American and Bermudan Options in Currency Markets with Proportional Transaction Costs

Abstract

The pricing and hedging of a general class of options (including American, Bermudan and European options) on multiple assets are studied in the context of currency markets where trading is subject to proportional transaction costs, and where the existence of a risk-free numéraire is not assumed. Constructions leading to algorithms for computing the prices, optimal hedging strategies and stopping times are presented for both long and short option positions in this setting, together with probabilistic (martingale) representations for the option prices.

Appendix: Proof of Lemma 2

Lemma 2 in Sect. 5 depends on the following technical result.

Lemma 3

Fix some i=1,…,d, and let A ₁,…A _n be non-empty closed convex sets in $\mathbb{R}^{d}$ such that $\operatorname {dom}\delta^{\ast}_{A_{k}}$ is compactly i-generated for all k. Define $A:=\bigcap_{k=1}^{n} A_{k}\neq \emptyset$; then

$$\begin{aligned} \delta^\ast_A = \operatorname {conv}\bigl\{ \delta^\ast_{A_1}, \ldots,\delta^\ast_{A_n}\bigr\} , \end{aligned}$$

and for each $x\in\sigma_{i}(\operatorname {dom}\delta^{\ast}_{A})$ there exist α ₁,…,α _n ≥0 and x ₁,…,x _n with $x_{k}\in\sigma_{i}(\operatorname {dom}\delta^{\ast}_{A_{k}})$ for all k such that

$$\begin{aligned} \delta^\ast_A(x) = \sum_{k=1}^n\alpha_k\delta^\ast_{A_k}(x_k), \quad \sum_{k=1}^n\alpha_k = 1, \ \sum_{k=1}^n\alpha_kx_k = x. \end{aligned}$$

The cone $\operatorname {dom}\delta^{\ast}_{A}$ is moreover compactly i-generated and

$$\begin{aligned} \operatorname {dom}\delta^\ast_{A} = \operatorname {conv}\Biggl[\bigcup _{k=1}^n\operatorname {dom}\delta^\ast _{A_k}\Biggr]. \end{aligned}$$

(A.1)

Proof

Let $f:=\operatorname {conv}\{\delta^{\ast}_{A_{1}}, \ldots, \delta^{\ast}_{A_{n}}\}$. Then $\operatorname {cl}f = \delta^{\ast}_{A}$; see [23, Corollary 16.5.1]. Since $\delta^{\ast}_{A}$ is proper it follows that f is proper and

$$\begin{aligned} \overline{\operatorname {epi}f} = \operatorname {epi}\delta^\ast_A \end{aligned}$$

(A.2)

by (2.1), so that $\delta^{\ast}_{A}=\operatorname {cl}f\le f$.

For any k=1,…,n, the compact i-generation of $\operatorname {dom}\delta^{\ast}_{A_{k}}$ means that $\sigma_{i}(\operatorname {dom}\delta^{\ast}_{A_{k}})$ is compact and non-empty. Thus the positive homogeneity of $\delta^{\ast}_{A_{k}}$ guarantees the existence of a closed proper convex function g _k with $\operatorname {dom}g_{k}=\sigma_{i}(\operatorname {dom}\delta^{\ast}_{A_{k}})$ compact such that $\delta ^{\ast}_{A_{k}}$ is generated by g _k , i.e.

$$\begin{aligned} \delta^\ast_{A_k}(y) = \begin{cases} \lambda g_k(x) &\mbox{if there exists } \lambda\ge0\mbox{ and }x\in \operatorname {dom}g_k\mbox{ such that }y=\lambda x,\cr \infty& \mbox{otherwise}. \end{cases} \end{aligned}$$

Let $g:=\operatorname {conv}\{g_{1},\ldots,g_{n}\}$; then

$$\begin{aligned} \operatorname {dom}g = \operatorname {conv}\Biggl[\bigcup_{k=1}^n \sigma_i\bigl(\operatorname {dom}\delta^\ast_{A_k}\bigr)\Biggr] \end{aligned}$$

is compact [23, Corollary 9.8.2]. Moreover, g is closed and proper, and for each $x\in \operatorname {dom}g$ there exist α ₁,…,α _n ≥0 and x ₁,…,x _n such that $x_{k}\in\sigma _{i}(\operatorname {dom}\delta^{\ast}_{A_{k}})$ for all k and

$$\begin{aligned} g(x) = \sum_{k=1}^n\alpha_kg_k(x_k), \quad\sum_{k=1}^n\alpha_k = 1, \ \sum_{k=1}^n\alpha_kx_k = x; \end{aligned}$$

(A.3)

see [23, Corollary 9.8.3] (the common recession function is $\delta^{\ast}_{\mathbb{R}^{d}}$ since $\operatorname {dom}g_{k}$ is compact for all k).

Let h be the positively homogeneous function generated by g, i.e.

$$\begin{aligned} h(y) := \begin{cases} \lambda g(x) &\mbox{if there exists } \lambda\ge0\mbox{ and }x\in \operatorname {dom}g\mbox{ such that }y=\lambda x,\cr \infty& \mbox{otherwise.} \end{cases} \end{aligned}$$

Clearly, h is a proper convex function and $\operatorname {dom}h = \operatorname {cone}(\operatorname {dom}g)$ is compactly i-generated. The function h is moreover closed since

$$\begin{aligned} \operatorname {epi}h = \bigl(\operatorname {cone}(\operatorname {epi}g)\bigr) \cup\bigl\{ (0,\lambda):\lambda\ge0\bigr\} = \overline {\operatorname {epi}h}; \end{aligned}$$

see [23, Theorem 9.6], and it is majorised by $\delta^{\ast}_{A_{1}}, \ldots, \delta^{\ast}_{A_{n}}$, hence h≤f. Since h is closed, it then follows from (A.2) that

$$\begin{aligned} h \le\delta^\ast_A \le f. \end{aligned}$$

(A.4)

Fix any $y\in \operatorname {dom}h$. There exist λ≥0 and $x\in\sigma_{i}(\operatorname {dom}h)=\operatorname {dom}g$ such that y=λx. Fix any α ₁,…,α _n ≥0 and x ₁,…,x _n satisfying (A.3) and where $x_{k}\in\sigma_{i}(\operatorname {dom}\delta^{\ast}_{A_{k}})$ for all k. Let y _k :=λx _k for all k. Then

$$\begin{aligned} \sum_{k=1}^n \alpha_k y_k = \lambda\sum_{k=1}^n \alpha_k x_k = \lambda x = y \end{aligned}$$

and

$$\begin{aligned} \sum_{k=1}^n\alpha_k\delta^\ast_{A_k}(y_k) = \lambda\sum_{k=1}^n\alpha _kg_k(x_k) = \lambda g(x) = h(y). \end{aligned}$$

By the definition of the convex hull, this means that f(y)≤h(y). Combining this with (A.4) gives

$$\begin{aligned} f = h = \delta^\ast_A. \end{aligned}$$

The properties of $\operatorname {dom}\delta^{\ast}_{A}$, in particular (A.1), then follow upon observing that

$$\begin{aligned} \operatorname {dom}g=\sigma_i(\operatorname {dom}h)=\sigma_i\bigl(\operatorname {dom}\delta^\ast_{A}\bigr). \end{aligned}$$

 □

The paper concludes with the proof of Lemma 2.

Proof of Lemma 2

For each t, since $\mathcal{K}_{t}$ is a cone, the support function of $-\mathcal{K}_{t}$ is

$$\begin{aligned} \delta^\ast_{-\mathcal{K}_t}(x) = \begin{cases} 0 &\mbox{if } x\cdot y \le0 \mbox{ for all } y\in-\mathcal{K}_t\cr \infty&\mbox{otherwise} \end{cases} = \begin{cases} 0 &\mbox{if } x\in\mathcal{K}^\ast_t,\cr \infty&\mbox{otherwise}. \end{cases} \end{aligned}$$

(A.5)

Thus $\operatorname {dom}\delta^{\ast}_{-\mathcal{K}_{t}}=\mathcal{K}_{t}^{\ast}$, and so $\operatorname {dom}\delta^{\ast}_{-\mathcal{K}_{t}}$ is compactly i-generated.

For any t we have $U^{a}_{t}=\delta^{\ast}_{\mathbb{R}^{d}}$ on $\varOmega \setminus\mathcal{E}_{t}$, together with

$$\begin{aligned} U^a_t(y) = \delta^\ast_{\{-\xi_t\}-\mathcal{K}_t}(y) = \delta^\ast_{\{ -\xi_t\}}(y) + \delta^\ast_{-\mathcal{K}_t}(y) = -y\cdot\xi_t + \delta ^\ast_{-\mathcal{K}_t}(y) \end{aligned}$$

for $y\in\mathbb{R}^{d}$ on $\mathcal{E}_{t}$ [23, p. 113]. Similarly,

$$\begin{aligned} V^a_t = \delta^\ast_{-\mathcal{W}^a_t-\mathcal{K}_t} = \delta^\ast _{-\mathcal{W}^a_t} + \delta^\ast_{-\mathcal{K}_t} = W^a_t + \delta^\ast _{-\mathcal{K}_t}. \end{aligned}$$

Equalities (5.8) and (5.9) then follow from (2.2) and (A.5).

We now turn to claims (b) and (c). Note first that the sets $\mathcal {U}^{a}_{t}$, $\mathcal{V}^{a}_{t}$, $\mathcal{W}^{a}_{t}$ and $\mathcal{Z}^{a}_{t}$ are non-empty for all t. This is easy to check by taking the trivial superhedging strategy for the seller defined by (5.5) and following the backward induction argument in the proof of Proposition 3.

We show below by backward induction that $\operatorname {dom}Z^{a}_{t}$ is compactly i-generated on $\mathcal{E}^{\ast}_{t}$. While doing so we will establish claims (b) and (c) for all t. At time T, using $Z^{a}_{T}=U^{a}_{T}$ and (4.3), the set $\operatorname {dom}Z^{a}_{T} = \mathcal{K}_{T}^{\ast}$ is compactly i-generated on $\mathcal{E}^{\ast}_{T}=\mathcal{E}_{T}$, while $Z^{a}_{T}=\delta^{\ast}_{\mathbb{R}^{d}}$ on $\varOmega\setminus\mathcal {E}^{\ast}_{T}$. This establishes claim (b) for t=T since $\mathcal{E}^{\ast}_{T+1}=\emptyset$.

At any time t<T, suppose that $\operatorname {dom}Z^{a}_{t+1}$ is compactly i-generated on $\mathcal{E}^{\ast}_{t+1}$. For any μ∈Ω _t there are now two possibilities:

• If $\mu\subseteq\mathcal{E}^{\ast}_{t+1}$, then Lemma 3 applies to the sets $\{-\mathcal {Z}^{a\nu}_{t+1}:\nu\in \operatorname {succ}\mu\}$ since

  $$\begin{aligned} \bigcap_{\nu\in \operatorname {succ}\mu}\mathcal{Z}^{a\nu}_{t+1} = \mathcal {W}^{a\mu}_t \neq\emptyset; \end{aligned}$$

  this immediately gives claim (c). Moreover, the compact i-generation of $\operatorname {dom}W^{a\mu}_{t}$ in combination with

  $$\begin{aligned} \operatorname {dom}V^{a\mu}_t = \operatorname {dom}W^{a\mu}_t \cap\mathcal{K}^{\ast\mu}_t \end{aligned}$$

  shows that $\operatorname {dom}V^{a\mu}_{t}$ is also compactly i-generated. There are now two possibilities:

  • If $\mu\subseteq\mathcal{E}_{t}$, then Lemma 3 applies to the sets $-\mathcal {U}^{a\mu}_{t}$ and $-\mathcal{V}^{a\mu}_{t}$. This gives claim (b)(i) after noting that

    $$\begin{aligned} \operatorname {dom}Z^{a\mu}_t = \operatorname {conv}\bigl(\operatorname {dom}V^{a\mu}_t \cup\mathcal{K}^{\ast\mu }_t\bigr)=\mathcal{K}^{\ast\mu}_t \end{aligned}$$

    by (A.1).

  • If $\mu\nsubseteq\mathcal{E}_{t}$, then $Z^{a\mu}_{t}=V^{a\mu}_{t}$ by Remark 6, which gives claim (b)(iii).

• If $\mu\nsubseteq\mathcal{E}^{\ast}_{t+1}$, then $Z^{a\mu}_{t}=U^{a\mu }_{t}$ by Remark 6. There are again two possibilities:

  • If $\mu\subseteq\mathcal{E}_{t}$, then (5.8) gives $\operatorname {dom}Z^{a\mu}_{t}=\mathcal {K}^{\ast\mu}_{t}$. This is claim (b)(ii).

  • If $\mu\nsubseteq\mathcal{E}_{t}$, then (5.8) immediately gives claim (b)(iv).

In summary, we have shown that $\operatorname {dom}Z^{a}_{t}$ is compactly i-generated whenever

$$\begin{aligned} \mu\subseteq\bigl[\mathcal{E}^\ast_{t+1}\cap \mathcal{E}_t\bigr] \cup\bigl[\mathcal {E}^\ast_{t+1} \setminus\mathcal{E}_t\bigr] \cup\bigl[\mathcal{E}_t \setminus \mathcal{E}^\ast_{t+1}\bigr] = \mathcal{E}^\ast_t. \end{aligned}$$

This concludes the inductive step, and completes the proof of Lemma 2. □