E� Derman ma yhave had suc hprosp ect in

Hedging exotic derivatives through stochastic optimization

P. Hena

Banque Internationale de Placement

Dresdner Bank Group

108, Bd Haussmann

75008 Paris

June 17, 1996

Abstract

In this paper, we develop methods for hedging nancial instruments through stochastic optimization. We rst concentrate on the proper formulation of such problems. Indeed one may consider arbitrage-free prices, which are theoretical prices consistent with a model of the risk factors in the economy, or the observed market prices, which may not be consistent with any theoretical model. We dene the roles that these two sets of prices should play in order to obtain a well-formulated model, which yield an optimal wealth prole rather than optimal holdings. Next, we present ecient solution algorithms which are naturally derived from the formu-lations presented earlier. Finally, we show how optimal holdings may be computed.

1 Introduction

An \exotic" option, diers from a plain European or American option in the sense that it is perceived as complicated to replicate. When writing an exotic option, a hedger's dream would be to immediately buy back the risk. Ideally, this would be done through the purchase of a portfolio of options which replicate exactly the pay-o prole of the exotic instrument. A sure prot would be secured if the hedge portfolio could be constructed at a lower cost than the option premium. E. Derman may have had such prospect in mind while developing a method for constructing a static replication of barrier options 2].

If one excludes gross market distortions, such static replication does not exist when the exotic option is written (otherwise, no educated customer would buy the exotic option in the rst place). Indeed, the replicating portfolio computed in Derman's paper is more expensive than the exotic option it replicates.

In this paper, we develop a dierent approach to hedging exotic options:

instead of looking for a perfect replication, we will calculate hedges which have

good chances of performing well under a variety of scenarios, given the available budget,

we take into account opportunities to periodically re-balance the hedge portfolio,

and nally,

we look for a formulation which allows us to take advantage of current market

imperfections.

The basic framework is the stochastic optimization of the expected utility of the hedger's residual wealth after expiry of the exotic option. The paper is organized as follows. Section 2 discusses the choice of price system to be used in stochastic optimization models. We conclude from this discussion that arbitrage-free prices must be used in order to prevent biases in the model. This in turn provides the opportunity to greatly simplify the expression of the optimization models, by formulating them as the optimization of a wealth prole, rather than the optimization of the trading strategy. This is a signicant advantage when considering a multi-period optimization model. Section 5 addresses the issue of the determination of optimal holdings. This is where market information comes into play. At each stage of the presentation, we provide numerical illustrations of the algorithms.

2 Price system

The information structure of the economy is represented by an event tree spanning two time periods. There are

S

+ 1 states of the world, state 0 corresponding to the observation date, and the balance corresponding to the horizon date. At each date, there are

K

securities available for trading. A classical formulation of the hedging problem is as follows:

maximizeX

s

p

s

u

(

w

s) (P1)

such that

Px

;

l

w

(P1-1)

P

0

x

w

0 (P1-2)

where:

u

() utility function

w

s wealth at horizon

T

, under scenario

s

p

s subjective probability of scenario

s

l

vector of liability values at horizon

T

P

0 vector of asset prices at time 0

w

0 available budget

The nature of the price system used in this formulation is not often discussed. It is however a crucial issue when considering such models.

Let's rst consider the case where \observed" prices are used to form the vector

P

0.

This seems like a natural choice, since the object of this model is to identify market imperfections. In such cases, arbitrage opportunities often exist in the model.

An arbitrage opportunity appears if there exists

such that

P

0,

P

6= 0 and

P

0

= 0. If this is a case,

is a direction of innite ascent for all monotonously

increasing utility functions, and the problem is unbounded. Note that these arbitrage opportunities appear because the model only takes into account a limited number of risk factors. Market professionals may not consider these opportunities as true arbitrage. Thus, we will refer to these conditions as \pseudo-arbitrage" opportunities, since they only appear because the model does not fully take into account all the risk factors which are reected in the observed prices. Arbitrary constraints must be imposed on such model to prevent unboundedness. We conclude that, this form, portfolio optimization models are mispecied.

The fact that wealth proles and budget constraints must be computed from arbitrage-free prices, in order to avoid pseudo-arbitrages in the model, actually conforms to com-mon sense. It would be preposterous to specify a model where we would have to forecast, for each scenario and future time period, how asset prices deviate from arbitrage-free prices. Using arbitrage-free prices, however, may cause a problem of its own, resulting in multiple solutions in terms of holdings, even when a unique solution in terms of wealth prole does exist.

This comes from the fact that a resonable set of replicating instruments will often include perfect substitutes, when arbitrage-free prices are used. A well known such case stems from the call-put parity, which states:

C

t;

P

t=

S

t;

KB

tT

with:

C

t price at time

t

of a European call, strike

K

, expiry

T

P

t price at time

t

of a European put, strike

K

, expiry

T

S

t value of the underlying asset at

t

B

tT value at

t

of a zero coupon maturing at

T

The formulation of the hedging problem must therefore have the following properties: rstly, horizon residual wealth and the current budget constraint must be expressed in terms of arbitrage-free prices. Secondly the variables of the problem should be the wealth prole rather than the holdings. The next section is devoted to the analysis of such models. The calculation of optimal holdings will be dealt with in a later section.

3 Solving for an optimal wealth prole

In this section, we develop an algorithm for determining the optimal wealth prole. Although the primariry motivation of this approach is to explicitly enforce non-arbitrage conditions, we nd that the use of arbitrage-free prices leads to an ecient algorithm. In particular, we show that optimal wealth proles can be calculated without explicit reference to holdings.

We use a one-period model to introduce the notation and provide simple numerical illustrations.

3.1 Algorithm

We consider the following portfolio optimization problem:

maximizeX

s

p

s

u

(

w

s) (P2)

such that

Px

;

l

w

(P2-1)

P

0

x

l

0 (P2-2)

Ax

b

(P2-3)

w

2

W

where

A

represents a set of constraints on holdings. The rest of the notation is as (P1). This problem may be written as:

maximizeX

s

p

s

u

(

w

s) (P3)

such that

w

2

R

with:

R

=f

w

2

W

j9

xPx

;

l

w

;

P

0

x

;

l

0

Ax

b

g

Lemma 1 (Farka)

There exists a vector

x

0 satisfying

Bx

=

a

if and only if

a

0

v

0

for all

v

0 satisfying

B

0

v

0.

Applied to the system

0 B @

P

;

P

;

P

0

P

0

A

;

A

1 C A

x

+

x

; !

;

s

= 0 B @

w

+

l

;

l

0

b

1 C A (1)

we get that

w

is feasible if and only if

0 B @

w

+

l

;

l

0

b

1 C A 0

v

0 (2)

for all

v >

0 such that

0 B @

P

;

P

;

P

0

P

0

A

;

A

1 C A 0

v

0 (3)

or, 0 B @

P

;

P

0

A

1 C A 0

v

= 0 (4)

Let C be the set of such vectors

v

. Since the price system (

PP

0) is arbitrage-free, there

exist a set of state prices

such that

P

=

P

0. As a result,

C

is not empty, since the

vector (

0)0 belongs to C. For simplicity, we consider the case where C is bounded. It

is thus has a nite number of generators, and an element

v

2

C

can be written

v

=Xn

j=1

j

v

j j 0 (5)

A decomposition algorithm 1] for solving (P2) is then:

Initial step:

Find one vector

v

2

C

. Since the price system (

PP

0) is arbitrage free,

we know that (

0)0

2

C

.

may be obtained from the pricing model used to

compute the theoretical prices of the various instruments. Let

v

1 be this vector.

Set

J

= 1.

maximizeX

s

p

s

u

(

w

s) (P4)

such that

0 B @

w

+

l

;

l

0

b

1 C A 0

v

j 0

8

j

= 1

:::J

(P4-1)

Let

w

be the solution.

Optimality test:

Solve:

maximize

0 B @

w

+

l

;

l

0

b

1 C A 0

v

(P5)

such that

0 B @

P

;

P

0

A

1 C A 0

v

= 0 (P5-1)

v

0

Since

C

is bounded, the solution will be an extreme point of

C

. Let

v

be the

solution, and

z

be the corresponding objective value. . If

z

= 0, then stop:

w

is the optimal solution. Otherwise, a new constraint to the master problem has been identied. Set

v

J+1 =

v

,

J

J

+ 1, and go back to the master problem.

3.2 Utility function

We use a linear-quadratic approximation to the utility function, as introduced by King 4]. Let

u

2 be a second order Taylor expansion of

u

about ^

w

:

u

2(

w

) =

u

( ^

w

) +

u

0

( ^

w

)(

w

;

w

^) + 12

u

00( ^

w

)(

w

;

w

^)

2

In order to insure that this approximation is non decreasing, the quadratic term is replaced by

u

00( ^

w

)

q(

w

;

w

^), where:

q(

t

) =

(

(

w

;

w

^)

2 if (

w

;

w

^)

q

q

(

w

;

w

^) otherwise

(6)

with:

q

;

u

0( ^

w

)

Since our objective is to control the down-side risk as much as possible, a reasonable choice is to set

q

= ;

u

0( ^

w

)

u

00( ^

w

) (8)

That is, marginal utility is null in any scenario as soon as the wealth level is such that (

w

s;

w

^s)

> q

. It is easy to verify that this utility function is maximized by the following

program:

maximizeX

s

p

s(

u

0

s(

w

s;

w

^s) + 12

u

00

s(

w

s;

w

^s)

2) (P6)

such that

Px

;

l

w

(P6-1)

P

0

x

l

0 (P6-2)

Ax

b

(P6-3)

where:

u

0

s rst derivative of the utility function, estimated at ^

w

s

u

00

s second derivative of the utility function, estimated at ^

w

s

3.3 Illustration

Let's use this model to compute the optimal wealth prole when hedging a digital option in complete and incomplete markets. A European digital option pays $100 if the the value of the underlying asset is above the strike at maturity, and nothing otherwise. Table 3.3 summarizes the target to be hedged as well as the available instruments in two replicating portfolios.

The optimization is carried over a set of 7 scenarios, corresponding to various values of the underlying asset at maturity of the options. Figure 1 represents the optimal arbitrage-free prole of residual wealth

w

, when the budget available for replication is equal to the arbitrage-free value of the target. With 7 replicating instruments, we have a complete market. It comes as no surprise that a perfect replication can be obtained. If the optimization is restricted to 5 instruments, the market is incomplete, and a perfect replication is not atainable.

In all cases. the algorithm is initiated with

v

1= (

0) 0.

This procedure has two interesting features:

The nonlinear problem appears as a one-stage problem in the dimension of the

Type Strike Price 7 assets 5 assets Target

Digital 100 72.34 Replicating instruments

Money market .96 p p

Euro Call 80 23.11 p

Euro Call 90 14.01 p

Euro Call 100 6.66 p p

Euro Call 110 2.32 p

Euro Put 120 16.00 p

Euro Put 100 2.81 p

Euro Put 90 .55 p p

Table 1: Replicating instruments and target

1 2 3 4 5 6 7

-1.000 -0.667 -0.333 0.000 0.333 0.667 1.000 1.333 1.667 2.000

5 assets

⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕

7 assets

⊕

1 2 3 4 5 6 7 -5.000

-4.333 -3.667 -3.000 -2.333 -1.667 -1.000 -0.333 0.333 1.000

5 assets

⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕

7 assets

⊕

Figure 2: Optimal wealth with limited budget

as it substantialy aleviate to \curse of dimentionality" which traditionaly plagues these formulations.

from one iteration to the next, the optimality test only changes in its cost

co-ecients. The previous optimal solution is still feasible. LP solvers can take advantage of such situation.

A multi-period model without transaction cost is now considered. The solution method is a straight-forward extension of the principle presented above.

4 Multi-periods without transaction costs

4.1 Model formulation

We follow, in this section, a formulation introduced by He and Pearson 3], and adapt it to the problem at hand. There are

T

+ 1 time periods

t

= 0

:::T

. A multi-period trading strategy is a

K

dimentional process

x

= f

x

t

t

= 0

T

;1g, where

x

t is the

vector of holdings in the portfolio between

t

and

t

+ 1.

Let

L

be the number of nodes in the event tree. To each node

l

we associate the set

C

(

l

), which is the set of nodes having node

l

as parent. A wealth prole is a vector in

R

L_{. A trading strategy may be characteriszed by holdings at each node in the event}

tree, i.e. a vector

x

2

R

M, where

M

=

K

L

. Finaly, let

P

lk be the price of security

k

at node

l

.

Let

y

_lk represent the investment in security

k

at node

l

. Let

t

be the time period associated with this node. We associate column

m

of

A

to such transaction, with

A

lm =;

P

lk and

A

l

0m=

P

l0k for all

l

0

2

C

(

l

). With this notation, the residual weath at

the nodes of the event tree, produced by strategy

x

is:

Ax

; 0 B B B B @

l

0

l

1 ...

l

T 1 C C C C A = 0 B B B B @

w

0

w

1 ...

w

T 1 C C C C A (9)

where

w

t and

l

t are vectors of residual wealth and liability for the nodes occuring at

time

t

. In order to highlight the parallel with the formulation of the previous section, equation (9) is split into two, with the rst row forming the budget constraint. The multi-period portfolio optimization problem is then formulated as follows:

maximizeX

s

X

t

p

ts

u

(

w

ts) (P7)

such that

A

1

x

; 0 B @

l

1 ...

l

T 1 C A 0 B @

w

1 ...

w

T 1 C A (P7-1)

P

0

x

0

l

0 (P7-2)

A

2

x

b

(P7-3)

where:

A

1 is matrix

A

of (9) less its rst row,

A

2 represents a set of constraints on the trading strategies,

x

represents the trading strategy for periods

t

= 1

:::T

.

The rest of the notation is as in (P1). The arguments presented in the previous section clearly apply to this formulation, and so does the decomposition algorithm.

4.2 Illustration

We use in this section a three-period model, constructed with a trinomial tree. Decisions are made at the beginning of periods 1 and 2. The targets are always options expiring in period 3. The nodes in the tree are numbered as in gure 4.2.

In this experiment, we replicate a barrier-like option which has the following payo at maturity:

L

(

S

T) =

(

0 if

S

T

>

120

max(0

S

T ;90) otherwise

1 3 2 3 Q Q Q Q Q Q Q s 4 3 5 6 Q Q Q Q Q Q Q s 7 3 8 9 Q Q Q Q Q Q Q s 10 3 11 12 Q Q Q Q Q Q Q s 13

Figure 3: Nodes of trinomial tree

Node Underlying Target 1 100.00 10.32 2 119.72 -3 100.00

-4 83.52

-5 143.33 0 6, 8 119.72 29.72 7, 9, 11 100.00 10.00 10, 12 83.52 0 13 69.72 0

Iteration 1 2 Feasibility test -.297 0.00 Objective value 2.5 2.04

node

w

1

w

2

5 0 .18 -.15 .18 6 0 .12 -.15 .12 7 0 .09 -.15 .09 8 0 .12 -.15 .12 9 0 .08 -.15 .08 10 0 .06 -.15 .06 11 0 .09 0.81 .12 12 0 .06 2.00 .00 13 0 .05 -1.0 .08

Table 3: Iterations for replication of barrier-like option

Three instruments are available for replication: a money-market instrument, a european call 100, and a put 110. The available budget is the theoretical value of the target. An optimal solution is obtained after two iterations. The progress of the iterations is summarized in Table 4.2.

The rst iteration uses the state prices as constraint of the master problem. Thus the optimal prole is a perfect replication of the target, since the available budget is the arbitrage-free price of the liability. The feasibility test shows however that this prole is not atainable. An aditional constraint is added, which yields the solution.

5 Determination of optimal holdings

For all practical purposes, it is only relevant to determine the instruments to be held at the current time. The detail of all future transactions is of no interest. It is sucient to know that the available instruments, along with the constraints of the problem, have determined the optimal wealth prole

w

, which can be achieved through some trading

strategy, based on arbitrage-free prices. We now turn to the determination of current holdings. This is the time to incorporate in our optimization model the information about market prices, which may dier from arbitrage-free prices. We are led to consider the trade-o between two aspects of the investment decision: on one hand we want to take advantage of current market imperfections (i.e. to buy underpriced assets and to sell overpriced ones), but this may lead, on the other hand, to a departure from the optimal arbitrage-free wealth prole. The optimal wealth prole computed in the previous section provides a robust benchmark to assess this trade-o. Many formulations of the trade-o can be constructed, depending upon the context. We provide below an illustration.

maximize 1_{1 +}

_r

X

s

p

s

v

(

w

s;

w

s) + (

P

0 ;

P

^

0)

0

x

(P8)

such that

Px

;

l

w

(P8-1)

^

P

0

x

b

(P8-2)

^

P

0

x

+

b

(P8-3)

x

;

x

+

0 (P8-4)

Ax

b

(P8-5)

w

2

W

where:

v

() utility of deviation from optimal wealth ^

P

0 vector of market prices at time 0

b

available budget

scalar,

0

x

+ identical to

x

when

x >

0, 0 otherwise

Constraint P8-3 controls the amount of leverage in the solution. The introduction of market prices could otherwise generate unbounded solutions.

The target is a barrier-like option with the following pay-o at maturity of 1 year:

L

(

S

T) =

(

10 if

S

T

>

120

max(0

S

T ;90) otherwise

(11)

Seven instruments, all maturing in 1 year, are available for replicating the option. The horizon is the maturity. Two replications are calculated: in the rst one, all market prices are identical to the arbitrage-free prices. The optimal portfolio is described in table 5. In the second calculation, the market price of the Call 90 is set .5% above the arbitrage-free price. As expected, the optimal portfolio now shorts this option. It is worth noting, moreover, that this portfolio provides a higher expected wealth at horizon than in the previous case. The expected increase in terminal wealth is .018. It is often the case that stochastic optimization models manage to extract value from market discrepancies, irrespective of the nature of these discrepancies.

6 Conclusion

Instrument All market prices at

arbitrage-free prices Call 90 overpriced

Money market 0 13

C 90 1.29 -.34

C 100 -.83 .96

P 100 .31 -1.47

P 90 -.15 1.48

P 120 -.147 0

C 120 -1.54 -1.69

Table 4: Optimal holdings for replicating a barrier-like option

method which has two advantages: it explicitly enforces arbitrage-free conditions, and provides a computationally ecient decomposition method. Both issues were illustrated by simple examples.

References

1] J.F. Benders. Partitioning procedures for solving mixed variables programming prob-lems. Numerishe Mathematik, 4:238{252, 1962.

2] E. Derman. Forever hedged. Risk Magazine, 1994.

3] H. He and N.D. Pearson. Consumption and portfolio policies with incomplete mar-kets and short-sale constraints. Mathematical Finance, 1(3):1{10, July 1991. 4] A.J. King. Asymmetric risk measures and tracking models for portfolio optimization