Cornelis Springer (1817-1891) was a Dutch landscape painter.
The topics are
We shall examine various types of market imperfection. In particular, incompleteness of the market, restrictions on short-selling of stocks and borrowing of cash, and the case of different borrowing and lending rates. For simplicity, we assume that both parties to a contract are faced with the same kind of market imperfection. We shall also focus on a one-period case. In most cases, the standard no-arbitrage arguments cannot be applied under market imperfections. Therefore, the arbitrage price is not well-defined. Recall that the arbitrage price (also known as the fair price) is the value of a security which appears to be equally satisfactory for all parties no matter what their assessment of future market behavior and/or their respective long or short positions. As we shall see in what follows, under the market incompleteness or in the presence of frictions, the determination of a fair price by means of no-arbitrage arguments is no longer possible in general, even if the market model is arbitrage-free.
\begin{equation}{\label{a}}\tag{A}\mbox{}\end{equation}
Perfect Hedging.
Generally speaking, existing approaches to imperfect security markets can be classified as follows:
- methods that are invariant with respect to an equivalent change of the underlying probability measure;
- methods that are sensitive with respect to an equivalent change of the underlying probability measure.
Methods from the first class are based on certain almost-sure properties of the wealth process of a trading strategy. Therefore, they are invariant with respect to the choice of an element in a given class of mutually equivalent probability measures. In this sense, the resulting valuation procedure is independent of the subjective assessments of market behavior. This does not mean that a fair price for any contingent claim can be derived by means of such a procedure. Usually, one obtains only the no-arbitrage bounds for the price. Methods belonging to the second class concentrate on the minimization of the expected risk under a given probability measure. For this reason, the results obtained by any of these methods will exhibit the rather undesirable (at least from the viewpoint of valuation) feature of being explicitly dependent on a subjective assessment of the market.
Incomplete Markets.
We shall first consider the case of a frictionless market which is incomplete, meaning that not all contingent claims are attainable. However, all remaining assumptions relative to the notion of a perfect market remain in force. In an incomplete market model, the risk exposure cannot always be eliminated completely by means of a judicious trading strategy. We shall focus on a one-period case with only two dates \(0\) and \(T\). We assume that the interest rate \(r\) is constant, the initial stock price is a strictly positive constant \(S_{0}>0\), and its terminal price \(S_{T}\) admits \(p\) strictly positive values, which satisfy
\[s_{1}>\cdots >s_{l}>S_{0}\cdot (1+r)>s_{l+1}>\cdots >s_{p}>0\]
for some \(p\geq 3\). The stock price \(S_{T}\) can be seen as a random variable on a finite probability space \((\Omega ,{\cal F},P)\), where \(\Omega =\{\omega_{1},\cdots ,\omega_{p}\}\). The model of the stock price adopted may seem somewhat artificial at the first glance. We could have started by first constructing a complete finite market model with a bond and \(p-1\) traded stocks, and then imposing the constraint that an investor has no access to \(p-2\) stocks. In other words, trading in all but one stock is prohibited. We restrict ourselves here to such a special situation, when a contingent claim we wish to value depends only on the terminal price of the stock available for investment.
Since the notion of a perfect hedging (also known as super-hedging) is invariant with respect to an equivalent change of probability measure, we do not need to specify, at this stage, the probability law \(\mathbb{P}\) that governs the terminal stock price \(S_{T}\). We do assume \(\mathbb{P}\{S_{T}=s_{i}\}>0\) for every \(i=1,\cdots ,p\), where \(\mathbb{P}\) represents the subjective probability. Let us consider an arbitrary contingent claim \(X\) which settles at time \(T\) and has the form \(h(S_{T})\) for a certain function \(h:\mathbb{R}\rightarrow\mathbb{R}\). Let the vector \(\boldsymbol{\phi}=(\alpha_{0},\beta_{0})\in \mathbb{R}^{2}\) represent the portfolio held at time \(0\). The terminal value of \(\boldsymbol{\phi}\) equals
\[V_{T}(\boldsymbol{\phi})=\alpha_{0}\cdot S_{T}+\beta_{0}(1+r),\]
or equivalently,
\begin{equation}{\label{museq2026}}\tag{1}
V_{T}(\boldsymbol{\phi})=\alpha_{0}(S_{T}-S_{0}\cdot (1+r))+c\cdot (1+r),
\end{equation}
where \(c=\alpha_{0}\cdot S_{0}+\beta_{0}\) stands for the initial wealth of the portfolio. Suppose first that the aim is to replicate the claim \(X\), i.e., \(h(S_{T})=X=V_{T}(\boldsymbol{\phi})\). The replicating strategy is determined by the system of \(p\) linear equations
(ref. (\ref{museq2026}))
\[\alpha_{0}\cdot (s_{i}-S_{0}\cdot (1+r))+c\cdot (1+r)=h(s_{i})\mbox{ for }i=1,\cdots ,p.\]
Since \(p\geq 3\), it is obvious that there exist contingent claims that are not attainable, which also means that we do not admit replicating portfolios. More specifically, a contingent claim \((h(s_{1}),\cdots ,h(s_{p}))\in \mathbb{R}^{p}\) is attainable if and only if it belongs to the linear subspace of \(\mathbb{R}^{p}\) generated by two linearly independent vectors \((s_{1},\cdots ,s_{p)}\) and \((1,\cdots ,1)\). The concept of perfect hedging is based on the observation that the risk exposure can always be completely eliminated by means of a dynamic portfolio, leaving perhaps a surplus of wealth in some states, after the terminal liability represented by a claim \(X\) is met.
Definition. A perfect hedging strategy against a contingent claim \(X\) which settles at time \(T\) is an arbitrary self-financing trading strategy \(\boldsymbol{\phi}\) for which \(V_{T}(\boldsymbol{\phi})\geq X\). The seller’s price \(\Pi_{0}^{s}(X)\) at time \(0\) of a contingent claim \(X\) is the minimal initial investment for which there exists a perfect hedging strategy against \(X\); that is,
\[\Pi_{0}^{s}(X)=\inf\left\{V_{0}(\boldsymbol{\phi}):V_{T}(\boldsymbol{\phi})\geq X\right\},\]
or more explicitly
\begin{equation}{\label{museq41}}\tag{2}
\Pi_{0}^{s}(X)=\inf\left\{c\in \mathbb{R}:\alpha_{0}\cdot (S_{T}-
S_{0}\cdot (1+r))+c\cdot (1+r)\geq X\mbox{ for }\alpha_{0}\in \mathbb{R}\right\}.
\end{equation}
Any trading strategy that realizes the infimum in (\ref{museq41}) will be referred to as a minimal perfect hedging strategy against the
short position in \(X\). \(\sharp\)
Observe that the definition of the minimal hedging assumes implicitly that the infimum (\ref{museq41}) can actually be attained by means of some trading strategy. In the case of a finite market, this is easily seen to hold. However, in the case of continuous-time markets, such a nice property is far from being trivial. Intuitively, the value \(\Pi_{0}^{s}(X)\) represents the minimal amount of cash that the seller of \(X\) needs to invest at time \(0\) in order to make sure that his/her portfolio will enable him/her to meet the liabilities at the terminal date \(T\). For a given contingent claim \(X\), we can alternatively consider a party who assumes a long position in the contingent claim \(X\), i.e., a short position in \(-X\). We shall call such a party a buyer of \(X\). The terminological distinction between the seller and the buyer of \(X\) is rather formal. It does not imply a priori that the party termed the “buyer” is actually ready to pay the seller a positive price in exchange for a claim \(X\). On the other hand, a buyer of \(X\) may be seen as a seller of \(-X\).
Definition. The buyer’s price \(\Pi_{0}^{b}(X)\) of a contingent claim \(X\) at time \(0\) is given by the following equality
\[\Pi_{0}^{b}(X)=-\Pi_{0}^{s}(-X);\]
that is, it equals the opposite of the seller’s price of the claim \(-X\). More explicitly,
\begin{equation}{\label{museq42}}\tag{3}
\Pi_{0}^{b}(X)=-\inf\left\{V_{0}(\boldsymbol{\phi}):V_{T}(\boldsymbol{\phi})\geq -X\right\}. \sharp
\end{equation}
The minus sign is put on the right-hand side of (\ref{museq42}) in order to make the cash flow corresponding to the seller’s and buyer’s prices directly comparable. The value \(\Pi_{0}^{b}(X)\) is the maximal amount that the buyer of \(X\) is ready to pay for \(X\) while still being sure that, by a judicious choice of a portfolio, he/she will terminate with a nonnegative wealth at time \(T\) in all states, after receiving \(X\). As usual, \({\cal P}(S^{*})\) stands for the collection of all (equivalent) martingale measures for the process \(S^{*}\). We emphasize that we employ here martingale measures which are merely absolutely continuous with respect to the underlying probability \(\mathbb{P}\), i.e., generalized martingale measures. As before, we write \(\bar{\cal P}(S^{*})\) to denote the class of all generalized martingale measures. Note that any probability measure \(\mathbb{P}^{*}\) on \(\Omega\) for which \(\mathbb{E}_{\mathbb{P}^{*}}[S_{T}]=S_{0}\cdot (1+r)\) belongs to the class \(\bar{{\cal P}}(S^{*})\). We may and do identify an underlying probability space \(\Omega\) with the set of terminal values of stock price, i.e., \(\Omega =\{s_{1},\cdots ,s_{p}\}\). In particular, for any \(m\leq l\) and \(n>l\), we denote by \(\mathbb{P}_{m,n}^{*}\) the unique generalized martingale measure for \(S^{*}\) which charges only two values of the terminal stock prices \(s_{m}\) and \(s_{n}\). More explicitly,
\[\mathbb{P}_{m,n}^{*}\{s_{m}\}=1-\mathbb{P}_{m,n}^{*}\{s_{n}\}=\frac{S_{0}\cdot (1+r)-s_{n}}{s_{m}-s_{n}}\]
for an arbitrary choice of \(m\leq l\) and \(n>l\). As usual, we denote \(X^{*}=X/(1+r)\).
\begin{equation}{\label{musp411}}\tag{4}\mbox{}\end{equation}
Proposition \ref{musp411}. The following assertions are valid for the seller’s price \(\Pi_{0}^{s}(X)\) and the buyer’s price \(\Pi_{0}^{b}(X)\).
(i) There exists a pair \(m_{0}\leq l\) and \(n_{0}>l\) such that \(\Pi_{0}^{s}(X)\) satisfies
\begin{equation}{\label{museq43}}\tag{5}
\Pi_{0}^{s}(X)=\mathbb{E}_{\mathbb{P}^{*}_{m_{0},n_{0}}}[X^{*}].
\end{equation}
The following formulas are valid
\begin{equation}{\label{museq44}}\tag{6}
\Pi_{0}^{s}(X)=\max_{m\leq l,n>l} \mathbb{E}_{\mathbb{P}^{*}_{m,n}}[X^{*}]=\sup_{\mathbb{P}^{*}\in\bar{\cal P}(S^{*})}\mathbb{E}_{\mathbb{P}^{*}}[X^{*}].
\end{equation}
(ii) The buyer’s price \(\Pi_{0}^{b}(X)\) equals
\begin{equation}{\label{museq45}}\tag{7}
\Pi_{0}^{b}(X)=\mathbb{E}_{\mathbb{P}^{*}_{m_{1},n_{1}}}[X^{*}].
\end{equation}
for some pair \(m_{1}\leq l\) and \(n_{1}>l\). Moreover, we have
\begin{equation}{\label{museq46}}\tag{8}
\Pi_{0}^{b}(X)=\min_{m\leq l,n>l} \mathbb{E}_{\mathbb{P}^{*}_{m,n}}[X^{*}]=
\inf_{\mathbb{P}^{*}\in\bar{\cal P}(S^{*})}\mathbb{E}_{\mathbb{P}^{*}}[X^{*}].
\end{equation}
Proof. For the first formula, it is sufficient to observe that the seller’s price \(\Pi_{0}^{s}(X)\) solves the linear programming problem, which means to minimize \(c\in \mathbb{R}\) subject to the following constraints
\[\left\{\begin{array}{l}
\alpha_{0}\cdot (s_{1}-S_{0}\cdot (1+r))+c\cdot (1+r)\geq h_{1}\\
\vdots\\
\alpha_{0}\cdot (s_{p}-S_{0}\cdot (1+r))+c\cdot (1+r)\geq h_{p}\\
\alpha_{0}\in \mathbb{R}.
\end{array}\right .\]
This problem is easily seen to admit a unique solution \(c^{*}\), which is determined by the following systems of linear equations
\[\left\{\begin{array}{l}
\alpha_{0}\cdot (s_{m_{0}}-S_{0}\cdot (1+r))+c^{*}\cdot (1+r)=h_{m_{0}}\\
\alpha_{0}\cdot (s_{n_{0}}-S_{0}\cdot (1+r))+c^{*}\cdot (1+r)=h_{n_{0}}
\end{array}\right .\]
for some pair \((m_{0},n_{0})\) with \(m_{0}\leq l\) and \(n_{0}>l\). This proves (\ref{museq43}). Furthermore, a direct inspection shows that, for every \(m\leq l\) and \(n>l\) satisfying \((m,n)\neq (m_{0},n_{0})\), we have
\[\mathbb{E}_{\mathbb{P}^{*}_{m,n}}[X^{*}]\leq \mathbb{E}_{\mathbb{P}^{*}_{m_{0},n_{0}}}[X^{*}]=c^{*}.\]
This immediately yields the first equality in (\ref{museq44}). To justify the second, it is enough to observe that the class \(\bar{\cal P}(S^{*})\) of all generalized martingale measures for \(S^{*}\) can be identified with a bounded convex subset of \(\mathbb{R}^{p}\) with the probability measures of the form \(\mathbb{P}^{*}_{m,n}\), for \(m\leq l\) and \(n>l\), being its extremal points.
The proof of part (ii) goes along the same lines. Indeed, it is enough to note that the value \(-\Pi_{0}^{b}(X)=\Pi_{0}^{s}(-X)\) solves the following linear programming problem, which means to minimize \(c\in \mathbb{R}\) subject to the following constraints
\[\left\{\begin{array}{l}
\alpha_{0}\cdot (s_{1}-S_{0}\cdot (1+r))+c\cdot (1+r)\geq -h_{1}\\
\vdots\\
\alpha_{0}\cdot (s_{p}-S_{0}\cdot (1+r))+c\cdot (1+r)\geq -h_{p}\\
\alpha_{0}\in \mathbb{R}.
\end{array}\right .\]
Equalities (\ref{museq45}) and (\ref{museq46}) can be established using arguments similar to those in the first part of the proof. This completes the proof. \(\blacksquare\)
Suppose that \(X\) is a nonnegative contingent claim. Then, the following chain of inequalities is valid
\begin{equation}{\label{museq47}}\tag{9}
0\leq\Pi_{0}^{b}(X)\leq\Pi_{0}^{s}(X).
\end{equation}
Note that the equality \(\Pi_{0}^{b}(X)=\Pi_{0}^{s}(X)\) holds if and only if \(X\) is attainable. Therefore, for any non-attainable claim \(X\), the open interval \(J=(\Pi_{0}^{b}(X),\Pi_{0}^{s}(X))\) is nonempty. It is easily seen that if we take an arbitrary value \(x\in J\) as the price of \(X\), the market would remain arbitrage-free. To be more explicit, neither selling at the price \(x\) nor buying at this price would allow an investor to construct a portfolio that would be an arbitrage opportunity. In this sense, the arbitrage-free interval \(J\) may be seen as a set of potential rational prices in the absence of arbitrage. On the other hand, for any choice of \(x\in J\), both parties involved in a deal at this price would be unable to eliminate their risk exposures; the (random) size of potential losses would be different for each party and would depend on the particular value of \(x\). In a slightly more general situation, when a claim \(X\) takes both positive and negative values, it may happen that \(\Pi_{0}^{b}(X)<0<\Pi_{0}^{s}(X)\), which means that both parties involved in a deal require a positive initial investment if each is willing to establish a perfect hedging portfolio for his/her position (Recall that the minimal initial investment required by the buyer of \(X\) to completely eliminate risk is \(-\Pi_{0}^{b}(X)\), and not \(\Pi_{0}^{b}(X)\)). This somewhat pathological feature of the seller’s and buyer’s prices is a major deficiency of the concept of perfect hedging as a general pricing method. In particular case, for instance when dealing with standard options, the no-arbitrage bounds provided by (\ref{museq47}) appear to be useful.
Proposition. The seller’s price \(\Pi_{0}^{s}(X)\) agrees with the buyer’s price \(\Pi_{0}^{b}(X)\) if and only if the contingent claim \(X\) is attainable. The seller’s and buyer’s prices are monotone applications of the terminal payoff \(X\) in the following sense: for arbitrary contingent claims \(X_{1},X_{2}\), the following implication is valid
\[X_{2}\geq X_{1}\Rightarrow\Pi_{0}^{s}(X_{2})\geq\Pi_{0}^{s}(X_{1}),\Pi_{0}^{b}(X_{2})\geq\Pi_{0}^{b}(X_{1}).\]
The prices defined by means of perfect hedging are not additive functions of the terminal payoff in general. Neither are they strictly
increasing functions; that is, the implication
\[X_{2}\geq X_{1},X_{2}\neq X_{1}\Rightarrow\Pi_{0}^{s}(X_{2})>\Pi_{0}^{s}(X_{1}),\Pi_{0}^{b}(X_{2})>\Pi_{0}^{b}(X_{1}).\]
is not necessarily satisfied.
Example. Let us examine the case of call and put options with the expiration date \(T\) and strike price \(K\) with \(s_{p}<K<s_{1}\). Since the payoff functions
\[h_{1}(s)=(s-K)^{+}\mbox{ and }h_{2}(s)=(K-s)^{+}\]
are convex, it is clear that the seller’s prices of call and put options equal
\[\Pi_{0}^{s}(h_{j}(S_{T}))=E_{P^{*}_{1,p}}\left [\frac{h_{j}(S_{T})}{1+r}\right ]\]
for \(j=1,2\). This may be rewritten in a more explicit way as follows
\begin{equation}{\label{museq48}}\tag{10}
\Pi_{0}^{s}(h_{j}(S_{T}))=\frac{1}{1+r}\cdot\left (h_{j}(s_{1})\cdot
\frac{S_{0}\cdot (1+r)-s_{p}}{s_{1}-s_{p}}+h_{j}(s_{p})\cdot\frac{s_{1}-S_{0}\cdot (1+r)}{s_{1}-s_{p}}\right )
\end{equation}
for \(j=1,2\). It appears that the seller’s prices of both call and put options are strictly positive numbers: \(\Pi_{0}^{s}(C_{T})>0\) and \(\Pi_{0}^{s}(P_{T})>0\). On the other hand, once again by the convexity of \(h_{1}\) and \(h_{2}\), we obtain
\[\Pi_{0}^{b}(h_{j}(S_{T}))=E_{P^{*}_{l,l+1}}\left [\frac{h_{j}(S_{T})}{1+r}\right ]\]
for \(j=1,2\), or more explicitly
\[\Pi_{0}^{b}(h_{j}(S_{T}))=\frac{1}{1+r}\cdot\left (h_{j}(s_{l})\cdot
\frac{S_{0}\cdot (1+r)-s_{l+1}}{s_{l}-s_{l+1}}+h_{j}(s_{l+1})\cdot\frac{s_{l}-S_{0}\cdot (1+r)}{s_{l}-s_{l+1}}\right ).\]
This means that \(\Pi_{0}^{b}(C_{T})=0\) whenever the strike price \(K\geq s_{l+1}\). On the other hand, \(\Pi_{0}^{b}(P_{T})=0\) if and only if \(K\leq s_{l}\). \(\sharp\)
Let us now comment briefly on a multi-period case. The monotonicity of the seller’s and buyer’s prices allow for a direct extension of Proposition \ref{musp411} to the case of a multi-period finite market (as well as to the case of American-style contingent claims). Working backward in time from the settlement date \(T\) to the initial date \(0\), it is possible to check that there exists a unique self-financing trading strategy \(\boldsymbol{\phi}_{\min}(X)\) satisfying \(\Pi_{0}^{s}(X)=V_{0}(\boldsymbol{\phi}_{\min}(X))\), where by definition
\[\Pi_{0}^{s}(X)=\inf\left\{V_{0}(\boldsymbol{\phi}):V_{T}(\boldsymbol{\phi})\geq X\right\}.\]
More generally, for any date \(t\leq T\), the value of the seller’s price \(\Pi_{t}^{s}(X)\) (resp. the buyer’s price \(\Pi_{t}^{b}(X)\)) can be defined in terms of the current wealth of the minimal hedging strategy \(\boldsymbol{\phi}_{\min}\) against \(X\) (resp. against \(-X\)), namely
\[\Pi_{t}^{s}(X)=V_{t}(\boldsymbol{\phi}_{\min}(X))\mbox{ and }\Pi_{t}^{b}(X)=-V_{t}(\boldsymbol{\phi}_{\min}(-X)).\]
More explicitly, the price \(\Pi_{t}^{s}(X)\) and \(\Pi_{t}^{b}(X)\) equal
\[\Pi_{t}^{s}(X)=\inf\left\{V_{t}(\boldsymbol{\phi}):V_{T}(\boldsymbol{\phi})\geq X\mbox{ for }\boldsymbol{\phi}\in
\boldsymbol{\Phi}_{t,T}\right\}\]
and
\[\Pi_{t}^{b}(X)=-\inf\left\{V_{t}(\boldsymbol{\phi}):V_{T}(\boldsymbol{\phi})\geq -X\mbox{ for }\boldsymbol{\phi}\in
\boldsymbol{\Phi}_{t,T}\right\}\]
for every \(t\leq T\), where \(\boldsymbol{\Phi}_{t,T}\) stands for the class of all self-financing trading strategies restricted to the time interval \([t,T]\). Furthermore, for any \(t\leq T\), the following equalities are satisfied
\[\Pi_{t}^{s}(X)=\sup_{P^{*}\in\bar{\cal P}(S^{*})}E_{P^{*}}\left [\left .(1+r)^{-(T-t)}\cdot X\right |{\cal F}_{t}\right ]\]
and
\[\Pi_{t}^{b}(X)=\inf_{P^{*}\in\bar{\cal P}(S^{*})}E_{P^{*}}\left [\left . (1+r)^{-(T-t)}\cdot X\right |{\cal F}_{t}\right ].\]
In explicit calculations of the buyer’s and seller’s prices in a multi-period model, we may proceed by backward induction. In the first step, we find the terminal hedging cost at each node which corresponds to the date \(T-1\). Subsequently, these minimal costs are interpreted as a contingent claim which settles at time \(T-1\), and so forth. By proceeding in this way, we are able to find the buyer’s and seller’s prices at any date.
Constraints on Short-Selling and Borrowing of Cash.
We examine various forms of restrictions imposed on share trading and borrowing of cash. Let us assume that the short-selling of shares is
prohibited. We may still define the seller’s and buyer’s prices by setting (ref. (\ref{museq41}))
\[\widehat{\Pi}_{0}^{s}(X)=\inf\left\{c\in \mathbb{R}:\alpha_{0}\cdot (S_{T}-
S_{0}\cdot (1+r))+c\cdot (1+r)\geq X\mbox{ for }\alpha_{0}\geq 0\right\}\]
and
\[\widehat{\Pi}_{0}^{b}(X)=-\widehat{\Pi}_{0}^{s}(-X).\]
This means that in order to determine the seller’s price, one needs to examine the optimization problem, which means to minimize \(c\in \mathbb{R}\) subject to the slightly modified constraints
\[\left\{\begin{array}{l}
\alpha_{0}\cdot (s_{1}-S_{0}\cdot (1+r))+c\cdot (1+r)\geq h_{1}\\
\vdots\\
\alpha_{0}\cdot (s_{p}-S_{0}\cdot (1+r))+c\cdot (1+r)\geq h_{p}\\
\alpha_{0}\geq 0.
\end{array}\right .\]
Let us state that the following clain of inequalities
\[\widehat{\Pi}_{0}^{b}(X)\leq\Pi_{0}^{b}(X)\leq\Pi_{0}^{s}(X)\leq\widehat{\Pi}_{0}^{s}(X).\]
If \(X\) is an attainable (but not constant) contingent claim, we have either
\[\widehat{\Pi}_{0}^{b}(X)<\Pi_{0}^{b}(X)=\Pi_{0}^{s}(X)\leq\widehat{\Pi}_{0}^{s}(X)\mbox{ or }
\widehat{\Pi}_{0}^{b}(X)=\Pi_{0}^{b}(X)=\Pi_{0}^{s}(X)<\widehat{\Pi}_{0}^{s}(X).\]
Under short-selling (and/or borrowing) restrictions, the representation of the seller’s (buyer’s) price by means of the class of all martingale measures for the process \(S^{*}\) is not available in general. Let us finally mention that the inequality \(\alpha_{0}\geq 0\) may be replaced by a weaker condition \(\alpha_{0}\geq -d_{0}\) for some strictly positive real \(d_{0}\) representing the maximal number of shares which can be shorten (note that in this case it is implicitly assumed that the proceeds from short-selling are available at time \(0\); that is, they may be deposited in an interest-earning account).
Let us examine very succinctly the case in which restrictions are imposed on the amount of borrowed cash. For simplicity, we assume that the same kind of restrictions apply to all parties involved in a trade (this assumption is not essential). It is clear that the borrowing of cash is subject to limitations, then in order to determine the seller’s price of \(X\), one needs to solve the following optimization problem: minimize \(c=\alpha_{0}\cdot S_{0}+\beta_{0}\) subject to the following set of linear constraints
\[\left\{\begin{array}{l}
\alpha_{0}\cdot s_{1}+\beta_{0}\cdot (1+r)\geq h_{1}\\
\vdots\\
\alpha_{0}\cdot s_{p}+\beta_{0}\cdot (1+r)\geq h_{p}\\
\alpha_{0}\geq 0,\beta_{0}\geq d_{0},
\end{array}\right .\]
where \(d_{0}\leq 0\) represents the limit imposed on the amount of borrowed cash. We conclude that so long as the considered market model is finite, no matter what specific set of restrictions is imposed on trading strategies involved, the seller’s and buyer’s prices can be found by solving the corresponding linear programming problem.
Different Lending and Borrowing Rates.
Let \(r\) and \(R\) stand for constant lending and borrowing rates, respectively. To avoid simple arbitrage we have to assume that the equality \(R\geq r\) holds, otherwise a simple strategy of borrowing cash and replacing it in a savings account would generate a risk-free profit. On the other hand, when \(R\geq r\), it is not rational to borrow money just to invest in risk-free bonds; such an investment would give rise to a sure loss. Therefore, we may and do assume that \(\beta_{0}=V_{0}(\boldsymbol{\phi})-\alpha_{0}\cdot S_{0}\) is either
the amount of cash borrowed (when \(\beta_{0}<0\)) or the amount of cash invested in riskless bonds (this holds when \(\beta_{0}>0\)). Let us find the terminal wealth of a portfolio \(\alpha_{0},\beta_{0}\) established at time \(0\). We have
\[V_{T}(\boldsymbol{\phi})=\alpha_{0}\cdot S_{T}=\beta_{0}^{+}\cdot (1+r)+\beta_{0}^{-}\cdot (1+R),\]
where \(\beta_{0}^{-}=\min\{\beta_{0},0\}\) represents the amount of borrowed cash. Using the initial wealth equality \[c=\alpha_{0}\cdot S_{0}+\beta_{0}^{+}+\beta_{0}^{-},\]
we have
\[V_{T}(\boldsymbol{\phi})=\alpha_{0}\cdot (S_{T}-S_{0}\cdot (1+r))+c\cdot (1+r)+(R-r)(c-\alpha_{0}\cdot S_{0})^{-}\]
since
\[\beta_{0}^{+}\cdot (1+r)+\beta_{0}^{-}\cdot (1+r)=\beta_{0}\cdot (1+r)+(R-r)\cdot (c-\alpha_{0}\cdot S_{0})^{-}.\]
Therefore, the seller’s price \(\widetilde{\Pi}_{0}^{s}(X)\) of a contingent claim \(X\) is determined by the following expression
\[\widetilde{\Pi}_{0}^{s}(X)=\inf\left\{c\in \mathbb{R}:\alpha_{0}\cdot (S_{T}-
S_{0}\cdot (1+r))+c\cdot (1+r)+(R-r)\cdot (c-\alpha_{0}\cdot S_{0})^{-}\geq X\right\}.\]
The buyer’s price, denoted by \(\widetilde{\Pi}_{0}^{b}(X)\), is defined, as usual, by setting \(\widetilde{\Pi}_{0}^{b}(X)=-\widetilde{\Pi}_{0}^{s}(-X)\). The perfect hedging problem can be reformulated as follows: minimize \(c\in \mathbb{R}\) subject to the following nonlinear constraints
\[\left\{\begin{array}{l}
\alpha_{0}\cdot (s_{1}-S_{0}\cdot (1+r))+c\cdot (1+r)+(R-r)\cdot
(c-\alpha_{0}\cdot S_{0})^{-}\geq h_{1}\\
\vdots\\
\alpha_{0}\cdot (s_{p}-S_{0}\cdot (1+r))+c\cdot (1+r)+(R-r)\cdot
(c-\alpha_{0}\cdot S_{0})^{-}\geq h_{p}.
\end{array}\right .\]
Let us find the seller’s and buyer’s prices of a call option under different lending and borrowing rates. To find the seller’s price, we have to determine the minimal value of the initial investment \(c=\alpha_{0}\cdot S_{0}+\beta_{0}\) over all pairs \((\alpha_{0},\beta_{0})\) for which the following conditions are satisfied
\[\left\{\begin{array}{l}
\alpha_{0}\cdot s_{1}+\beta_{0}^{+}\cdot (1+r)+\beta_{0}^{-}\cdot (1+R)
\geq (s_{1}-K)^{+}\\
\vdots\\
\alpha_{0}\cdot s_{p}+\beta_{0}^{+}\cdot (1+r)+\beta_{0}^{-}\cdot (1+R)
\geq (s_{p}-K)^{+}.
\end{array}\right .\]
By considering separately the case \(\beta_{0}\geq 0\) and \(\beta_{0}<0\), it is possible to check that the solution to the above minimization problem corresponds to the latter case, hence the seller’s price of the call option equals (ref. (\ref{museq48}))
\[\widetilde{\Pi}_{0}^{s}(C_{T})=\frac{s_{1}-K}{1+R}\cdot\frac {S_{0}\cdot (1+R)-s_{p}}{s_{1}-s_{p}},\]
where, as usual \(C_{T}=(S_{T}-K)^{+}\). This means that the implementation of the perfect hedging strategy with minimal cost for the short party demands additional borrowing of cash. On the other hand, using similar arguments, one checks that the buyer’s price of the options is
\[\widetilde{\Pi}_{0}^{b}(C_{T})=\frac{s_{1}-K}{1+r}\cdot\frac{S_{0}\cdot (1+r)-s_{p}}{s_{1}-s_{p}}\]
so that the perfect hedging strategy with minimal cost for the long party involves investing in risk-free bonds and short-selling of shares. For the put option, the situation is just the opposite. The seller’s option price equals
\[\widetilde{\Pi}_{0}^{s}(P_{T})=\frac{K-s_{p}}{1+r}\cdot\frac{s_{1}-S_{0}\cdot (1+r)}{s_{1}-s_{p}},\]
where \(P_{T}=(K-S_{T})^{+}\), while the buyer’s price is
\[\widetilde{\Pi}_{0}^{b}(P_{T})=\frac{K-s_{p}}{1+R}\cdot\frac{s_{1}-S_{0}\cdot (1+R)}{s_{1}-s_{p}}.\]
Therefore, the perfect hedging strategy of the short call (or the long put) demands the borrowing of cash, while the perfect hedging strategy
related to the long call (or to the short put) involves the short-selling of shares and the investment of proceeds in risk-free bonds. It is not
difficult to check that
\[\widetilde{\Pi}_{0}^{s}(C_{T})>\widetilde{\Pi}_{0}^{b}(C_{T})\mbox{ and }
\widetilde{\Pi}_{0}^{s}(P_{T})>\widetilde{\Pi}_{0}^{b}(P_{T}),\]
unless the lending and borrowing rates are identical; that is, unless \(r=R\).
\begin{equation}{\label{b}}\tag{B}\mbox{}\end{equation}
Mean-Variance Hedging.
By mean-variance hedging, we mean any methodology for the hedging of non-attainable contingent claims that is based on expectations and variances of the relevant random variables. Basically, there are two kinds of mean-variance hedging. The first assumes that a trading strategy is necessarily self-financing, and focuses on the minimization of the tracking error at the terminal date only. The second, more flexible method of mean-variance hedging considers trading strategies that are not necessarily self-financing.
Variance-Minimizing Hedging.
We start by focusing on those methods of mean-variance hedging in which the portfolio revisions are done in a self-financing manner. The
following optimization problems are considered.
- ( MV1). Given a contingent claim \(X\) and a real number \(c\in \mathbb{R}\), minimize
\[J_{1}(\boldsymbol{\phi})=E_{P}\left [X^{*}-V_{T}^{*}(\boldsymbol{\phi})\right ]^{2}\]
over all self-financing trading strategies \(\boldsymbol{\phi}\) which satisfy \(V_{0}(\boldsymbol{\phi})=c\). - (MV2). Given a contingent claim \(X\), minimize
\[J_{2}(\boldsymbol{\phi})=E_{P}\left [X^{*}-V_{T}^{*}(\boldsymbol{\phi})\right ]^{2}\]
over all self-financing trading strategies \(\boldsymbol{\phi}\).
In the first problem above, we search for the optimal (in the sense of the expected quadratic terminal risk) self-financing trading strategy with a pre-specified initial investment. In the second problem, the minimization is taken with respect to the initial cost also. Notice that the quadratic terminal risk is simply the expected quadratic cost of revising the terminal portfolio in order to replicate a given claim. In both cases, the optimal strategy is referred to as the minimal variance hedging of \(X\) under \(\mathbb{P}\). The optimization problems (MV1) and (MV2) depend on the choice of the underlying probability measure \(\mathbb{P}\). Therefore, either of the functionals above can be at best interpreted as a subjective measure of an intrinsic risk (which means the specific risk of a non-attainable contingent claim such that it cannot be eliminated using self-financing trading strategies) determined by an investor who aims to replicate the claim \(X\) by means of a self-financing trading strategy. Hence, in this case, the intrinsic risk can be represented by the minimal incremental “cost” associated with the rebalancement of the portfolio at the terminal stae \(T\). Since the “cost” of the terminal rebalancement equals \(X-V_{T}(\boldsymbol{\phi})\), it represents either the amount of funds that should be injected at time \(T\), or the amount of funds that can be withdrawn at this date (the term “cost” is thus purely conventional). On the other hand, in the case of a perfect hedging, the incermental “cost” of terminal rebalancement is nonpositive with probability one; that is, it always represents a surplus of funds. Recall also that perfect hedging is invariant with respect to the choice of an underlying probability measure in the class of mutually equivalent probability measures, which is no longer the case of solutions of (MV1) and (MV2). Problems (MV1) and (MV2) are examined in a general discrete-time setup (without assuming that the underlying probability space is finite) by Sch\”{a}l \cite{sch} and Schweizer \cite{sch95b,sch96}. Since we work in a finite setting, we omit certain technical assumptions, which are relevant in a general discrete-time framework. Given a contingent claim \(X\) which settles at time \(T\), we write \(\tilde{\boldsymbol{\phi}}(c,X)\) and \((\tilde{c}(X),\tilde{\boldsymbol{\phi}}(X))\) to denote the solutions of problems (MV1) and (MV2), respectively. The number \(\tilde{c}(X)\) is termed the variance-minimizing value of \(X\) under \(\mathbb{P}\). Recall first that we have
\[V_{T}^{*}(\boldsymbol{\phi})=V_{0}(\boldsymbol{\phi})+\sum_{t=0}^{T-1}\boldsymbol{\phi}_{t}\cdot\Delta_{t}S^{*},\]
where \(\Delta_{t}S^{*}=S_{t+1}^{*}-S_{t}^{*}\). Let us consider the followingclass of \({\cal F}_{T}\)-measurable random variables
\[{\cal G}_{T}(\boldsymbol{\Phi})=\left\{\sum_{t=0}^{T-1}\boldsymbol{\phi}_{t}\cdot\Delta_{t}S^{*}:\boldsymbol{\phi}\in
\boldsymbol{\Phi}\right\}.\]
Basically, to prove the existence of a solution to any of the optimization problems above, it is enough to show that the class \({\cal G}_{T}(\boldsymbol{\Phi})\) is a closed subspace of the Hilbert space \(L^{2}(P)\) of square-integrable random variables (in the present setup, all random variables are manifestly square-integrable under any probability measure). Though it turns out that problem (MV1) has no solution in general, it is shown in Schweizer \cite{sch95b} that under mild technical assumptions, the closeness of \({\cal G}_{T}(\boldsymbol{\Phi})\) holds. Consequently, for any claim \(X\) and any initial investment \(c\in \mathbb{R}\), the optimization problem (MV1) admits a unique solution \(\tilde{\boldsymbol{\phi}}(c,X)\). Formally, it is then enough to project the random variable \(X-c\) on \({\cal G}_{T}(\boldsymbol{\Phi})\), and to find a self-financing strategy which replicates this projection. To describe this solution in a more explicit way, it is convenient to introduce the signed measure \(\tilde{P}\) on \((\Omega, {\cal F}_{T})\), referred to as the variance-minimizing measure associated with \(\mathbb{P}\), by setting (let us make clear that \(\tilde{P}\) is absolutely continuous with respect to \(\mathbb{P}\), but not necessarily equivalent to \(\mathbb{P}\))
\begin{equation}{\label{museq410}}\tag{11}
\frac{d\tilde{P}}{dP}=\frac{\tilde{Z}}{\mathbb{E}_{\mathbb{P}}[\tilde{Z}]}
\end{equation}
provided that \(\mathbb{E}_{\mathbb{P}}[\tilde{Z}]>0\), where the random variable \(\tilde{Z}\) equals
\begin{equation}{\label{museq411}}\tag{12}
\tilde{Z}=\prod_{t=1}^{T}(1-\tilde{\delta}_{t}\cdot\Delta_{t-1}S^{*})
\end{equation}
and the predictable process \(\tilde{\delta}\) is defined by the recurrence relation
\[\tilde{\delta}_{t}=\frac{\mathbb{E}_{\mathbb{P}}\left [\left .\Delta_{t}S^{*}\cdot\prod_{s=t+1}^{T}(1-\tilde{\delta}_{s}\cdot\Delta_{s-1}S^{*})\right |{\cal F}_{t}\right ]}{\mathbb{E}_{\mathbb{P}}\left [\left .\Delta_{t}(S^{*})^{2}\cdot
\prod_{s=t+1}^{T}(1-\tilde{\delta}_{s}\cdot\Delta_{s-1}S^{*})^{2}\right |{\cal F}_{t}\right ]}\]
for \(t=1,\cdots ,T\) with \(\tilde{\delta}_{T+1}=0\), where we write
\[\Delta_{t}(S^{*})^{2}=(S_{t+1}^{*})^{2}-(S_{t}^{*})^{2}.\]
In a particular case, when \(\mathbb{P}\) is a martingale measure for \(S^{*}\), it is easily seen that \(\tilde{\delta}_{t}=0\) for every \(t\), and thus the measure \(\tilde{\mathbb{P}}\) coincides with \(\mathbb{P}\). It is also possible to show that if \(\tilde{\mathbb{P}}\) is a probability measure, then it belongs to the class of generalized martingale measures for the process \(S^{*}\). More generally, if \(\tilde{\mathbb{P}}\) is merely a signed measure, then the discounted stock price \(S^{*}\) has under \(\tilde{\mathbb{P}}\) the “weak martingale property” which reads
\[\mathbb{E}_{\mathbb{P}}\left [\left .\tilde{Z}\cdot (S_{t+1}^{*}-S_{t}^{*})\right |
{\cal F}_{t}\right ]=0\mbox{ for all }t=1,\cdots ,T-1.\]
As one might expect, the measure \(\tilde{P}\) is closely related to optimization problems (MV1) and (MV2). Indeed, it is possible to prove that the pair \((\tilde{c}(X),\tilde{\boldsymbol{\phi}}(\tilde{c}(X),X)\), where \(\tilde{c}(X)=\mathbb{E}_{\tilde{\mathbb{P}}}[X^{*}]\), solves (MV2). Furthermore, the strategy \(\tilde{\boldsymbol{\phi}}(\tilde{c}(X),X)\) is also a solution to the following optimization problem
- ( MV3). Given a contingent claim \(X\), minimize the variance
\[J_{3}(\boldsymbol{\phi})=\mbox{Var}_{\mathbb{P}}[X^{*}-V_{T}^{*}(\boldsymbol{\phi})]\]
over all self-financing trading strategies \(\boldsymbol{\phi}\).
The last property supports the name variance-minimizing hedging given to a solution of (MV2). Finally, let us observe that if \(\mathbb{P}\) is a martingale measure then, for any \(c\in \mathbb{R}\), the optimal strategy \(\tilde{\boldsymbol{\phi}}(c,X)\) for problem (MV1) is given by an explicit formula
\[\tilde{\boldsymbol{\phi}}_{t}(c,X)=\frac{\mathbb{E}_{\mathbb{P}}[X\cdot\Delta_{t}S^{*}|{\cal F}_{t}]}{\mathbb{E}_{\mathbb{P}}[\Delta_{t}(S^{*})^{2}|{\cal F}_{t}]}.\]
It is apparent that the optimal strategy \(\tilde{\boldsymbol{\phi}}(c,X)\) does not depend on \(c\), therefore it is easily seen to solve also problems (MV2) and (MV3).
Risk-Minimizing Hedging.
From a practical viewpoint, a major deficiency of variance-minimizing hedging is a somewhat counter-intuitive assumption than an investor is unwilling to make additional borrowings or withdrawals of funds before the terminal date. A conceivable, and perhaps more appealing, approach would be to allow for fund transfers at any time with no restrictions whatsoever. This means that the usual assumption that a trading strategy should be self-financing is simply abandoned. To make such an approach nontrivial, we need to impose instead a specific optimality criterion, which in this case focuses on the minimization of the future risk exposure at any time.
One-Period Market.
We find it convenient to analyze first the case of a one-period market with two dates \(0\) and \(T=1\). For any initial portfolio \(\boldsymbol{\phi}_{0}(\alpha_{0},\beta_{0})\), we have
\[V_{0}(\boldsymbol{\phi})=\alpha_{0}\cdot S_{0}+\beta_{0}\equiv C_{0}(\boldsymbol{\phi}),\]
where \(\alpha_{0}\) and \(\beta_{0}\) are real numbers, and \(C_{0}(\boldsymbol{\phi})\) represents the {\bf initial cost} of \(\boldsymbol{\phi}\). If \(\boldsymbol{\phi}\) were self-financing, its terminal wealth, denoted by \(\tilde{V}_{1}(\boldsymbol{\phi})\),
would equal
\[\tilde{V}_{1}(\boldsymbol{\phi})=\alpha_{0}\cdot S_{1}+ \beta_{0}\cdot (1+r).\]
However, since this is not necessarily the case, we may and do assume that the terminal wealth \(V_{1}(\boldsymbol{\phi})\) of \(\boldsymbol{\phi}\) just an arbitrary \({\cal F}_{1}\)-measurable random variable. It is obvious that \(V_{1}(\boldsymbol{\phi})\) admits the representation \(V_{1}(\boldsymbol{\phi})=\alpha_{0}\cdot S_{1}+\beta_{1}\), where \(\beta_{1}\) is also \({\cal F}_{1}\)-measurable. We conclude that, given \(\alpha_{0}\), there is a one-to-one correspondence between the terminal wealth \(V_{1}(\boldsymbol{\phi})\) and terminal cash endowment \(\beta_{1}\). In other words, the cash component \(\beta_{1}\) can be chosen in such a way that the terminal wealth of \(\boldsymbol{\phi}\) reaches any prespecified level. The implementation of \(\boldsymbol{\phi}\) may thus involve an additional transfer of funds at time \(T\). To measure this additional cash flow, it will be convenient to introduce a \({\cal F}_{1}\)-measurable random variable \(C_{1}(\boldsymbol{\phi})\) which satisfies
\begin{equation}{\label{museq413}}\tag{13}
C_{1}(\boldsymbol{\phi})-C_{0}(\boldsymbol{\phi})\cdot (1+r)=\beta_{1}-\beta_{0}\cdot (1+r)=V_{1}(\boldsymbol{\phi})-
\tilde{V}_{1}(\boldsymbol{\phi}).
\end{equation}
Notice that the quantity \(\delta (\boldsymbol{\phi})\equiv C_{1}(\boldsymbol{\phi})-C_{0}(\boldsymbol{\phi})\cdot (1+r)\) represents the additional cash flow associated with the strategy \(\boldsymbol{\phi}\) at time \(T\). Solving (\ref{museq413}) for \(C_{1}(\boldsymbol{\phi})\) and discounting, we get
\begin{equation}{\label{museq414}}\tag{14}
C_{1}^{*}(\boldsymbol{\phi})=C_{0}(\boldsymbol{\phi})+\Delta_{0}C^{*}(\boldsymbol{\phi})=V_{1}^{*}(\boldsymbol{\phi})
-\alpha_{0}\cdot (S_{1}^{*}-S_{0}^{*}),
\end{equation}
where
\[C_{1}^{*}(\boldsymbol{\phi})=\frac{C_{1}(\boldsymbol{\phi})}{1+r},V_{1}^{*}(\boldsymbol{\phi})=\frac{V_{1}(\boldsymbol{\phi})}{1+r},\mbox{ and }\Delta_{0}C^{*}(\boldsymbol{\phi})=
C_{1}^{*}(\boldsymbol{\phi})-C_{0}^{*}(\boldsymbol{\phi}).\]
Put another way, the discounted terminal wealth of any trading strategy \(\boldsymbol{\phi}\) admits the following representation
\begin{equation}{\label{museq415}}\tag{15}
V_{1}^{*}(\boldsymbol{\phi})=c_{0}+\Delta_{0}C^{*}(\boldsymbol{\phi})+\alpha_{0}\cdot (S_{1}^{*}-S_{0}^{*}),
\end{equation}
where \(c_{0}=C_{0}(\boldsymbol{\phi})=\alpha_{0}\cdot S_{0}+\beta_{0}\). For any claim \(X\) and any initial portfolio \((\alpha_{0},\beta_{0})\), there exists a unique \({\cal F}_{1}\)-measurable random variable \(\Delta_{0}C^{*}(\boldsymbol{\phi})\) such that the terminal wealth \(V_{1}(\boldsymbol{\phi})\) of the strategy \(\boldsymbol{\phi}\) replicates \(X\). In terms of discounted values, for any trading strategy \(\boldsymbol{\phi}\) which replicates \(X\)
\begin{equation}{\label{museq416}}\tag{16}
X^{*}=c_{0}+\alpha_{0}\cdot (S_{1}^{*}-S_{0}^{*})+\Delta_{0}C^{*}(\boldsymbol{\phi}).
\end{equation}
We now introduce a suitable criterion which measures the intrinsic risk. For a given claim \(X\), we wish to minimize the expected quadratic risk \(R_{\boldsymbol{\phi}}(0)\), which equals
\[R_{\boldsymbol{\phi}}(0)\equiv \mathbb{E}_{\mathbb{P}}\left [\left (\frac{\delta (\boldsymbol{\phi})}{1+r}\right )^{2}\right ]=\mathbb{E}_{\mathbb{P}}\left [\left (\Delta_{0}C^{*}(\boldsymbol{\phi})\right )^{2}\right ]\]
over all choices of initial portfolio \((\alpha_{0},\beta_{0})\). This means that we do not intend to minimize the total cost of replication, but only the incremental cost of trading after the portfolio is established at the initial date. In view of (\ref{museq416}), it is clear that the aim is to minimize the expectation (We are in fact very close to the already examined variance-minimizing problem (MV2). Such coincidence is no longer valid if a multi-period market model is considered.)
\[R_{\boldsymbol{\phi}}(0)=\mathbb{E}_{\mathbb{P}}\left [\left (X^{*}-c_{0}-\alpha_{0}\cdot (S_{1}^{*}-S_{0}^{*})\right )^{2}\right ]\]
over all real numbers \(c_{0}\) and \(\alpha_{0}\). Formally, we are thus searching for the best linear estimate (in the mean-square sense under \(\mathbb{P}\)) of the random variable \(X^{*}\) based on the increment \(S_{1}^{*}-S_{0}^{*}\). The well-known explicit solution to the classic linear regression problem is given by the formula
\begin{equation}{\label{museq417}}\tag{17}
\widehat{\alpha}_{0}=\frac{\mbox{Cov}_{\mathbb{P}}(X^{*},S_{1}^{*}-S_{0}^{*})}{\mbox{Var}_{\mathbb{P}}[S_{1}^{*}-S_{0}^{*}]}=\frac{\mbox{Cov}_{\mathbb{P}}(X^{*},S_{1}^{*})}{\mbox{Var}_{\mathbb{P}}[S_{1}^{*}]}
\end{equation}
and
\begin{equation}{\label{museq418}}\tag{18}
\widehat{c}_{0}=\mathbb{E}_{\mathbb{P}}[X^{*}]-\widehat{\alpha}_{0}\cdot \mathbb{E}_{\mathbb{P}}[S_{1}^{*}-S_{0}^{*}].
\end{equation}
The optimal strategy \(\boldsymbol{\phi}^{*}(X)\), referred to as risk-minimizing hedging under \(\mathbb{P}\), satisfies \(\boldsymbol{\phi}_{0}^{*}=(\widehat{\alpha}_{0},\widehat{\beta}_{0})\) and \(\boldsymbol{\phi}^{*}_{1}=(\widehat{\alpha}_{0},X-\widehat{\alpha}_{0}\cdot S_{0})\), where \(\widehat{\beta}_{0}= \widehat{c}_{0}-\widehat{\alpha}_{0}\cdot S_{0}\). It follows from (\ref{museq414}), combined with (\ref{museq418}), that for the optimal portfolio \(\boldsymbol{\phi}^{*}\) we have
\[\mathbb{E}_{\mathbb{P}}\left [C_{1}^{*}(\boldsymbol{\phi}^{*})\right ]=C_{0}^{*}(\boldsymbol{\phi}^{*}),\]
that is, the discounted cost process follows a martingale under \(\mathbb{P}\). We shall refer to this peculiar property of the optimal portfolio by saying that the strategy \(\boldsymbol{\phi}^{*}\) is mean self-financing under \(\mathbb{P}\).
Case of Martingale Measure.
Suppose that \(\mathbb{P}=\mathbb{P}^{*}\) is a martingale measure for \(S^{*}\). From (\ref{museq418}), we get \(C_{0}(\boldsymbol{\phi}^{*})=\mathbb{E}_{\mathbb{P}^{*}}[X^{*}]\), and thus the initial cost associated with risk-minimizing hedging is given by the risk-neutral valuation formula. Furthermore, it is possible to find the risk-minimizing strategy \(\boldsymbol{\phi}^{*}\) using a martingale approach. Indeed, given a \(\mathbb{P}^{*}\)-martingale \(S^{*}\), any \({\cal F}_{1}\)-measurable random variable
$X^{*}$ admits the unique representation (which can be seen as a primitive version of the Kunita-Watanabe decomposition)
\begin{equation}{\label{museq419}}\tag{19}
X^{*}=c+\alpha\cdot (S_{1}^{*}-S_{0}^{*})+L_{X}(1)
\end{equation}
for some constants \(c\) and \(\alpha\), where the process \(L_{X}(0)=0\), \(L_{X}(1)\) follows a martingale under \(\mathbb{P}^{*}\). In addition, \(L_{X}\) is strongly orthogonal to the \(\mathbb{P}^{*}\)-martingale \(S^{*}\) under \(\mathbb{P}^{*}\), meaning that
\[\mathbb{E}_{\mathbb{P}^{*}}[S_{1}^{*}\cdot L_{X}(1)]=S_{0}^{*}\cdot L_{X}(0)=0.\]
Equivalently, we have
\begin{equation}{\label{museq420}}\tag{20}
\mathbb{E}_{\mathbb{P}^{*}}\left [(S_{1}^{*}-S_{0}^{*})\cdot L_{X}(1)\right ]=0
\end{equation}
so that \(S_{1}^{*}-S_{0}^{*}\) and \(L_{X}(1)\) are uncorrelated random variables. We claim that representation (\ref{museq419}) determines the risk-minimizing replicating strategy of \(X\) under \(P^{*}\). Firstly, by taking expectations in (\ref{museq419}), we find that \(c=\mathbb{E}_{\mathbb{P}^{*}}[X^{*}]=\widehat{c}_{0}\). Next, multiplying (\ref{museq419}) by \(S_{1}^{*}-S_{0}^{*}\), taking expectations, and using (\ref{museq420}), we arrive at the following equality
\[\mathbb{E}_{\mathbb{P}^{*}}\left [X^{*}\cdot (S_{1}^{*}-S_{0}^{*})\right ]=\alpha\cdot\mathbb{E}_{\mathbb{P}^{*}}\left [\left (S_{1}^{*}-S_{0}^{*}\right )^{2}\right ].\]
This shows immediately that \(\alpha =\widehat{\alpha}_{0}\). Concluding, representation (\ref{museq416}), with \(c_{0}=\widehat{c}_{0}\) and \(\alpha_{0}=\widehat{\alpha}_{0}\), is a direct consequence of (\ref{museq419}). In particular, the martingale \(L_{X}\), which is strongly orthogonal to \(S^{*}\) under \(P^{*}\), corresponds to the cost process \(C^{*}-\widehat{c}_{0}\).
Multi-Period Market.
In a multi-period case, we define a trading strategy as an arbitrary two-dimensional proces \(\boldsymbol{\phi}=(\alpha ,\beta )\), where for every \(t\) the random variables \(\alpha_{t}\) and \(\beta_{t}\) are assumed to be \({\cal F}_{t}\)-measurable. The wealth process $V_{\boldsymbol{\phi}}$ associated with a trading strategy \(\boldsymbol{\phi}\) (not necessarily self-financing) equals
\[V_{0}(\boldsymbol{\phi})=\alpha_{0}\cdot S_{0}+\beta_{0}\equiv C_{0}(\boldsymbol{\phi})\]
and
\[V_{t}(\boldsymbol{\phi})=\alpha_{t-1}\cdot S_{T}+\beta_{t}\mbox{ for }t=1,\cdots ,T.\]
We assume also that \(V_{T}(\boldsymbol{\phi})=X\); that is, \(\boldsymbol{\phi}\) replicates \(X\). Given the wealth prices \(V(\boldsymbol{\phi})\), the discounted cost \(C^{*}(\boldsymbol{\phi})\) of \(\boldsymbol{\phi}\) is introduced through a recurrence relation (ref. (\ref{museq414}))
\begin{equation}{\label{museq421}}\tag{21}
C^{*}_{t+1}(\boldsymbol{\phi})-C^{*}_{t}(\boldsymbol{\phi})\equiv V_{t+1}^{*}(\boldsymbol{\phi})-V_{t}^{*}
(\boldsymbol{\phi})-\alpha_{t}\cdot (S_{t+1}^{*}-S_{t}^{*})
\end{equation}
for \(t=0,1,\cdots T-1\), the difference \(C^{*}_{t+1}(\boldsymbol{\phi})-C^{*}_{t}(\boldsymbol{\phi})\) represents the discounted value of the additional transfers of funds that take place at time \(t\). It follows immediately from (\ref{museq421}) that the process \(C^{*}_{t}(\boldsymbol{\phi})\) admits the following representation
\begin{equation}{\label{museq422}}\tag{22}
C^{*}_{t}(\boldsymbol{\phi})=V_{t}^{*}(\boldsymbol{\phi})-
\sum_{i=1}^{t}\alpha_{i-1}\cdot (S^{*}_{i}-S^{*}_{i-1})\mbox{ for }t\leq T.
\end{equation}
In a multi-period setting, we are searching for a trading strategy \(\boldsymbol{\phi}\) which solves the following optimization problem.
- (RM1). Minimize, for every \(t\), the conditional local risk
\[R_{t}(\boldsymbol{\phi})\equiv \mathbb{E}_{\mathbb{P}}\left [\left .\left (
C^{*}_{t+1}(\boldsymbol{\phi})-C^{*}_{t}(\boldsymbol{\phi})\right )^{2}\right |{\cal F}_{t}\right ]\]
over all trading strategies which replicate a given contingent claim \(X\).
Definition. A trading strategy \(\boldsymbol{\phi}^{*}(X)\) which solves optimization problem (MV1) is called a locally risk-minimizing hedging for \(X\) under \(\mathbb{P}\). The initial cost \(V_{0}(\boldsymbol{\phi}^{*})=C_{0}(\boldsymbol{\phi}^{*})\)
is called the {\bf risk-minimizing value of \(X\) under \(\mathbb{P}\). \(\sharp\)
To solve problem (RM1), note that, in view of (\ref{museq421}), \(R_{t}(\boldsymbol{\phi})\) can be written as
\begin{equation}{\label{museq423}}\tag{23}
R_{t}(\boldsymbol{\phi})=\mathbb{E}_{\mathbb{P}}\left [\left .\left (V_{t+1}^{*}(\boldsymbol{\phi})-V_{t}^{*}(\boldsymbol{\phi})-
\alpha_{t}\cdot (S_{t+1}^{*}-S_{t}^{*})\right )^{2}\right |{\cal F}_{t}\right ]
\end{equation}
for every \(t=0,1,\cdots ,T-1\) with the terminal condition \(V_{T}^{*}(\boldsymbol{\phi})=X^{*}\). In view of representation of (\ref{museq423}), it is rather clear that the optimal strategy \(\boldsymbol{\phi}^{*}\) can be found by means of backward induction.
Let us now comment on the way in which the minimization problem can be solved recursively. Observe first that for any \(t\), given a random variable \(V_{t+1}^{*}(\boldsymbol{\phi}^{*})\), we need to find \({\cal F}_{t}\)-measurable random variables \(c_{t}\) and \(\alpha_{t}\) which minimize the following conditional expectation
\[R_{t}(\boldsymbol{\phi})=\mathbb{E}_{\mathbb{P}}\left [\left .\left (V_{t+1}^{*}(\boldsymbol{\phi}^{*})-c_{t}-\alpha_{t}\cdot
(S_{t+1}^{*}-S_{t}^{*})\right )^{2}\right |{\cal F}_{t}\right ].\]
The unique solutions \(\widehat{\alpha}_{t}\) and \(\widehat{c}_{t}\) to this problem are known to be (ref. (\ref{museq417}) and (\ref{museq418}))
\begin{equation}{\label{museq424}}\tag{24}
\widehat{\alpha}_{t}=\frac{\mbox{Cov}_{\mathbb{P}}\left (\left . V_{t+1}^{*}(\boldsymbol{\phi}^{*}),\Delta_{t}S^{*}\right |
{\cal F}_{t}\right )}{\mbox{Var}_{\mathbb{P}}\left [\left .\Delta_{t}S^{*}\right |{\cal F}_{t}\right ]},
\end{equation}
where \(\Delta_{t}S^{*}=S_{t+1}^{*}-S_{t}^{*}\), and
\begin{equation}{\label{museq425}}\tag{25}
\widehat{c}_{t}=\mathbb{E}_{\mathbb{P}}\left [\left .V_{t+1}^{*}(\boldsymbol{\phi}^{*})
\right |{\cal F}_{t}\right ]-\widehat{\alpha}_{t}\cdot \mathbb{E}_{\mathbb{P}}\left [\left .\Delta_{t}S^{*}\right |{\cal F}_{t}\right ].
\end{equation}
In view of (\ref{museq421}), the cost process of an optimal strategy \(\boldsymbol{\phi}^{*}\) satisfies
\[C_{t+1}^{*}(\boldsymbol{\phi}^{*})-C_{t}^{*}(\boldsymbol{\phi}^{*})=
V_{t+1}^{*}(\boldsymbol{\phi}^{*})-\widehat{c}_{t}-\widehat{\alpha}_{t}\cdot\Delta_{t}S^{*}.\]
Combining the last equality with (\ref{museq425}), we deduce that the process \(C^{*}\) follows a martingale under \(\mathbb{P}\).
To solve recursively problem (RM1), we start with \(t=T-1\). Using (\ref{museq424}) and (\ref{museq425}) with \(t=T-1\) and \(V_{t+1}^{*}(\boldsymbol{\phi}^{*})=X^{*}\), we find the random variables \(\widehat{\alpha}_{T-1}\) and \(\widehat{c}_{T-1}\). The wealth of the optimal portfolio \(\boldsymbol{\phi}^{*}\) at time \(T-1\) thus equals \(V_{\boldsymbol{\phi}^{*}}(T-1)=\widehat{c}_{T-1}\), and the optimal portfolios at the sates \(T-1\) and \(T\) are
\[\boldsymbol{\phi}^{*}_{T-1}=(\widehat{\alpha}_{T-2},\widehat{\beta}_{T-1})\mbox{ and }\boldsymbol{\phi}^{*}_{T}=
(\widehat{\alpha}_{T-1},X-\widehat{\alpha}_{T}\cdot S_{T}).\]
Notice that \(\widehat{\alpha}_{T-2}\) and \(\widehat{\beta}_{T-1}\) are not yet known at this stage, but can be easily found by solving the optimization problem for \(t=T-2\) is solved, and making use of the equality
\[V_{T-1}^{*}(\boldsymbol{\phi})=\widehat{c}_{T-1}=\widehat{\alpha}_{T-2}\cdot S^{*}_{T-1}+\widehat{\beta}_{T-1}.\]
We shall now focus on the martingale approach.
Case of Martingale Measure.
Assume that \(\mathbb{P}=\mathbb{P}^{*}\) is a martingale measure for \(S^{*}\). In this case, we deduce from (\ref{museq422}) that the discounted wealth process \(V_{\boldsymbol{\phi}^{*}}^{*}\) of the risk-minimizing hedging strategy is a martingale under \(\mathbb{P}^{*}\)
\begin{equation}{\label{museq426}}\tag{26}
V_{t}^{*}(\boldsymbol{\phi}^{*})=\mathbb{E}_{\mathbb{P}^{*}}\left [\left .X^{*}\right |{\cal F}_{t}\right ]\mbox{ for all }t=0,1,\cdots ,T.\end{equation}
Moreover, we can now find the optimal trading strategy by applying the Kunita-Watanabe decomposition of \(X^{*}\) with respect to the martingale \(S^{*}\).
\begin{equation}{\label{musp421}}\tag{27}\mbox{}\end{equation}
Proposition \ref{musp421}. Suppose that the Kunita-Watanabe decomposition of \(X^{*}\) with respect to \(S^{*}\) under \(\mathbb{P}^{*}\) is
\begin{equation}{\label{museq427}}\tag{28}
X^{*}=c+\sum_{s=0}^{T-1}\alpha_{s}\cdot\Delta_{s}S^{*}+L_{X}(T),
\end{equation}
where \(L_{X}\) is a martingale strongly orthogonal to \(S^{*}\) with \(L_{X}(0)=0\). Then the locally risk-minimizing hedging strategy \(\boldsymbol{\phi}^{*}(X)\) for \(X\) under \(\mathbb{P}^{*}\) is given by the formula
\[\boldsymbol{\phi}^{*}_{t}=(\alpha_{t},\beta_{t})\mbox{ for all }t=1,\cdots ,T,\]
where
\[\beta_{t}=V_{t}(\boldsymbol{\phi}^{*})-\alpha_{t-1}\cdot S_{T}=
(1+r)^{t}\cdot \mathbb{E}_{\mathbb{P}^{*}}\left [\left .X^{*}\right |{\cal F}_{t}\right ]-\alpha_{t-1}\cdot S_{t-1}\]
and
\[\boldsymbol{\phi}^{*}_{0}=(\alpha_{0},\mathbb{E}_{\mathbb{P}^{*}}[X^{*}]-\alpha_{0}\cdot S_{0}).\]
Furthermore, the discounted cost process of \(\boldsymbol{\phi}^{*}\) equals
\[C_{t}^{*}(\boldsymbol{\phi}^{*})=c+L_{X}(t)\mbox{ for all }t=1,\cdots ,T.\]
Finally, the discounted wealth process of the locally risk-minimizing hedging strategy associated with a claim \(X\) satisfies
\begin{equation}{\label{museq428}}\tag{29}
V_{t}^{*}(\boldsymbol{\phi}^{*})=\mathbb{E}_{\mathbb{P}^{*}}\left [\left .X^{*}\right |{\cal F}_{t}\right ]\mbox{ for all }t=0,1,\cdots ,T.
\end{equation}
In particular, the risk-minimizng value of \(X\) under \(\mathbb{P}^{*}\) equals \(\mathbb{E}_{\mathbb{P}^{*}}[X^{*}]\).
Proof. Combining (\ref{museq426}) with (\ref{museq427}), we find that for every \(t=0,1,\cdots ,T\)
\[V_{t}^{*}(\boldsymbol{\phi}^{*})=\mathbb{E}_{\mathbb{P}^{*}}\left [\left .X^{*}\right |{\cal F}_{t}\right ]=c+\sum_{s=0}^{t-1}\alpha_{s}\cdot\Delta_{s}S^{*}+L_{X}(t).\]
Substituting this into (\ref{museq423}) with \(\boldsymbol{\phi}=\boldsymbol{\phi}^{*}\), we obtain
\[R_{t}(\boldsymbol{\phi})=\mathbb{E}_{\mathbb{P}^{*}}\left [\left .\left (
\Delta_{t}L_{X}-(\widehat{\alpha}_{t}-\alpha_{t})\cdot\delta_{t}S^{*}\right )^{2}\right |{\cal F}_{t}\right ],\]
where \(\Delta_{t}L_{X}=L_{X}(t+1)-L_{X}(t)\) for \(t\leq T-1\). After simple manipulations involving conditional expectations, we arrive at the following representation for the conditional local risk
\[R_{t}(\boldsymbol{\phi})=\mathbb{E}_{\mathbb{P}^{*}}\left [\left .(\Delta_{t}L_{X})^{2}\right |{\cal F}_{t}\right ]-(\widehat{\alpha}_{t}-\alpha_{t})^{2}\cdot \mathbb{E}_{\mathbb{P}^{*}}\left [\left .(\Delta_{t}S^{*})^{2}\right |{\cal F}_{t}\right ].\]
It is apparent that the last expression is minimized for \(\alpha_{t}=\widehat{\alpha}_{t}\). All statements are now rather straightforward consequences of properties of the Kunita-Watanabe decomposition. \(\blacksquare\)
Let us denote by \(\tilde{R}_{t}(\boldsymbol{\phi})\) the conditional total risk, which equals
\[\tilde{R}_{t}(\boldsymbol{\phi})\equiv \mathbb{E}_{\mathbb{P}^{*}}\left [\left .\left (C^{*}_{T}(\boldsymbol{\phi})-C^{*}_{t}(\boldsymbol{\phi})\right )^{2}\right |{\cal F}_{t}\right ],\]
and let us introduce the following optimization problem.
- (RM2). Given a contingent claim \(X\), minimize for every \(t=0,1,\cdots ,T-1\) the conditional total risk \(\tilde{R}_{t}(\boldsymbol{\phi})\) over all trading strategies which replicates \(X\).
It can be shown that if \(P\) is a martingale measure for \(S^{*}\), then the locally risk-minimizing hedging strategy \(\boldsymbol{\phi}^{*}\) solves also the optimization problem (RM2). For a continuous-time counterpart of this result, we may consult Schweizer \cite{sch90}.
Let us now comment on the practical interpretation of Proposition \ref{musp421}. Though (\ref{museq428}) resembles the standard risk-neutral valuation formula, the initial cost of a locally risk-minimizing strategy which replicates \(X\) cannot be interpreted as the arbitrage price of \(X\) in general. Indeed, if a contingent claim \(X\) is not attainable, then the discounted wealth process of the locally risk-minimizing hedging strategy (given by the right-hand side of equality (\ref{museq428})) depends on the choice of a martingale measure \(\mathbb{P}^{*}\), and thus it is a subjective notion. To overcome this deficiency, it seems justified to perform the risk-minimization under the actual probability measure \(\mathbb{P}\) (of course, in this case the result will in turn depend on the choice of \(\mathbb{P}\)).
General Case.
We follow Schweizer \cite{sch95b}. In a general case (that is, in a multi-period market, when \(\mathbb{P}\) is not a martingale measure for \(S^{*}\)) the problem of finding a locally risk-minimizing hedging strategy through the martingale approach becomes rather involved, and thus we do not go into details here. Let us simply observe that one way to find \(\boldsymbol{\phi}^{*}\) is to associate with any probability measure \(\mathbb{P}\), a unique martingale measure \(\hat{\mathbb{P}}\) such that
\[C_{0}(\boldsymbol{\phi}^{*})=V_{0}(\boldsymbol{\phi}^{*})=\mathbb{E}_{\hat{\mathbb{P}}}[X^{*}],\]
where \(\boldsymbol{\phi}^{*}=\boldsymbol{\phi}^{*}(X)\) is the locally risk-minimizing hedging for \(X\) under \(P\). To this end, we make use of the Doob-Meyer decomposition of \(S^{*}\) under \(\mathbb{P}\), which states that \(S^{*}=S_{0}+M+A\), where \(M\) follows a martingale under \(\mathbb{P}\), and \(A\) is a predictable process with \(M_{0}=A_{0}=0\). More explicitly, we have
\[\Delta_{t}A=A_{t+1}-A_{t}=\mathbb{E}_{\mathbb{P}}\left [\left .\Delta_{t}S^{*}\right |
{\cal F}_{t}\right ]=\mathbb{E}_{\mathbb{P}}[S_{t+1}^{*}|{\cal F}_{t}]-S_{t}^{*}\]
for \(t=1,\cdots ,T\). It is now possible to show that any contingent claim \(X\) admits the following representation
\begin{equation}{\label{museq430}}\tag{30}
X^{*}=c+\sum_{s=0}^{T}\alpha_{s}\cdot\Delta_{s}S^{*}+L_{X}(T),
\end{equation}
where \(L_{X}\) follows a martingale that is strongly orthogonal to \(M\) under \(\mathbb{P}\) with \(_{X}(0)=0\). Using representation (\ref{museq430}), Schweizer shows that the locally risk-minimizing problem can be solved by introducing the concept of a minimal martingale measure \(\hat{\mathbb{P}}\) associated with the underlying probability measure \(\mathbb{P}\). Roughly, the minimal martingale measure is that martingale measure for \(S^{*}\) under which any martingale orthogonal to \(M\) under \(\mathbb{P}\) remains a martingale under \(\hat{\mathbb{P}}\). However, it is a signed measure in general. In a discrete-time framework, the minimal martingale measure \(\hat{\mathbb{P}}\) is known to be given by the formula
\[\frac{d\hat{P}}{dP}=\prod_{t=1}^{T}\left (1-\frac{\hat{\delta}_{t}\cdot\Delta_{t-1}M}{1-\hat{\delta}_{t}\cdot\Delta_{t-1}A}\right )\]
provided that the process \(\hat{\delta}\), which equals
\[\hat{\delta}_{t}=\frac{\Delta_{t-1}A}{\mathbb{E}_{\mathbb{P}}[\Delta_{t-1}(S^{*})^{2}|{\cal F}_{t}]},\]
satisfies \(\hat{\delta}_{t}\Delta_{t}A<1\). If \(\hat{\mathbb{P}}\) is a probability measure, then it belongs to the class \(\bar{{\cal P}}(S^{*})\) of generalized martingale measures. As one might expect, \(\hat{\mathbb{P}}\) coincides with \(\mathbb{P}\) if the underlying probability \(\mathbb{P}\) is chosen to be martingale measure for the discounted stock price \(S^{*}\). It is important to note that for any fixed \(\mathbb{P}\), the associated minimal martingale measure \(\hat{\mathbb{P}}\) is unique, but \(\hat{\mathbb{P}}\) depends on the choice of \(\mathbb{P}\). Let us finally mention that the measure \(\tilde{\mathbb{P}}\) given by (\ref{museq410}) and (\ref{museq411}) does not necessarily coincide with the minimal martingale measure \(\hat{\mathbb{P}}\). This feature corresponds to the fact that the solutions to risk-minimizing and variance-minimizing problems are associated with different trading strategies.
