A stochastic semidefinite programming approach for bounds on option pricing under regime switching

Abstract

We consider bounds for the price of a European-style call option under regime switching. Stochastic semidefinite programming models are developed that incorporate a lattice generated by a finite-state Markov chain regime-switching model as a representation of scenarios (uncertainty) to compute bounds. The optimal first-stage bound value is equivalent to a Value at Risk quantity, and the optimal solution can be obtained via simple sorting. The upper (lower) bounds from the stochastic model are bounded below (above) by the corresponding deterministic bounds and are always less conservative than their robust optimization (min-max) counterparts. In addition, penalty parameters in the model allow controllability in the degree to which the regime switching dynamics are incorporated into the bounds. We demonstrate the value of the stochastic solution (bound) and computational experiments using the S&P 500 index are performed that illustrate the advantages of the stochastic programming approach over the deterministic strategy.

Appendix 1

1.1 Estimation results

The following estimation results are based on the option data with maturities in multiple of five weeks and the time series of S&P500 index from October 2004 to March 2008. The transition probability matrix $P$ and the filtered probability of the regimes are estimated based on the time series using Maximum Likelihood Estimation (MLE), whereas the volatility of each regime is further calibrated by minimizing the quantity: $\$RMSE = \sqrt{\frac{1}{N_t}\sum _{i}(C_{i,t}-C_{i,t}(h_t))^2}$, where $C_{i,t}$ is the market price of contract $i$ at time $t$, $C_{i,t}(h_t)$ is the respective model price, and $N_t$ is the total number of contracts available at time $t$ (Tables 3, 4, 5, 6).

Table 3 Estimation result for the case of 2 regimes

Table 4 Estimation result for the case of 3 regimes

Table 5 Estimation result for the case of 4 regimes

Table 6 Estimation result for the case of 5 regimes

1.2 Proofs

1.3 Theorem 1.

Proof

Suppose that $\mathrm{UB}_{\mathrm{SSDP}}(q) < \mathrm{UB}_{\mathrm{SDP}}(q)$ for some $q$. Since the optimal solution of (8) is feasible for the problem (6), and the form of the function $\mathrm{UB}_{\mathrm{SSDP}}(q): = \sum _{r=0}^{n}q_ry_r^{opt}$ is identical to the objective function of (6), this contradicts the fact that $\mathrm{UB}_{\mathrm{SDP}}(q)$ is the optimal value of (6). $\square$

1.4 Theorem 2.

Proof

Based on Theorem 1, $\mathrm{UB}_{\mathrm{SSDP}}(q) \ge \mathrm{UB}_{\mathrm{SDP}}(q)$ for any $q$. Here, we further show that $\mathrm{UB}_{\mathrm{SSDP}}(q) \le \mathrm{UB}_{\mathrm{SDP}}(q)$ for any $q$ given that $b^+ \le 1$. First, let $y^a:=(y_0^a,\ldots ,y_n^a)$ be the optimal solution of the problem (8) and let $y^b:=(y_0^b,\ldots ,y_n^b)$ be the optimal solution of the problem (6). Suppose now that $\mathrm{UB}_{\mathrm{SSDP}}(q) > \mathrm{UB}_{\mathrm{SDP}}(q)$ given that $b^+ \le 1$, which can be equivalently written as

$$\begin{aligned} \sum _{r=0}^{n}q_ry_r^a = \sum _{r=0}^{n}q_ry_r^b + \delta ,\;\; \delta > 0. \end{aligned}$$

(16)

Then, by substituting the right-hand-side of (16) as the optimal-value quantity associated with the solution $y^a$ into the objective function of (8), and further re-arranging the objective function based on the following partitions of $w_s$

$$\begin{aligned}&\mathcal{I}^1 := \{w_s\;|\; h(w_s) \ge \sum _{r=0}^{n}q_ry_r^a\} \\&\mathcal{I}^2 := \{w_s\;|\; \sum _{r=0}^{n}q_ry_r^b \le h(w_s) < \sum _{r=0}^{n}q_ry_r^a\} \\&\mathcal{I}^3 := \{w_s\;|\; h(w_s) < \sum _{r=0}^{n}q_ry_r^b\}, \end{aligned}$$

we obtain the following quantity

$$\begin{aligned} \sum _{r=0}^{n}q_ry_r^b + \delta + \sum _{w_s \in \mathcal{I}^1} p(w_s) \cdot s(w_s) + \sum _{w_s \in \mathcal{I}^2} p(w_s) \cdot s(w_s) + \sum _{w_s \in \mathcal{I}^3} p(w_s) \cdot s(w_s),\nonumber \\ \end{aligned}$$

(17)

where

$$\begin{aligned} s(w_s) = \left\{ \begin{array}{l l} b^+ (h(w_s) - \sum _{r=0}^{n}q_ry_r^b-\delta ) &{} \quad w_s \in \mathcal{I}^1\\ b^- (\sum _{r=0}^{n}q_ry_r^b +\delta - h(w_s) ) &{} \quad w_s \in \mathcal{I}^2\\ b^- (\sum _{r=0}^{n}q_ry_r^b +\delta - h(w_s) ) &{} \quad w_s \in \mathcal{I}^3. \end{array} \right. \end{aligned}$$

The quantity (17) can be re-written as

$$\begin{aligned} \sum _{r=0}^{n}q_ry_r^b+\mathcal{Q}(y^b) + \Delta + \delta (1-\sum _{w_s \in \mathcal{I}^1}p(w_s)b^+ + \sum _{w_s \in \mathcal{I}^2}p(w_s)b^- + \sum _{w_s \in \mathcal{I}^3}p(w_s)b^-),\nonumber \\ \end{aligned}$$

(18)

where $\Delta : = b^- \sum _{w_s \in \mathcal{I}^2} p(w_s) \cdot (\sum _{r=0}^{n}q_ry_r^b-h(w_s))- b^+ \sum _{w_s \in \mathcal{I}^2} p(w_s) \cdot (h(w_s)-\sum _{r=0}^{n}q_ry_r^b)$. Due to (16), $h(w_s)-\sum _{r=0}^{n}q_ry_r^b \le \delta$ for $w_s \in \mathcal{I}^2$ holds, and thus $\Delta \ge \delta (-b^+ \sum _{w_s \in \mathcal{I}^2}p(w_s)-b^- \sum _{w_s \in \mathcal{I}^2}p(w_s))$. Based on this and some algebraic manipulation, it is easy to see that $\Delta + \delta (1-\sum _{w_s \in \mathcal{I}^1}p(w_s)b^+ + \sum _{w_s \in \mathcal{I}^2}p(w_s)b^- + \sum _{w_s \in \mathcal{I}^3}p(w_s)b^-) > 0$ if $b^+ \le 1$, which leads to the conclusion that $y^b$ is more optimal than $y^a$ for the problem (8). This is a contradiction, and thus if $b^+ \le 1$, $\mathrm{UB}_{\mathrm{SSDP}}(q) \le \mathrm{UB}_{\mathrm{SDP}}(q)$ must hold. $\square$

1.5 Lemma 2.

Lemma 2

The function $\bar{f} : y \rightarrow \sum _{r=0}^{n}q_ry_r$ given any $q:=(q_0,\ldots ,q_n)$ is continuous and unbounded above over the feasible set $\{(X,Z,y) \in \mathcal{G} \subset (\mathcal{J}^{n+1},\mathcal{J}^{n+1},{\mathbb R}^{n+1})\;|\; X,Z \succeq 0\}$, where $y:=(y_0,\ldots ,y_n)$.

Proof

The continuity of $\bar{f}$ is obvious since the feasible set is a convex set. To see the unboundness, consider maximizing instead of minimizing the objective function in (6), equivalently, in (4). The dual problem of the maximization form of (4) is as follows.

$$\begin{aligned} \mathrm{minimize}_{\pi (s_T)}&\int \limits ^{\infty }_0 -\max (0,x-\overline{K})\pi (x)dx,\nonumber \\ \mathrm{subject\ to}&\int \limits ^{\infty }_0 x^j\pi (x)dx = -q_j,\ j=0,1,\ldots ,n,\nonumber \\&\pi (x) \ge 0, \end{aligned}$$

(19)

where $q_0=1$ and $\overline{K}=e^{-\gamma \tau }K$. Clearly, no feasible solution exists for the above problem. Based on duality theory, this implies that the primal problem, the maximization form of (4), is unbounded. This completes the proof. $\square$

1.6 Theorem 3.

Proof

Consider the problem (8) with the first stage parameter $q:=(q_0,\ldots ,q_n) \in \mathcal{C}$. Let $y^a:=(y_0^a,\ldots ,y_n^a)$ be the optimal solution of the problem (8). Suppose now that $\mathrm{UB}_{\mathrm{SSDP}}(q) > \mathrm{WUB}_{\mathrm{SDP}}$. This implies that

$$\begin{aligned} \sum _{r=0}^{n}q_ry_r^a > \mathrm{WUB}_{\mathrm{SDP}}&\ge \max _{q(w) \in \mathcal{C}}\min _{y,X,Z: (X,Z,y) \in \mathcal{G}, X,Z \succeq 0} \sum _{r=0}^{n}q(w)_ry_r. \end{aligned}$$

(20)

To see why the last inequality in (20) is true, let $q_{opt}$ and $y_{opt}$ denote the optimal $q(w)$ and $y$ in the optimization problem in (20). Then, in the worst-case upper bound problem (13), if the optimal $y^*$ in (13) equals to $y_{opt}$, the inequality in (20) must hold. Consider now if the optimal $y^*$ in (13) does not equal to $y_{opt}$, the following must hold by the definition of the optimization problem in (20)

$$\begin{aligned} \sum _{r=0}^{n} (q_{opt})_ry^*_r \ge \sum _{r=0}^{n} (q_{opt})_r(y_{opt})_r. \end{aligned}$$

Thus, the below inequalities follow immediately

$$\begin{aligned} \min _{y,X,Z: (X,Z,y) \in \mathcal{G}, X,Z \succeq 0}\max _{q(w) \in \mathcal{C}}\sum _{r=0}^{n}q(w)_ry_r&= \max _{q(w) \in \mathcal{C}}\sum _{r=0}^{n}q(w)_ry^*_r\\&\ge \sum _{r=0}^{n} (q_{opt})_ry^*_r \ge \sum _{r=0}^{n} (q_{opt})_r(y_{opt})_r. \end{aligned}$$

This completes the verification of the last inequality in (20). The inequality in (20) implies the following two inequalities

$$\begin{aligned}&\sum _{r=0}^{n}q_ry_r^a > \max _{q(w) \in \mathcal{C}}\min _{y,X,Z: (X,Z,y) \in \mathcal{G}, X,Z \succeq 0} \sum _{r=0}^{n}q(w)_ry_r \ge \min _{y,X,Z: (X,Z,y) \in \mathcal{G}, X,Z \succeq 0} \sum _{r=0}^{n}q_ry_r\nonumber \\&\quad \qquad \qquad \qquad = \sum _{r=0}^{n}q_ry_r^b \end{aligned}$$

(21)

$$\begin{aligned}&\sum _{r=0}^{n}q_ry_r^a > \max _{q(w) \in \mathcal{C}}\min _{y,X,Z: (X,Z,y) \in \mathcal{G}, X,Z \succeq 0} \sum _{r=0}^{n}q(w)_ry_r \ge \mathrm{UB}_{\mathrm{SDP}}(q(w)),\;\; \forall q(w) \in \mathcal{C},\nonumber \\ \end{aligned}$$

(22)

where $y^b:=(y_0^b,\ldots ,y_n^b)$ denotes the optimal solution of the last optimization problem in (21). The inequalities (21) imply that there exists a feasible $y^b$ such that

$$\begin{aligned} \sum _{r=0}^{n}q_ry_r^a > \sum _{r=0}^{n}q_ry_r^b. \end{aligned}$$

(23)

Thus, based on Lemma 2 there must exists $y^c:=(y_0^c,\ldots ,y_n^c)$ such that

$$\begin{aligned} \sum _{r=0}^{n}q_ry_r^a > \sum _{r=0}^{n}q_ry_r^c = \mathrm{WUB}_{\mathrm{SDP}}, \;\;\mathrm{and\ thus}\; \sum _{r=0}^{n}q_ry_r^c \ge \mathrm{UB}_{\mathrm{SDP}}(q(w)),\;\; \forall q(w) \in \mathcal{C}.\nonumber \\ \end{aligned}$$

(24)

Finally, based on (22) and (24), it is easy to verify that $y^c$ is more optimal than $y^a$ for the problem (8), which is a contradiction. Thus, $\mathrm{UB}_{\mathrm{SSDP}}(\tilde{q}) \le \mathrm{WUB}_{\mathrm{SDP}}$ must hold for any $\tilde{q} \in \mathcal{C}$. $\square$

1.7 Theorem 4.

Proof

To prove the equivalency, it suffices to show that for any $x$ that satisfies $x \ge \mathrm{UB}_{\mathrm{SDP}}(q)$, there exists a solution $y := (y_0,\ldots ,y_n)$ that is feasible with respect to both the equality $\sum _{r=0}^{n}q_ry_r = x$ and constraints in (8). Given that $\mathrm{UB}_{\mathrm{SDP}}(q)<\infty$, i.e. $\exists (y_0,\ldots ,y_n)$ such that $\sum _{r=0}^{n}q_ry_r \le x$ for any $x \ge l$, the existence of a feasible $y$ that satisfies $\sum _{r=0}^{n}q_ry_r = x$ can be proven by showing that the function $\sum _{r=0}^{n}q_ry_r$ is continuous with respect to the feasible set $\{(X,Z,y) \in \mathcal{G} \subset (\mathcal{J}^{n+1},\mathcal{J}^{n+1},{\mathbb R}^{n+1})\;|\; X,Z \succeq 0\}$ and it is unbounded above, which is the result of Lemma 2. $\square$

See Tables 7, 8 and 9.

Table 7 Upper/lower bounds and prices for different strike prices $K$, $b^+(b^-)$-values and time to maturity under 2 regimes

Table 8 Upper/lower bounds and prices for different $b^+(b^-)$-values and time to maturity under different number of regimes

Table 9 The value of stochastic solutions