Mean-Variance Hedging on Uncertain Time Horizon in a Market with a Jump

Abstract

In this work, we study the problem of mean-variance hedging with a random horizon T∧τ, where T is a deterministic constant and τ is a jump time of the underlying asset price process. We first formulate this problem as a stochastic control problem and relate it to a system of BSDEs with a jump. We then provide a verification theorem which gives the optimal strategy for the mean-variance hedging using the solution of the previous system of BSDEs. Finally, we prove that this system of BSDEs admits a solution via a decomposition approach coming from filtration enlargement theory.

Appendix

1.1 A.1 Proof of Proposition 2.1

We first suppose that X is a nonnegative $\mathcal{P}(\mathbb {G})$-measurable process. For n≥1, we define the process X ⁿ by

$$\begin{aligned} X^n_t = & X_t\wedge n ,\quad t\in[0,T] . \end{aligned}$$

Then X ⁿ is a bounded $\mathbb{G}$-predictable process, and from Lemma 4.4 in [10], there exist a $\mathcal{P}(\mathbb{F})$-measurable process X ^n,b and a $\mathcal{P}(\mathbb{F})\otimes\mathcal{B}(\mathbb{R}_{+})$-measurable process X ^n,a such that

(A.1)

Since the sequence (X ⁿ) _n is nondecreasing, we can assume w.l.o.g. that the sequences (X ^a,n) _n and (X ^b,n) _n are also nondecreasing. Define the processes X ^a and X ^b by

$$X^a = \lim_{n\rightarrow\infty} X^{n,a} \quad \mbox{and}\quad X^b = \lim_{n\rightarrow\infty} X^{n,b} . $$

Then X ^a is $\mathcal{P}(\mathbb{F})\otimes\mathcal{B}(\mathbb {R}_{+})$-measurable and X ^b is $\mathcal{P}(\mathbb{F})$-measurable and sending n to infinity in (A.1), we get

(A.2)

For a general $\mathcal{P}(\mathbb{G})$-measurable process X, we write X=X ⁺−X ⁻ where X ⁺=max(X,0) and X ⁻=max(−X,0) and we apply the previous result to the nonnegative processes X ⁺ and X ⁻. From the linear stability of the decomposition (A.2) we get the result.  □

1.2 A.2 BMO Stability

Theorem A.1

Let $\mathbb{Q}_{1}$ and $\mathbb{Q}_{2}$ be two probability measures on $(\varOmega,\mathcal{G})$. Let M and N be two continuous $(\mathbb{F},\mathbb{Q}_{1})$-local martingales with $N\in\rm{BMO}(\mathbb{Q}_{1})$. Suppose that $\mathbb{Q}_{1}$ and $\mathbb{Q}_{2}$ are equivalent with ${d\mathbb{Q}_{2}\over d\mathbb{Q}_{1}} |_{\mathcal{F}_{T}}={\mathcal{E}}(N)_{T}$. If $M\in\rm {BMO}(\mathbb{Q}_{1})$ then $M-\langle M,N\rangle\in\rm{BMO}(\mathbb{Q}_{2})$.

Proof

This result is a direct consequence of Theorem 3.6 in [12]. □

1.3 A.3 An Estimate for Conditional Moments

Proposition A.1

Let A be a continuous increasing $\mathbb{F}$-adapted process. Fix a t≥0 such that there exists a constant C>0 satisfying

$$\begin{aligned} \mathbb{E} [A_t-A_s | \mathcal{F}_s ] \leq& C , \end{aligned}$$

for any s∈[0,t]. Then, we have for any s∈[0,t] and any p≥1

$$\begin{aligned} \mathbb{E} \bigl[{|A_t-A_s|}^{p} | {\mathcal{F}}_s \bigr]\leq p! |C|^{p} \end{aligned}$$

and

$$\begin{aligned} \mathbb{E} \bigl[\exp \bigl(\delta(A_t-A_s) \bigr) \big| \mathcal {F}_s \bigr] \leq& \frac{1}{1-\delta C} , \end{aligned}$$

for any $\delta\in(0,{1\over C})$.

Proof

Let A be a continuous increasing $\mathbb {F}$-adapted process satisfying $\mathbb{E}[A_{t}-A_{s} | \mathcal{F}_{s}] \leq C$ for any s∈[0,t]. We first prove by iteration that $\mathbb{E} [{|A_{t}-A_{s}|}^{p} | {\mathcal{F}}_{s}]\leq p! |C|^{p}$ for any p≥1.

• For p=1, we have by assumption $\mathbb{E}[A_{t}-A_{s} | \mathcal{F}_{s}] \leq C$.

• Suppose that for some p≥2, we have $\mathbb{E}[{|A_{t}-A_{s}|}^{p-1} | {\mathcal{F}}_{s}]\leq(p-1)! |C|^{p-1}$. Since A is a continuous increasing $\mathbb{F}$-adapted process we have

  $$\begin{aligned} {|A_t-A_s|}^{p}=p \int_s^t {|A_t-A_u|}^{p-1} dA_u , \end{aligned}$$

  for any s∈[0,t]. Consequently we get

  $$\begin{aligned} \mathbb{E} \bigl[|A_t-A_s|^{p} | \mathcal{F}_s \bigr] = & p \mathbb{E} \biggl[ \int_s^t |A_t-A_u|^{p-1}dA_u \bigg| \mathcal{F}_s \biggr] \\ = & p \mathbb{E} \biggl[ \int_s^t \mathbb{E} \bigl[|A_t-A_u|^{p-1} | \mathcal{F}_u \bigr] dA_u \bigg| \mathcal{F}_s \biggr] \\ \leq& p! |C|^{p-1} \mathbb{E}[ A_t - A_s |\mathcal{F}_s] \\ \leq& p! |C|^p . \end{aligned}$$

• Since the result holds true for p=1 and for any p≥2 as soon as it holds for p−1, it holds for p, we get

  $$\begin{aligned} \mathbb{E} \bigl[{|A_t-A_s |}^{p} | { \mathcal{F}}_s\bigr]\leq p! |C|^{p} , \end{aligned}$$

  for any p≥1.

From this last inequality, we get for any $\delta\in(0, \frac{1}{C})$

$$\begin{aligned} \mathbb{E} \biggl[\sum_{p\geq0}{1\over p!}| \delta|^p {|A_t-A_s|}^p \bigg| \mathcal {F}_s \biggr] \leq& \sum_{p\geq0} |\delta C|^p = {1\over1-\delta C} , \end{aligned}$$

which is the expected result. □