1. APPROXIMATING STATIC PORTFOLIO FOR
EXOTIC
PAYOFFS
A project submitted to the Department of Finance
of
Università
Commerciale Luigi Bocconi
Author: Marios Aspris
Supervisor: Professor Andrea Roncoroni
Milano, Italy
July 2015
3. Abstract
In a simple economic period, I assume the existence of three asset
classes. A risk free Bond, a Forward Contract and a range of
derivative instruments, Calls and Put options with different strike
prices. I construct a portfolio in this period using these instruments
with the assumption of continuous time mathematics. I construct a
portfolio for two different increments, and the portfolio consists of
all of the three asset classes for each one of the increment
considered. In this setting of continuity, I take as assumption a
complete market. Since the derivatives market is incomplete,
causes to develop an approximation of the strategy for the
portfolio. With the inability to make use of continuity, I make an
approximation for the portfolio with the use of discrete time
mathematics. In this way I find a portfolio for exotic payoffs with a
small approximation error.
4. Table of Contents
1. Introduction…………………………………………………………….….1
2. Replication of exotic payoffs……………………………………….….2
3. Discretization of Process…………………………………………..….7
4. Approximation Scheme…...………………………………………….10
5. Summary…...…………………………………………………………….12
6. References…..…………………………………………………………..13
5. 1
Introduction
Yumi Oum and Shmuel Oren [2] proposed a way of a hedging portfolio that
maximizes the expected profit subject to price and volumetric risk with a VaR
constraint. They represent the hedging portfolio as a general exotic option
with a nonlinear payoff contingent to the price of electricity. They replicate the
desired payoff function with a portfolio. The purpose of this paper is to portray
the payoff function that consists of two increments that will eventually be used
to derive the quantities of forwards and options at different strike prices for
each increment. Peter Carr and Dilip Madan [3] demonstrated, with the use of
continuous time mathematics, a way of replicating the payoff function with one
increment. I extend the calculations in the two dimensional case. Because the
market is incomplete and does not allow assumptions of continuity, to
implement the replicating strategy I make use of discrete set providing an
approximation to the replicating strategy, with a risk free bond, a forward
contract and a range of call and put options with different strike prices.
6. 2
1. Replication of exotic payoffs
Consider a load service entity whose revenue is determined by a fixed retail
price r and the uncertain demand q. Denoting uncertain wholesale electricity
price per unit as p, the profit 𝑦(𝑝, 𝑞) from retail sales at time 1 depends on the
two random variable p and q, i.e., 𝑦(𝑝, 𝑞) = (𝑟 − 𝑝)𝑞.
Suppose the load service entity hedges the profit through an exotic electricity
option maturing at time 1. Let 𝑌(𝑥) be the hedged profit, then
𝑌(𝑥) = 𝑦(𝑝, 𝑞) + 𝑥(𝑝), where 𝑥(𝑝)is a payoff function of the exotic option,
which is contingent on the price p.
Now considering the payoff function of the exotic option to be contingent on
two prices, 𝑆1 and 𝑆2. To construct the portfolio that replicates the two
dimensional function, it is necessary to make use the theory in the one
dimensional case.
Suppose there exists a twice continuously differentiable function x. As stated
in the Fundamental Theorem of Calculus I can write the function as follows:
First I state the Theorem.
Theorem Statement: If x’ is an integrable function on a closed interval [a,b] ,
then 𝑥(𝑏) − 𝑥(𝑎) = ∫ 𝑥′(𝑢)𝑑𝑢 .
𝑏
𝑎
Notice that if I assign to a=F and b=S, then I can write the formula in the form:
𝑥(𝑆) = 𝑥(𝐹) + ∫ 𝑥′(𝑢)𝑑𝑢 .
𝑆
𝐹
If I expand the formula by adding logical components, I have:
𝑥(𝑆) = 𝑥(𝐹) + 1 𝑆>𝐹 ∗ ∫ 𝑥′(𝑢)𝑑𝑢 − 1 𝑆<𝐹 ∗ ∫ 𝑥′(𝑢)𝑑𝑢
𝐹
𝑆
.
𝑆
𝐹
Suppose now that the function depends on two variables: 𝑥(𝑆1, 𝑆2) . To apply
the same logic and use the Fundamental Theorem of Calculus I will consider
the function fixed in the second component and apply the theorem on the first
component, and taking the partial derivative. Assuming that the function is
twice continuously differentiable with respect to 𝑆1and 𝑆2. I derive the
following:
𝑥(𝑆1, 𝑆2) − 𝑥(𝐹1, 𝑆2) + 𝑥(𝑆1, 𝑆2) − 𝑥(𝑆1, 𝐹2) = 1 𝑆1>𝐹1
∫
𝜕𝑥(𝑢1,𝑆2)
𝜕𝑆1
𝑆1
𝐹1
𝑑𝑢1 −
1 𝑆1<𝐹1
∫
𝜕𝑥(𝑢1,𝑆2)
𝜕𝑆1
𝐹1
𝑆1
𝑑𝑢1 + 1 𝑆2>𝐹2
∫
𝜕𝑥(𝑆1,𝑢2)
𝜕𝑆2
𝑆2
𝐹2
𝑑𝑢2 − 1 𝑆2<𝐹2
∫
𝜕𝑥(𝑆1,𝑢2)
𝜕𝑆2
𝐹2
𝑆2
𝑑𝑢2
7. 3
Now I consider the first two components of the right hand side of the equation
and the same calculations apply for the other two components. If I apply the
Fundamental Theorem inside the integral I derive:
1 𝑆1>𝐹1
∫
𝜕𝑥(𝑢1, 𝑆2)
𝜕𝑆1
𝑆1
𝐹1
𝑑𝑢1 − 1 𝑆1<𝐹1
∫
𝜕𝑥(𝑢1, 𝑆2)
𝜕𝑆1
𝐹1
𝑆1
𝑑𝑢1=
= 1 𝑆1>𝐹1
∫[
𝜕𝑥(𝐹1, 𝑆2)
𝜕𝑆1
𝑆1
𝐹1
+ ∫
𝜕2
𝑥(𝑣1, 𝑆2)
𝜕𝑆1
2
𝑢1
𝐹1
𝑑𝑣1 ]𝑑𝑢1
− 1 𝑆1<𝐹1
∫[
𝜕𝑥(𝐹1, 𝑆2)
𝜕𝑆1
𝐹1
𝑆1
− ∫
𝜕2
𝑥(𝑣1, 𝑆2)
𝜕𝑆1
2
𝐹1
𝑢1
𝑑𝑣1 ]𝑑𝑢1
Since now the component
𝜕𝑥(𝐹1,𝑆2)
𝜕𝑆1
does not depend on 𝑢1, then
=
𝜕𝑥(𝐹1, 𝑆2)
𝜕𝑆1
(1 𝑆1>𝐹1
∗ (𝑆1 − 𝐹1)) + 1 𝑆1>𝐹1
∫(
𝑆1
𝐹1
∫
𝜕2
𝑥(𝑣1, 𝑆2)
𝜕𝑆1
2
𝑢1
𝐹1
𝑑𝑣1 )𝑑𝑢1
−
𝜕𝑥(𝐹1, 𝑆2)
𝜕𝑆1
(1 𝑆1<𝐹1
∗ (𝐹1 − 𝑆1))
+ 1 𝑆1<𝐹1
∫
𝐹1
𝑆1
( ∫
𝜕2
𝑥(𝑣1, 𝑆2)
𝜕𝑆1
2
𝐹1
𝑢1
𝑑𝑣1 )𝑑𝑢1
Since the intervals [𝐹1, 𝑆1] and [𝑆1, 𝑢1] are finite measurable spaces and
𝜕2 𝑥(𝑣1,𝑆2)
𝜕𝑆1
2 is a measurable function and ∫
𝐹1
𝑆1
(∫ l
𝜕2 𝑥(𝑣1,𝑆2)
𝜕𝑆1
2 l
𝑆1
𝑢1
𝑑𝑣1 ) 𝑑𝑢1 < ∞ ,
since it is a continuous function by assumption in a finite space, then I can
apply Fubini theorem:
Theorem Statement: Let (X, P, μ) and (Y, L, λ) be σ-finite measure spaces,
and let f be an (P x L)-measurable function on X x Y. If 0≤f≤∞, and if
φ(x) = ∫ 𝑓𝑥𝑌
𝑑𝜆 , 𝜓(x) = ∫ 𝑓𝑌𝑋
𝑑𝜇 (x ε X, y ε Y),
8. 4
then φ is P –measurable, 𝜓 is L –measurable, and
∫ φ𝑋
𝑑𝜇 = ∫ 𝑓𝑋 𝑥 𝑌
𝑑(𝜇 𝑥 𝜆) = ∫ 𝜓𝑦
𝑑𝜆. The first and last integrals can also be
written in the more usual form ∫ 𝑑𝜇(𝑥) ∫ f(x, y)𝑌
𝑑𝑋
𝜆(𝑦) =
∫ 𝑑𝜆(𝑦) ∫ f(x, y)𝑋
𝑑𝜇(𝑥)𝑌
. These are the so called iterated integrals of f.
Continuing the calculations:
=
𝜕𝑥(𝐹1, 𝑆2)
𝜕𝑆1
(𝑚𝑎𝑥(𝑆1 − 𝐹1, 0)) + 1 𝑆1>𝐹1
∫(
𝑆1
𝐹1
∫
𝜕2
𝑥(𝑣1, 𝑆2)
𝜕𝑆1
2
𝑆1
𝑣1
𝑑𝑢1 )𝑑𝑣1
−
𝜕𝑥(𝐹1, 𝑆2)
𝜕𝑆1
(𝑚𝑎𝑥(𝐹1 − 𝑆1, 0))
+ 1 𝑆1<𝐹1
∫
𝐹1
𝑆1
(∫
𝜕2
𝑥(𝑣1, 𝑆2)
𝜕𝑆1
2
𝑣1
𝑆1
𝑑𝑢1)𝑑𝑣1
=
𝜕𝑥(𝐹1, 𝑆2)
𝜕𝑆1
(𝑆1 − 𝐹1)+
+ 1 𝑆1>𝐹1
∫
𝑆1
𝐹1
∫
𝜕2
𝑥(𝑣1, 𝑆2)
𝜕𝑆1
2
𝑆1
𝑣1
𝑑𝑢1 𝑑𝑣1
−
𝜕𝑥(𝐹1, 𝑆2)
𝜕𝑆1
(𝐹1 − 𝑆1)+
− 1 𝑆1<𝐹1
∫
𝐹1
𝑆1
∫
𝜕2
𝑥(𝑣1, 𝑆2)
𝜕𝑆1
2
𝑣1
𝑆1
𝑑𝑢1 𝑑𝑣1
Notice that with the change of integration the function does not depend on 𝑢1
and the integration space [𝐹1, 𝑆1] equals to [𝑣1 𝑆1,] when I change position. So I
can easily solve it. Now (𝑆1 − 𝐹1)+
− (𝐹1 − 𝑆1)+
= max(𝑆1 − 𝐹1, 0) + 𝑚𝑖𝑛(𝐹1 −
𝑆1, 0) = 𝑆1 − 𝐹1 , and with the change of integration I arrive at:
=
𝜕𝑥(𝐹1, 𝑆2)
𝜕𝑆1
(𝑆1 − 𝐹1) + 1 𝑆1>𝐹1
∫
𝜕2
𝑥(𝑣1, 𝑆2)
𝜕𝑆1
2
𝑆1
𝐹1
(𝑆1 − 𝑣1)𝑑𝑣1
+ 1 𝑆1<𝐹1
∫
𝜕2
𝑥(𝑣1, 𝑆2)
𝜕𝑆1
2
𝐹1
𝑆1
(𝑣1 − 𝑆1)𝑑𝑣1
Applying the logical component on the space of integration and in the function
I arrive at the final form:
9. 5
=
𝜕𝑥(𝐹1, 𝑆2)
𝜕𝑆1
(𝑆1 − 𝐹1) + ∫
𝜕2
𝑥(𝑣1, 𝑆2)
𝜕𝑆1
2
∞
𝐹1
(𝑆1 − 𝑣1)+
𝑑𝑣1
+ ∫
𝜕2
𝑥(𝑣1, 𝑆2)
𝜕𝑆1
2
𝐹1
0
(𝑣1 − 𝑆1)+
𝑑𝑣1
Now I can apply the same steps for the second variable therefore calculations
are omitted. So the function will have the final form:
𝑥(𝑆1, 𝑆2) =
1
2
[ 𝑥(𝐹1, 𝑆2) +
𝜕𝑥(𝐹1, 𝑆2)
𝜕𝑆1
(𝑆1 − 𝐹1) + ∫
𝜕2
𝑥(𝑣1, 𝑆2)
𝜕𝑆1
2
∞
𝐹1
(𝑆1 − 𝑣1)+
𝑑𝑣1
+ ∫
𝜕2
𝑥(𝑣1, 𝑆2)
𝜕𝑆1
2
𝐹1
0
(𝑣1 − 𝑆1)+
𝑑𝑣1 ] +
1
2
[ 𝑥(𝑆1, 𝐹2)
+
𝜕𝑥(𝑆1, 𝐹2)
𝜕𝑆2
(𝑆2 − 𝐹2) + ∫
𝜕2
𝑥(𝑆1, 𝑣2)
𝜕𝑆2
2
∞
𝐹2
(𝑆2 − 𝑣2)+
𝑑𝑣2
+ ∫
𝜕2
𝑥(𝑆1, 𝑣2)
𝜕𝑆2
2
𝐹2
0
(𝑣2 − 𝑆2)+
𝑑𝑣2 ]
If I consider the above function as a function that replicates the exotic payoff
function and denoting as 𝑆1, 𝑆2 the uncertain price per unit and 𝐹1, 𝐹2 be a
forward price for a delivery at time 1 then I construct a portfolio composed of
standard instruments. Rewriting the equation:
𝑥(𝑆1, 𝑆2) =
1
2
[ 𝑥(𝐹1, 𝑆2) +
𝜕𝑥(𝐹1, 𝑆2)
𝜕𝑆1
(𝑆1 − 𝐹1) + ∫
𝜕2
𝑥(𝐾1, 𝑆2)
𝜕𝑆1
2
∞
𝐹1
(𝑆1 − 𝐾1)+
𝑑𝐾1
+ ∫
𝜕2
𝑥(𝐾1, 𝑆2)
𝜕𝑆1
2
𝐹1
0
(𝐾1 − 𝑆1)+
𝑑𝐾1 ] +
1
2
[ 𝑥(𝑆1, 𝐹2)
+
𝜕𝑥(𝑆1, 𝐹2)
𝜕𝑆2
(𝑆2 − 𝐹2) + ∫
𝜕2
𝑥(𝑆1, 𝐾2)
𝜕𝑆2
2
∞
𝐹2
(𝑆2 − 𝐾2)+
𝑑𝐾2
+ ∫
𝜕2
𝑥(𝑆1, 𝐾2)
𝜕𝑆2
2
𝐹2
0
(𝐾2 − 𝑆2)+
𝑑𝐾2 ]
10. 6
Notice that 1,(𝑆𝑖 − 𝐹𝑖), (𝐾𝑖 − 𝑆𝑖)+
, (𝑆𝑖 − 𝐾𝑖)+
for i=1,2 in the above expression
represents payoffs at time 1 of a Bond, Forward Contract, European Put
options, and European Calls respectively. Therefore an exact replication can
be obtained from a long cash position of size 𝑥(𝐹1, 𝑆2) for the first asset and
𝑥(𝑆1, 𝐹2) for the second asset, a long forward position of size
𝜕𝑥(𝐹1,𝑆2)
𝜕𝑆1
,
𝜕𝑥(𝑆1,𝐹2)
𝜕𝑆2
of the first and second asset respectively, long positions
𝜕2 𝑥(𝐾1,𝑆2)
𝜕𝑆1
2 𝑑𝐾1,
𝜕2 𝑥(𝑆1,𝐾2)
𝜕𝑆2
2 𝑑𝐾2 in puts struck at 𝐾𝑖, for a continuum 𝐾𝑖 < 𝐹𝑖 ,i=1,2 , and long
positions
𝜕2 𝑥(𝐾1,𝑆2)
𝜕𝑆1
2 𝑑𝐾1,
𝜕2 𝑥(𝑆1,𝐾2)
𝜕𝑆2
2 𝑑𝐾2 in calls struck at 𝐾𝑖, for a continuum 𝐾𝑖 >
𝐹𝑖 ,i=1,2.
11. 7
2. Discretization of the process
Notice that the above I make use of the assumption of a continuum of strike
prices. The optimal strategy involves purchasing a spectrum of Calls and Puts
for both increments with continuum strike prices. This result demonstrates that
it should be purchased a portfolio of options. With the assumption of continuity
is the counterpart of the standard assumption of continuous trading.
Therefore, I consider a complete market. In practice the derivatives market is
incomplete and does not offer options for the full continuum of strike prices.
Consequently, I take our assumption as a reasonable approximation when
there are a large but finite number of option strike prices offered and where
investors can trade frequently.
To implement the above replicating strategy using a discrete set of standard
plain Vanilla options, I need to discretize the strike prices and approximate the
payoff function using a set of discrete options at available strike prices.
Therefore, I am taking the summation of integrals.
The strike prices are discretized as follows: Suppose there are put options
with 𝐾11
< 𝐾12
< ⋯ < 𝐾1 𝑛
= 𝐹 strikes and call options with strike prices 𝐹 =
𝐾′
11
< ⋯ < 𝐾′
1 𝑚
in the market for the first increment of the approximation
function and with the same procedure for the second increment. I set 𝐾1 𝑛+1
=
𝐾1 𝑛
, 𝐾10
= 0, 𝐾′
10
= 𝐾′
11
, and 𝐾′
1 𝑚+1
= ∞. As a result, I consider the
following:
∫
𝜕2 𝑥(𝐾1,𝑆2)
𝜕𝑆1
2
𝐹1
0
(𝐾1 − 𝑆1)+
𝑑𝐾1 + ∫
𝜕2 𝑥(𝐾1,𝑆2)
𝜕𝑆1
2
∞
𝐹1
(𝑆1 − 𝐾1)+
𝑑𝐾1=
= ∑ ∫
𝜕2
𝑥(𝐾1, 𝑆2)
𝜕𝑆1
2
𝐾1 𝑖+1
𝐾1 𝑖
(𝐾1 − 𝑆1)+
𝑑𝐾1
𝑛−1
𝑖=0
+ ∑ ∫
𝜕2
𝑥(𝐾1, 𝑆2)
𝜕𝑆1
2
𝐾′
1 𝑖+1
𝐾′
1 𝑖
(𝑆1 − 𝐾1)+
𝑑𝐾1
𝑚
𝑖=1
I consider what applies for the first increment 𝑆1 of the function because the
calculations are similar for the second increment and as a result are omitted.
To proceed to the next step, I will make use of the Mean Value Theorem for
integrals.
12. 8
Theorem Statement: Suppose that f is a continuous function on [a, b] and
that g is integrable and nonnegative on [a, b]. Then ∫ 𝑓(𝑥)
𝑏
𝑎
𝑔(𝑥)𝑑𝑥 =
𝑓(𝜉) ∫ 𝑔(𝑥)
𝑏
𝑎
𝑑𝑥 for some ξ in [a, b].
If I denote as 𝑔 =
𝜕2 𝑥(𝐾1,𝑆2)
𝜕𝑆1
2 and 𝑓 = (𝐾1 − 𝑆1)+
in each different part in the
equation above then I can apply it. Another thing is the integration limits
where I have to consider if 𝑆1 < 𝐾1 𝑖
𝑜𝑟 𝐾1 𝑖
< 𝑆1 < 𝐾1 𝑖+1
𝑜𝑟 𝑆1 > 𝐾1 𝑖+1
. By taking
the 𝑚𝑎𝑥(𝑆1, 𝐾1 𝑖
) 𝑎𝑛𝑑 𝑚𝑖𝑛(𝑆1, 𝐾′
1 𝑖
) for the payoff of the Put and Call option
respectively, then I can take an approximation of the series. I also consider as
𝑓(𝜉) =
1
2
{ (𝐾1 𝑖
− 𝑆1)+
+ (𝐾1 𝑖+1
− 𝑆1)+
} (appropriately for the Call payoff).
Therefore:
≈ ∑ ∫
𝜕2
𝑥(𝐾1, 𝑆2)
𝜕𝑆1
2
𝑚𝑎𝑥(𝑆1,𝐾1 𝑖+1
)
𝑚𝑎𝑥(𝑆1,𝐾1 𝑖
)
𝑑𝐾1
𝑛−1
𝑖=0
∗
1
2
{ (𝐾1 𝑖
− 𝑆1)+
+ (𝐾1 𝑖+1
− 𝑆1)+
}
+ ∑ ∫
𝜕2
𝑥(𝐾1, 𝑆2)
𝜕𝑆1
2
𝑚𝑖𝑛(𝑆1,𝐾′
1 𝑖+1
)
𝑚𝑖𝑛(𝑆1,𝐾′
1 𝑖
)
𝑑𝐾1
𝑚
𝑖=0
∗
1
2
{ (𝑆1−𝐾′
1 𝑖
)+
+ (𝑆1−𝐾′
1 𝑖+1
)+
}
Now I can take another approximation in the calculations integrating from
𝐾1 𝑖
𝑡𝑜 𝐾1 𝑖+1
:
≈ ∑ ∫
𝜕2
𝑥(𝐾1, 𝑆2)
𝜕𝑆1
2
𝐾1 𝑖+1
𝐾1 𝑖
𝑑𝐾1
𝑛−1
𝑖=0
∗
1
2
{ (𝐾1 𝑖
− 𝑆1)+
+ (𝐾1 𝑖+1
− 𝑆1)+
}
+ ∑ ∫
𝜕2
𝑥(𝐾1, 𝑆2)
𝜕𝑆1
2
𝐾′
1 𝑖+1
𝐾′
1 𝑖
𝑑𝐾1 ∗
1
2
{ (𝑆1−𝐾′
1 𝑖
)+
+ (𝑆1−𝐾′
1 𝑖+1
)+
}
𝑚
𝑖=1
13. 9
Now I defined as 𝐾11
< 𝐾12
< ⋯ < 𝐾1 𝑛
= 𝐹 as the strikes of the Put options
and 𝐹 = 𝐾′
11
< ⋯ < 𝐾′
1 𝑚
the strike prices of the Call options. Also 𝐾1 𝑛+1
=
𝐾1 𝑛
, 𝐾10
= 0, 𝐾′
10
= 𝐾′
11
, and 𝐾′
1 𝑚+1
= ∞.
In this way if I write the limits of the series and changing appropriately the
integration limits I arrive at the final form.
= ∑ ∫
𝜕2
𝑥(𝐾1, 𝑆2)
𝜕𝑆1
2
𝐾1 𝑖+1
𝐾1 𝑖−1
𝑑𝐾1
𝑛
𝑖=1
∗
1
2
{ (𝐾1 𝑖
− 𝑆1)+
}
+ ∑ ∫
𝜕2
𝑥(𝐾1, 𝑆2)
𝜕𝑆1
2
𝐾′
1 𝑖+1
𝐾′
1 𝑖−1
𝑑𝐾1 ∗
1
2
{ (𝑆1−𝐾′
1 𝑖
)+
}
𝑚
𝑖=1
= ∑
1
2
[
𝜕𝑥(𝐾1 𝑖+1
, 𝑆2)
𝜕𝑆1
−
𝜕𝑥(𝐾1 𝑖−1
, 𝑆2)
𝜕𝑆1
] ∗ { (𝐾1 𝑖
− 𝑆1)+
}
𝑛
𝑖=1
+ ∑
1
2
[
𝜕𝑥(𝐾′
1 𝑖+1
, 𝑆2)
𝜕𝑆1
−
𝜕𝑥(𝐾′
1 𝑖−1
, 𝑆2)
𝜕𝑆1
] ∗ { (𝑆1−𝐾′
1 𝑖
)+
}
𝑚
𝑖=1
15. 11
𝑥(𝑆1, 𝑆2) = [
1
2
𝑥(𝐹1, 𝑆2) +
1
2
𝜕𝑥(𝐹1, 𝑆2)
𝜕𝑆1
(𝑆1 − 𝐹1)
+ ∑
1
4
[(
𝜕𝑥(𝐾1 𝑖+1
, 𝑆2)
𝜕𝑆1
−
𝜕𝑥(𝐾1 𝑖−1
, 𝑆2)
𝜕𝑆1
)] ∗ { (𝐾1 𝑖
− 𝑆1)+
}
𝑛
𝑖=1
+ ∑
1
4
[(
𝜕𝑥(𝐾′
1 𝑖+1
, 𝑆2)
𝜕𝑆1
−
𝜕𝑥(𝐾′
1 𝑖−1
, 𝑆2)
𝜕𝑆1
)] ∗ { (𝑆1−𝐾′
1 𝑖
)+
}
𝑚
𝑖=1
]
+ [
1
2
𝑥(𝑆1, 𝐹2) +
1
2
𝜕𝑥(𝑆1, 𝐹2)
𝜕𝑆2
(𝑆2 − 𝐹2)
+ ∑
1
4
[ (
𝜕𝑥(𝑆1, 𝐾2 𝑖+1
)
𝜕𝑆2
−
𝜕𝑥(𝑆1, 𝐾2 𝑖−1
)
𝜕𝑆2
) ] ∗ { (𝐾2 𝑖
− 𝑆2)+
}
𝑛
𝑖=1
+ ∑
1
4
[ (
𝜕𝑥(𝑆1, 𝐾′
2 𝑖+1
)
𝜕𝑆2
−
𝜕𝑥(𝑆1, 𝐾′
2 𝑖−1
)
𝜕𝑆2
)] ∗ { (𝑆2−𝐾′
2 𝑖
)+
}
𝑚
𝑖=1
]
Here is provided an approximate replication of an exotic payoff function using
the existing plain vanilla options so that the total payoff from those options is
close to the exotic payoff. As a result, the strategy consists of:
a long cash position of size
1
2
∗ 𝑥(𝐹1, 𝑆2) for the price of 𝑆1 and
1
2
𝑥(𝑆1, 𝐹2) for the price 𝑆2,
a long forward position of
1
2
𝜕𝑥(𝐹1,𝑆2)
𝜕𝑆1
and
1
2
𝜕𝑥(𝑆1,𝐹2)
𝜕𝑆2
for 𝑆1 and 𝑆2
respectively,
long positions of size
1
4
[(
𝜕𝑥(𝐾1 𝑖+1
,𝑆2)
𝜕𝑆1
−
𝜕𝑥(𝐾1 𝑖−1
,𝑆2)
𝜕𝑆1
)] in put options struck
at 𝐾1 𝑖
𝑓𝑜𝑟 𝑖 = 1, … , 𝑚 . Respectively for the price 𝑆2 long position of size
1
4
[ (
𝜕𝑥(𝑆1,𝐾2 𝑖+1
)
𝜕𝑆2
−
𝜕𝑥(𝑆1,𝐾2 𝑖−1
)
𝜕𝑆2
) ] in puts with strike price 𝐾2 𝑖
,
long positions of size
1
4
[(
𝜕𝑥(𝐾′
1 𝑖+1
,𝑆2)
𝜕𝑆1
−
𝜕𝑥(𝐾′
1 𝑖−1
,𝑆2)
𝜕𝑆1
)] in calls struck at
𝐾′
1 𝑖
𝑓𝑜𝑟 𝑖 = 1, … , 𝑚 for the price 𝑆1 and of size
1
4
[ (
𝜕𝑥(𝑆1,𝐾′
2 𝑖+1
)
𝜕𝑆2
−
𝜕𝑥(𝑆1,𝐾′
2 𝑖−1
)
𝜕𝑆2
)] for price 𝑆2 struck at 𝐾′
2 𝑖
𝑓𝑜𝑟 𝑖 = 1, … , 𝑚.
16. 12
In this approximation form, I will have a small error if
𝜕2 𝑥(𝐾1,𝑆2)
𝜕𝑆1
2 , for appropriate
index, is constant in each interval between two consecutive strike prices. I can
also reduce the error by refining the strike price discretization in the range
where there is high probability that 𝑆𝑖 ,for i=1,2 , will fall.
Summary
To obtain a realistic hedging portfolio for two dimensions, I solved for the
payoff function what represents the payoff of two exotic options as a function
of price. Then it is shown how we can replicate the function using a portfolio of
forward contracts and options and how can we take an approximation of the
function with a small approximation error.
17. 13
References
[1] Michael D. Spivak, Calculus 3rd
Edition (1994), 282-288.
[2] Y. Oum and S. Oren, VaR constrained hedging of fixed price load-following
obligations in competitive electricity markets, Risk and Decision Analysis 1
(2009),
50-54.
[3] P. Carr and D.Madan, Optimal Positioning in derivative securities,
Quantitative Finance 1 (2001), 34-36.
[4] Walter Rudin, Real and Complex Analysis 3rd
Edition (1987), 164-165.
