# A Time-Series Approach to Non-Self-Financing Hedging in a Discrete-Time Incomplete Market

Authors: N. Josephy; L. Kimball; V. Steblovskaya
Journal: Journal of Applied Mathematics and Stochastic Analysis
 (2008)

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## Abstract

We present an algorithm producing a dynamic non-self-financing hedging strategy in an incomplete
market corresponding to investor-relevant risk criterion. The optimization is a two-stage process
that first determines market calibrated model parameters that correspond to the market price of the
option being hedged. In the second stage, an optimal set of model parameters is chosen from the
market calibrated set. This choice is based on stock price simulations using a time-series model
for stock price jump evolution. Results are presented for options traded on the New York Stock
Exchange.


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## Body

## 1. Introduction

This paper  is a
continuation of  the work originally presented in [1] where we developed an
algorithm producing a dynamic non-self-financing hedging strategy in an
incomplete market optimizing a suitable investor-relevant risk criterion. The
algorithm expanded on theoretical investigations of A. V. Nagaev et al. (see
[2–5]) where the authors studied
asymptotic behavior of the residual value of a minimum cost super-hedge. The
residual value occurs as a result of a non-self-financing dynamic hedging
strategy of an option seller in an incomplete market introduced and discussed
in [2].The financial market model considered by Nagaev et al.
consists of a bond, a stock, and a European type derivative security with a convex
payoff function. The stock price jumps are assumed to be distributed over a
bounded interval. This model is a natural extension of the classical binomial
model of Cox-Ross-Rubinstein as well as its further  multinomial extensions
(see, e.g., [6–8], and references
therein).The model considered in this paper as well as its multinomial
predecessors produces an incomplete market. A significant proportion of
research on option pricing and hedging in incomplete markets constructs
self-financing trading strategies that satisfy both a primary no-arbitrage
condition and secondary conditions on portfolio risk and return. A
comprehensive survey of modern methodologies can be found in [9]. A number of articles that
deal with frictions in markets, shortfall risks, and quadratic hedging (all
producing incomplete markets) can be found in the recent compendium [10].Less prevalent is the study of non-self-financing
trading strategies in similar economic environments. The encyclopedic reference
[11] and the more
modest [12] both
illuminate option pricing with consumption, the model which is similar to the
work presented here. The application of constrained quadratic programming to
the pricing of options by constructing non-self-financing portfolios in an
incomplete market can be found in [13]. We note that the alternative approach of specifying
a deterministic model of incomplete markets is an active area of research, as
illustrated by [14–17].In [1] as well as in the present work, we study the
short-term behavior of the residual value of a dynamic non-self-financing
hedging strategy, whose long-term behavior was studied by Nagaev in [5]. Our goal is to develop and
evaluate an algorithm that will determine a dynamic non-self-financing hedging
strategy consisting of a portfolio of our stock and bond assets in an
incomplete market. The portfolio will approximately hedge the derivative
security and will satisfy additional criteria that are meaningful to the
investor, based on the deviation of the portfolio value from the required
hedging value.Our algorithm is a two-stage process that first
determines a set of market calibrated model parameters that correspond to the
market price of the option being hedged. In the second stage, an optimal set of
model parameters is chosen from this set. This choice is based on stock price
simulations. In our initial work [1], we used a bootstrap method to simulate future stock
price paths assuming that the stock price jumps are independent identical
distributed random variables. In the present work, we relax the i.i.d.
assumption on the stock price jumps and use a time-series model for stock price
jump evolution.We fit an ARMA model to historical stock price jump
series and extract the correlation structure in the stock price jump process.
Further, we simulate ARMA model driving noise to construct future values of the
price jump process by using the nonparametric bootstrap method of sampling with
replacement from the ARMA model residual sequence. The above method (unlike
common parametric methods based on an assumed Gaussian distribution) captures
long tails observed in typical stock price jump distributions.The remainder of the paper is organized as follows. We
develop the discrete time financial model in Section2. The notion of the
residual value of a minimum cost super-hedge is described in Section 3.
Non-self-financing dynamic hedging strategies and residual values associated
with them are discussed in Section 4. Section 5 contains a brief overview of
our hedging algorithm. The first stage of the algorithm, where we find the set
of market calibrated model parameters, is presented in Section 6. Section 7
describes the second stage of the algorithm: the numerical optimization over the
set of market calibrated model parameters in order to determine an optimal
hedging strategy based on a risk criterion chosen by an investor. Section 8 is
devoted to the ARMA model based stock price simulation. The results of applying
our algorithm to a variety of stock options are presented in Section 9. Section
10 presents the comparison of our optimal hedging strategy with the
Black-Scholes hedging strategy. We conclude with some remarks in Section 11.


## 2. Discrete-time market model and no-arbitrage option prices

Our discrete-time financial market model consists of
two fundamental assets and a derivative security.(1)
A risk-free bond with fixed periodic interest
rater,
evolving from an initial value b0>0 at time t=0 to bk at time t=k as(2.1)bk=b0(1+r)k.(2)
A risky stock evolving from an initial values0 at time t=0 to sk at time t=k as(2.2)sk=s0ξ1ξ2⋯ξk,where the stock price jumps ξk=sk/sk−1 are assumed to be random variables distributed
over a bounded interval [D,U], D<U at every time step k. No further assumptions are made on the
distribution for the ξk.(3)
A European type option with expiryn and convex payoff function f. We will require
the condition(2.3)D<1+r<Uto be fulfilled to guarantee
no-arbitrage in this market.This discrete-time market model is incomplete. Indeed
it is well known that the famous binomial model is complete while its trinomial
(as well as more general multinomial) extensions are not (see, e.g., [6]). Our model generalizes the
multinomial models allowing the stock price jumps ξk at every time step k to be distributed over a bounded interval [D,U], D<U. The incompleteness of the model is manifested
in an open interval (x¯k,X¯k) (k=0,…,n−1) of no-arbitrage option prices (see, e.g.,
[2, 18]). It is shown in [18] that the end points of the
interval are given by the following formulas:(2.4)X¯k(U,D)=gk(U,D,sk),(2.5)x¯k(U,D)=(1+r)−(n−k)f(sk(1+r)n−k),where D and U define the support of the stock price jump
distribution (see above), f is a convex payoff function of the option, n is the number of periods to the option
expiration, sk is the stock price at time k, and gk is defined as follows:(2.6)gk(U,D,s)=(1+r)−(n−k)∑j=0n−k(n−kj)[p(U,D)]j[1−p(U,D)]n−k−jf(sUjDn−k−j),(2.7)p(U,D)=(1+r)−DU−D.(Here, (n−kj) is the binomial coefficient.)For the option seller, the upper boundX¯k(U,D) is the demarcation between risk sharing with
the option buyer (if the option sale price is below X¯k(U,D)) and the potential for arbitrage profit (if
the option sale price is at or above X¯k(U,D)).


## 3. Minimum cost super hedge

For the rest of the paper, we take the position of an
option seller who wishes to hedge the potential liability of the sold option
being exercised. In this section, we will consider an extreme case: suppose the
option with the convex payoff functionf is sold at time t=0 for the upper bound price X¯0(U,D). The option seller uses the amount X¯0(U,D) to finance the setup cost of a portfolio
consisting of γ0 stocks and β0 bonds with the intention of hedging the short
position in the option:(3.1)X¯0(U,D)=γ0s0+β0b0.  The seller
rebalances the portfolio at each time instant t=k (k=1,…,n−1) creating a dynamic trading strategy (γk,βk) (k=0,…,n−1).We will choose the quantitiesγk and βk according to the formulas(3.2)γk=γk(U,D)=gk+1(U,D,skU)−gk+1(U,D,skD)sk(U−D),(3.3)βk=βk(U,D)=Ugk+1(U,D,skD)−Dgk+1(U,D,skU)(1+r)bk(U−D),where gk is defined in (2.6). S. A. Nagaev  and A. V. Nagaev [2] showed (based on a
convexity argument) that the dynamic trading strategy (γk,βk) (k=0,…,n−1) represents a minimum cost super-hedging
strategy in the following sense: the associated portfolio value at every time
instant t=k is greater than or equal to the value of the
option. It is worth explaining the above statement in more detail.Suppose at each time instantk,
the option seller liquidates the portfolio constructed in the prior period [k−1,k) and uses the proceeds to construct a new
portfolio for the current period [k,k+1).
Let us denote by vk(U,D) the liquidation value of the prior period
portfolio:(3.4)vk(U,D)=γk−1(U,D)sk+βk−1(U,D)bk=γk−1(U,D)sk−1ξk+βk−1(U,D)bk−1(1+r).Combining (3.4) with (3.2) and (3.3),
one gets(3.5)vk(U,D)=U−ξkU−Dgk(U,D,sk−1D)+ξk−DU−Dgk(U,D,sk−1U).On the other hand, the funds
required to finance the current period portfolio, or setup cost, are given by
the upper bound X¯k(U,D) of the no-arbitrage price interval
corresponding to the time moment t=k:  (3.6)X¯k(U,D)=gk(U,D,sk−1ξk),(see (2.4)). The difference
between the liquidation value (3.5) and the setup cost (3.6) is a residual amount δk:  (3.7)δk(U,D)=U−ξkU−Dgk(U,D,sk−1D)+ξk−DU−Dgk(U,D,sk−1U)−gk(U,D,sk−1ξk).Since we assume that D≤ξk≤U, it follows from the convexity of the payoff
function f that the residual is
nonnegative:(3.8)δk(U,D)≥0,k=1,…,n.In this fashion, at every time stepk, the option seller obtains a nonnegative
residual δk(U,D),
which is withdrawn after each portfolio liquidation prior to the construction
of the next time period super-hedge. The above constructed super-hedging
strategy is in general non-self-financing.The accumulated value of the withdrawn residuals at
maturity, which we will refer to as the minimum cost super-hedge residual, is
given by(3.9)Δn(U,D)=δ1(U,D)(1+r)n−1+δ2(U,D)(1+r)n−2+⋯+δn(U,D).


## 4. Non-self-financing hedging strategies and their residuals

In the previous section, we considered a hypothetical
situation where the option has been sold at timet=0 for the price equal to the upper bound X¯0(U,D) of the no-arbitrage option price interval. We
saw how the option seller could use this option premium to construct a minimum
cost super hedge based on a non-self-financing strategy.In this section, we consider a more realistic
situation where the initial (time zero) option price is lower than the upper
boundX¯0(U,D), but still falls within the open interval of
no-arbitrage option prices (x¯0(U,D),X¯0(U,D)). In this case, the option seller cannot in
general construct a super hedge, but it is possible to construct a
non-self-financing trading strategy that will produce a possibly nonnegative
residual amount.In order to explain how such trading strategy can be
constructed, we need to present the following short mathematical introduction.
Let us consider the quantityxk(u,d) given as follows:(4.1)xk(u,d)=gk(u,d,sk),k=0,…,n−1,where sk is the stock price at time k,
and gk is defined by (2.6) with boundary parameters U, D replaced with the values u, d such that(4.2)D≤d≤1+r≤u≤U.The next proposition follows
from the convexity arguments.Proposition 4.1.
Letf be a convex function. The function xk(u,d) maps the set of (u,d) pairs satisfying (4.2) onto the option price
interval [x¯k(U,D),X¯k(U,D)] defined in (2.4) and (2.5). When (u,d)=(U,D),xk(u,d)=X¯k(U,D) and when (u,d)=(1+r,1+r), then xk(u,d)=x¯k(U,D).The above proposition infers that for any choice ofd and u satisfying(4.3)D<d<1+r<u<U,the quantity (4.1) falls within
the no-arbitrage option price interval(4.4)x¯k(U,D)<xk(u,d)<X¯k(U,D),k=0,…,n−1.Conversely, every no-arbitrage
market option price y (every point y∈(x¯k(U,D),X¯k(U,D))) can be put into correspondence with at least
one pair (u,d) satisfying (4.3): y=xk(u,d).Suppose the option with the convex payoff functionf was sold at time k=0 for the price x¯0. If we assume that there is no arbitrage on the
market, then there is at least one (u,d) pair satisfying (4.3) that allows the amount x¯0 to be identified with the no-arbitrage option
price x0(u,d) satisfying (4.4) with k=0. In other words,(4.5)x¯0=x0(u,d).We remark here that the exact
values of the boundary parameters D and U are not important for practical purposes of
option hedging, since they play purely theoretical role in our setting.Let us return to (4.5). There are an infinite number of (u,d) pairs satisfying (4.5). For a fixed observed
market option price x¯0, we will call the set of (u,d) pairs satisfying (4.5) the market calibrated set of model parameters.
Our goal will be to choose the best market calibrated (u,d) pair based on an optimization criterion
explained in Section 7.The option seller uses the amountx¯0 to setup the hedging portfolio
(γ0(u,d),β0(u,d)):(4.6)x¯0=γ0(u,d)s0+β0(u,d)b0,which will be rebalanced at
every time instant t=k following the dynamic portfolio
strategy(4.7)(γk(u,d),βk(u,d)),k=0,…,n−1,where γk(u,d) and βk(u,d) are defined in (3.2) and (3.3), respectively, with
the boundary parameters U, D replaced with the values u, d:  (4.8)γk(u,d)=gk+1(u,d,sku)−gk+1(u,d,skd)sk(u−d),βk(u,d)=ugk+1(u,d,skd)−dgk+1(u,d,sku)(1+r)bk(u−d).We remark that in fact there is an infinite number of
dynamic portfolio strategies (4.7) defined by the formulas (4.8). These
strategies are distinguished by the values of the market calibrated parameters (u,d). We will call them the market calibrated dynamic portfolio
strategies, or simply market calibrated hedging strategies.By analogy with the arguments of Section3, for every k=1,…,n, one can calculate the difference between the
liquidation value of the portfolio constructed in the period [k−1,k) and the setup cost of the portfolio
constructed for the period [k,k+1). This difference constitutes a residual
value(4.9)δk(u,d)=u−ξku−dgk(u,d,sk−1d)+ξk−du−dgk(u,d,sk−1u)−gk(u,d,sk−1ξk).The latter formula
(4.9) was
obtained by analogy with the formula (3.7), which represents the residual amount
corresponding to the boundary case of the minimum cost super hedge.It is
straightforward to show that(i)
δk(u,d)>0, if d<ξk<u;(ii)
δk(u,d)=0, if ξk=d or ξk=u;(iii)
δk(u,d)<0, if D<ξk<d or u<ξk<U. In order to
maintain the dynamic portfolio strategy defined by (4.7), at each time step k=1,…,n, the investor will either withdraw the residual
(4.9) from the liquidated proceeds when δk(u,d)>0 or add the amount when δk(u,d)<0.
The local residuals δk(u,d) produce an accumulated residual at option
expiration:(4.10)Δn(u,d)=δ1(u,d)(1+r)n−1+δ2(u,d)(1+r)n−2+⋯+δn(u,d).Note that the accumulated
residual defined in (4.10) differs from the minimum cost super-hedge residual
defined in (3.9) as illustrated by the characterizations of δk given above. S. A. Nagaev  and A. V. Nagaev studied asymptotic
properties of the minimum cost super-hedge residual (3.9) (or riskless profit of
the investor, in his terminology) extensively in [2–4]. That work was extended to
the asymptotic properties of the accumulated residual (4.10) in [5]. Our work investigates the
short-term behavior of the accumulated residual and its usefulness in
constructing practical hedging strategies for this market
model.Remark 4.2.
We would
like to stress here that the dynamic portfolio strategy constructed in (4.7) is
in general non-self-financing. Also, although the strategy provides an
approximate hedging of the short position in an option, for the sake of
simplicity we will still call it a hedging strategy.Given the market option pricex¯0, theoretically one has an infinite choice of
market calibrated model parameters ((u,d) pairs) each determining a market calibrated
hedging strategy. Each market calibrated hedging strategy produces a residual
sequence. An investor (an option seller) will want to choose values for d and u that determine a residual sequence with
additional desirable risk/return characteristics. It is the choice of the model
parameter values d and u based on the risk/return characteristics of
the residual sequence that constitutes our algorithm design.For the remainder of this paper, we will assume a
European call option payoff functionf:  (4.11)f(s)=(s−K)+,where K is the option strike price. Also, for all our
computations we will use the value r=0 for the risk-free interest rate. The latter is
justified by the results of our computational experiments confirming that a
realistic variation in values of r has a minimal effect on the results for the
case of short-term option contracts studied in this paper.


## 5. Brief overview of the hedging algorithm

We will start with the brief summary of the previous
sections. As was explained in Section2, the incompleteness of our market model
results in an infinite number of possible no-arbitrage option prices located
within the open interval (x¯0(U,D),X¯0(U,D)). Assuming that there is no arbitrage on the
market, we can associate a market option price x¯0 with one of the points within the theoretical
interval of no-arbitrage option prices: x¯0=x0(u,d)∈(x¯0(U,D),X¯0(U,D)).
The infinite set of (u,d) pairs for which the equality x¯0=x0(u,d) holds is called a market calibrated set of
model parameters.Each no-arbitrage option pricex0(u,d) gives rise to a non-self-financing dynamic
hedging strategy (γk(u,d),βk(u,d)), k=0,…,n−1,
that is distinguished by the value of the parameters (u,d). If the no-arbitrage option price x0(u,d) under consideration is identified with the
actual market option price x¯0 so that x¯0=x0(u,d), we will call the associated hedging strategies (γk(u,d),βk(u,d)) market calibrated.Each market calibrated hedging strategy(γk(u,d),βk(u,d)) produces a residual sequence and an
accumulated residual amount Δn(u,d), when applied to a given stock price path. By
imposing a risk criterion on the residual sequence, we can choose a unique
optimal pair of parameters (u*,d*) by optimizing the risk criterion over the set
of market calibrated parameters. This results in the choice of the
corresponding optimal market calibrated hedging strategy (γk(u*,d*),βk(u*,d*)).This optimization is accomplished computationally by
selecting a representative finite subset from the infinite set of market
calibrated model parameters ((u,d) pairs) and simulating a large number of stock
price paths for each selected (u,d) pair. Each stock price path determines a
residual sequence, from which we can estimate a criterion value using a
suitable sample statistic.Figure1 gives a brief scheme of our proposed
two-stage algorithm.Figure 1
Two-stage algorithm.We will now proceed with a more detailed explanation
of each stage of the algorithm.


## 6. Market calibrated set of the model parameters: stage 1 of the algorithm

Let us place ourselves in the setting described in
Section4. At time t=0, the option seller receives a premium x¯0, which he/she is willing to use in order to
build a dynamic hedging strategy that depends on the unknown parameters u and d. Our goal at the first stage of the algorithm
is to choose a set of market calibrated parameters u and d. In order to do so we will identify the market
option premium x¯0 with a no-arbitrage option price x0(u,d):  (6.1)x¯0=x0(u,d)=g0(u,d,s0),(see
(4.1)). The market
calibrated parameters (u,d) must satisfy (6.1), where g0(u,d,s0) is given by(6.2)g0(u,d,s0)=(1+r)−n∑j=0n(nj)[p(u,d)]j[1−p(u,d)]n−j(s0ujdn−j−K)+p(u,d)=(1+r)−du−d.
It is more convenient to deal
with the normalized quantity g0(u,d,s0)/s0 which we will denote by c0(u,d)(6.3)c0(u,d)=g0(u,d,s0)s0=(1+r)−n∑j=0n(nj)[p(u,d)]j[1−p(u,d)]n−j(ujdn−j−R)+,where(6.4)R=Ks0.We will be interested in the
level curves Σ of the surface c0(u,d) (we will call them contours):(6.5)Σc∗={(u,d):c0(u,d)=c∗}.The set  of market calibrated
model parameters ((u,d) pairs) is represented by the contour
with(6.6)c∗=x¯0s0.We recall  that x¯0 is the market option price at time t=0, and s0 is the time zero stock price.Several contours for a representative option are shown
in Figure2. The option BAC+JV is a Bank of America call option expiring on
October 15, 2004 with a strike price of $42.50 and a current stock price of
$44.16. The contours shown each correspond to a value of c∗ (indicated on the curve scaled by a factor of
100), and the contour shown in bold is the set of market calibrated model
parameters described by (6.5) and (6.6) with a value of c∗=.0453.
The value of R=0.9624 (being less than 1) indicates that the call
option is currently in the money. The shape of the market calibrated contour is
similar for all data sets examined.Figure 2
Contours for
BAC+JV option.For the sake of comparison, Figure3 depicts the set
of contours for an Intel call option (INTC+JE) that is out of the money (R=1.199). The option expired October 15, 2004 with a
strike price of $25. The market calibrated contour (indicated in bold)
corresponds to the value c∗=0.0024. The shape of the contour differs slightly from
the market calibrated contour shown in Figure 2. The shape of this contour is
similar for other out of the money options.Figure 3
Contours for
INTC+JE option.The first stage of our algorithm is accomplished
computationally by utilizing contour construction software to compute a finite
number of market calibrated(u,d) pairs satisfying (6.5) and (6.6). There are
typically between 90 and 100 (u,d) pairs identified on the market calibrated
contour. It is this set of (u,d) pairs that is used by the second stage of our
algorithm.


## 7. Choosing the optimal hedging strategy: stage 2 of the algorithm

Let us recall that each(u,d) pair on the market calibrated contour Σ defined in (6.5) and (6.6) determines a market
calibrated dynamic hedging strategy (γk(u,d),βk(u,d)), k=0,…,n−1.
For a given stock price path {sk},
this hedging strategy produces a sequence of residuals δk(u,d), k=1,…,n, each representing a residual profit/loss for
an investor. This sequence is the economic consequence of choosing model
parameters (u,d) and the associated dynamic hedging portfolio.The second stage of the two-stage algorithm selects a
unique(u,d) pair on the market calibrated contour Σ. This chosen pair of parameters (u*,d*) uniquely defines a hedging strategy (γk(u*,d*),βk(u*,d*)), k=0,…,n−1, that will numerically optimize a risk
criterion chosen by the investor.We consider three possible criteria to choose from.
These criteria convert the residual sequence into a scalar measure of investor
risk, each reflecting some aspect of the option seller attitude towards risk.(i)
Maximize the likelihood of a positive
accumulated residual:(7.1)max(u,d)∈Σprob(Δn(u,d)>0).(ii)
Minimize expected shortfall:(7.2)min(u,d)∈ΣE(short  fall)=min(u,d)∈ΣE(max1≤k≤n(−δk(u,d))).(iii)
Maximize the expected accumulated
profit:(7.3)max(u,d)∈ΣE(Δn(u,d)).The first criterion (7.1) interprets a positive
residual as a profit and chooses a (u,d) pair that has the highest probability of a net
profit. In the absence of arbitrage, a large accumulated profit is not
attainable with high probability. There is the possibility, however, of an
investor achieving a small positive profit. The optimization problem presented
here produces a market calibrated hedging strategy that maximizes the
likelihood of a positive accumulated profit.The second
criterion (7.2) reflects an investor's desire to minimize the amount of
single period additional funding needed to rebalance the portfolio over the
life of the option.  A negative residual δk(u,d) represents the cash shortfall in rebalancing
the portfolio at time k. The largest negative δk(u,d) is the largest shortfall value.  Optimizing this criterion produces a hedging
portfolio with minimal expected single period additional funding.Our final criterion (7.3) maximizes the expected
accumulated residual, which reflects total net profit from using the dynamic
portfolio strategy based on the chosen (u,d).
It was shown in [2]
that the expected accumulated profit is asymptotically constant on contours of
constant no-arbitrage price. We thus anticipate minimal differences in the
expected accumulated profit at each (u,d) pair on our market calibrated contours when n is large. For small n, empirical results show that it is possible
to  have a market contour with
nonconstant expected accumulated profit.Let us choose one of the listed criteria. The process
of selecting a unique optimal pair of parameters consists of the following. For
each(u,d) pair in the market calibrated subset, we
simulate a number of stock price time series {sk}, k=1,…,n.
We then apply the hedging strategy associated with the given (u,d) pair to each of the simulated paths in order
to determine a sequence of residuals {δk(u,d),k=1,…,n}.
A particular choice of a risk criterion reduces the sequence {δk(u,d)} to a single scalar value of risk (described in
(7.1) through (7.3)). For example, the appropriate scalar value of risk for the
criterion (7.3) is the accumulated residual Δn(u,d). For a given (u,d) pair, we collect the corresponding sample of
scalar risk values (the sample size equals the number of the simulated stock
price paths). An appropriate sample statistic (mean value or probability of a
desirable event) is then computed from the sample as the utility value of the (u,d) pair. The best value of the sample statistic,
as the (u,d) pair is varied over the market calibrated
contour, is chosen as the optimal (u,d).
For example, the optimal (u,d) pair for criterion (7.3) will correspond to the
largest averaged accumulated residual Δn(u,d).A description of the modeling process for the stock
price time series is described in the following section.


## 8. Stock price process modeling

In our initial
study (see [1]), we
examined hedging portfolios produced by a risk minimization algorithm that
simulated a stock price process under the assumption of independence of stock
price jumps. A bootstrap procedure was used to sample stock price jumps with
replacement from historical data to generate stock price paths. This approach
assumed the independence of the sample stock price jumps. In this paper, we
consider a more sophisticated model of the stock price process. We use
time-series model to extract the correlation structure in the stock price jump
process, producing a more structured model of the stock price process that
accounts for autocorrelation in the stock price jump data.Time-series models
are developed for stock price jumpsξk.
We fit an ARMA (p,q) model of the form(8.1)ξk=α1ξk−1+α2ξk−2+⋯+αpξk−p+θ1ϵk−1+θ2ϵk−2+⋯+θqϵk−q+ϵkk=1,…,nto historical price jump series.
Here, α1,…,αp and θ1,…,θq are the parameters of the model, while ϵk,ϵk−1,…,ϵk−q are the error terms. Examining the data for
the stock price jumps, it is apparent that the jump series is stationary having
approximately constant mean and variance. It is also important to note that we
are dealing with short-term behavior, so long-term trends and seasonality are
not an issue.The appropriate model order is determined individually
for each data set based on the standard analysis of several factors (see
[19]). A preliminary
estimate of (p,q) is made by examining the autocorrelation (ACF)
and partial autocorrelation (PACF) functions. The model fit for several values
of (p,q) is evaluated using diagnostic statistics. The
evaluation criteria include the statistical significance level of individual
ARMA terms, Akaike information criterion (AIC) value associated with the model,
and validation of the noise assumptions. The lack of correlation in the
residual noise sequence is established using the Box-Ljung statistics. Although
many of the data sets considered indicate a p=1,q=1 model, each data set exhibits unique patterns,
and higher order models are appropriate in some cases. The model parameters are
evaluated based on the model order. The time-series model for the stock price
jump process and an appropriate residual noise process can be used to simulate
future values of the stock price jumps (and then stock prices).Common parametric methods for generating residual
noise values include random number generation using an assumed underlying
distribution. Examining the residuals in our time-series jump data, we see that
the distribution has a long tail indicative of a nonnormal distribution (see
Figure4). To guarantee the appropriate inclusion of such tail valued price
jumps in our simulation, the method chosen for simulating ARMA model driving
noise for future values of the price jump process is the nonparametric
bootstrap method of sampling with replacement from the residual sequence {ϵk}.Figure 4
ARMA residual
density function.Using the ARMA residual sequence as noise, we simulate
future values of the stock jump process. The jump process is accumulated to
form a stock price process, which can be used to calculate a residual sequenceδk or an accumulated residual Δn using our risk minimization algorithm. This
process is summarized in the flow chart shown in
Figure 5.Figure 5
Simulation of
stock price paths.We conclude this section with a brief overview of the
existing literature on autoregressive processes with given marginal
distributions. In our model, we assume that the stock price jump distribution
at every time instantk has a bounded support. A natural question
arises: is it possible to construct an ARMA (or AR) type process with marginal
distributions of bounded support? This issue is addressed in a number of recent
works (see, e.g., [20–23], and references therein). Namely, in [20] the author uses a special
random coefficient autoregression to model a first-order AR process with beta
marginal distributions. In [21], the authors extend on the earlier works [22, 23] and present a first-order
autoregressive time-series model with the uniform (0,1) marginal distributions. Autoregressive
time-series processes with marginal distributions of other types (not
necessarily with bounded supports) have been studies extensively in the
literature. Namely, [24] discusses non-Gaussian ARMA processes with marginal distributions
of the Laplace and l-Laplace types. The first-order autoregressive models
producing time series with logistic, hyperbolic secant, exponential, Laplace,
and Gamma marginal distributions are considered in [25]. In [26], by means of special choice
of noise, autoregressive processes with Student type marginal distributions are
constructed. An extensive survey of non-Gaussian conditional linear
autoregressive models can be found in [27].


## 9. Numerical results

## 9.1. Market environment and option characteristics

In order to test the algorithm under varying market
conditions and with a range of option characteristics, data were collected for
55 call options traded on the New York Stock Exchange. The volatility index, or
VIX, was used as an indicator of the market environment. The volatility index
for a period from September 2002 through April 2007 is shown in
Figure6. Two
time periods were selected with differing VIX characteristics. The selected
time periods are indicated by the darker sections of the graph between the
dashed vertical lines. The first period encompasses dates ranging from October
2002 through October 2004. As shown in the figure, the volatility index was high
and widely varying at the beginning of the time period, with lower values
towards the end of the period. The second chosen time period was July 2006
through March 2007, with smaller and less variable volatility index values.Figure 6
Volatility
index.Option data was collected for options in the money, at
the money and out of the money, from varying industrial sectors and time to
expiration. Option characteristics are detailed in Table4. The data collected
for each option was strike price and expiration date, historical stock price
data, and historical option prices. Over 500 daily stock prices were recorded.
To capture the market behavior corresponding to the volatility index
fluctuations occurring at the beginning of 2003, all 500 data values were used
in constructing time-series models for the stock price jump process for the
options expiring on October 15, 2004. We used 150 historical stock prices in
constructing time-series models for the stock price jump process for the
options expiring on March 17, 2007 since the volatility index was lower and
more stable in this regime. Tests using larger amounts of historical data
produced similar results, indicating that a large amount of historical data was
not required.For all options in this study, 100 stock price paths
were simulated. The market option price data for a period of 40 days prior to
expiration were collected, and the market option pricen days prior to expiration was used in
identifying the appropriate contour for each of the reported values of n (as described in Section 6).


## 9.2. Discussion of algorithm results

The risk criteria values given in (7.1) through (7.3)
were evaluated for each (u,d) pair on the market calibrated contour for each
option listed in Table 4. Table 1 presents the optimal risk criteria values for
all options with n=30 days to expiration. In addition, the minimum
value for each risk criterion is presented in order to gauge the magnitude of
the investor's gain associated with choosing the optimal hedging strategy
produced by the algorithm. Note that the number at the end of each option
ticker indicates the number of days to expiration.Table 1
Algorithmic results for options with 30 days
to expiration.
Resid-minResid-maxShortfall-minShortfall-maxPosProb-minPosProb-maxBAC+CJ300.17010.3446−0.2167−0.02260.711.00BAC+CW300.24460.3592−0.0952−0.00350.981.00BAC+CX300.07760.3253−0.3301−0.08650.520.87BAC+JV300.13680.4406−0.4111−0.04250.740.98CYQ+CE300.03010.5328−0.2833−0.12260.600.97CYQ+CX300.12800.2995−0.1182−0.03010.831.00CYQ+CY30−0.07501.0162−0.5349−0.24800.550.86GE+CF300.15420.1778−0.0308−0.00041.001.00GE+CG300.09590.1862−0.2934−0.04410.600.94GE+CY300.14820.1776−0.00730.00001.001.00GE+CZ300.07990.1805−0.0964−0.00490.921.00GE+JF300.02590.0600−0.1824−0.01590.350.95HNZ+CI300.31180.6032−0.2572−0.01660.951.00HNZ+CJ300.12430.6220−0.3183−0.04420.771.00INTC+JE30−0.35300.0502−0.3145−0.03440.060.83MER+CP300.09450.3211−0.2597−0.01010.851.00MQF+JX300.16650.1921−0.0504−0.00200.991.00MS+CO300.40380.7941−0.4197−0.03190.891.00MSQ+CK300.02660.5451−0.2825−0.04190.451.00MSQ+CY300.00620.1225−0.0918−0.00720.551.00MSQ+JE300.09150.1234−0.1689−0.02010.690.98NQ+CD30−0.12160.1762−0.3961−0.08850.180.93WMT+CH300.29450.3740−0.1248−0.00191.001.00WMT+CJ30−0.08720.0787−0.3912−0.07770.290.84WMT+JJ30−0.18920.2226−0.4898−0.05140.540.90XOM+CN300.00580.5323−0.5517−0.05420.310.98XOM+CO30−0.07481.2703−0.9711−0.12130.350.89XOM+JI30−0.13730.1527−0.2978−0.05450.150.93XOM+JV300.05280.2231−0.1508−0.01170.591.00Considering the first option in Table1 (BAC+CJ30), we
see that the optimal value of E(Δn) or the expected accumulated residual is 0.3446
as reported in column 3 (Resid-max), which is over twice as large as the
smallest possible value of 0.1701 reported in column 2 (Resid-min). Similarly,
we can examine the improvement in shortfall associated with the algorithmic
results. The minimum expected shortfall is −0.0226 (Shortfall-min),
approximately 10% of the worst case scenario with a shortfall of −0.2167
(Shortfall-max). The probability of a positive accumulated residual increases
from a possible low of 0.71 (PosProb-min) presented in column 6 to 1 (PosProb-max)
using the optimal portfolio identified by the algorithm.In most cases, the expected accumulated residual value
is improved by a factor between 1.5 and 3 when the optimal hedging portfolio is
chosen. Several cases produced much more dramatic improvements in the objective
value. The most significant improvement in expected accumulated residual is
found in the results for the XOM+CN30 option. The value associated with the
optimal hedging portfolio is 0.5323, almost 100 times as large as the smallest
possible value of .0058. Other notable cases include the Microsoft options
MSQ+CK30 and MSQ+CY30 where the optimal accumulated residual is approximately
20 times as large as the smallest possible value and the Cisco option CYQ+CY30
where the optimal is more than 13 times as large as the smallest possible
value.On average, the shortfall is reduced to approximately
13% of the largest possible shortfall by following the hedging strategy
associated with the optimal hedging portfolio. While the shortfall is reduced
to just over 1% of the worst case for the GE option GE+CF30, the optimal is
approximately 46% of the worst case for the Cisco option CYQ+CY30. It is
important to note that the hedging portfolio associated with minimizing
shortfall is not the same portfolio choice for maximizing expected accumulated
residual.The Intel option INTC+JE30 provides an interesting
example for the benefit of choosing the hedging strategy produced by the
algorithm to maximize the probability of a positive accumulated residual. The
objective value associated with the optimal strategy is 0.85 as seen in column
7 (ProbPos-max) as compared to values as low as 0.06 (PosProb-min) associated
with other strategies.To investigate the implications of a shorter time
horizon, results are summarized for two options (BAC+JV and GE+CF) in Table2
for n=30,20,15,10 days to expiration. The number of days to
expiration is indicated by the value at the end of the option ticker in the
first column. Examining the first row of the table, we see that for the option
BAC+JV with ten days to expiration, the expected accumulated residual has
optimal value associated with algorithmic hedging portfolio of 0.3202
(Resid-max), minimum shortfall of −0.0085 (Shortfall-min), and the probability
of a positive accumulated residual of 1.0 (PosProb-max). Following the hedging
strategy associated with the optimal choice of (u,d) chosen by the algorithm still provides some
advantage as compared to other possible hedging portfolios, although not as
much as with a longer time to expiration, as would be expected. The largest
improvement in expected accumulated residual for less than 30 days to
expiration occurs with the BAC+JV option with 15 days to expiration. Choosing
the optimal hedging strategy provides a value 1.4 times as large as the
smallest possible expected accumulated residual. For all cases presented, the
minimum shortfall produced by the algorithm is very close to zero and the
probability of a positive accumulated residual is very close to 1.Table 2
Algorithmic results for options with varying
time to expiration.
Resid-maxShortfall-minPosProb-maxBAC+JV100.3202−0.00851.00BAC+JV150.3044−0.02600.99BAC+JV200.2881−0.01881.00BAC+JV250.1876−0.03340.99BAC+JV300.4406−0.04250.98GE+CF100.13020.00001.00GE+CF150.16120.00001.00GE+CF200.0615−0.00011.00GE+CF250.1179−0.00061.00GE+CF300.1778−0.00041.00


## 10. Evaluation of numerical results

The results given in Section9 document the advantages
of using the optimal hedging strategy in comparison to choosing from the range
of other market calibrated hedging strategies. To further evaluate the results
of our risk minimization algorithm, we take the position of an investor
possessing a hedging portfolio based on the optimal strategy identified by the
algorithm and compute the accumulated residual value using a representative
actual stock price path. To obtain representative stock price values, we divide
the collected stock price data into two sets: a large set of historical values
used to compute the optimal strategy and a set of n+1 values, where n is the number of days to expiration of the
option, to be interpreted as the actual values of the stock (we use n=30 days to expiration for this study).Figure7 illustrates this data splitting for a set of
Bank of America stock prices from October 15, 2002 through October 15, 2004. To
evaluate the optimal hedging portfolio produced by the risk minimization
algorithm for an option n=30 days to expiration, the stock prices for
October 15, 2002 through September 1, 2004 are used in the algorithm to compute
the optimal hedging strategy, and stock prices from September 2, 2004 through
October 15, 2004 (highlighted in gray) are used as the actual stock price
values. The stock price and option price on the first day of the actual stock
price path are used as the current stock and option price. The data used as the
actual stock path is shown in more detail in the lower graph in
Figure 7.Stock price
data.
(a)(b)Having computed the accumulated residual for the
single actual path, we compare these to the hedging cost of maintaining the
classic Black-Scholes hedge when rebalancing is done daily. Thus, in parallel
with the computation of our residuals, we compute the accumulated rebalancing
cost of the Black-Scholes hedge, as described in [28]. To be consistent with the
method described in [28], we account for the future value (value at
expiration) of the option premium received when we take the position of the
option seller. It consists of adding the future value of the option premium to
the accumulated residuals at expiration. We call the resulting value the accumulated
hedging residual. Positive accumulated hedging residuals indicate a gain for
the portfolio holder, while a negative accumulated hedging residual indicates a
loss for the portfolio holder.Accumulated residual for both the optimal hedging
strategy produced by the algorithm and the Black-Scholes hedge is presented in
Table3. Of the twenty-seven options listed in the table, there are sixteen for
which our accumulated hedge residual exceeds that of the Black-Scholes hedge.
The results illustrate that following the hedging portfolio strategy produced
by the algorithm is beneficial to an investor, producing residual values that
exceed the classic Black-Scholes value in 59% of the test cases. The number of
favorable cases provides support for the use of our algorithm.Table 3
Model
evaluation results.
OptionAHRBSAHRBAC+CJ−0.199−0.137BAC+CW0.3540.278BAC+CX−1.977−0.444BAC+JV0.4760.177CYQ+CE0.5570.282CYQ+CX0.300.281CYQ+CY0.0160.186GE+CG0.4380.102GE+CY0.2020.170GE+CZ0.1800.168GE+JF0.0750.066HNZ+CI0.5900.053HNZ+CJ−0.2830.144INQ+JE−0.024−0.073INQ+CD−0.5130.059MER+CP0.0440.254MS+CO−0.433−0.097MQF+JX0.1850.180MSQ+CK−1.793−0.101MSQ+JE0.1320.123WMT+CH0.2180.318WMT+CJ−0.016−0.166WMT+JJ0.3150.286XOM+CN0.4340.506XOM+CO−2.90−0.034XOM+JI0.2540.004XOM+JV0.2340.199AHR: Algorithmic
Accumulated Hedge Residual.BSAHR: Black-Scholes Accumulated Hedge Residual.Table 4
Options used in
data analysis.
Company nameOptionDays to expiration (n)Strike priceExpiration dataBank
of America Corp.BAC+CJ3050March 17, 2007Bank
of America Corp.BAC+CW3047.50March 17, 2007Bank
of America Corp.BAC+CX3052.50March 17, 2007Bank
of America Corp.BAC+JV10,
15, 20, 25, 3042.50October 15, 2004Cisco
Systems, Inc.CYQ+CE3025March 17, 2007Cisco
Systems, Inc.CYQ+CX3022.50March 17, 2007Cisco
Systems, Inc.CYQ+CY3027.50March 17, 2007Exxon
Mobil Corp.XOM+CN3070March 17, 2007Exxon
Mobil Corp.XOM+CO3075March 17, 2007Exxon
Mobil Corp.XOM+JI30, 2045October 15, 2004Exxon
Mobil Corp.XOM+JV30,
20, 1042.5October 15, 2004General
Electric Co.GE+CF10,
15, 20, 25, 3030March 17, 2007General
Electric Co.GE+CG10,
20,  3035March 17, 2007General
Electric Co.GE+CY3027.5March 17, 2007General
Electric Co.GE+CZ3032.50March 17, 2007General
Electric Co.GE+JF3030October 15, 2004HJ
Heinz Co.HNZ+CI3045March 17, 2007HJ
Heinz Co.HNZ+CJ3050March 17, 2007Intel
Corp.INQ+CD30, 20, 1020March 17, 2007IntelCorp.INQ+JE30, 20, 1025October 15, 2004Merrill
Lynch& Co., Inc.MER+CP3080March 17, 2007Morgan
StanleyMS+CO3075March 17, 2007Microsoft
Corp.MSQ+CK3030March 17, 2007Microsoft
Corp.MSQ+CY20,
3027.50March 17, 2007Microsoft
Corp.MSQ+JE3025October 15, 2004Microsoft
Corp.MQF+JX3022.50October 15, 2004Walmart
Stores, Inc.WMT+CH10, 15, 20, 25, 3040March 17, 2007Walmart
Stores, Inc.WMT+CJ10, 20, 3050March 17, 2007Walmart
Stores, Inc.WMT+JJ30, 20, 750October 15, 2004


## 11. Conclusions

We have
developed an algorithm based on an ARMA time-series model for a stock price
jump process that produces a non-self-financing hedging strategy in an
incomplete market corresponding to one of several investor risk criteria. The
two-stage algorithm optimizes an investor chosen statistical property of the
portfolio residual profit or shortfall. The algorithm was tested on a number of
options traded on the New York Stock Exchange.



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*Source: 275217-2008-09-02.xml*