Bounds for path-dependent options

Abstract

We develop new semiparametric bounds on the expected payoffs and prices of European call options and a wide range of path-dependent contingent claims. We first focus on the trinomial financial market model in which, as is well-known, an exact calculation of derivative prices based on no-arbitrage arguments is impossible. We show that the expected payoff of a European call option in the trinomial model with martingale-difference log-returns is bounded from above by the expected payoff of a call option written on an asset with i.i.d. symmetric two-valued log-returns. We further show that the expected payoff of a European call option in the multiperiod trinomial option pricing model is bounded by the expected payoff of a call option in the two-period model with a log-normal asset price. We also obtain bounds on the possible prices of call options in the (incomplete) trinomial model in terms of the parameters of the asset’s distribution. Similar bounds also hold for many other contingent claims in the trinomial option pricing model, including those with an arbitrary convex increasing payoff function as well as for path-dependent ones such as Asian options. We further obtain a wide range of new semiparametric moment bounds on the expected payoffs and prices of path-dependent Asian options with an arbitrary distribution of the underlying asset’s price. These results are based on recently obtained sharp moment inequalities for sums of multilinear forms and U-statistics and provide their first financial and economic applications in the literature. Similar bounds also hold for many other path-dependent contingent claims.

Appendix: Probabilistic foundations for the analysis

Let $(\Omega , \mathfrak {I}, P)$ be a probability space equipped with a filtration $\mathfrak {I}_0=(\Omega , \emptyset )\subseteq \mathfrak {I}_1\subseteq \ldots \mathfrak {I}_t\subseteq \ldots \subseteq \mathfrak {I}.$ Further, let $(a_t)_{t=1}^{\infty }$ and $(b_t)_{t=1}^{\infty }$ be arbitrary sequences of real numbers such that $a_t\ne b_t$ for all t.

The key to the analysis in Sects. 2 and 3 is provided by the following theorems. These theorems are consequences of more general results obtained in Sharakhmetov and Ibragimov (2002) that show that r.v.’s taking $k+1$ values form a multiplicative system of order k if and only if they are jointly independent (see also de la Peña et al. 2006). These results imply, in particular, that r.v.’s each taking two values form a martingale-difference sequence if and only if they are jointly independent.

To illustrate the main ideas of the proof, we first consider the case of r.v.’s taking values $\pm 1$.

Theorem 6.1

If r.v.’s $U_t$, $t=1, 2, \ldots ,$ form a martingale-difference sequence with respect to a filtration $(\mathfrak {I}_t)_t$ and are such that $P(U_t=1)=P(U_t=-1)=\frac{1}{2}$ for all t,  then they are jointly independent.

For completeness, the proof of the theorem is provided below.

Proof

It is easy to see that, under the assumptions of the theorem, one has that, for all $1\le l_1<l_2<\cdots <l_k$, $k=2, 3, \ldots ,$

$$\begin{aligned} EU_{l_1}\ldots U_{l_{k-1}}U_{l_k}=E\left( U_{l_1}\ldots U_{l_{k-1}}E(U_{l_k}|\mathfrak {I}_{l_k-1})\right) =E\left( U_{l_1}\ldots U_{l_{k-1}}\times 0\right) =0.\nonumber \\ \end{aligned}$$

(6.1)

It is easy to see that, for $x_t\in \{-1, 1\}$, $I(X_t=x_t)=(1+x_tU_t)/2.$ Consequently, for all $1\le j_1<j_2<\cdots <j_m$, $m=2, 3, \ldots ,$ and any $x_{j_k}\in \{-1, 1\}$, $k=1, 2, \ldots , m,$ we have

$$\begin{aligned}&P(U_{j_1}=x_{j_1}, U_{j_2}=x_{j_2}, \ldots , U_{j_m}=x_{j_m})\\\\&\quad = EI(U_{j_1}=x_{j_1})I(U_{j_2}=x_{j_2})\ldots I( U_{j_m}=x_{j_m}) \\&\quad =\frac{1}{2^m} E(1+x_{j_1}U_{j_1})(1+x_{j_2}U_{j_2})\ldots (1+x_{j_m}U_{j_m})\\&\quad =\frac{1}{2^m} \left( 1+\sum _{c=2}^m \sum _{i_1<\cdots <i_c\in \{j_1, j_2, \ldots , j_m\}} EU_{i_1}\ldots U_{i_c}\right) = \frac{1}{2^m}\\&\quad =P(U_{j_1}=x_{j_1})P(U_{j_2}=x_{j_2}) \ldots P(U_{j_m}=x_{j_m}) \end{aligned}$$

by (6.1). $\square$

The proof of the analogue of the result in the case of r.v.’s each of which takes arbitrary two values is completely similar and the following more general result holds.

Theorem 6.2

If r.v.’s $X_t$, $t=1, 2, \ldots ,$ form a martingale-difference sequence with respect to a filtration $(\mathfrak {I}_t)_t$ and each of them takes two (not necessarily the same for all t) values $\{a_t, b_t\}$, then they are jointly independent.

Proof

Let the r.v. $X_t$ take the values $a_t$ and $b_t$, $a_t\ne b_t,$ with probabilities $P(X_t=a_t)=p_t$ and $P(X_t=b_t)=q_t,$ respectively. It not difficult to check that, for $x_t\in \{a_t, b_t\},$

$$\begin{aligned} I(X_t=x_t)= & {} P(X_t=x_t)\left( 1+\frac{(X_t-a_tp_t-b_tq_t)(x_t-a_tp_t-b_tq_t)}{(a_t-b_t)^2p_tq_t}\right) \\= & {} P(X_t=x_t)\left( 1+\frac{(X_t-EX_t)(x_t-EX_t)}{(a_t-b_t)^2p_tq_t}\right) \\= & {} P(X_t=x_t)\left( 1+\frac{X_tx_t}{(a_t-b_t)^2p_tq_t}\right) =P(X_t=x_t)\left( 1+\frac{X_tx_t}{Var(X_t)}\right) , \end{aligned}$$

where $EX_t=a_tp_t+b_tq_t=0$ and $Var(X_t)=(b_t-a_t)^2p_tq_t$ are the mean and the variance of $X_t.$ Since the r.v.’s $X_t$ satisfy property (6.1) with $U_{l_j}$ replaced by $X_{l_j}$, $j=1, \ldots , k$, similar to the proof of Theorem 6.1 we have that, for all $1\le j_1<j_2<\cdots <j_m$, $m=2, 3, \ldots ,$ and any $x_{j_k}\in \{a_{j_k}, b_{j_k}\}$, $k=1, 2, \ldots , m,$

$$\begin{aligned}&P(X_{j_1}=x_{j_1}, X_{j_2}=x_{j_2}, \ldots , X_{j_m}=x_{j_m})\\&\quad = EI(X_{j_1}=x_{j_1})I(X_{j_2}=x_{j_2})\ldots I( X_{j_m}=x_{j_m})\\&\quad =\prod _{s=1}^m P(X_{j_s}=x_{j_s}) E\left( 1+\frac{X_{j_1}x_{j_1}}{Var(X_{j_1 })}\right) \ldots \left( 1+\frac{X_{j_m}x_{j_m}}{Var(X_{j_m})}\right) \\&\quad =\prod _{s=1}^m P(X_{j_s}=x_{j_s})\\&\qquad \times \left( 1+\sum _{c=2}^m \sum _{i_1<\cdots <i_c\in \{j_1, j_2, \ldots , j_m\}} EX_{i_1}\ldots X_{i_c}x_{i_1} \ldots x_{i_c}/(Var(X_{i_1})\ldots Var(X_{i_c})) \right) \\&\quad = P(X_{j_1}=x_{j_1})P(X_{j_2}=x_{j_2}) \ldots P(X_{j_m}=x_{j_m}). \end{aligned}$$

$\square$

Let $X_t$, $t=1, 2, \ldots ,$ be an $(\mathfrak {I}_t)$-martingale-difference sequence consisting of r.v.’s each of which takes three values $\{-a_t, 0, a_t\}.$ Denote by $\epsilon _t$, $t=1, 2, \ldots ,$ a sequence of i.i.d. symmetric Bernoulli r.v.’s independent of $(X_t)_{t=1}^{\infty }.$ The following theorem provides an upper bound for the expectation of arbitrary convex function of $X_t$ in terms of the expectation of the same function of the r.v.’s $\epsilon _t.$

Theorem 6.3

If $f:\mathbf{R}^n\rightarrow \mathbf{R}$ is a function convex in each of its arguments, then the following inequality holds:

$$\begin{aligned} Ef(X_1, \ldots , X_n)\le Ef(a_1\epsilon _1, \ldots , a_n\epsilon _n). \end{aligned}$$

(6.2)

Proof

Let ${\tilde{\mathfrak {I}}}_0=\mathfrak {I}_n.$ For $t=1, 2, \ldots , n,$ denote by ${\tilde{\mathfrak {I}}}_t$ the $\sigma$-algebra spanned by the r.v.’s $X_1, X_2, \ldots , X_n$, $\epsilon _1, \ldots , \epsilon _t.$ Further, let, for $t=0, 1, \ldots , n$, $E_{t}$ stand for the conditional expectation operator $E(\cdot |{\tilde{\mathfrak {I}}}_t)$ and let $\eta _t$, $t=1, \ldots , n,$ denote the r.v.’s $\eta _t=X_t+\epsilon _tI(X_t=0).$

Using conditional Jensen’s inequality, we have

$$\begin{aligned} Ef(X_1, X_2, \ldots , X_n)= & {} Ef(X_1+E_0[\epsilon _1I(X_1=0)], X_2, \ldots , X_n)\nonumber \\\le & {} E[E_0f(X_1+\epsilon _1I(X_1=0), X_2, \ldots , X_n)]\nonumber \\= & {} Ef(\eta _1, X_2, \ldots , X_n). \end{aligned}$$

(6.3)

Similarly, for $t=2, \ldots , n,$

$$\begin{aligned}&Ef(\eta _1, \eta _2, \ldots , \eta _{t-1}, X_t, X_{t+1}, \ldots , X_n) \nonumber \\&\quad =Ef(\eta _1, \eta _2, \ldots , \eta _{t-1}, X_t+E_{t-1}[\epsilon _tI(X_t=0)], X_{t+1}, \ldots , X_n) \nonumber \\&\quad \le E[E_{t-1}f(\eta _1, \eta _2, \ldots , \eta _{t-1}, X_t+\epsilon _tI(X_t=0), X_{t+1}, \ldots , X_n)]\nonumber \\&\quad =Ef(\eta _1, \eta _2, \ldots , \eta _{t-1}, \eta _t, X_{t+1}, \ldots , X_n). \end{aligned}$$

(6.4)

From equations (6.3) and (6.4) by induction it follows that

$$\begin{aligned} Ef(X_1, X_2, \ldots , X_n)\le Ef(\eta _1, \eta _2, \ldots , \eta _n). \end{aligned}$$

(6.5)

It is easy to see that the r.v.’s $\eta _t$, $t=1, 2, \ldots , n,$ form a martingale-difference sequence with respect to the sequence of $\sigma$-algebras ${\tilde{\mathfrak {I}}}_0\subseteq {\tilde{\mathfrak {I}}}_1\subseteq \cdots \subseteq {\tilde{\mathfrak {I}}}_t \subseteq \cdots ,$ and each of them takes two values $\{-a_t, a_t\}.$ Therefore, from Theorems 6.1 and 6.2 we get that $\eta _t$, $t=1, 2, \ldots , n,$ are jointly independent and, therefore, the random vector $(\eta _1, \eta _2, \ldots , \eta _n)$ has the same distribution as $(a_1\epsilon _1, a_2\epsilon _2, \ldots , a_n\epsilon _n).$ This and (6.5) implies estimate (6.2). $\square$