are, by definition, bounded at 1 and have already reached their

maximum to 910 accuracy at 5x .

The quality of numerical integration using (83) – (85) was tested up to six observation dates by

applying a mere 16-point Gauss-Legendre quadrature rule. A first set of tests were performed using

analytical benchmarks which consisted of products of univariate normal cumulative distribution

functions, by assuming that all correlation coefficients were equal to zero. In every dimension from 3

to 6, some 500 integral convolutions were computed by means of Proposition 1 and Proposition 2 with

randomly drawn parameters. The results always matched the analytical benchmarks to at least 810

accuracy. Then, option prices with non-zero, arbitrary values of the correlation coefficients were

computed and compared with numerical values obtained by Monte Carlo simulation using antithetic

variates and the Mersenne Twister random number generator. In terms of efficiency, it always took

less than one second to compute prices of 4-year autocallable notes with yearly observation dates by

means of Proposition 2 on a mainstream commercial PC, which is very fast in absolute terms and

dramatically more efficient than simulation techniques, which required from four to ten minutes to

yield accurate approximations in dimensions 5 and 6. Moreover, the efficiency gains are even greater

when it comes to the computation of the option sensitivities or greeks. A clear pattern of convergence

of the Monte Carlo estimates to the analytical values was noticed, as more and more simulations were

performed. For autocallable notes with 4 observation dates, 400,000 simulations along with a time

discretization factor of 8 quotations per business day in the time interval1,

n nt t

were necessary to

achieve 410 convergence, and it took 1,000,000 simulations to achieve 510 convergence.

Thus, in moderate dimension, the analytical formulae provided by Proposition 1 and Proposition 2

give very efficient and accurate numerical results, while the computational time required for a Monte

Carlo simulation to return reasonably precise approximations is clearly not satisfactory for practical

purposes. It must be stressed that the valuation of the majority of autocallable notes does not involve

more than moderate dimension, as most traded products include between four and eight possible exit

dates, being typically annual observation dates on 4 to 8 – year maturity notes. In higher dimension, as

more and more exit dates might be added, the quality of a plain Gauss-Legendre implementation of

46

(85) will deteriorate. One solution, then, would be to implement an adaptive Gauss-Legendre

quadrature, in order to control the error resulting from numerical integration. Combining this with a

Kronrod rule will reduce the number of required iterations (Davis and Rabinowitz, 2007).

Alternatively, another solution is to notice that the function that needs to be computed in increasing

dimension has the attractive feature that it matches the special structure of Gaussian convolutions

handled by the efficient Broadie-Yamamoto algorithm (Broadie and Yamamoto, 2005).

Regarding Proposition 3, the quadruple infinite series can be truncated in a simple manner by setting a

convergence threshold such that no further terms are added once the difference between two

successive finite sums of the quadruple series becomes smaller than that prespecified level. The speed

of convergence is inversely related to the intensity of the Poisson process. Numerical experiments

were carried out that showed that, when was equal to 0.1 , only three or four terms had to be

computed in each summation operator for the percentage variation in two successive approximations

of the quadruple infinite series to decrease below 0.0001%,; when was equal to 1 , seven terms were

required to achieve the same level of convergence. It should be pointed out that the rate of decay of the

denominator in (30) is “fast”, so that numerical errors may arise for “large” values of 1n , 2n , 3n and

4n . Fortunately, finance applications do not require the level of accuracy that might raise such

concerns, i.e. a small number of terms in the infinite series will suffice to achieve adequate

convergence for all practical purposes. The total computational time required by Proposition 3, on a

mainstream commercial PC, ranges between 0.8 and 18 seconds, depending on the number of terms to

be computed in each summation operator (from 4 to 8) . Such an efficiency is in sharp contrast with

the slowness of a Monte Carlo simulation approximation. For only a moderate level of accuracy to be

achieved ( 310 divergence from the analytical formula) , the latter requires between one and two

hours of computational time, depending on the level of (from 0.1 to 1), and assuming that the

maximum expiry of the autocallable structured product under consideration is 4t four years. It must

be emphasized that, compared with a standard geometric Brownian motion model with constant

parameters, the introduction of additional stochastic factors such as jumps and a multi-factor term

structure of interest rates heavily deteriorates the quality of Monte Carlo simulation approximations, as

the payoff under consideration is path-dependent and requires time discretization. Hence the

usefulness of accurate, quickly computed analytical formulae.

autocallable structured products · they have to sell longer-dated zero coupon bonds and buy more...

Documents