underwriting fees and power derivatives

Initial version: June 1998 Current version: August 1999

UNDERWRITING FEES AND POWER DERIVATIVES

by E. Barone(*) and A. Castagna(*)

(*) SanPaolo IMI. (**) IMI Bank The authors wish to thank Vittorio Serafino whose suggestions and encouragement stimulated this paper and Stefano Risa for his helpful comments.

SUMMARY

In this paper we analyze the problem of determining standby underwriting fees within the framework of option-pricing theory. Financial institutions that provide standby underwriting for a stock placement bear the risk of having to buy unplaced stocks if the offered quantity is not completely absorbed by the market. We describe a simple model in which the equilibrium price of the stock at the end of the placement period is log-normal and the demand curve for stocks can shift according to stochastic shocks. We show that the value of the guarantee offered by the financial institutions is proportional to the value of a �quadratic� power put, that is a derivative whose payoff is the square of an ordinary put. This option can be priced within the Black & Scholes theoretical framework. The closed-form formula shows that the value of a quadratic put is much greater than that of an ordinary put and is not simply the square of the latter�s value. The paper also analyzes the problem of �re-insurance�, which arises when the underwriting fee is partially transferred to another financial institution that provides a subsidiary guarantee. We analyze three different contractual arrangements between two financial institutions, and accordingly show how to allocate the total fee consistently between them. A simulated application of the model is also presented.

CONTENTS

1. INTRODUCTION 1

2. THE GENERAL MODEL 1 2.1 Stock price dynamics 1 2.2 Demand curve 2 2.3 An example 3 2.4 Insurer�s loss 4 2.5 A quadratic put 5 2.6 Comparative statics 5

3. INSURANCE AND RE-INSURANCE 6 3.1 Shadow loss 7 3.2 Agreement A 8 3.3 Agreement B 9 3.4 Agreement C 9

4. A SIMULATION 9

5. CONCLUSIONS 11

REFERENCES 14

APPENDIX 15 Quadratic calls and puts 15

Quadratic calls 15 Quadratic puts 16

1. INTRODUCTION

The purpose of this paper is twofold. In the first place it presents an analytical model with which to determine the fair level of the standby underwriting fee in a stock placement. Specifically, we show that this depends on the value of a �quadratic� power put option (that is, a derivative promising at maturity the square of the payoff of an ordinary put) and derive a closed-form formula for this option in a Black & Scholes economy.

Secondly, we examine the case of a placement guaranteed by two banks (A and B). Bank A, acts as an �insurer� and underwrites the stocks offered to the market up to a given maximum quantity; Bank B acts as a �re-insurer� and pledges, for a fraction of the fee, to underwrite the remaining stocks. Bank B buys stocks only if the unplaced quantity is greater than that guaranteed by Bank A.

The allocation of the total fee between the �insurer� and the �re-insurer� depends on the contractual arrangements. We describe three possible agreements and provide closed-form formulas for the allocation of the fees in each case.

2. THE GENERAL MODEL

As a first step we describe a simple model that enables us to analyze the results of the placement in a probabilistic setting. The notation that we use is the following:

f: underwriting fee per share offered, to be split between Bank A and Bank B; L: total loss for Bank A and Bank B; l: loss per share offered; m: loss per share underwritten; η: price elasticity at the initial equilibrium price S; QO: quantity of stocks offered; QM: quantity of stocks demanded by the market at price K; QA: quantity of stocks underwritten by A; QB: quantity of stocks underwritten by B; k: maximum share of offered (or unplaced) quantity underwritten by A; K: placement price, set at time t; u: under-pricing factor of the offer; S: equilibrium price at which the market would absorb, at t, the entire offer Qc; ST: equilibrium price at which the market would absorb, at T, the entire offer Qc; T � t: length of the placement period measured according to calendar (or civil) time; φ(T � t): length of the placement period measured according to market (or process) time; r: risk-free interest rate; σ: stock price volatility.

2.1 Stock price dynamics

We assume that an equilibrium shadow price exists at which the market would absorb the entire offer. This price, which depends on the value of the company and the quantity of stocks to be placed, is equal to S at time t (the beginning of the placement period) and to ST at time T (the end of the placement period).1

The placement price of the stock, K, can be set at a discount to the fair price S in order to reduce the likelihood of an unsuccessful placement.2 We denote the under pricing factor by u (u > 0), defined as

1 If the company is listed, the fair price S can be put equal to the market quote at the beginning of the placement period. 2 A placement is considered successful if the market takes up all the stocks offered. On the other hand, the placement is considered unsuccessful if Bank A and (eventually) Bank B have to take up the residual quantity.

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=

KSu ln

(1)

so that .uKeS = (2)

In our model, market time does not have to be equal to calendar time; it can be linked to the latter by some function. For the sake of simplicity, we assume a proportionality factor φ.

Normally, market time and calendar time are the same, that is φ = 1. During a placement, however, it is probably better to assume that market time is not equal to φ calendar time, since the volume of traded stocks is somehow forced to be higher than usual. This implies φ ≠ 1.

The equilibrium price, S, is a random variable. Specifically, in market time S follows a geometric Brownian motion with drift rate µ and variance rate σ²φ

.SdzSdtdS φσµ += (3)

Consequently, given the stock price at time t, the equilibrium market price ST at time T > t is log-normally distributed (in market time):

−−−−+ )(),(

2)()ln('~)ln(

2tTtTtTSNST φσφσµ

(4)

where N' is the standardized normal density function.

2.2 Demand curve

The quantity of stocks demanded by the market is inversely proportional to the placement price and can only shift in a parallel manner. The demand schedule is linearly downward sloping. In particular, at time t

OQS βα −= (5)and at time T

MT QK βα −= (6)

OTT QS βα −= (7)

where β > 0 and α t(S, t) is a real-valued stochastic process.3 It should be stressed that there is a placement price high enough (K = αΤ) to make the market

demand, QM, equal to zero. Moreover, since:

ββ

α KQ TM −=

(8)

the maximum quantity of stocks demanded by the market, when K = 0, is QM = αΤ /β. Specifically, if η is the price elasticity at the initial equilibrium price S

S

QQS O

O∂∂−=η

(9)

then, from (5), the slope of the demand curve is given by the angular coefficient

.OQ

Sηβ =

(10)

3 According to the assumptions stated above, the quantity QM of stocks demanded by the market is also a random variable.

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Using equations (6) and (7) and applying Ito�s lemma, we can derive the dynamics for α during the placement period for a given placement price. In fact it is easy to check that

αα −=− TT SS (11)

and hence that

.αddS = (12)

It follows from (11) that αΤ is displaced log-normally distributed:

.)(),(2

)()ln('~)ln(2

StTtTtTSNT −+

−−−−+ αφσφσµα

(13)

The actual distribution of QM can be obtained by using (8).

2.3 An example

Let us consider the placement of 3,000,000 shares at a unit price of $100 (20% less than the current equilibrium price). The offering lasts for one week. The stock volatility is 40 per cent and the risk-free interest rate is 5 per cent, both on an annual basis (calendar time). The price elasticity is 0.25 and the wheels of market time run 20 times faster than calendar time. Therefore, using our notation, QO = 3,000,000, K = $100, u = 20%, T � t = 7 days, σ = 40%, r = 5%, η = 0.25, φ = 20.

The demand curve can shift downwards or upwards in a parallel manner depending on the shocks affecting the equilibrium price during the placement period. Figure 1 shows two possible shifts, leading respectively to d � d and d� � d�. The figure also shows the probability density functions for equilibrium prices ST and quantities QM.

If the demand curve is d � d, the equilibrium price (ST) is higher than the placement price (K), the quantity of stocks (QM) demanded by the market is greater than the offered quantity (QO) and an allotment is necessary (Figure 1, point A). The unsatisfied demand raises the market price immediately after the end of the placement period.

100

130

160

190

220

0 3 6 9 12 15 18

Price ($)

Quantity (mln.)

Two possibleequilibrium prices

A

B

d

d

d�

d�

(QO = 3,000,000, K = $100, u = 20%, r = 5%, σ = 40%, T � t = 1 week, φ = 20, η = 0.25)

Figure 1 Shadow equilibrium prices

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If the demand curve turns out to be d� � d�, the equilibrium price (ST) is lower than the placement price (K), the quantity of stocks (QM) demanded by the market is lower than the offered quantity (QO) and the guarantee covenant is triggered. The market price stays at the level set for the placement as long as the insurer continues to sterilize some of the offer by keeping stocks on its books.

2.4 Insurer’s loss

What is the insurer�s loss when the placement is unsuccessful? How do we measure it? If the stocks underwritten by the insurer were valued at the market price, then no loss would be reported, because (as already pointed out) at the end of the placement period the market price will be equal to the placement price. Nevertheless, given the assumptions about the price/quantity relationship, a loss would arise in the event of disposal. The shadow loss, L, is equal to the area of the triangle ABC (Figure 2), that is

( ) ( ).0,max0,max21

MOT QQSKL −×−=

(14)

The loss is null in the event of a successful placement (QO ≤ QM and K ≤ ST) and is non-null otherwise (QO > QM and K > ST).

Since from (6) and (7)

( )MOT QQSK −=− β (15)

so that

( )TMO SKQQ −=−β1

(16)

and equation (14) is equal to

C

Insurer

70

100

130

0 3 6

Price ($)

Quantity (mln.)

A B Equilibriumprice

Market

(QO = 3,000,000, K = $100, u = 20%, r = 5%, σ = 40%, T � t = 1 week, φ = 20, η = 0.25)

Figure 2 Insurer�s intervention.

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( ) .0,max21 2

TSKL −=β

(17)

By using (10), the total loss, L, is

( )20,max2 T

O SKS

QL −=

η (18)

and the loss per share offered, l, is

( ) .0,max2

1 2T

OSK

SQLl −=≡

η (19)

2.5 A quadratic put

The underwriting fee per share offered, f, should reward the present value of the expected loss:

( ) ( )lEef tTr �−−= (20)

where E� is the expectation operator under the risk-neutral probability measure. Therefore

qS

fη21=

(21)

where q

( ) ( )[ ]20,max�T

tTr SKEeq −= −− (22)

is the value of a �quadratic� power put, with strike price equal to the placement price K and maturity at time T, written on a stock with terminal price ST.

Hence the problem concerns the pricing of a power derivative, i.e. a derivative whose terminal payoff depends on the n-th power of the value of an underlying financial variable. More precisely, the underwriting fee is 1/(2ηS) times the value of a derivative paying at maturity the squared payoff of an ordinary put option. The value of this derivative is not simply the square of an ordinary put, just as the value of a derivative paying at maturity the squared price of a stock is not simply the square of the stock�s current price.4

The closed-form formula for a quadratic put, q = pp, is derived in the Appendix. By setting K1 = K2 = K in (a22), we have

( )( ) ( )[ ]{ }( )[ ]{ }

( ) ).(

22

2

2 2

dNKe

tTdSKNtTdNeSq

tTr

tTr

−+

+−+−−

−+−=

−−

−+

φσφσφσ

(23)

where

( )

( ).)5.0()/ln( 2

tTtTrKSd

−−−+=

ϕσφσ

(24)

2.6 Comparative statics

The underwriting fee is affected by five parameters: r (the risk-free interest rate), σ (the volatility), u (the under-pricing factor), φ (the ratio between �market time� and �calendar time�) and η (the price

4 See questions 10.11, 11.14, 11.15 in Hull (1997) and the answers in the Instructor�s Manual.

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elasticity at the initial equilibrium price S). The level of the underwriting fee rises as σ and φ increase, whereas it falls as r, u and η increase.

The risk-free interest rate has the usual effect on the put�s value: if it rises, the put�s value diminishes, together with the underwriting fee.

Quadratic options are even more sensitive than ordinary options to variations in the parameter σ: a high volatility implies a higher probability that the terminal equilibrium price will wander away from the placement price. This greater risk is rewarded by a larger fee.5

The under-pricing parameter u enters into the formula for the calculation of the fee through the exercise price of the quadratic put, as can be seen from equation (2). Usually the put is out of the money at the start, since the exercise price K is lower than the initial equilibrium price S. A high level of u implies a high degree of under-pricing, with a placement price far below the starting equilibrium price. In this case the quadratic put in (21) will be (deep) out of the money and it will not be very valuable. Therefore, the more the stock is under-priced for placement purposes, the lower is the level of the fee.

If the quantity of stocks to place is large, in absolute terms or with respect to market conditions (e.g. because other placements are on the run), and the absorption by the market is difficult, the level of φ (determining the market time) is high and vice versa.6 Therefore, the higher the value of φ, the higher the level of the fee.

The insurer�s loss depends crucially on the slope of the demand curve. As can be seen in Figure 2, given a certain discount (BC) of the equilibrium price ST with respect to the placement price K, the lower the slope (and hence the elasticity η), the bigger the area of triangle ABC, which measures the insurer�s loss. Therefore, the lower the elasticity, the higher the underwriting fee.

3. INSURANCE AND RE-INSURANCE

Big stock placements are usually carried out in cooperation by several financial institutions, which share the revenues according to fairness criteria. We assume that two banks, A and B, are involved in the placement.

In general, the agreement could be one of the following:

Bank A underwrites a share k of the offered quantity; Bank B underwrites the remaining share 1 � k. Bank A underwrites a share k of the unplaced quantity; Bank B underwrites the remaining share 1 � k. Bank A underwrites the unplaced quantity up to a maximum quantity equal to a share k of the

offered quantity; Bank B underwrites the remaining share 1 � k.

Analytically, the total quantity underwritten by Bank A and Bank B is equal to

)0,max( MOBA QQQQ −=+ (25)

The quantity underwritten by Bank A under Agreement A is

).0,max( MOA QkQQ −= (26)

Under Agreement B it is

).0,max( MOA QQkQ −×= (27)

Under Agreement C it is

5 In order to determine the right level for the volatility, σ, of the stock to be placed, the implied volatilities of short-term options written on similar stocks can be considered. 6 In order to determine the value of φ, the daily trading volume can be considered (if the stock is already traded on an exchange). Specifically, the higher the volume, the lower the φ.

It�s likely that φ and the under-pricing parameter u will be calibrated jointly, since they are negatively correlated.

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[ ]

( ) ( )[ ].0,1max0,max),0,max(min

MOMO

OMOA

QkQQQkQQQQ

−−−−==−=

(28)

The quantity underwritten by Bank B can be determined as a residual:

)0,max( AMOB QQQQ −−= (29)

In order to stress the differences between the three different contractual arrangements let us suppose that:

1. the number of offered stocks is 100; 2. the share pertaining to Bank A is 60%; 3. the number of stocks bought by the market is 50.

The roles of Bank A and Bank B under the three different agreements are as follows: a) Bank A buys 10 stocks and Bank B buys 40; b) Bank A buys 30 stocks and Bank B buys 20; c) Bank A buys 50 stocks and Bank B buys 0.

The least burdensome agreement for Bank A, is the one sub a), followed by those sub b) and c) in decreasing order.

3.1 Shadow loss

In order to determine the allocation of the underwriting fee between the insurer and the re-insurer under the three agreements, we have to make some assumptions about how the shadow loss is borne in the event of an unsuccessful placement. The total loss is given by the area of the triangle ADE (Figure 3). We assume that the shadow loss, m, for each share underwritten by Bank A or Bank B is equal on average to

( ).0,max21

TSKm −=

(30)

Re-insurer

70

100

130

0 3 6

Price ($)

Quantity (mln.)

A B

CEquilibrium

price

Market Insurer

D

E

Figure 3 Insurer�s and re-insurer�s intervention.

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This means that the two banks have no agreement concerning the disposal of the unplaced stocks, so that they can both be assumed to start selling them at an average price between the equilibrium price and the placement price.

3.2 Agreement A

Let us begin with Agreement A, under which Bank A underwrites a share k of the offered quantity, and Bank B underwrites the remaining share 1 � k . Given (26) and (30), the total loss borne by Bank A is:

( ) ( ).0,max210,max TMOAA SKQkQmQL −×−=×=

(31)

From (10) and (16), we have

( )TO

MO SXS

QQkQ −=−

η (32)

where

.)1( SkKX η−−= (33)

Therefore, equation (31) can be written as

( ) ( )[ ]0,max0,max2

0TTA SKSX

SQ

L −×−=η

(34)

and the loss for each share offered, lA, is

( ) ( )[ ].0,max0,max2

1TT

O

AA SKSX

SQL

l −×−=≡η

(35)

It is worth noting that if only Bank A provides a guarantee (k = 1), then from (33) X = K, so that lA in (35) equals l in (19). By contrast, if Bank A provides no guarantee (k = 0), then X = K � ηS = ST � βQM and lA in (35) equals 0.

The underwriting fee for Bank A can be determined by discounting the expected loss lA. Therefore

wS

f A η21=

(36)

where w

( ) ( ) ( )[ ]0,max0,max�TT

tTr SKSXEew −−= −− (37)

is the value of a quadratic put, with strike prices K1 = X and K2 = K (K1 ≤ K2), written on the equilibrium terminal price ST.

The value of w = pq can be calculated using formula (a22):

( )( ) ( )[ ]{ }( )[ ]{ }

( ) ).(

)(

2

121

121

12 2

dNKKe

tTdNKKS

tTdNeSw

tTr

tTr

−+

+−+−+−

+−+−=

−−

−+

φσφσφσ

(38)

where

( ) ( )( )

( )tTtTrKS

d−

−−+=

φσφσ 2

11

5.0/ln

(39)

The underwriting fee for Bank B, fB, can be obtained as the difference between (21) and (36).

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3.3 Agreement B

Under Agreement B, Bank A underwrites a share k of the unplaced stocks, and Bank B underwrites the remaining share 1 � k. Given (27) and (30), the total loss borne by Bank A is

( ) ( ).0,max210,max TMOAA SKQQkmQL −×−×=×=

(40)

Since LA is just a share k of L, the fee for Bank A is easily obtained as

qS

kf A η2=

(41)

where q is given by (23).

3.4 Agreement C

Under Agreement C, Bank A underwrites unplaced stocks up to a share k of the offered quantity, and the remaining share is underwritten by Bank B. Given (28) and (30), the total loss borne by Bank A is

( ) ( )[ ]{ } ( ).0,max210,1max0,max TMOMOAA SKQkQQQmQL −×−−−−=×=

(42)

Recalling (10) and (16), equation (42) can be written as

( ) ( ) ( )[ ]0,max0,max0,max2

2TTT

OA SKSXSK

SQ

L −×−−−=η

(43)

where X is now defined by

.SkKX η−= (44)

The loss for each share offered, lA, is

( ) ( ) ( )[ ].0,max0,max0,max2

1 2TTTA SKSXSK

Sl −×−−−=

η (45)

If bank A provides no guarantee (k = 0), then X = K and lA shrinks to 0. By contrast, if only bank A provides a guarantee (k = 1), then X = K � ηS = ST � βQM and lA in (45) equals l in (19). The underwriting fee for Bank A can be determined by discounting the expected loss lA. Therefore

( )wqS

f A −=η21

(46)

where q is given by (23) and w by (38).

4. A SIMULATION

We now present a simulated application of the model. The parameters are those used in Figure 1, Figure 2 and Figure 3: QO = 3,000,000, K = $100, T � t = 7 days, r = 5%, σ = 40%. In addition, φ = 10 or 20 or 30, η = 0.30 or 0.25 or 0.20, u = 15% or 20% or 25% and k = 50% or 70% or 90%.

Table 1 shows the underwriting fee f calculated using formula (21). The gray zone contains the underwriting fees lying between the 0.5 and 2.5 per cent. In more details, with a �market time� set to 20 times the calendar value, the elasticity set equal to 0.25 and an under-pricing assumed to be 20%, the underwriting fee is found to be 1.08 per share to be placed.

Table 2 shows the expected losses in the event of an unsuccessful placement as a percentage of the placement value. For example, given the same combination of input parameters and variables as above, the expected loss (conditioned to the �unsuccessful placement� event) is 4.38% of the placement value.

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Table 3 shows the probability of an unsuccessful placement. It can be checked that the likelihood of a failure decreases (together with the underwriting fees) as the under-pricing parameter u increases.7

7 The probabilities shown refer to risk-neutral operators.

TABLE 1 � Underwriting fees

Ratio of the underwriting fees to the value of the stocks to be placed (%)

φ = 10 φ = 20 φ = 30

k η = 0.30 η = 0.25 η = 0.20 η = 0.30 η = 0.25 η = 0.20 η = 0.30 η = 0.25 η = 0.20

50% 0.43 0.52 0.65 1.37 1.65 2.06 2.45 2.94 3.68 70% 0.43 0.52 0.65 1.37 1.65 2.06 2.45 2.94 3.68

u =

15%

90% 0.43 0.52 0.65 1.37 1.65 2.06 2.45 2.94 3.68 50% 0.23 0.27 0.34 0.90 1.08 1.35 1.75 2.10 2.63 70% 0.23 0.27 0.34 0.90 1.08 1.35 1.75 2.10 2.63

u =

20%

90% 0.23 0.27 0.34 0.90 1.08 1.35 1.75 2.10 2.63 50% 0.11 0.13 0.17 0.57 0.68 0.86 1.23 1.47 1.84 70% 0.11 0.13 0.17 0.57 0.68 0.86 1.23 1.47 1.84

u =

25%

90% 0.11 0.13 0.17 0.57 0.68 0.86 1.23 1.47 1.84

Note: it is assumed that: QO = 3 millions, K = 100, T � t = 7 days (calendar time), r = 5% (continuously compounded ) and σ = 40%.

TABLE 2 � Expected loss in the event of an unsuccessful placement

Ratio of the total loss to the value of the stocks to be placed (%)

φ = 10 φ = 20 φ = 30

k η = 0.30 η = 0.25 η = 0.20 η = 0.30 η = 0.25 η = 0.20 η = 0.30 η = 0.25 η = 0.20

50% 1.97 2.37 2.96 4.38 5.25 6.57 6.72 8.07 10.09 70% 1.97 2.37 2.96 4.38 5.25 6.57 6.72 8.07 10.09

u =

15%

90% 1.97 2.37 2.96 4.38 5.25 6.57 6.72 8.07 10.09 50% 1.56 1.88 2.34 3.65 4.38 5.47 5.75 6.90 8.62 70% 1.56 1.88 2.34 3.65 4.38 5.47 5.75 6.90 8.62

u =

20%

90% 1.56 1.88 2.34 3.65 4.38 5.47 5.75 6.90 8.62 50% 1.25 1.50 1.87 3.05 3.67 4.58 4.92 5.91 7.38 70% 1.25 1.50 1.87 3.05 3.67 4.58 4.92 5.91 7.38

u =

25%

90% 1.25 1.50 1.87 3.05 3.67 4.58 4.92 5.91 7.38


TABLE 3 � Probability of an unsuccessful placement

Probability of the stocks not being completely placed (%)

φ = 10 φ = 20 φ = 30

k η = 0.30 η = 0.25 η = 0.20 η = 0.30 η = 0.25 η = 0.20 η = 0.30 η = 0.25 η = 0.20

50% 21.94 21.94 21.94 31.37 31.37 31.37 36.47 36.47 36.47 70% 21.94 21.94 21.94 31.37 31.37 31.37 36.47 36.47 36.47

u =

15%

90% 21.94 21.94 21.94 31.37 31.37 31.37 36.47 36.47 36.47 50% 14.47 14.47 14.47 24.59 24.59 24.59 30.48 30.48 30.48 70% 14.47 14.47 14.47 24.59 24.59 24.59 30.48 30.48 30.48

u =

20%

90% 14.47 14.47 14.47 24.59 24.59 24.59 30.48 30.48 30.48 50% 8.93 8.93 8.93 18.70 18.70 18.70 24.97 24.97 24.97 70% 8.93 8.93 8.93 18.70 18.70 18.70 24.97 24.97 24.97

u =

25%

90% 8.93 8.93 8.93 18.70 18.70 18.70 24.97 24.97 24.97


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Once the total fee has been calculated, it has to be allocated between Bank A and Bank B. As an example, let us assume that they agreed on a �re-insurance� plan of the �Agreement C� type. The allocation of the underwriting fee to Bank A has been determined using equation (46) (Table 4 and Table 7). Specifically, with the combination of parameters adopted earlier the portion of the underwriting fee due to Bank A is 65.19%, 81.44% and 91.51% where the maximum guarantee provided by Bank A is, respectively, 50%, 70% and 90% of the offered quantity.

Table 5 and Table 8 show the expected loss for Bank A and Bank B (as a percentage of the value of the offered stocks) in the event of an unsuccessful placement. For example, with the same combination of parameters the expected loss for Bank A (conditioned to the �unsuccessful placement� event) is 3.57% of the value of the offered stocks and that for Bank B (conditioned to the triggering of the �reinsurance� clause) is 4.11%.

The probabilities of Bank A and Bank B having to intervene are given in Table 6 and Table 9. For example, the probability of Bank A having to step in is 24.59%, whereas the probability of the supplementary guarantee of bank B being needed is 4.87%. Clearly, the probability of a failure decreases (together with the underwriting fee) as the level of the under-pricing, measured by the parameter u, increases.

5. CONCLUSIONS

In this paper we presented a simple model for consistently determining the underwriting fee in a stock placement: the fee turns out to be proportional to the value of an exotic option paying at maturity the square of the payoff of an ordinary put. The closed-form formula we derived allows the fee to be calculated easily.

In this setting we also analyzed the problem of the allocation of the total fee between two banks acting as an �insurer� and a �re-insurer� in the placement. Here again, the model allows the total fee to be allocated consistently between two institutions under different contractual arrangements.

The model could be extended to a bond placement, although in this case a term structure model is needed and the analysis would be more complex and it would probably not be possible to arrive at closed-form formulas. This issue, and that concerning the estimation of the relevant parameters, are left for future research.

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TABLE 4 � Insurance

Portion of the underwriting fee due to Bank A (%)

φ = 10 φ = 20 φ = 30

k η = 0.30 η = 0.25 η = 0.20 η = 0.30 η = 0.25 η = 0.20 η = 0.30 η = 0.25 η = 0.20

50% 85.56 77.56 66.90 69.30 60.33 50.08 59.87 51.34 42.06 70% 96.07 91.22 82.67 85.33 76.92 65.87 76.76 67.54 56.55

u =

15%

90% 99.29 97.32 92.11 94.32 88.17 78.28 88.35 80.15 68.97 50% 89.79 82.76 72.65 74.13 65.19 54.61 64.28 55.50 45.73 70% 97.83 94.35 87.31 89.07 81.44 70.75 80.96 72.01 60.89

u =

20%

90% 99.72 98.63 95.03 96.42 91.51 82.70 91.46 84.13 73.42 50% 93.14 87.26 78.03 78.72 69.98 59.24 68.70 59.75 49.57 70% 98.91 96.60 91.13 92.19 85.52 75.46 84.82 76.33 65.27

u =

25%

90% 99.90 99.37 97.08 97.89 94.19 86.67 94.02 87.71 77.70


TABLE 5 � Expected loss of Bank A in the event of an unsuccessful placement

Ratio of the loss of Bank A to the value of the stocks to be placed (%)

φ = 10 φ = 20 φ = 30

k η = 0.30 η = 0.25 η = 0.20 η = 0.30 η = 0.25 η = 0.20 η = 0.30 η = 0.25 η = 0.20

50% 1.69 1.84 1.98 3.03 3.17 3.29 4.03 4.14 4.24 70% 1.90 2.16 2.45 3.74 4.04 4.33 5.16 5.45 5.70

u =

15%

90% 1.96 2.30 2.73 4.13 4.63 5.14 5.94 6.47 6.96 50% 1.40 1.55 1.70 2.71 2.86 2.99 3.69 3.83 3.94 70% 1.53 1.77 2.05 3.25 3.57 3.87 4.65 4.97 5.25

u =

20%

90% 1.56 1.85 2.23 3.52 4.01 4.53 5.26 5.80 6.33 50% 1.16 1.31 1.46 2.40 2.57 2.71 3.38 3.53 3.66 70% 1.23 1.45 1.71 2.82 3.13 3.46 4.18 4.51 4.82

u =

25%

90% 1.25 1.49 1.82 2.99 3.45 3.97 4.63 5.18 5.74


TABLE 6 � Probability of an unsuccessful placement

Probability of intervention by Bank A (%)

φ = 10 φ = 20 φ = 30

k η = 0.30 η = 0.25 η = 0.20 η = 0.30 η = 0.25 η = 0.20 η = 0.30 η = 0.25 η = 0.20

50% 21.94 21.94 21.94 31.37 31.37 31.37 36.47 36.47 36.47 70% 21.94 21.94 21.94 31.37 31.37 31.37 36.47 36.47 36.47

u =

15%

90% 21.94 21.94 21.94 31.37 31.37 31.37 36.47 36.47 36.47 50% 14.47 14.47 14.47 24.59 24.59 24.59 30.48 30.48 30.48 70% 14.47 14.47 14.47 24.59 24.59 24.59 30.48 30.48 30.48

u =

20%

90% 14.47 14.47 14.47 24.59 24.59 24.59 30.48 30.48 30.48 50% 8.93 8.93 8.93 18.70 18.70 18.70 24.97 24.97 24.97 70% 8.93 8.93 8.93 18.70 18.70 18.70 24.97 24.97 24.97

u =

25%

90% 8.93 8.93 8.93 18.70 18.70 18.70 24.97 24.97 24.97


- 13 -

TABLE 7 � Reinsurance

Portion of the underwriting fee due to Bank B (%)

φ = 10 φ = 20 φ = 30

k η = 0.30 η = 0.25 η = 0.20 η = 0.30 η = 0.25 η = 0.20 η = 0.30 η = 0.25 η = 0.20

50% 14.44 22.44 33.10 30.70 39.67 49.92 40.13 48.66 57.94 70% 3.93 8.78 17.33 14.67 23.08 34.13 23.24 32.46 43.45

u =

15%

90% 0.71 2.68 7.89 5.68 11.83 21.72 11.65 19.85 31.03 50% 10.21 17.24 27.35 25.87 34.81 45.39 35.72 44.50 54.27 70% 2.17 5.65 12.69 10.93 18.56 29.25 19.04 27.99 39.11

u =

20%

90% 0.28 1.37 4.97 3.58 8.49 17.30 8.54 15.87 26.58 50% 6.86 12.74 21.97 21.28 30.02 40.76 31.30 40.25 50.43 70% 1.09 3.40 8.87 7.81 14.48 24.54 15.18 23.67 34.73

u =

25%

90% 0.10 0.63 2.92 2.11 5.81 13.33 5.98 12.29 22.30


TABLE 8 - Expected loss of Bank B in the event of an unsuccessful placement

Ratio of the loss of Bank B to the value of the stocks to be placed (%)

φ = 10 φ = 20 φ = 30

k η = 0.30 η = 0.25 η = 0.20 η = 0.30 η = 0.25 η = 0.20 η = 0.30 η = 0.25 η = 0.20

50% 2.02 2.46 3.09 4.05 4.97 6.33 5.99 7.38 9.44 70% 1.92 2.38 3.05 3.79 4.74 6.13 5.56 6.98 9.08

u =

15%

90% 1.77 2.26 2.97 3.47 4.45 5.89 5.08 6.53 8.68 50% 1.72 2.08 2.60 3.50 4.28 5.43 5.24 6.44 8.21 70% 1.65 2.04 2.60 3.29 4.11 5.30 4.87 6.11 7.93

u =

20%

90% 1.53 1.95 2.54 3.02 3.88 5.11 4.44 5.73 7.60 50% 1.49 1.78 2.21 3.05 3.72 4.69 4.60 5.64 7.18 70% 1.44 1.77 2.23 2.88 3.59 4.61 4.29 5.37 6.96

u =

25%

90% 1.33 1.70 2.20 2.64 3.39 4.46 3.90 5.04 6.68


TABLE 9 - Probability of an unsuccessful placement

Probability of intervention by Bank B (%)

φ = 10 φ = 20 φ = 30

k η = 0.30 η = 0.25 η = 0.20 η = 0.30 η = 0.25 η = 0.20 η = 0.30 η = 0.25 η = 0.20

50% 3.09 4.75 6.95 10.41 13.16 16.25 16.43 19.41 22.58 70% 0.89 1.91 3.69 5.32 8.03 11.47 10.25 13.68 17.59

u =

15%

90% 0.17 0.61 1.73 2.25 4.38 7.60 5.63 8.94 13.15 50% 1.34 2.25 3.57 6.63 8.75 11.25 11.95 14.53 17.36 70% 0.30 0.75 1.66 2.98 4.87 7.43 6.84 9.62 12.95

u =

20%

90% 0.04 0.19 0.66 1.07 2.36 4.56 3.37 5.82 9.19 50% 0.51 0.95 1.66 3.98 5.53 7.44 8.37 10.53 12.96 70% 0.08 0.26 0.66 1.55 2.76 4.56 4.35 6.50 9.20

u =

25%

90% 0.01 0.05 0.22 0.46 1.17 2.56 1.88 3.60 6.16


- 14 -

REFERENCES

HART, I. and ROSS, M., �Striking Continuity�, in Over the Rainbow � Developments in Exotic Options and Complex Swaps, (Ed. R. Jarrow), London: Risk Books, 1995.

HULL, J. C., Options, Futures, and Other Derivatives, 3rd edition, Prentice Hall, 1997.

- 15 -

APPENDIX

Quadratic calls and puts

In this appendix we derive the pricing formulas for quadratic calls and puts. We define a quadratic option as a derivative paying at maturity the product of the payoff of two standard options (two calls or two puts).

Quadratic calls

The payoff of a quadratic call is

( ) ( ) ( )0,max0,max, 21 KSKSTSc TTTq −×−= (a1)

where ST is the stock price at expiration time T and K1 and K2 (K1 ≤ K2) are the strike prices of two standard calls.

The evaluation can be performed using the standard risk-neutrality approach. Let us assume that the dynamics of the stock price (under the equivalent martingale probability measure) is given by the following stochastic differential equation

.SdzrSdtdS φσ+= (a2)

The current value, cq, of the quadratic call is equal to the expected terminal payoff discounted at the risk-free rate

( ) ( ) ( )[ ]0,max0,max�21 KSKSEec TT

tTrq −×−= −−

(a3)

where E� is the expectation operator under risk neutrality. In other words

( ) ( ) ( )[ ].�221 KSKSKSEec TTT

tTrq ≥−×−= −−

(a4)

The expectation is conditioned to ST ≥ K2, since for ST < K2 the second call has no value and the quadratic call is also worth nothing. Expanding (a4) we have

( ) ( )[ ]( ) ( ) ( )[ ] ( ){ }.��

�

22122122

221212

KSKKEKSKKSEKSSEe

KSKKKKSSEec

TTTTTtTr

TTTtTr

q

≥+≥+−≥=

=≥++−=−−

−−

(a5)

First we compute the expected squared price, conditioned to the stock price being higher than the exercise price K2. Since, by Ito�s lemma:

( ) ( ) dzSdtSrSd φσφσ 2222 22 ++= (a6)

we have

( ) ( ) ( ) ( )[ ].2,2ln'~ln 22 tTtTrSNST −−+ φσ (a7)

where N' is the standardized normal density function. The expected squared price can be written as

( ) ( ) ( )∫∞ −

−+−=≥2

2

2�

5.0222

22

K

tTtTrTT deeSKSSE ε

π

εεφσ

(a8)

Using the trick of completing the squared exponent, and integrating for the stock price being greater than the strike price K2, formula (a8) becomes

( ) ( ) ( )( )[ ]

∫∞

−

−−−−+−=≥

2

22

2�

25.022

22

d

tTtTtTr

TT deeSKSSE επ

φσεφσ

(a9)

- 16 -

where

( ) ( )( )

( ).

5.0/ln 22

2 tTtTrKS

d−

−−+=

φσφσ

(a10)

By the change of variable

tT −−= σευ 2 (a11)

and recalling that for a normal distribution

)Pr()Pr( 22 dd <=−> εε (a12)

we have:

( ) ( ) ( ) ( )[ ].2�2

222

2 2tTdNeSKSSE tTtTr

TT −+=≥ −+− φσφσ (a13)

The remaining two expectations in (a5) can be computed by a slight modification of the Black & Scholes formula:

[ ] ( ) ( )[ ]tTdNKKSeKSKKSE tTrTT −++=≥+ − φσ221221 )()(�

(a14)

( ) ( ).�221221 dNKKKSKKE T =≥ (a15)

Thus the pricing formula for a quadratic call is:8

( )( ) ( )[ ]( )[ ]

( ) ).(

)(

2

221

221

22 2

dNKKe

tTdNKKS

tTdNeSc

tTr

tTrq

−−

−+

+

+−++−

+−+=

φσ

φσφσ

(a16)

Quadratic puts

The pricing formula for a quadratic put is easily derived from the previous results. The payoff of a quadratic put is

( ) )0,max()0,max(, 21 TTTq SKSKTSp −×−= (a17)

where ST is the stock price at expiration time T and K1 and K2 (K1 ≤ K2) are the strike prices of two standard puts.

The current value, pq, of the quadratic put is equal to the expected terminal payoff discounted at the risk-free rate

( ) [ ]121 )()(� KSSKSKEep TTTtTr

q ≤−×−= −−

(a18)

The expectation is conditioned to ST ≤ K1, since for ST > K1 the first put has no value and the quadratic put is also worth nothing. Expanding (a18) we have

( ) ( )[ ]( ) ( ) ( )[ ] ( ){ }.��

�

12112112

121212

KSKKEKSKKSEKSSEe

KSKKKKSSEep

TTTTTtTr

TTTtTr

q

≤+≤+−≤=

=≤++−=−−

−−

(a19)

Since

)()(1)Pr( 111 dNdNKST −=−=≤ (a20)

where 8 Apart from the scale factor ½, when K = K1 = K2 Formula (a16) is equal to the analytic solution for pricing a continuous strike option given by Hart and Ross (1995) in the case where δ = 0.

- 17 -

( ) ( )( )

( )tTtTrKS

d−

−−+=

φσφσ 2

11

5.0/ln

(a21)

the pricing formula is

( )( ) ( )[ ]{ }( )[ ]{ }

( ) ).(

)(

2

121

121

12 2

dNKKe

tTdNKKS

tTdNeSp

tTr

tTrq

−+

+−+−+−

+−+−=

−−

−+

φσ

φσφσ

(a22)

underwriting fees and power derivatives

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