2. The setup

We first introduce the setup for the problem. Let $(\Omega, \mathcal F, \mathbb{F}, \mathbb{P})$ be a filtered probability space satisfying the usual conditions. The probability measure $\mathbb{P}$ is referred to as the physical measure. We consider the processes $X^0, X^1\colon [0, T] \times \Omega \to \mathbb R_+$ , known as the bond and stock prices respectively, satisfying the stochastic differential equations (SDE)

(2.1) \begin{align} \mathrm{d} X^0_t & = rX^0_t \, \mathrm{d} t, \nonumber \\[5pt] \mathrm{d} X^1_t & = X^1_t (\mu\, \mathrm{d} t + \sigma \, \mathrm{d} W_t), \end{align}

with $X^0_0 = 1$ , $X^1_0 = x_0$ almost surely (a.s.) for some $x_0 \in \mathbb R_+$ , and $W=\{W_t\}_{t\in[0,T]}$ a standard Brownian motion with T denoting a fixed and finite time horizon. Here, $r, \mu, \sigma \in \mathbb R$ are positive constants, respectively known as the risk-free interest rate, expected return rate, and volatility of the stock. The filtration $\mathbb{F} = \{\mathcal F_t\}_{t \geq 0}$ is the natural filtration generated by the Brownian motion W. We assume that $\mathbb F$ satisfies the usual conditions, that is, it is right continuous and contains all $\mathbb P$ -null sets.

To describe the randomness of the exercise time of life-contingent options, we let $\tau$ be an a.s. finite random variable independent of $\mathcal F$ taking finitely many values $0 < t_1 < \dots < t_n < t_{n+1} = T$ , representing the possible exercise and expiry times. We interpret the times $t_1 \dots t_n$ as being the possible occurrence times of the event of interest (e.g. the death of the policy holder) at which time the option may be exercised. On the other hand, if $\tau = t_{n+1} = T$ , we interpret it as the option having expired before the event of interest occurs. We denote by $\mathcal G$ the natural filtration of the process $G\colon[0, T]\times \Omega \to \mathbb R$ given by $G(t, \cdot) = \textbf{1}_{\{\tau \leq t\}}$ . By construction, $\tau$ is then a bounded stopping time of $\mathcal G$ .

We now introduce some key definitions.

Definition 2.1. (T-year life-contingent option.) Let T be the fixed expiration date of the life-contingent option, and $0\leq \tau \leq T$ be the stopping time of an event of interest. The payoff of a life-contingent option is an $(\mathcal F_T \vee \mathcal G_{\tau})$ -measurable random variable, where $\mathcal G_\tau$ denotes the $\sigma$ -algebra of the stopping time $\tau$ , and $\mathcal F_T \vee \mathcal G_{\tau}$ denotes the $\sigma$ -algebra generated by $\mathcal F_T$ and $\mathcal G_\tau$ . The time- $\tau$ payoff of the life-contingent option is of the form

\begin{align*}f_\tau \;:\!=\; \textbf{1}_{\{\tau < T\}} b(X^1_\tau) + \textbf{1}_{\{\tau = T\}} c(X^1_T),\end{align*}

where $X^1$ is the underlying price process satisfying (2.1) and $\tau$ is the random exercise time. $b,c\colon \mathbb R_+ \to \mathbb R_+$ are bounded Borel-measurable functions called the death benefit payoff and compensation payoff, respectively.

Thus, the policy pays death benefit $b(X^1_\tau)$ if the policy holder dies at time $\tau$ but before time $T = t_{n+1}$ , and otherwise the compensation payoff is $c(X^1_T)$ at time T.

1. (C1) The portfolio must be self-financing, i.e. $X^0_t\,\mathrm{d} H^0_t + X^1_t\,\mathrm{d} H^1_t = 0$ a.s.

2. (C2) The portfolio must have a nonnegative value at all times: $X^0_t H^0_t + X^1_t H^1_t \geq 0$ a.s. for all $t \in [0, T]$ .

We denote by $V_t (H) \;:\!=\; X^0_t H^0_t + X^1_t H^1_t$ the value process of the portfolio H at time t, and $V_0 (H)$ the initial investment of the portfolio. The set of self-financing portfolios is denoted by $\textit{SF}$ .

Definition 2.3. (Super-replication portfolio.) A super-replication portfolio for the lifetime-contingent option f is a self-financing portfolio whose associated value process V satisfies $V_\tau \geq f_\tau$ almost surely. More precisely, a super-replication portfolio is an element of the set $\mathcal S(f, \tau)$ defined by

\begin{align*}\mathcal S(f_,\tau) \;:\!=\; \{H\mid H\in\textit{SF};\; X^0_\tau H^0_\tau + X^1_\tau H^1_\tau \geq f_\tau\ \mathrm{a.s.}\}.\end{align*}

Definition 2.4. (Minimal super-replication price.) We define the minimal super-replication price $\pi_0(f)$ for the life-contingent option $(f,\tau)$ to be the infimal value of a super-replication portfolio for $(f, \tau)$ , i.e.

\begin{align*}\pi_0(f) \;:\!=\; \mathrm{inf}\{V_0 (H) \mid H = (H^0, H^1) \in \mathcal S(f, \tau)\},\end{align*}

where $V_0(H)$ is the initial investment for the portfolio H.

Recall that the market involving the two assets $X^0$ and $X^1$ is complete. This implies that there exists a unique equivalent martingale measure $\mathbb Q$ , which is a probability measure equivalent to $\mathbb P$ such that the discounted asset prices are martingales.

3. Existence of a minimal super-replication portfolio

We are now ready to state the first main theorem of the paper.

Theorem 3.1. Let b, c, f, and $\tau$ be as in the setup in Section 2. There exists a minimal hedge for the life-contingent option $(f,\tau)$ whose associated initial investment $\pi_0$ is

(3.1) \begin{equation} \pi_0 = D_0 \dots D_{n} (U), \end{equation}

where the random variable U is defined by

\begin{align*} U \;:\!=\; \begin{cases} c(X^1_T) & \textit{if }\; \mathbb P(\tau = T) \neq 0, \\[5pt] 0 & \textit{if }\; \mathbb P(\tau = T) = 0, \\[5pt] \end{cases} \end{align*}

and the operators $D_k\colon L^1(\Omega) \to \mathbb R$ are defined as follows. For any random variable $Y \in L^1(\Omega)$ , $D_0 (Y) \;:\!=\; \mathbb E^{\mathbb Q} [\mathrm{e}^{-rt_1} Y]$ ; for $1 \leq k \leq n$ ,

\begin{align*}D_{k}(Y) \;:\!=\; \max\big(b(X^1_{t_{k}}),\mathbb E^{\mathbb Q}\big[\mathrm{e}^{-r(t_{k+1}-t_{k})}Y\mid\mathcal F_{t_{k}}\big]\big).\end{align*}

Here, $\mathbb E^{\mathbb Q}$ denotes the expectation under the probability measure $\mathbb Q$ , and we recall that $t_1, \dots, t_{n+1}$ are the possible exercise or expiry times of the option.

For the proof of Theorem 3.1, we will need the following lemma on independence of filtrations, as well as a standard result from the theory of option pricing.

Proof. Let $H \in \mathcal H_t$ and $K \in K_r$ for some $t, r \geq 0$ . Then

\begin{align*} \mathbb Q(H \cap K) & = \mathbb E^{\mathbb Q}[\textbf{1}_H\textbf{1}_K] = \mathbb E^{\mathbb P}[Z\textbf{1}_H\textbf{1}_K] \\[5pt] & = \mathbb E^{\mathbb P}[Z\textbf{1}_H]\mathbb E^{\mathbb P}[\textbf{1}_K] \\[5pt] & = \mathbb E^{\mathbb P}[Z\textbf{1}_H]\mathbb E^{\mathbb P}[\textbf{1}_K]\mathbb E^{\mathbb P}[Z] \\[5pt] & = \mathbb E^{\mathbb P}[Z\textbf{1}_H]\mathbb E^{\mathbb P}[Z\textbf{1}_K] = \mathbb E^{\mathbb Q}[\textbf{1}_H]\mathbb E^{\mathbb Q}[\textbf{1}_K] = \mathbb Q(H)\mathbb Q(K), \end{align*}

where the equality on the third line is valid because $E^{\mathbb P}[Z] = \int_{\Omega}Z\,\mathrm{d}\mathbb P = \int_{\Omega}\mathrm{d}\mathbb Q = 1$ .

We now recall a key result in the theory of option pricing, which can be found in references such as [Reference Elliott and Kopp1, Theorem 7.5.10, p. 190]: given a geometric Brownian motion market model with two assets, the minimal initial investment for a hedge of a European option with payoff $h_T$ is $\mathbb E^{\mathbb Q}[\mathrm{e}^{-rT}h_T]$ , where $\mathbb Q$ denotes an equivalent martingale measure and T is the exercise time. Stated more precisely, the form of this result we will need is as follows.

Proposition 3.1. Let W be a standard Brownian motion on a filtered probability space. Suppose $X^0$ and $X^1$ are solutions to the SDE

(3.2) \begin{align} \mathrm{d} X^0_t & = rX^0_t\,\mathrm{d} t , \end{align}

(3.3) \begin{align} \mathrm{d} X^1_t & = X^1_t(\mu\,\mathrm{d} t + \sigma\,\mathrm{d} W_t) \end{align}

for $t \in [q, s]$ with initial conditions $X^0_q = x_0$ and $X^1_q = x_1$ for some $x_0, x_1 \in \mathbb R_{+}$ . Let $f_s$ be an $\mathcal F_s$ -measurable random variable. Suppose $(Z^0, Z^1)$ is a super-replication portfolio consisting of $X^0$ and $X^1$ on [q, s]. Then

\begin{align*}Z_q^0 X_q^0 + Z_q^1 X_q^1\geq\mathbb E^{\mathbb Q}[\mathrm{e}^{r(q-s)}f_s\mid\mathcal F_q]\ \mathrm{a.s.},\end{align*}

where $\mathbb Q$ is a probability measure equivalent to $\mathbb P$ under which the discounted process $\mathrm{e}^{-r(t-q)}X_t^1$ is a martingale on [q, s] with respect to the filtration $\mathbb F$ . Further, there exists a self-financing portfolio $H = (H^0, H^1)$ with $X^0_s H^0_s + X^1_s H^1_s = f_s$ a.s., in which case

\begin{align*}X^0_q H^0_q + X^1_q H^1_q = \mathbb E^{\mathbb Q}[\mathrm{e}^{r(q-s)}f_s\mid\mathcal F_q].\end{align*}

As a first step to proving Theorem 3.1, we obtain a lower bound on the value of any $(f, \tau)$ hedge at the times $t_1, \dots, t_{n+1}$ . For convenience, we write $t_0 = 0$ .

Proof. Let V(H) be the value process of a hedging portfolio $H = (H^0, H^1)$ . Assuming first that $\mathbb P(\tau = T) \neq 0$ , we will show that $V_T (H) \geq c(X^1_T)$ a.s. Indeed, assume otherwise; then the event $A \;:\!=\; \{V_T (H) < c(X^1_T)\}$ has nonzero probability. Since the two variables involved in the defining inequality are $\mathcal F_T$ -measurable, A is $\mathcal F_T$ -measurable.

But then, since $\tau$ is independent of the filtration generated by the Brownian motion, we have that $A \cap \{\tau = T\}$ has nonzero probability as well. Thus $V_\tau < c(X_\tau)$ with nonzero probability, contradicting the definition of a hedge.

By exactly the same reasoning, we conclude that

(3.4) \begin{equation} V_{t_i}(H) \geq b(X^1_{t_i}) \ \mathrm{a.s}, \end{equation}

for each $1 \leq i \leq n$ .

It remains to show the inequalities

(3.5) \begin{equation} V_{t_i} (H) \geq D_{i} \ldots D_{n} (U)\; \mathrm{ a.s}, \end{equation}

for each $0 \leq i \leq n$ . We treat the cases $1 \leq i \leq n$ and $i = 0$ separately. For the former, we take a dynamic programming approach and induct backwards on i.

For the base case, we must show that

(3.6) \begin{equation} V_{t_{n}} (H) \geq D_{n} (U)\; \mathrm{ a.s.} \end{equation}

We first note that by the Markov property of Itô SDEs, conditional on $\mathcal F_{t_{n}}$ , we find that $X^0, X^1$ , and the portfolio $H^0, H^1$ restricted to $[t_{n}, T]$ satisfy the hypotheses of Proposition 3.1.

Indeed, $X^0$ and $X^1$ satisfy the SDEs (3.2) and (3.3) on the interval $[t_n, T]$ , and, from above, $V_T (H) = X^0_T H^0_T + X^1_T H^1_T \geq c(X^1_T)$ a.s. Thus $V_{t_{n}}(H)\geq\mathbb E^{\mathbb Q}[\mathrm{e}^{r(t_n-T)}c(X^1_T)\mid\mathcal F_{t_{n}}]$ . Combining this with the fact that $V_{t_{n}}(H)\geq b(X_{t_{n}})$ from (3.4), we conclude the inequality (3.6) as desired.

For the induction step, let $2 \leq k \leq n$ , and assume that $V_{t_k} (H) \geq D_{k} \dots D_{n} (U)$ a.s. We must show that $V_{t_{k-1}}(H) \geq D_{k-1} \dots D_{n} (U)$ a.s. However, conditional on $\mathcal F_{t_{k-1}}$ , we again find that $X^0, X^1$ , and the portfolio $H^0, H^1$ restricted to $[t_{k-1}, t_k]$ , satisfy the hypotheses of Proposition 3.1. Indeed, we again check that $X^0$ and $X^1$ satisfy the SDEs (3.2) and (3.3) on the interval $[t_{k-1}, t_k]$ and, by the induction hypothesis,

\begin{align*}V_{t_k}(H) = X^0_{t_{k}}H^0_{t_{k}} + X^1_{t_{k}}H^1_{t_{k}} \geq D_{k} \dots D_{n} (U), \mathrm{ a.s.}\end{align*}

Together with (3.4), we deduce that

\begin{equation*} V_{t_{k-1}}(H) \geq \mathrm{max}\big(b(X^1_{t_{k-1}}),\mathbb E^{\mathbb Q} \big[\mathrm{e}^{r(t_{k-1}-t_{k})}D_{k}\dots D_{n}(U)\mid\mathcal F_{t_{k-1}}\big]\big) = D_{k-1} \dots D_{n} (U)\; \mathrm{ a.s}, \end{equation*}

which proves the case $1 \leq i \leq n$ in (3.5) as required. Finally, one more application of Proposition 3.2 proves the case $i = 0$ in (3.5).

In the case where $\mathbb P(\tau = T) = 0$ , we note that by similar considerations to earlier, we still have the inequality $V_{t_n}(H) \geq b(X^1_{t_n})$ a.s., whence the rest of the proof proceeds verbatim.

Now we set out to construct a minimal hedge. Before we do so, we need the following generalities on regular conditional probabilities. The definitions below are largely based on [Reference Leao, Fragoso and Ruffino7].

1. (i) $\nu(x, \cdot)$ is a probability measure on $(\Omega, \mathcal F)$ for all $x \in E$ ;

2. (ii) $\nu(\cdot, A)$ is an $\mathcal E$ -measurable function on E for all $A \in \mathcal F$ .

Definition 3.2. (Regular conditional probability.) Let $T \colon \Omega \to E$ be a measurable function. A regular conditional probability with respect to T is a transition probability $\nu \colon E \times \mathcal F \to [0, 1]$ from $(E,\mathcal E)$ to $(\Omega,\mathcal F)$ such that

\begin{align*}\mathbb P[A \cap T^{-1} (B)] = \int_B \nu(x, A)\,T_*\mathbb P(\mathrm{d} x)\end{align*}

for all $x \in E$ , $A \in \mathcal F$ , and $B \in \mathcal E$ , where $T_* \mathbb P$ denotes the image measure of $\mathbb P$ under T.

The following is [Reference Yong and Zhou11, Proposition 1.9], and gives sufficient conditions for regular conditional probabilities to exist. We then state a proposition which ensures that regular conditional probabilities exist in our setting.

We shall need the following technical proposition, whose proof we relegate to the Appendix.

We are now ready to construct our minimal hedge and, in doing so, prove Theorem 3.1.

Proof of Theorem 3.1. Denote by $0 < t_1 < \dots < t_n < t_{n+1} = T$ the values of $\tau$ that occur with nonzero probability, with the exception of $\mathbb P(\tau = t_{n+1})$ which is permitted to possibly be 0. In the notation introduced in the statement of the theorem, write, for convenience, $J_i \;:\!=\; D_{i} \dots D_{n} (U)$ for $1 \leq i \leq n$ .

We define our hedging process H on $[0, t_1]$ by $H^0 = Y^0$ , $H^1 = Y^1$ , where $Y^0, Y^1$ are defined as follows. By Proposition 3.2, given $X^0, X^1$ as in the setup, there exists an $\mathbb F$ -adapted solution $(Y^0, Y^1, V)$ to the following system (FBSDE 1) of forward–backward stochastic differential equations,

(3.7) \begin{align} \mathrm{d} X^0_t & = rX^0_t \, \mathrm{d} t, \end{align}

(3.8) \begin{align} \mathrm{d} X^1_t & = X^1_t(\mu\,\mathrm{d} t + \sigma\,\mathrm{d} W_t), \end{align}

(3.9) \begin{align} \mathrm{d} V_t & = Y^0_t\,\mathrm{d} X^0_t + Y^1_t\,\mathrm{d} X^1_t, \end{align}

(3.10) \begin{align} 0 & = X^0_t\,\mathrm{d} Y^0_t + X^1_t\,\mathrm{d} Y^1_t \end{align}

for $t \in [0, t_1]$ under $\mathbb P$ with initial conditions $X_0^0 = 1$ , $X_0^1 = x$ , and terminal condition $V_{t_1} = J_1$ almost surely.

Suppose now inductively our processes $(H^0, H^1, V(H))$ have already been defined on $[0,t_{i}]$ for some $1 \leq i \leq n$ and satisfy (3.7)–(3.10), and further that $V_{t_k} (H) \geq J_k$ for each $1 \leq k \leq i$ . Consider the regular conditional probability $\xi$ of $\mathbb P$ given $\mathcal F_{t_i}$ .

Now fix $x \in \mathbb R_+$ and $\omega \in \Omega$ such that $X^1_{t_i} (\omega) = x$ , and consider the system (FBSDE 2)

\begin{align*} \mathrm{d} X^0_t & = r X^0_t \, \mathrm{d} t, \\[5pt] \mathrm{d} X^1_t & = X^1_t(\mu\,\mathrm{d} t + \sigma\,\mathrm{d} W_t), \\[5pt] \mathrm{d} R_t & = Z^0_t\,\mathrm{d} X^0_t + Z^1_t\,\mathrm{d} X^1_t, \\[5pt] 0 & = X^0_t\,\mathrm{d} Z^0_t + X^1_t\,\mathrm{d} Z^1_t \end{align*}

for $t \in [t_i, t_{i+1}]$ under the probability measure $\xi(\omega, \cdot)$ with initial conditions $X_{t_i}^0 = \mathrm{e}^{rt_i}$ , $X_{t_i}^1 = x$ , and terminal condition $R_{t_{i + 1}} = J_{i+1}$ , $\xi(\omega, \cdot)$ -a.s. Note that $\xi(\omega, \cdot)$ is supported on the event $\{X^1_{t_i} = x\}$ , since $X^1_{t_i}$ is $\mathcal F_{t_i}$ -measurable, and $X^1_{t_i} (\omega) = x$ .

By Proposition 3.1, for each such $\omega \in \Omega$ there exists a $\xi(\omega,\cdot)$ -a.s. well-defined solution $(Z^0,Z^1,R) \;=\!:\; (Z^{0,x},Z^{1,x},R^x)$ to (FBSDE 2). We define our process for $t \in (t_i, t_{i+1}]$ by

\begin{align*} H^0_t & = Z^{0, x}_t + \mathrm{e}^{-rt_i}(V_{t_i} (H) - Z^{0, x}_{t_i} X^0_{t_i} - Z^{1, x}_{t_i} X^1_{t_i}), \\[5pt] H_t^1 & = Z_t^{1, x}, \\[5pt] V_t(H) & = \mathrm{e}^{-rt_i}(V_{t_i} (H) - Z^{0, x}_{t_i} X^0_{t_i} - Z^{1, x}_{t_i} X^1_{t_i}) X_{t}^0 + Z_t^{0, x} X_t^0 + Z_t^{1, x} X_t^1 = H^0_t X^0_t + H^1_t X^1_t \end{align*}

on the event $\{X^1_{t_i} = x\}$ . This defines $(H^0, H^1, V)$ $\mathbb P$ -a.s. up to $[0,t_{i+1}]$ .

Indeed, denoting by E the set on which $(H^0, H^1, V)$ is well defined up to $[0,t_{i+1})$ , we have

\begin{align*}\mathbb P(E) = \int_{\Omega}\xi(\omega,E)\,\mathbb P(\mathrm{d}\omega) = \int_{\Omega}1\,\mathbb P(\mathrm{d}\omega) = 1.\end{align*}

We now show that $(H^0, H^1, V(H))$ satisfies the system (FBSDE 1) on $[0, t_{n+1}]$ under $\mathbb P$ with initial conditions $X_0^0 = 1$ , $X_0^1 = x$ , and terminal condition $V_{t_{i + 1}}(H) \geq J_{i+1}$ , almost surely. That (3.7) and (3.8) are satisfied has already been established a priori in the construction of the market model. For $t\in[t_i,t_{i+1}]$ , (3.9) is satisfied. Indeed, writing $E_{t_i} \;:\!=\; \mathrm{e}^{-rt_i}(V_{t_i} (H) - Z^{0, x}_{t_i} X^0_{t_i} - Z^{1, x}_{t_i} X^1_{t_i})$ , we get

\begin{align*} \mathrm{d} V_t(H) & = E_{t_i}\,\mathrm{d} X^0_t + \mathrm{d} Z_t^{0, x}X^0_t + \mathrm{d} Z_t^{1, x}X^1_t + Z_t^{0, x}\,\mathrm{d} X^0_t + Z_t^{1, x}\,\mathrm{d} X^1_t \\[5pt] & = E_{t_i}\,\mathrm{d} X^0_t + Z_t^{0, x}\,\mathrm{d} X^0_t + Z_t^{1, x}\,\mathrm{d} X^1_t \\[5pt] & = H_t^0\,\mathrm{d} X^0_t + H_t^1\,\mathrm{d} X^1_t. \end{align*}

But we may also compute, for $s \in [0, t_i]$ and $t \in [t_i, t_{i+1}]$ ,

\begin{align*} V_t (H) - V_s(H) & = (V_t (H) - V_{t_i} (H)) + (V_{t_i} (H) - V_s (H)) \\[5pt] & = \int_{t_i}^t H^0_r\,\mathrm{d} X^0_r + \int_{t_i}^{t}H^1_r\,\mathrm{d} X^1_r + \int_{s}^{t_i}H^0_r\,\mathrm{d} X^0_r + \int_s^{t_i}H^1_r\,\mathrm{d} X^1_r \\[5pt] & = \int_{s}^t H^0_r\,\mathrm{d} X^0_r + \int_{s}^t H^1_r\,\mathrm{d} X^1_r, \end{align*}

which shows that (3.9) holds on $[0, t_{i+1}]$ . Finally, we check the self-financing condition (3.10). Recall that we have assumed inductively that (3.10) holds on $[0, t_i]$ . Thus we need only check that (3.10) is satisfied for times s, t in the interval $[0, t_{i+1}]$ , with $s < t$ , and $t \in (t_i, t_{i+1}]$ .

Assume first that $s \leq t_{i}$ . Then we have

\begin{align*} \int_s^t X^0_r\,\mathrm{d} H^0_r + \int_s^t X^1_r\,\mathrm{d} H^1_r & = \int_{[s, t_i)}X^0_r\,\mathrm{d} H^0_r + \int_{[s, t_i)}X^1_r\,\mathrm{d} H^1_r + \int_{(t_i, t]}X^0_r\,\mathrm{d} H^0_r + \int_{(t_i, t]}X^1_r\,\mathrm{d} H^1_r \\[5pt] & \quad + X^0_{t_i}(H^{0+}_{t_i} - H^{0}_{t_i}) + X^1_{t_i}(H^{1+}_{t_i} - H^{1}_{t_i}). \end{align*}

The first two terms are 0 by the induction hypothesis. We claim that the third and fourth terms are also 0. Indeed, on $[t_i, t_{i+1})$ , we have $H^0 = Z_t^{0, x} + E_{t_i}$ and $H^1 = Z_t^{1, x}$ , so that for, $t \in [t_i, t_{i+1})$ , $\mathrm{d} H^0_t = \mathrm{d} Z_t^{0, x}$ and $\mathrm{d} H^1_t = \mathrm{d} Z_t^{1, x}$ , and thus, by (FBSDE 2),

\begin{equation*} \int_{(t_i, t]}X^0_r\,\mathrm{d} H^0_r + \int_{(t_i, t]}X^1_r\,\mathrm{d} H^1_r = \int_{(t_i, t]}X^0_r\,\mathrm{d} Z^0_r + \int_{(t_i, t]}X^1_r\,\mathrm{d} Z^1_r = 0. \end{equation*}

Now, for the last two terms, we note that $X^0_{t_i} H^{0}_{t_i} + X^1_{t_i}H_{t_i}^{1} = V_{t_i}$ a.s., so

\begin{align*} \int_s^t X^0_r\,\mathrm{d} H^0_r + \int_s^t X^1_r\,\mathrm{d} H^1_r & = X^0_{t_i}H^{0+}_r + X^1_{t_i}H^{1+}_r - V_{t_i} (H) \\[5pt] & = X^0_{t_i}\bigg[Z_{t_i}^{0,x}+\frac{1}{X^0_{t_i}}(V_{t_i}(H)-Z_{t_i}^{0,x}X^0_{t_i}-Z^{1,x}_{t_i}X^1_{t_i})\bigg] + X^1_{t_i}Z_{t_i}^{1, x} - V_{t_i}(H) \\[5pt] & = X^0_{t_i}Z^{0,x} + V_{t_i}(H) - X^0_{t_i}Z^{0,x}_{t_i} - X^1_{t_i}Z^{1,x}_{t_i} + X^1_{t_i}Z_{t_i}^{1,x} - V_{t_i}(H) = 0. \end{align*}

Next, assume $s \in (t_i, t_{i+1}]$ . Then we simply have

\begin{equation*} \int_s^t X^0_r\,\mathrm{d} H^0_r + \int_s^t X^1_r\,\mathrm{d} H^1_r = \int_s^t X^0_r\,\mathrm{d} Z^0_r + \int_s^t X^1_r\,\mathrm{d} Z^1_r = 0. \end{equation*}

Thus, $X^0_t\,\mathrm{d} H^0_t + X^1_t\,\mathrm{d} H^1_t = 0$ for $t \in [0, t_{i+1}]$ , and so the self-financing condition (3.10) is verified.

It remains to check that the terminal condition $V_{t_{i + 1}} \geq J_{i+1}$ holds almost surely. But, by construction, $V_{t_{i + 1}} \geq J_{i+1}$ , $\mu(\omega, \cdot)$ -a.s. for each $\omega$ , so we have, denoting by F the event $\{V_{t_{i + 1}} \geq J_{i+1}\}$ ,

\begin{align*}\mathbb P(F) = \int_{\Omega}\xi(\omega,F)\,\mathbb P_{|\mathcal F_{t_i}}(\mathrm{d}\omega) = \int_{\Omega}1\,\mathbb P_{|\mathcal F_{t_i}}(\mathrm{d}\omega) = 1,\end{align*}

where $\mathbb P_{|\mathcal F_{t_i}}$ denotes the pushforward measure $I_*\mathbb P$ of $\mathbb P$ under the identity map $I\colon (\Omega, \mathcal F_{t_i}) \to (\Omega, \mathcal F)$ . Hence, we conclude that the FBSDE holds on the interval $[0, t_{i+1}]$ . Inductively, we obtain a solution $(V, H^0, H^1)$ on the whole interval [0, T].

To see that this is a minimal hedge, we note that non-negativity holds since on each interval $[t_i, t_{i+1}]$ the value process is the sum of the two non-negative portfolios $(Z^0, Z^1)$ and $\big(\mathrm{e}^{-rt_i}(V_{t_i}(H) - Z^0_{t_i}X^0_{t_i} - Z^1_{t_i}X^1_{t_i}), 0\big)$ , where the non-negativity of the second portfolio follows from the fact that $V_{t_i} (H) \geq J_{t_i}$ . Further, the hedging property holds since $V_{t_i} (H) \geq J_i \geq b(X^1_{t_i})$ for all i, so $V_{\tau} (H) \geq b(X_{\tau})$ almost surely. And finally, since $V_{t_1} = J_1$ , the initial investment $V_0$ is $V_0 = D_0(J_1) = D_0 \dots D_{n}(U)$ . So the hedge achieves the infimal hedging price, and is thus a minimal hedge.

Next, we derive an explicit expression for the minimal super-replicating hedge and hedging price associated to a particular life-contingent option. Due to the iterated conditional expectations in (3.1), it is in general difficult or impossible to obtain closed-form solutions for the super-replication price and minimal hedge. However, we are able to do so here for a very simple case.

Proposition 3.5. Let the market model be as in Section 2. Consider a life-contingent option $f_\tau = b(X_\tau)\textbf{1}_{\{\tau < T\}} + c(X_\tau)\textbf{1}_{\{\tau = T\}}$ with only two exercise times $0 < t_1 < t_2 = T$ , where we assume $\mathbb P(\tau = t_i) \neq 0$ for each $i = 1, 2$ . We suppose that the payoffs b, c are given by $b(x) = \max(K, x)$ and $c(x) = x$ respectively, where $K > 0$ is a strike price. We note that $b(x) = x + (K - x)^+$ , i.e. it is the combination of a long position in the stock and a put option on the stock. Then the minimal initial investment for a super-replicating portfolio, $\pi_0$ , is given by $\pi_0 = X^1_0 + \mathbb E^{\mathbb Q}\big[\mathrm{e}^{-r t_1}\big(K - X^1_{t_1}\big)^+\big]$ , where we recall that $\mathbb Q$ is an equivalent probability measure under which the discounted asset prices are martingales.

A minimal super-replicating hedge $H \;:\!=\; (H^0, H^1)$ is given by

\begin{align*} H^0_t & \;:\!=\; \begin{cases} K\mathrm{e}^{-rt_1}\bigg(1 - \Phi\bigg(\dfrac{\log({X^1_t}/{K}) + (t_1 - t)(r - ({\sigma^2}/{2}))} {\sigma\sqrt{t_1 - t}}\bigg)\bigg) & \textit{for }\; 0 \leq t \leq t_1, \\[5pt] \mathrm{e}^{-rt_1}\big(K - X^1_{t_1}\big)^+ & \textit{for }\; t_1 < t \leq T, \\[5pt] \end{cases} \\[5pt] H^1_t & \;:\!=\; \begin{cases} \Phi\bigg(\dfrac{\log({X^1_t}/{K}) + (t_1 - t)(r + ({\sigma^2}/{2}))} {\sigma\sqrt{t_1 - t}}\bigg) & \textit{for }\; 0 \leq t \leq t_1, \\[5pt] 1 & \textit{for }\; t_1 < t \leq T, \\[5pt] \end{cases} \end{align*}

where $\Phi(y) \;:\!=\; ({1}/{\sqrt{2 \pi}})\int_{-\infty}^y\mathrm{e}^{-{z^2}/{2}}\,\mathrm{d} z$ denotes the cumulative distribution function of the standard normal random variable.

Proof. Recall the notation

\begin{align*}D_0(Y) \;:\!=\; \mathbb E^{\mathbb Q}[\mathrm{e}^{-rt_1}Y], \qquad D_{k}(Y) \;:\!=\; \max\big(b(X^1_{t_{k}}),\mathbb E^{\mathbb Q} \big[\mathrm{e}^{-r(t_{k+1} - t_{k})}Y\mid\mathcal F_{t_{k}}\big]\big)\end{align*}

for any random variable Y. By Theorem 3.1, the minimal super-replication price $\pi_0$ is then given by $\pi_0 = D_0 D_1\big[c\big(X^1_T\big)\big]$ . First, note that

\begin{align*} D_{1}\big[c\big(X^1_T\big)\big] = D_1\big[X^1_T\big] & = \max\big(b\big(X^1_{t_{1}}\big),\mathbb E^{\mathbb Q}\big[\mathrm{e}^{-r(T - t_1)}X^1_T\mid\mathcal F_{t_{1}}\big]\big) \\[5pt] & = \max\big(b\big(X^1_{t_{1}}\big), X^1_{t_1}\big) \\[5pt] & = \max\big(\max\big(K, X^1_{t_1}\big), X^1_{t_1}\big) = \max\big(K , X^1_{t_1}\big) , \end{align*}

where on the second line we have used the martingale property of the discounted stock price under $\mathbb Q$ . To compute $\pi_0$ , we have

\begin{align*} \pi_0 = D_0D_{1}\big[c\big(X^1_T\big)\big] & = D_0\big[\max\big(K , X^1_{t_1}\big)\big] \\[5pt] & = E^{\mathbb Q}\big[\mathrm{e}^{-rt_1}\max\big(K, X^1_{t_1}\big)\big] \\[5pt] & = E^{\mathbb Q}\big[\mathrm{e}^{-rt_1}X^1_{t_1} + \mathrm{e}^{-rt_1}\big(K - X^1_{t_1}\big)^+\big] \\[5pt] & = X^1_0 + E^{\mathbb Q}\big[\mathrm{e}^{-rt_1}\big(K - X^1_{t_1}\big)^+\big], \end{align*}

where again in the third line we have used the martingale property of the discounted stock price. Note that the minimal super-replication portfolio can be viewed as a combination of a long position in the underlying and a long position of an European put option with exercise time $t_1$ .

Next, we derive the expression for the hedge. First, we show that the given portfolio $H \;:\!=\;(H^0, H^1)$ is well defined, i.e. it satisfies assumptions (C1) and (C2) in Definition 2.2. To this end, consider a European put option on the underlying with exercise time $t_1$ and strike price K, with corresponding payoff $g\big(X^1_{t_1}\big) = \big(K - X^1_{t_1}\big)^+$ . By standard results (see [Reference Elliott and Kopp1, Theorem 7.6.2]), a replicating portfolio $R \;:\!=\; (R^0, R^1)$ of the European put option with exercise time $t_1$ for $t \in [0, t_1]$ is given by

\begin{align*} R^0_t & \;:\!=\; K\mathrm{e}^{-rt_1}\bigg(1 - \Phi\bigg(\frac{\log({X^1_t}/{K}) + (t_1 - t)(r - ({\sigma^2}/{2}))} {\sigma\sqrt{t_1 -t}}\bigg)\bigg), \\[5pt] R^1_t & \;:\!=\; \Phi\bigg(\frac{\log({X^1_t}/{K}) + (t_1 - t)(r + ({\sigma^2}/{2}))} {\sigma\sqrt{t_1 - t}}\bigg) - 1. \end{align*}

Noting this, we may write our portfolio H as

\begin{align*} H^0_t & \;:\!=\; \begin{cases} R^0_t + J^0_t & \mathrm{for }\; 0 \leq t \leq t_1, \\[5pt] \mathrm{e}^{-rt_1}\big(K - X^1_{t_1}\big)^+ & \mathrm{for }\; t_1 < t \leq T, \\[5pt] \end{cases} \\[5pt] H^1_t & \;:\!=\; \begin{cases} R^1_t + J^1_t & \mathrm{for }\; 0 \leq t \leq t_1, \\[5pt] 1 & \mathrm{for }\; t_1 < t \leq T, \\[5pt] \end{cases} \end{align*}

where $J^0_t = 0$ , $J^1_t = 1$ . Now we check that assumption (C2) holds. Indeed, H is clearly non-negative on $(t_1, T]$ , since the holdings in both the riskless and underlying assets are non-negative, while on $(0, t_1)$ it is the sum of the two non-negative portfolios R and $J = (J^0, J^1) \;:\!=\; (0, 1)$ .

Next, we check the self-financing condition (C1). On $(0, t_1]$ , it is the sum of the two self-financing portfolios R and J, and thus is self-financing on this interval. On the other hand, on the interval $(t_1, T]$ , $\mathrm{d} H^0 = \mathrm{d} H^1 = 0$ , and thus we need only check the self-financing condition for times s, t with $s < t_1 < t$ . To this end, we compute

\begin{align*} \int_s^t X^0_t\,\mathrm{d} H^0_t + \int_s^t X^1_t\,\mathrm{d} H^1_t & = \int_s^{t_1}X^0_t\,\mathrm{d} H^0_t + \int_s^{t_1}X^1_t\,\mathrm{d} H^1_t + X^0_{t_1}\big(H^{0+}_{t_1} - H^0_{t_1}\big) + X^1_{t_1}\big(H^{1+}_{t_1} - H^1_{t_1}\big) \\[5pt] & = X^0_{t_1}\big(H^{0+}_{t_1} - H^0_{t_1}\big) + X^1_{t_1}\big(H^{1+}_{t_1} - H^1_{t_1}\big) \\[5pt] & = X^0_{t_1}H^{0+}_{t_1} + X^1_{t_1}H^{1+}_{t_1} - V_{t_1}(H) \\[5pt] & = \mathrm{e}^{rt_1}\big(\mathrm{e}^{-rt_1}\big(K - X^1_{t_1}\big)^+\big) + X^1_{t_1} - \big[\big(K - X^1_{t_1}\big)^+ + X^1_{t_1}\big] = 0, \end{align*}

where we have written $H^{0+}_{t_1}$ to denote $\lim_{t \to t_1^+} H^0_t$ , and likewise for $H^{1+}_{t_1}$ .

Thus, the portfolio H is a self-financing portfolio. Now we check that it is indeed a super-replicating portfolio for the contingent option. It is sufficient to check that $V_{t_i}(H) \geq f_{t_i}$ a.s. for $i = 1, 2$ . But for $i = 1$ , we see that

\begin{equation*} V_{t_1}(H) = V_{t_1}(R) + V_{t_i}(J) = \big(K - X^1_{t_1}\big)^+ + X^1_{t_1} = \max\big(K, X^1_{t_1}\big) = f_{t_1}, \end{equation*}

while for $i = 2$ , i.e. at time $t_2 = T$ , we have

\begin{equation*} V_{T}(H) = X^1_{T} + X^0_T\mathrm{e}^{-rt_1}\big(K - X^1_{T}\big)^+ \geq X^1_{T} = f_T. \end{equation*}

This shows that the portfolio is super-replicating, as desired. Finally, we check that H is a minimal super-replicating portfolio. It is sufficient to check that H achieves the minimal super-replication price, which by Theorem 3.1 we know is

\begin{align*}\pi_0 \;:\!=\; X^1_0 + \mathbb E^{\mathbb Q}\big[\mathrm{e}^{-r t_1}\big(K - X^1_{t_1}\big)^+\big].\end{align*}

But, by writing

\begin{equation*} V_0(H) = V_0(J) + V_0(R) = X^1_0 + \mathbb E^{\mathbb Q}\big[\mathrm{e}^{-r t_1}\big(K - X^1_{t_1}\big)^+\big] = \pi_0, \end{equation*}

we verify this immediately. This concludes the proof.

4. Existence of a replicating portfolio

Given that a super-replication trading strategy exists, a natural question to ask is: when is this super-replication portfolio a replicating portfolio? That is, the payoff of the portfolio is exactly the same as the option payoff at the exercise time. To make the above precise, we record here a few initial definitions.

Now we state the main theorem of this section.

For the other direction, suppose the life-contingent option is universally replicable, and let $s, t \in [0, T]$ , $s < t$ , be arbitrary times. We want to show that $\tilde f_s = \mathbb E^{\mathbb Q}[\tilde f_t\mid \mathcal F_s]$ almost surely. To this end, let $\tau$ be a stopping time independent of the asset filtration taking value s or t with probability $\frac{1}{2}$ each. Since the life-contingent option is universally replicable, there exists a replicating portfolio H associated with this $\tau$ . By the same argument as in the proof of Proposition 3.2, we must have that the value process V of H satisfies $V_t = f_t$ a.s., and $V_s = f_s$ a.s.

From the equality $V_t = f_t$ a.s., we deduce that H is a replicating portfolio for the simple European-style claim with payoff $f_t$ at time t. By [Reference Pascucci8, Theorem 7.13], the replicating portfolio for such a payoff is unique, and hence the equality in Proposition 3.1 holds, i.e. $V_s = \mathbb E^{\mathbb Q}[\mathrm{e}^{r(s-t)}f_t\mid\mathcal F_s]$ . Since $V_s = f_s$ a.s., we have $f_s = \mathbb E^{\mathbb Q}[\mathrm{e}^{r(s-t)}f_t\mid\mathcal F_s]$ , and rearranging the equality leads to the desired result $\tilde f_s = \mathbb E^{\mathbb Q}[\tilde f_t\mid\mathcal F_s]$ a.s.