Discrete Bismut formula: Conditional integration by parts and a representation for delta hedging process

Abstract

The paper gives discrete conditional integration by parts formula using a Malliavin calculus approach in discrete-time setting. Then the discrete Bismut formula is introduced for asymmetric random walk model and asymmetric exponential process. In particular, a new formula for delta hedging process is obtained as an extension of the Malliavin derivative representation of the delta where the conditional integration by parts formula plays a role in the proof.

Keywords

Random walk delta hedging discrete Bismut formula conditional integration by parts

1. Introduction

The delta hedging initially proposed by Black and Scholes [1] plays an important role in finance in order to control market risk. It provides a basic technique in trading and risk management, actually monitoring the delta is a fundamental task in financial institutions in practice.

Theoretically, the delta is given as the number of units of risky asset we should hold to hedge a written derivative. From a viewpoint of mathematical finance, the delta is the ratio of the change in the option price to the change in the underlying asset price, and is represented by using the Clark–Ocone formula (Ocone [2], Malliavin [3], Nualart [4]) through the Malliavin derivative of option price. The method with Malliavin derivative is called the Ocone–Karatzas hedging (Ocone and Karatzas [5], Malliavin and Thalmaier [6]) which gives a kind of numerical differentiation method in numerical analysis. In discrete-time market, the corresponding hedging scheme is derived by Privault [7] using the discrete-time Malliavin derivative.

In the paper, we give a new representation of the delta in discrete-time market, which is closely related to the Bismut formula in stochastic calculus on Wiener space (Bismut [8], Malliavin [3]). We briefly review the Bismut formula in the following. Let $X=\{X_{t}^{x}\}_{t\geq 0}$ be the solution of the following stochastic differential equation driven by a Brownian motion $\{W_{t}\}_{t\geq 0}$ : $\begin{eqnarray}\displaystyle dX_{t}^{x}=b(X_{t}^{x})dt+{\sigma}(X_{t}^{x})dW_{t},\quad X_{0}^{x}=x, & & \displaystyle \nonumber\end{eqnarray}$ and consider the corresponding generator ${\mathcal{L}}=b(\cdot )\frac{\partial }{\partial x}+\frac{1}{2}{\sigma}^{2}(\cdot )\frac{\partial ^{2}}{\partial x^{2}}$ . Then the derivative $\frac{\partial }{\partial x}u(0,x)$ of the function u satisfying the following partial differential equation $\begin{eqnarray}\displaystyle (\partial _{t}+{\mathcal{L}})u(t,x)=0,\quad u(T,\cdot )=f(\cdot ), & & \displaystyle \nonumber\end{eqnarray}$ is given by $\begin{eqnarray}\displaystyle \hspace{-12.0pt}\frac{\partial }{\partial x}u(0,x)=E\left[f(X_{T}^{x})\frac{1}{T}\int _{0}^{T}{\sigma}^{-1}(X_{s}^{x})J_{s}^{x}dW_{s}\right], & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$ (1) where 𝜎(⋅) is assumed to be invertible and $J_{t}^{x}=\frac{\partial }{\partial x}X_{t}^{x}$ , t ≥ 0, is the Jacobian process. The formula (1.1) is known as the Bismut formula and is used for representing the delta in finance (see Fournie et al. [9]). We give the discrete version of the Bismut formula which we call the discrete Bismut formula.

In order to obtain the discrete Bismut formula as a “stochastic process”, we introduce a conditional integration by parts formula which is a discrete version of the conditional integration by parts on Wiener space and is an extension of the discrete integration by parts in the book of Privault [7], which provides another perspective for the works in [10–14]. In the proof of the discrete conditional integration by parts, we do not employ the chaos expansion approach as in Privault [7] but give a simple proof with the discrete Clark–Ocone formula. Also, as preparation, we show an elementary proof for the discrete Clark–Ocone formula where the chaos expansion approach is also not used.

The discrete Bismut formula is obtained for asymmetric random walk model where the discrete conditional integration by parts formula plays a role. Then we show a new formula for the delta hedging process for discrete-time market governed by asymmetric exponential process, as an extension of the discrete Ocone–Karatzas hedging. We give two original proofs for the new hedging formula where the conditional integration by parts and the discrete Itô formula of Fujita [15] are used.

The paper is organized as follows. Sections 2 and 3 give the user’s guide for discrete-time Malliavin calculus. As preparation, we give the discrete Clark–Ocone formula in Section 2 where a new simple proof is adopted. In Section 3, the conditional integration by parts formula is introduced. Then Section 4 gives the discrete Bismut formula for asymmetric random walk model. In Section 5, a new delta hedging formula is given as the extension of the discrete Ocone–Karatzas hedging. Section 6 concludes on the method of the paper.

2. Preliminaries: Discrete Clark–Ocone formula revisited

For $N\in \mathbb{N}$ , let ${\rm\Omega}:=\{1,-1\}^{N}$ and ${\mathcal{F}}:=2^{{\rm\Omega}}:=\{A;∼A\subset {\rm\Omega}\}$ . We will write 𝜔 = (𝜔₁, …, 𝜔_k, …, 𝜔_N) for 𝜔 ∈ Ω. Let $\{Z_{k}\}_{k=1}^{N}$ be a sequence of i.i.d. random variables given by Z _k(𝜔): = 𝜔_k, 𝜔 ∈ Ω, k = 1, …, N with the probability measure $\begin{eqnarray}\displaystyle \hspace{-18.0pt}\begin{array}{@{}l@{}}P(\{{\omega};Z_{k}({\omega})=1\})=p\in (0,1),\\[3.0pt] P(\{{\omega};Z_{k}({\omega})=-1\})=q=1-p,\quad k=1,\ldots ,N.\end{array} & & \displaystyle\end{eqnarray}$ (2) We prepare a calculus of Bernoulli sequence on $({\rm\Omega},{\mathcal{F}},P)$ . Let $\{Y_{k}\}_{k}$ be the Bernoulli sequence (i.e. P (Y _k =1) = p, P (Y _k = 0) = q) given by $\begin{eqnarray}\displaystyle Y_{k}=\frac{1+Z_{k}}{2},\quad k=1,\ldots ,N. & & \displaystyle \nonumber\end{eqnarray}$ We normalize the Bernoulli sequence $\{Y_{k}\}_{k}$ by $\begin{eqnarray}\displaystyle {\rm\Delta}W_{k}=\frac{Y_{k}-p}{\sqrt{pq}}=\frac{1+Z_{k}-2p}{2\sqrt{pq}},\quad k=1,\ldots ,N, & & \displaystyle \nonumber\end{eqnarray}$ where $\sqrt{\text{Var}[Y_{k}]}=\sqrt{pq}$ . Then, we have $\begin{eqnarray}\displaystyle \hspace{-12.0pt}{\rm\Delta}W_{k}({\omega})=\frac{q}{\sqrt{pq}}({\omega}_{k}=1)\quad \text{or}\quad -\frac{p}{\sqrt{pq}}({\omega}_{k}=-1), & & \displaystyle \nonumber\end{eqnarray}$ and then $E[{\rm\Delta}W_{k}]=(pq-pq)/\sqrt{pq}=0$ and E[(ΔW _k)²] = (q ² p + p ² q)∕(pq) = p + q = 1, for k = 1, …, N. Let $\{{\mathcal{F}}_{k}\}$ be the filtration given by ${\mathcal{F}}_{0}=\{\emptyset,{\rm\Omega}\}$ , ${\mathcal{F}}_{k}={\sigma}({\rm\Delta}W_{1},\ldots ,{\rm\Delta}W_{k})$ , k = 1, …, N. Here, 𝜎(ΔW ₁, …, ΔW _k) represents the 𝜎-field generated by ΔW ₁, …, ΔW _k. By the construction, we see that ${\mathcal{F}}_{n}={\sigma}(Z_{1},\ldots ,Z_{n})$ , n ≥ 1. We define the process $\{W_{n}\}_{n=0}^{N}$ given by W ₀ = 0, $W_{n}=\sum _{k=1}^{n}{\rm\Delta}W_{k}$ , n ≥ 1. Then $\{W_{n}\}_{n=0}^{N}$ is $\{{\mathcal{F}}_{n}\}$ -martingale, i.e. $E[W_{n}|{\mathcal{F}}_{n-1}]=W_{n-1}$ , n ≥ 1, where $E[\cdot |{\mathcal{F}}_{0}]\equiv E[\cdot ]$ .

We now define the “Malliavin derivative” of the functional of ΔW _k, k = 1, …, N. For $f:\mathbb{R}^{N}\rightarrow \mathbb{R}$ and F = f (ΔW ₁, …, ΔW _N), we write $\begin{eqnarray}\displaystyle F({\omega}_{k}^{+}) & := & \displaystyle f({\rm\Delta}W_{1}({\omega}_{k}^{+}),\ldots ,{\rm\Delta}W_{N}({\omega}_{k}^{+})),\nonumber\\ \displaystyle F({\omega}_{k}^{-}) & := & \displaystyle f({\rm\Delta}W_{1}({\omega}_{k}^{-}),\ldots ,{\rm\Delta}W_{N}({\omega}_{k}^{-})),\nonumber\end{eqnarray}$ where $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\omega}_{k}^{+}:=({\omega}_{1},\ldots ,{\omega}_{k-1},1,{\omega}_{k+1},\ldots ,{\omega}_{N}),\qquad \nonumber\\ \displaystyle \text{} & \text{} & \displaystyle {\omega}_{k}^{-}:=({\omega}_{1},\ldots ,{\omega}_{k-1},-1,{\omega}_{k+1},\ldots ,{\omega}_{N})\nonumber\end{eqnarray}$ for k = 1, …, N. Here, ${\omega}_{1}^{\pm }$ and ${\omega}_{N}^{\pm }$ are understood as ${\omega}_{1}^{\pm }=(\pm 1,{\omega}_{2},\ldots ,{\omega}_{N})$ and ${\omega}_{N}^{\pm }=$ (𝜔₁, …,𝜔_N−1, ±1).

TheoremA .

For $f:\mathbb{R}^{N}\rightarrow \mathbb{R}$ and F = f (ΔW ₁, …, ΔW _N). We define a map $\{1,\ldots ,N\}\times {\rm\Omega}\ni (k,{\omega})\mapsto D_{k}F({\omega})\in \mathbb{R}$ by $\begin{eqnarray}\displaystyle D_{k}F({\omega}):=\sqrt{pq}(F({\omega}_{k}^{+})-F({\omega}_{k}^{-})). & & \displaystyle \nonumber\end{eqnarray}$ We call D _k F the discrete-time Malliavin derivative.

The discrete Ocone–Clark formula is obtained with more elementary proof than Privault [7].

TheoremB . (Discrete Ocone–Clark formula).

Let $f:\mathbb{R}^{N}\rightarrow \mathbb{R}$ and F = f (ΔW ₁, …, ΔW _N). Then we have $\begin{eqnarray}\displaystyle F & = & \displaystyle E[F]+\mathop{\sum }_{k=1}^{N}D_{k}E[F|{\mathcal{F}}_{k}]{\rm\Delta}W_{k}\nonumber\\ \displaystyle & = & \displaystyle E[F]+\mathop{\sum }_{k=1}^{N}E[D_{k}F|{\mathcal{F}}_{k-1}]{\rm\Delta}W_{k}.\nonumber\end{eqnarray}$

TheoremC .
Let $M_{N}:=F=E[F|{\mathcal{F}}_{N}]$ , $M_{k}:=E[F|{\mathcal{F}}_{k}]$ , k = 0,1, …, N −1, then we have $\begin{eqnarray}\displaystyle M_{N} & = & \displaystyle M_{0}+\mathop{\sum }_{k=1}^{N}(M_{k}-M_{k-1})\nonumber\\ \displaystyle & = & \displaystyle M_{0}+\mathop{\sum }_{k=1}^{N}\frac{M_{k}-M_{k-1}}{{\rm\Delta}W_{k}}{\rm\Delta}W_{k}.\end{eqnarray}$ (3) For k = 1, …, N, since M _k − M _k−1 is ${\mathcal{F}}_{k}$ -measurable, there exists a function $d_{k}:\{-1,1\}^{k}\rightarrow \mathbb{R}$ such that d _k(Z ₁, …, Z _k−1, Z _k) = M _k − M _k−1. Then we have $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \hspace{-22.0pt}\frac{M_{k}({\omega})-M_{k-1}({\omega})}{{\rm\Delta}W_{k}({\omega})}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad \hspace{-20.0pt}=\frac{\sqrt{pq}d_{k}(Z_{1}({\omega}),\ldots ,Z_{k-1}({\omega}),1)}{q}∼({\omega}_{k}=1)\quad \text{or}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad \hspace{-10.0pt}\frac{-\sqrt{pq}d_{k}(Z_{1}({\omega}),\ldots ,Z_{k-1}({\omega}),-1)}{p}∼({\omega}_{k}=-1).\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$ (4) Since $\{M_{k}\}_{k=0}^{N}$ is $\{{\mathcal{F}}_{k}\}$ -martingale, we have $E[M_{k}-M_{k-1}|{\mathcal{F}}_{k-1}]=∼0$ or equivalently $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle p\times d_{k}(Z_{1},\ldots ,Z_{k-1},1)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad =-q\times d_{k}(Z_{1},\ldots ,Z_{k-1},-1),\end{eqnarray}$ (5) for k = 1, …, N. Therefore (2.3) actually becomes $\begin{eqnarray}\displaystyle \frac{M_{k}-M_{k-1}}{{\rm\Delta}W_{k}} & = & \displaystyle \frac{\sqrt{pq}d_{k}(Z_{1},\ldots ,Z_{k-1},1)}{q}\nonumber\\ \displaystyle & = & \displaystyle \frac{-\sqrt{pq}d_{k}(Z_{1},\ldots ,Z_{k-1},-1)}{p},\nonumber\end{eqnarray}$ which suggests that $\frac{M_{k}-M_{k-1}}{{\rm\Delta}W_{k}}$ is ${\mathcal{F}}_{k-1}$ -measurable, in other words, $\{\frac{M_{k}-M_{k-1}}{{\rm\Delta}W_{k}}\}_{k}$ is a predictable process. The discrete-time Malliavin derivative of $M_{k}=E[F|{\mathcal{F}}_{k}]$ is given by $\begin{eqnarray}\displaystyle D_{k}M_{k} & = & \displaystyle \sqrt{pq}d_{k}(Z_{1},\ldots ,Z_{k-1},1)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -\sqrt{pq}d_{k}(Z_{1},\ldots ,Z_{k-1},-1)\nonumber\\ \displaystyle & = & \displaystyle \sqrt{pq}d_{k}(Z_{1},\ldots ,Z_{k-1},1)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼\sqrt{pq}\frac{p}{q}d_{k}(Z_{1},\ldots ,Z_{k-1},1)\nonumber\\ \displaystyle & = & \displaystyle \frac{\sqrt{pq}d_{k}(Z_{1},\ldots ,Z_{k-1},1)}{q}\nonumber\\ \displaystyle & = & \displaystyle \frac{-\sqrt{pq}d_{k}(Z_{1},\ldots ,Z_{k-1},-1)}{p},\nonumber\end{eqnarray}$ where we use (2.4). Then we get $\begin{eqnarray}\displaystyle \frac{M_{k}-M_{k-1}}{{\rm\Delta}W_{k}}=D_{k}E[F|{\mathcal{F}}_{k}] & & \displaystyle\end{eqnarray}$ (6) and $F=E[F]+\sum _{k=1}^{N}D_{k}E[F|{\mathcal{F}}_{k}]{\rm\Delta}W_{k}$ by (2.2) and (2.5). Finally, we check the equality $D_{k}E[F|{\mathcal{F}}_{k}]=E[D_{k}F|{\mathcal{F}}_{k-1}]$ . Since we have $\begin{eqnarray}\displaystyle E[F|{\mathcal{F}}_{k}] & = & \displaystyle \mathop{\sum }_{x_{k+1},\ldots ,x_{N}\in \{q/\sqrt{pq},-p/\sqrt{pq}\}}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \times ∼f({\rm\Delta}W_{1},\ldots ,{\rm\Delta}W_{k},x_{k+1},\ldots ,x_{N})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \times ∼p^{\#\{\ell ;x_{\ell }=q/\sqrt{pq}\}}q^{\#\{\ell ;x_{\ell }=-p/\sqrt{pq}\}},\nonumber\end{eqnarray}$ one has $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \hspace{-24.0pt}D_{k}E[F|{\mathcal{F}}_{k}]=\mathop{\sum }_{x_{k+1},\ldots ,x_{N}\in \{q/\sqrt{pq},-p/\sqrt{pq}\}}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad \hspace{-24.0pt}\times ∼f ({\rm\Delta}W_{1},\ldots ,{\rm\Delta}W_{k-1},q/\sqrt{pq},x_{k+1},\ldots ,x_{N} )\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad \hspace{-24.0pt}\times ∼p^{\#\{\ell ;x_{\ell }=q/\sqrt{pq}\}}q^{\#\{\ell ;x_{\ell }=-p/\sqrt{pq}\}}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad \hspace{-24.0pt}-∼\mathop{\sum }_{x_{k+1},\ldots ,x_{N}\in \{q/\sqrt{pq},-p/\sqrt{pq}\}}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad \hspace{-24.0pt}\times ∼f ({\rm\Delta}W_{1},\ldots ,{\rm\Delta}W_{k-1},-p/\sqrt{pq},x_{k+1},\ldots ,x_{N} )\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad \hspace{-24.0pt}\times ∼p^{\#\{\ell ;x_{\ell }=q/\sqrt{pq}\}}q^{\#\{\ell ;x_{\ell }=-p/\sqrt{pq}\}}.\nonumber\end{eqnarray}$ Also, we have $\begin{eqnarray}\displaystyle D_{k}F & = & \displaystyle f({\rm\Delta}W_{1},\ldots ,{\rm\Delta}W_{k-1},q/\sqrt{pq},\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle {\rm\Delta}W_{k+1},\ldots ,{\rm\Delta}W_{N} )\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -∼f({\rm\Delta}W_{1},\ldots ,{\rm\Delta}W_{k-1},-p/\sqrt{pq},\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle {\rm\Delta}W_{k+1},\ldots ,{\rm\Delta}W_{N} ),\nonumber\end{eqnarray}$ and then $\begin{eqnarray}\displaystyle \hspace{-18.0pt}E[D_{k}F|{\mathcal{F}}_{k-1}] & = & \displaystyle \mathop{\sum }_{x_{k+1},\ldots ,x_{N}\in \{q/\sqrt{pq},-p/\sqrt{pq}\}}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-45.0pt}\times ∼f ({\rm\Delta}W_{1},\ldots ,{\rm\Delta}W_{k-1},q/\sqrt{pq},x_{k+1},\ldots ,x_{N} )\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-45.0pt}\times ∼p^{\#\{\ell ;x_{\ell }=q/\sqrt{pq}\}}q^{\#\{\ell ;x_{\ell }=-p/\sqrt{pq}\}}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-45.0pt}-\mathop{\sum }_{x_{k+1},\ldots ,x_{N}\in \{q/\sqrt{pq},-p/\sqrt{pq}\}}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-45.0pt}\times ∼f ({\rm\Delta}W_{1},\ldots ,{\rm\Delta}W_{k-1},-p/\sqrt{pq},x_{k+1},\ldots ,x_{N} )\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-45.0pt}\times ∼p^{\#\{\ell ;x_{\ell }=q/\sqrt{pq}\}}q^{\#\{\ell ;x_{\ell }=-p/\sqrt{pq}\}}.\nonumber\end{eqnarray}$ Therefore we obtain $D_{k}E[F|{\mathcal{F}}_{k}]=E[D_{k}F|{\mathcal{F}}_{k-1}]$ and the assertion of the theorem. □

The result and the proof of Theorem B . are also extensions and alternatives of those of the lemma in Section 15.1 of Williams [16] or Theorem 1.1 of Watanabe [14]. We note that the result of Theorem B . can be written by the language of stochastic integral. For a predictable process $\{h_{k}\}_{k}$ and the martingale $\{W_{k}\}_{k}$ , define the discrete-time stochastic integral $\begin{eqnarray}\displaystyle \int _{k}^{n}hdW:=\mathop{\sum }_{i=k+1}^{n}h_{i}{\rm\Delta}W_{i},\quad 0\leq k<n\leq N. & & \displaystyle \nonumber\end{eqnarray}$ Then F = f (ΔW ₁, …, ΔW _N) has the form $F=E[F]+\int _{0}^{N}{\varphi}dW$ with ${\varphi}_{k}=D_{k}E[F|{\mathcal{F}}_{k}]=E[D_{k}F|{\mathcal{F}}_{k-1}]$ , k = 1, …, N. The discrete-time stochastic integral $\{\int _{0}^{n}hdW\}_{n}$ with respect to the martingale $\{W_{k}\}_{k}$ is also a martingale, which is called the martingale transformation. As it is known, the martingale representation theorem in stochastic integration theory gives the inverse assertion, in other words, a martingale is represented by a stochastic integral. The discrete martingale representation theorem is immediately obtained by the proof of Theorem B . above where the integrand is given by the discrete-time Malliavin derivative of M _k.

TheoremD . (Discrete martingale representation theorem).

Let $M=\{M_{k}\}_{k=0}^{N}$ be a $\{{\mathcal{F}}_{k}\}$ -martingale. Then we have $\begin{eqnarray}\displaystyle M_{N}=M_{0}+\mathop{\sum }_{k=1}^{N}(D_{k}M_{k}){\rm\Delta}W_{k}. & & \displaystyle\end{eqnarray}$ (7)

Theorem B . and Corollary 1 play important roles in the proof of the discrete conditional integration by parts and the discrete Bismut formula in Sections 3 and 4.

3. Conditional integration by parts

We give conditional integration by parts formula in discrete-time setting with an elementary proof.

TheoremE . (Discrete conditional integration by parts).

Let $f:\mathbb{R}^{N}\rightarrow \mathbb{R}$ , F = f (ΔW ₁, …, ΔW _N) and $\{h_{i}\}_{i=1}^{N}$ be a predictable process. Then we have $\begin{eqnarray}\displaystyle \hspace{-12.0pt}E\left[\mathop{\sum }_{i=k+1}^{N}D_{i}Fh_{i}|{\mathcal{F}}_{k}\right]=E\left[F\int _{k}^{N}hdW|{\mathcal{F}}_{k}\right], & & \displaystyle\end{eqnarray}$ (8) for k = 0,1, …, N −1.

TheoremF .

For fixed k = 0,1, …, N −1, we consider the calculation of the following term: $\begin{eqnarray}\displaystyle E\left[F\mathop{\sum }_{i=k+1}^{N}h_{i}{\rm\Delta}W_{i}|{\mathcal{F}}_{k}\right]. & & \displaystyle \nonumber\end{eqnarray}$ By the discrete Clark–Ocone formula $F=E[F]+\sum _{j=1}^{N}E[D_{j}F|{\mathcal{F}}_{j-1}]{\rm\Delta}W_{j}$ in Theorem B ., we have $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \hspace{-22.0pt}E\left[F\mathop{\sum }_{i=k+1}^{N}h_{i}{\rm\Delta}W_{i}|{\mathcal{F}}_{k}\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad \hspace{-20.0pt}=E\left[\left(E[F]+\mathop{\sum }_{j=1}^{N}E[D_{j}F|{\mathcal{F}}_{j-1}]{\rm\Delta}W_{j}\right)\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \qquad \hspace{-20.0pt}\times \left.\mathop{\sum }_{i=k+1}^{N}h_{i}{\rm\Delta}W_{i}|{\mathcal{F}}_{k}\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad \hspace{-20.0pt}=E[F]E\left[\mathop{\sum }_{i=k+1}^{N}h_{i}{\rm\Delta}W_{i}|{\mathcal{F}}_{k}\right]\end{eqnarray}$ (9) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \qquad \hspace{-20.0pt}+∼E\left[\mathop{\sum }_{j=1}^{N}E[D_{j}F|{\mathcal{F}}_{j-1}]{\rm\Delta}W_{j}\mathop{\sum }_{i=k+1}^{N}h_{i}{\rm\Delta}W_{i}|{\mathcal{F}}_{k}\right].\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$ (10) For the first term of the right-hand side of (3.2), we easily see that $\begin{eqnarray}\displaystyle E[F]E\left[\mathop{\sum }_{i=k+1}^{N}h_{i}{\rm\Delta}W_{i}|{\mathcal{F}}_{k}\right]=0, & & \displaystyle\end{eqnarray}$ (11) by the martingale property of $\sum _{i=k+1}^{N}h_{i}{\rm\Delta}W_{i}$ . Then we focus on the term (3.3). Let $g_{j}:=E[D_{j}F|{\mathcal{F}}_{j-1}]$ for j = 1, …, N. Then $\{g_{j}\}_{j\geq 1}$ is a predictable process.

For j ≤ k < i, we have $\begin{eqnarray}\displaystyle \hspace{-18.0pt}E[g_{j}{\rm\Delta}W_{j}h_{i}{\rm\Delta}W_{i}|{\mathcal{F}}_{k}] & = & \displaystyle g_{j}{\rm\Delta}W_{j}E[h_{i}{\rm\Delta}W_{i}|{\mathcal{F}}_{k}]\nonumber\\ \displaystyle & = & \displaystyle g_{j}{\rm\Delta}W_{j}E[E[h_{i}{\rm\Delta}W_{i}|{\mathcal{F}}_{i-1}]|{\mathcal{F}}_{k}]\nonumber\\ \displaystyle & = & \displaystyle g_{j}{\rm\Delta}W_{j}E[h_{i}E[{\rm\Delta}W_{i}|{\mathcal{F}}_{i-1}]|{\mathcal{F}}_{k}]\nonumber\\ \displaystyle & = & \displaystyle g_{j}{\rm\Delta}W_{j}E[h_{i}E[{\rm\Delta}W_{i}]|{\mathcal{F}}_{k}]\nonumber\\ \displaystyle & = & \displaystyle 0.\end{eqnarray}$ (12) For j ≥ k +1 and j = i, we have $\begin{eqnarray}\displaystyle \hspace{-18.0pt}E[g_{j}{\rm\Delta}W_{j}h_{i}{\rm\Delta}W_{i}|{\mathcal{F}}_{k}] & = & \displaystyle E[E[g_{j}h_{j}({\rm\Delta}W_{j})^{2}|{\mathcal{F}}_{j-1}]|{\mathcal{F}}_{k}]\nonumber\\ \displaystyle & = & \displaystyle E[g_{j}h_{j}E[({\rm\Delta}W_{j})^{2}|{\mathcal{F}}_{j-1}]|{\mathcal{F}}_{k}]\nonumber\\ \displaystyle & = & \displaystyle E[g_{j}h_{j}E[({\rm\Delta}W_{j})^{2}]|{\mathcal{F}}_{k}]\nonumber\\ \displaystyle & = & \displaystyle E[g_{j}h_{j}|{\mathcal{F}}_{k}]\nonumber\\ \displaystyle & = & \displaystyle E[E[D_{j}F|{\mathcal{F}}_{j-1}]h_{j}|{\mathcal{F}}_{k}]\nonumber\\ \displaystyle & = & \displaystyle E[E[D_{j}Fh_{j}|{\mathcal{F}}_{j-1}]|{\mathcal{F}}_{k}]\nonumber\\ \displaystyle & = & \displaystyle E[D_{j}Fh_{j}|{\mathcal{F}}_{k}],\end{eqnarray}$ (13) using the tower property, the independence of 𝜎(ΔW _j) and ${\mathcal{F}}_{j-1}$ and E[(ΔW _j)²] = 1 with the predictability of $\{h_{j}\}_{j\geq 1}$ and $\{g_{j}\}_{j\geq 1}$ .

For j ≥ k +1 and j > i, $\begin{eqnarray}\displaystyle \hspace{-18.0pt}E[g_{j}{\rm\Delta}W_{j}h_{i}{\rm\Delta}W_{i}|{\mathcal{F}}_{k}] & = & \displaystyle E[E[g_{j}{\rm\Delta}W_{j}h_{i}{\rm\Delta}W_{i}|{\mathcal{F}}_{j-1}]|{\mathcal{F}}_{k}]\nonumber\\ \displaystyle & = & \displaystyle E[g_{j}h_{i}{\rm\Delta}W_{i}E[{\rm\Delta}W_{j}|{\mathcal{F}}_{j-1}]|{\mathcal{F}}_{k}]\nonumber\\ \displaystyle & = & \displaystyle E[g_{j}h_{i}{\rm\Delta}W_{i}E[{\rm\Delta}W_{j}]|{\mathcal{F}}_{k}]\nonumber\\ \displaystyle & = & \displaystyle 0.\end{eqnarray}$ (14) For j ≥ k +1 and j < i, we similarly have $\begin{eqnarray}\displaystyle E[g_{j}{\rm\Delta}W_{j}h_{i}{\rm\Delta}W_{i}|{\mathcal{F}}_{k}]=0 & & \displaystyle\end{eqnarray}$ (15) as in (3.7).

Therefore, by (3.2)–(3.8), we obtain $\begin{eqnarray}\displaystyle \hspace{-15.0pt}E\left[\mathop{\sum }_{i=k+1}^{N}D_{i}Fh_{i}|{\mathcal{F}}_{k}\right]=E\left[F\mathop{\sum }_{i=k+1}^{N}h_{i}{\rm\Delta}W_{i}|{\mathcal{F}}_{k}\right].\quad \Box & & \displaystyle \nonumber\end{eqnarray}$

TheoremG .

It is well known that on Wiener space one has $\begin{eqnarray}\displaystyle E\left[\int _{0}^{T}D_{s}Fh_{s}ds\right]=E\left[F\int _{0}^{T}h_{s}dW_{s}\right], & & \displaystyle \nonumber\end{eqnarray}$ for a smooth Wiener functional F (in the Malliavin sense) and a predictable L ²-vector field h, where DF is the Malliavin derivative of F and $\int _{0}^{T}h_{s}dW_{s}$ is the Itô integral, the L ²-divergence. See [3,4,6,9] for the detail.

The formula given in Theorem E . is the discrete version of the following conditional formula: $\begin{eqnarray}\displaystyle E\left[\int _{t}^{T}D_{s}Fh_{s}ds|{\mathcal{F}}_{t}\right]=E\left[F\int _{t}^{T}h_{s}dW_{s}|{\mathcal{F}}_{t}\right]. & & \displaystyle \nonumber\end{eqnarray}$ For the detail on the conditional integration by parts on Wiener space, see [17] for instance.

4. Discrete Bismut formula for asymmetric random walk

We now show the discrete Bismut formula. Let $X_{0}\in \mathbb{R}$ and 𝜎 > 0 and define a stochastic process $\begin{eqnarray}\displaystyle X_{k}:=X_{0}+{\sigma}Z_{1}+\cdots +{\sigma}Z_{k},\quad k=1,\ldots ,N, & & \displaystyle \nonumber\end{eqnarray}$ which is an asymmetric random walk driven by $\{Z_{k}\}_{k}$ . Let $f:\mathbb{R}\rightarrow \mathbb{R}$ and we introduce a notation: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}\partial _{X_{k}}E[f(X_{N})|{\mathcal{F}}_{k}]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad \hspace{-15.0pt}:={\displaystyle \frac{E[f(X_{N})|{\mathcal{F}}_{k}]({\omega}_{k}^{+})-E[f(X_{N})|{\mathcal{F}}_{k}]({\omega}_{k}^{-})}{X_{k}({\omega}_{k}^{+})-X_{k}({\omega}_{k}^{-})}},\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad k=1,\ldots ,N.\nonumber\end{eqnarray}$ Then $\partial _{X_{k}}E[f(X_{N})|{\mathcal{F}}_{k}]$ is represented as follows.

TheoremH .
(Discrete Bismut formula for asymmetric random walk model). It holds that $\begin{eqnarray}\displaystyle \hspace{-18.0pt}\partial _{X_{k}}E[f(X_{N})|{\mathcal{F}}_{k}] & = & \displaystyle E\left[f(X_{N}){\displaystyle \frac{1}{N-(k-1)}}\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad \hspace{-10.0pt}\times \left.\int _{k-1}^{N}(2{\sigma}\sqrt{pq})^{-1}dW|{\mathcal{F}}_{k-1}\right]\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$ (16) for k = 1, …, N.

TheoremI .
We note that $X_{k}({\omega}_{k}^{+})-X_{k}({\omega}_{k}^{-})=2{\sigma}$ and $E[f(X_{N})|{\mathcal{F}}_{k}]({\omega}_{k}^{+})-E[f(X_{N})|{\mathcal{F}}_{k}]({\omega}_{k}^{-})=\frac{1}{\sqrt{pq}}$ $D_{k}E[f(X_{N})|{\mathcal{F}}_{k}]=\frac{1}{\sqrt{pq}}∼E[D_{k}f(X_{N})|{\mathcal{F}}_{k-1}]$ . Then we have $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \hspace{10.0pt}\partial _{X_{k}}E[f(X_{N})|{\mathcal{F}}_{k}]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \qquad \quad =E[D_{k}f(X_{N})(2{\sigma}\sqrt{pq})^{-1}|{\mathcal{F}}_{k-1}].\nonumber\end{eqnarray}$ Since $\{Z_{k}\}_{k}$ is a sequence of i.i.d. such that P (Z _k = 1) = p, we have $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \hspace{-27.0pt}E[D_{k}f(X_{N})|{\mathcal{F}}_{k-1}]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad \hspace{-26.0pt}=\mathop{\sum }_{z_{k+1},\ldots ,z_{N}\in \{1,-1\}}f\left(X_{0}+\mathop{\sum }_{i=1}^{k-1}Z_{i}+\mathop{\sum }_{i=k+1}^{N}z_{i}+1\right)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \qquad \hspace{-26.0pt}\times ∼p^{\#\{\ell ;z_{\ell }=1\}}q^{\#\{\ell ;z_{\ell }=-1\}}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \qquad \hspace{-26.0pt}-\mathop{\sum }_{x_{k+1},\ldots ,x_{N}\in \{1,-1\}}\hspace{-3.0pt}f\left(X_{0}+\mathop{\sum }_{i=1}^{k-1}Z_{i}+\mathop{\sum }_{i=k+1}^{N}z_{i}-1\right)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \qquad \hspace{-26.0pt}\times ∼p^{\#\{\ell ;x_{\ell }=1\}}q^{\#\{\ell ;x_{\ell }=-1\}},\nonumber\end{eqnarray}$ and then $\begin{eqnarray}\displaystyle \hspace{-10.0pt}E[D_{k}f(X_{N})|{\mathcal{F}}_{k-1}] & = & \displaystyle E[D_{k+1}f(X_{N})|{\mathcal{F}}_{k-1}]\nonumber\\ \displaystyle & = & \displaystyle \cdots =E[D_{N}f(X_{N})|{\mathcal{F}}_{k-1}],\nonumber\end{eqnarray}$ i.e. $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}E[D_{k}f(X_{N})(2{\sigma}\sqrt{pq})^{-1}|{\mathcal{F}}_{k-1}]=\frac{1}{N-(k-1)}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad \hspace{-18.0pt}\times ∼E\left[\mathop{\sum }_{i=k}^{N}D_{i}f(X_{N})(2{\sigma}\sqrt{pq})^{-1}|{\mathcal{F}}_{k-1}\right].\nonumber\end{eqnarray}$ Therefore, by the conditional integration by parts (3.1) in Theorem E ., we obtain $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \hspace{-26.0pt}\partial _{X_{k}}E[f(X_{N})|{\mathcal{F}}_{k}]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad \hspace{-26.0pt}=E\left[{\displaystyle \frac{1}{N-(k-1)}}\mathop{\sum }_{i=k}^{N}D_{i}f(X_{N})(2{\sigma}\sqrt{pq})^{-1}|{\mathcal{F}}_{k-1}\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad \hspace{-26.0pt}=E\left[f(X_{N}){\displaystyle \frac{1}{N-(k-1)}}\int _{k-1}^{N}(2{\sigma}\sqrt{pq})^{-1}dW|{\mathcal{F}}_{k-1}\right].\quad \Box \nonumber\end{eqnarray}$

TheoremJ .
When p = q = 1∕2 (the symmetric case), the Bismut formula becomes $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \hspace{-22.0pt}\partial _{X_{k}}E[f(X_{N})|{\mathcal{F}}_{k}]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad \hspace{-18.0pt}=E\left[f(X_{N}){\displaystyle \frac{1}{N-(k-1)}}\int _{k-1}^{N}{\sigma}^{-1}dW|{\mathcal{F}}_{k-1}\right],\quad \nonumber\\ \displaystyle \text{} & \text{} & \displaystyle k=1,\ldots ,N.\nonumber\end{eqnarray}$

5. Delta hedging

5.1. Discrete-time market: asymmetric exponential process

We still adopt the probability space (and the definitions) in Section 2. Let $r,d,u\in \mathbb{R}$ be constants and $\{{\chi}_{k}\}_{k=1}^{N}$ be a sequence of i.i.d. random variables given by $\begin{eqnarray}\displaystyle {\chi}_{k}({\omega})=\left\{\begin{array}{@{}l@{}}u({\omega}_{k}=1)\quad \\ d({\omega}_{k}=-1),\quad \end{array}\right.\quad {\omega}=({\omega}_{1},\ldots ,{\omega}_{N})\in {\rm\Omega}, & & \displaystyle \nonumber\end{eqnarray}$ for k = 1, …, N. We define the process of an underlying asset $\{S_{k}\}_{k=1}^{N}$ given by $S_{k}=S_{0}\prod _{i=1}^{k}(1+{\chi}_{i})$ , k = 1, …, N, where S ₀ > 0. Note that $\{S_{k}\}_{k=1}^{N}$ can be written as $S_{k}=S_{0}\prod _{i=1}^{k}(1+{\rm\Delta}S_{i})$ when we define ΔS _k: = S _k − S _k−1 (k = 1, …, N), which is called asymmetric exponential process. Let $\{B_{k}\}_{k=1}^{N}$ be the money market account given by $\begin{eqnarray}\displaystyle \hspace{-10.0pt}B_{k}({\omega})=(1+r)B_{k-1}({\omega}),\quad {\omega}=({\omega}_{1},\ldots ,{\omega}_{N})\in {\rm\Omega}, & & \displaystyle \nonumber\end{eqnarray}$ for k = 1, …, N, where B ₀(⋅) ≡ B ₀ > 0. We assume no-arbitrage condition −1 < d < r < u.

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ and consider the replication/pricing problem of an European option with the payoff f(S _N). We define the stochastic process of the portfolio value given by $\begin{eqnarray}\displaystyle V_{k}({\omega}) & = & \displaystyle {\rm\Delta}_{k-1}({\omega})S_{k}({\omega})+{\Psi}_{k-1}({\omega})B_{k}({\omega}),\quad \nonumber\\ \displaystyle \text{} & \text{} & \displaystyle {\omega}\in {\rm\Omega},∼k=1,\ldots ,N\nonumber\end{eqnarray}$ where Δ_k−1 and 𝛹_k−1 are ${\mathcal{F}}_{k-1}$ -measurable random variables for k = 1, …, N, which represent the amounts of S _k and B _k (determined at time k −1) that we should hold. We suppose that the portfolio is self-financing, i.e. $\begin{eqnarray}\displaystyle \hspace{-18.0pt}V_{k}-V_{k-1} & = & \displaystyle {\rm\Delta}_{k-1}(S_{k}-S_{k-1})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼{\Psi}_{k-1}(B_{k}-B_{k-1}),\quad k=1,\ldots ,N,\nonumber\end{eqnarray}$ which is equivalent to $\begin{eqnarray}\displaystyle \widetilde{V}_{n}=\widetilde{V}_{0}+\mathop{\sum }_{k=1}^{n}{\rm\Delta}_{k-1}(\widetilde{S}_{k}-\widetilde{S}_{k-1}),\quad n=1,\ldots ,N, & & \displaystyle \nonumber\end{eqnarray}$ where $\widetilde{V}_{k}=\frac{1}{(1+r)^{k}}V_{k}$ and $\widetilde{S}_{k}=\frac{1}{(1+r)^{k}}S_{k}$ , k = 0,1, …, N. The purpose here is to construct the replicating portfolio, in other words, we will find the delta hedging process $\{{\rm\Delta}_{k}\}_{k}$ such that V _N = f(S _N) or $\widetilde{V}_{N}=\frac{1}{(1+r)^{N}}f(S_{N})$ . We specify the probability measure so that $\{\widetilde{S}_{k}\}_{k\geq 0}$ becomes a martingale as follows: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}P(\{{\omega}\in {\rm\Omega};{\chi}_{k}({\omega})=u\})=p:={\displaystyle \frac{r-d}{u-d}}>0,\qquad \nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}P(\{{\omega}\in {\rm\Omega};{\chi}_{k}({\omega})=d\})=q=1-p={\displaystyle \frac{u-r}{u-d}}>0,\nonumber\end{eqnarray}$ for k = 1, …, N, of which existence (i.e. p, q > 0) is ensured by the no-arbitrage condition. In this setting, 𝜒_k has the form ${\chi}_{k}({\omega})=r+(u-d)\sqrt{pq}{\rm\Delta}W_{k}({\omega})$ for 𝜔 ∈ Ω, k = 1, …, N. Then we have $S_{N}=S_{0}\prod _{i=1}^{N}(1+r+(u-d)\sqrt{pq}{\rm\Delta}W_{i})$ .

The delta hedging process is given using the Malliavin derivative of $E[f(S_{N})|{\mathcal{F}}_{k}]$ (Privault [7]) as follows.

TheoremK . (Discrete Ocone–Karatzas hedging).

$\begin{eqnarray}\displaystyle \hspace{-18.0pt}{\rm\Delta}_{k-1}={\displaystyle \frac{D_{k}E[f(S_{N})|{\mathcal{F}}_{k}]}{(1+r)^{N-k}S_{k-1}(u-d)\sqrt{pq}}},\quad k=1,\ldots ,N. & & \displaystyle \nonumber\end{eqnarray}$

TheoremL .
See Privault [7]. □

TheoremM .
We note that the formula above is a kind of numerical differentiation of stochastic process. Then we need two conditional expectations to compute Δ_k−1 by the definition of $D_{k}F=\sqrt{pq}(F({\omega}_{k}^{+})-F({\omega}_{k}^{-}))$ . Hereafter, we give a formula of Δ_k−1 with a single conditional expectation by using the discrete Bismut formula.

5.2. Hedging with discrete Bismut formula

We give a new representation for the delta hedging process $\{{\rm\Delta}_{k}\}_{k=0}^{N-1}$ where each random variable is given by a single conditional expectation.

TheoremN . (Discrete Bismut formula for delta).

It holds that $\begin{eqnarray}\displaystyle {\rm\Delta}_{k-1} & = & \displaystyle {\displaystyle \frac{1}{(1+r)^{N-k}}}E\left[f(S_{N}){\displaystyle \frac{1}{N-(k-1)}}\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \times \left.\int _{k-1}^{N}((u-d)\sqrt{pq}S_{k-1})^{-1}dW|{\mathcal{F}}_{k-1}\right],\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$ (17) for k = 1, …, N.

TheoremO .

Since we have $D_{k}E[f(S_{N})|{\mathcal{F}}_{k}]=E$ [D _k $f(S_{N})|{\mathcal{F}}_{k-1}$ ] in the discrete Ocone–Karatzas hedging of Theorem K ., it holds that $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \hspace{-22.0pt}{\rm\Delta}_{k-1}={\displaystyle \frac{1}{(1+r)^{N-k}}}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad \times ∼E[D_{k}f(S_{N})((u-d)\sqrt{pq}S_{k-1})^{-1}|{\mathcal{F}}_{k-1}].\nonumber\end{eqnarray}$ Note that $((u-d)\sqrt{pq}S_{k-1})^{-1}$ is ${\mathcal{F}}_{k-1}$ -measurable and $\begin{eqnarray}\displaystyle E[D_{k}f(S_{N})|{\mathcal{F}}_{k-1}] & = & \displaystyle E[D_{k+1}f(S_{N})|{\mathcal{F}}_{k-1}]\nonumber\\ \displaystyle & = & \displaystyle \cdots =E[D_{N}f(S_{N})|{\mathcal{F}}_{k-1}].\nonumber\end{eqnarray}$ Then we have $\begin{eqnarray} \displaystyle \text{} & \text{} & \displaystyle E[D_{k}f(S_{N})((u-d)\sqrt{pq}S_{k-1})^{-1}|{\mathcal{F}}_{k-1}]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad =\frac{1}{N-(k-1)}E\left[\mathop{\sum }_{i=k}^{N}D_{i}f(S_{N})\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \qquad \times \left.((u-d)\sqrt{pq}S_{k-1})^{-1}|{\mathcal{F}}_{k-1}\vphantom{\sum_{i=k}^{N}}\right].\nonumber \end{eqnarray}$ Therefore, by the conditional integration by parts (3.1) of Theorem E ., we obtain $\begin{eqnarray} \displaystyle \text{} & \text{} & \displaystyle \hspace{-25.0pt}{\rm\Delta}_{k-1}={\displaystyle \frac{1}{(1+r)^{N-k}}}E\left[\vphantom{\sum_{i=k}^{N}}{\displaystyle \frac{1}{N-(k-1)}}\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \qquad \hspace{-13.0pt}\times \left.\mathop{\sum }_{i=k}^{N}D_{i}f(S_{N})((u-d)\sqrt{pq}S_{k-1})^{-1}|{\mathcal{F}}_{k-1}\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad \hspace{-13.0pt}={\displaystyle \frac{1}{(1+r)^{N-k}}}E\left[f(S_{N}){\displaystyle \frac{1}{N-(k-1)}}\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \qquad \hspace{-13.0pt}\times \left.\int _{k-1}^{N}((u-d)\sqrt{pq}S_{k-1})^{-1}dW|{\mathcal{F}}_{k-1}\right].\quad \Box \nonumber \end{eqnarray}$

TheoremP .

We note that the weight $\int _{k-1}^{N}$ ( $(u-d)\sqrt{pq}$ S _k−1)⁻¹ dW in the formula in Theorem N . is computed using the same random variables constituting S _N.

TheoremQ .

When u = −d =: 𝜎 ∈ (0,1), we have $\begin{eqnarray}\displaystyle {\rm\Delta}_{k-1} & = & \displaystyle {\displaystyle \frac{1}{(1+r)^{N-k}}}E\left[f(S_{N}){\displaystyle \frac{1}{N-(k-1)}}\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \times \left.\int _{k-1}^{N}(2{\sigma}\sqrt{pq}S_{k-1})^{-1}dW|{\mathcal{F}}_{k-1}\right],\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad k=1,\ldots ,N.\nonumber\end{eqnarray}$ Furthermore, when r = 0, i.e. the case p = q = 1∕2, it holds that $\begin{eqnarray}\displaystyle \hspace{-18.0pt}{\rm\Delta}_{k-1} & = & \displaystyle E\left[f(S_{N}){\displaystyle \frac{1}{N-(k-1)}}\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \times \hspace{-3.0pt}\left.\int _{k-1}^{N}({\sigma}S_{k-1})^{-1}dW|{\mathcal{F}}_{k-1}\right],\quad \hspace{-3.0pt}k=1,\ldots ,N.\nonumber\end{eqnarray}$

Finally, we give another representation of the delta. Hereafter, we assume $\begin{eqnarray}\displaystyle u=-d=:{\sigma}\in (0,1) & & \displaystyle \nonumber\end{eqnarray}$ for simplicity. Before showing the result, we prepare the discrete Itô formula of Fujita [15].

TheoremR . (Discrete Itô formula (Fujita [15])).

Let $h:\{1,\ldots ,N\}\times \mathbb{R}\rightarrow \mathbb{R}$ and $\{\widetilde{Z}_{j}\}_{j=1}^{N}$ be a process given by $\widetilde{Z}_{j}=Z_{0}+\sum _{i=1}^{j}{\xi}Z_{i}$ , j = 1, …, N, where ${\xi}\in \mathbb{R}$ and $\{Z_{k}\}_{k=1}^{N}$ is a sequence of i.i.d. random variables given in Section 2 with (2.1). Then, we have $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \hspace{-26.0pt}h(j+1,\widetilde{Z}_{j+1})-h(j,\widetilde{Z}_{j})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad \hspace{-29.0pt}=(h(j+1,\widetilde{Z}_{j}+{\xi})-h(j+1,\widetilde{Z}_{j}-{\xi}))\sqrt{pq}{\rm\Delta}W_{j}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \qquad \hspace{-31.0pt}+∼ph(j+1,\widetilde{Z}_{j}+{\xi})+qh(j+1,\widetilde{Z}_{j}-{\xi})-h(j,\widetilde{Z}_{j}),\nonumber\end{eqnarray}$ for j = 0,1, …, N −1.

The delta has the following representation through the discrete Itô formula with the integration by parts, which can also be obtained as a consequence of Theorem N ..

TheoremS . (Discrete Bismut formula for delta II).

It holds that $\begin{eqnarray}\displaystyle {\rm\Delta}_{k-1} & = & \displaystyle {\displaystyle \frac{1}{(1+r)^{N-k}}}\left.E\left[f(S_{N-(k-1)}^{x}){\displaystyle \frac{1}{N-(k-1)}}\right.\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \times \left.\left.\int _{0}^{N-(k-1)}(2{\sigma}\sqrt{pq}x)^{-1}dW\right]\right|_{x=S_{k-1}},\nonumber\end{eqnarray}$ for k = 1, …, N.

TheoremT .
Let $S_{k}^{x}=x\prod _{i=1}^{k}(1+{\chi}_{i})=x\prod _{i=1}^{k}(1+{\sigma}Z_{i})$ and $(P_{k}f)(x):=E[f(S_{k}^{x})]$ for x > 0 and k = 0,1, …, N. Note that $S_{k}^{S_{0}}\equiv S_{k}$ . Also, we define 𝛻(ℓ, x): = $\frac{P_{\ell -1}f(x(1+{\sigma}))-P_{\ell -1}f(x(1-{\sigma}))}{2{\sigma}x}$ . For k = 1, …, N, the delta Δ_k−1 at k −1 is given by $\begin{eqnarray}\displaystyle \hspace{-18.0pt}{\rm\Delta}_{k-1} & = & \displaystyle {\displaystyle \frac{1}{(1+r)^{N-k}}}{\nabla}(N-k+1,S_{k-1})\nonumber\\ \displaystyle & = & \displaystyle {\displaystyle \frac{1}{(1+r)^{N-k}}}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \times ∼{\displaystyle \frac{P_{N-k}f(S_{k-1}(1+{\sigma}))-P_{N-k}f(S_{k-1}(1-{\sigma}))}{2{\sigma}S_{k-1}}}.\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$ (18) For ℓ = 1, …, N and k = 0,1, …, ℓ, we define g (k, y): = P _ℓ−k f (e ^y), $y\in \mathbb{R}$ , and $\widetilde{Z}_{\ell }^{z}=z+\sum _{p=1}^{\ell }{\xi}Z_{p}$ , where z = log x and ${\xi}=\frac{1}{2}(\log (1+{\sigma})-\log (1-{\sigma}))$ . Note that $g(\ell ,\widetilde{Z}_{\ell }^{z})=f(S_{\ell }^{x})$ , g (0, z) = P _ℓ f (x) and $E[g(\ell ,\widetilde{Z}_{\ell }^{z})|{\mathcal{F}}_{k}]=E[f(S_{\ell }^{x})|S_{k}^{x}]=E[f(S_{\ell -k}^{y})]|_{y=S_{k}^{x}}$ $=P_{\ell -k}f(S_{k}^{x})=g(k,\widetilde{Z}_{k}^{z})$ , which means that $\big\{$ g (k, $\widetilde{Z}_{k}^{z}$ ) $\big\}_{k}$ is a martingale. Then, by the discrete Itô formula in Lemma 1 with the definition of discrete-time Malliavin derivative, we have $\begin{eqnarray} \displaystyle \text{} & \text{} & \displaystyle \hspace{-24.0pt}g(k,\widetilde{Z}_{k}^{z})-g(k-1,\widetilde{Z}_{k-1}^{z})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad \hspace{-24.0pt}=(g(k,\widetilde{Z}_{k-1}^{z}+{\xi})-g(k,\widetilde{Z}_{k-1}^{z}-{\xi}))\sqrt{pq}{\rm\Delta}W_{k}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \qquad \hspace{-24.0pt}+∼(1-p)g(k,\widetilde{Z}_{k-1}^{z}+{\xi})+(-p)g(k,\widetilde{Z}_{k-1}^{z}-{\xi})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \qquad \hspace{-24.0pt}-∼g(k-1,\widetilde{Z}_{k-1}^{z})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad \hspace{-24.0pt}=(g(k,\widetilde{Z}_{k-1}^{z}+{\xi})-g(k,\widetilde{Z}_{k-1}^{z}-{\xi}))\sqrt{pq}{\rm\Delta}W_{k}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad \hspace{-24.0pt}=D_{k}P_{\ell -k}f(S_{k}^{x}){\rm\Delta}W_{k},\nonumber \end{eqnarray}$ for k = 1, …, ℓ and therefore we get $f(S_{\ell }^{x})=P_{\ell }f(x)+\sum _{k=1}^{\ell }D_{k}P_{\ell -k}f(S_{k}^{x}){\rm\Delta}W_{k}$ for ℓ = 1, …, N. Multiplying both sides by $\int _{0}^{\ell }dW=\sum _{i=1}^{\ell }{\rm\Delta}W_{i}$ and taking expectation, the integration by parts formula holds $\begin{eqnarray}\displaystyle \hspace{-18.0pt}E\left[f(S_{\ell }^{x})\int _{0}^{\ell }dW\right]=E\left[\mathop{\sum }_{k=1}^{\ell }D_{k}P_{\ell -k}f(S_{\ell }^{x})\right]. & & \displaystyle\end{eqnarray}$ (19) Furthermore, for k = 1, …, ℓ, one has $\begin{eqnarray}\displaystyle \hspace{-18.0pt}E[D_{k}P_{\ell -k}f(S_{k}^{x})] & = & \displaystyle \sqrt{pq}\{E[P_{\ell -k}f(S_{k-1}^{x}(1+{\sigma}))]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -∼E[P_{\ell -k}f(S_{k-1}^{x}(1-{\sigma}))]\}\nonumber\\ \displaystyle & = & \displaystyle \sqrt{pq}\{P_{k-1}P_{\ell -k}f(x(1+{\sigma}))\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -∼P_{k-1}P_{\ell -k}f(x(1-{\sigma}))\}\nonumber\\ \displaystyle & = & \displaystyle \sqrt{pq}\{P_{\ell -1}f(x(1+{\sigma}))\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -∼P_{\ell -1}f(x(1-{\sigma}))\}\nonumber\\ \displaystyle & = & \displaystyle 2{\sigma}x\sqrt{pq}{\nabla}(\ell ,x),\end{eqnarray}$ (20) where we used $(P_{k}P_{\ell }f)(x)=E[(P_{\ell }f)(S_{k}^{x})]=E$ [E $[f(S_{\ell }^{y})]|_{y=S_{k}^{x}}$ ] $=E[E[f(S_{k+\ell }^{x})|S_{k}^{x}]]=(P_{k+\ell }f)(x)$ . Therefore, by (5.2), (5.3) and (5.4), we obtain $\begin{eqnarray}\displaystyle \hspace{-18.0pt}{\rm\Delta}_{k-1} & = & \displaystyle {\displaystyle \frac{1}{(1+r)^{N-k}}}\left.E\left[f(S_{N-(k-1)}^{x}){\displaystyle \frac{1}{N-(k-1)}}\right.\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \times \left.\left.\int _{0}^{N-(k-1)}(2{\sigma}\sqrt{pq}x)^{-1}dW\right]\right|_{x=S_{k-1}}.\quad \Box \nonumber\end{eqnarray}$

The algorithm of the proposed scheme (with u = −d for simplicity) is given as follows:

_

Algorithm 1 Delta hedging process with discrete Bismut formula

_

Require: $N\in \mathbb{N}$ , S ₀ > 0, 𝜎 = u = −d > 0, r ∈ (d, u), $p=\frac{r-d}{u-d}$ ,q = 1 − p

Compute $\begin{eqnarray}\displaystyle {\rm\Delta}_{0} & = & \displaystyle (1+r)^{-(N-1)}\frac{1}{N}\mathop{\sum }_{j=0}^{N}\frac{N!}{(N-j)!j!}p^{j}(1-p)^{N-j}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle f(S_{0}(1+{\sigma})^{j}(1-{\sigma})^{N-j})\frac{{\mathcal{W}}_{0}+j\frac{q}{\sqrt{pq}}+(N-j)\frac{-p}{\sqrt{pq}}}{2{\sigma}\sqrt{pq}S_{0}}\nonumber\end{eqnarray}$

with ${\mathcal{W}}_{0}=∼0$

Let ${\mathcal{W}}_{0}({\omega}_{0})={\mathcal{W}}_{0}$ and S ₀(𝜔₀) = S ₀

for k = 1 to N −1 do

Update ${\mathcal{W}}_{k}({\omega}_{0}\cdots {\omega}_{k})={\mathcal{W}}_{k-1}({\omega}_{0}\cdots {\omega}_{k-1})+\frac{q}{\sqrt{pq}}$ $\mathbf{1}_{{\omega}_{k}=1}+\frac{-p}{\sqrt{pq}}\mathbf{1}_{{\omega}_{k}=-1}$ and S _k(𝜔₀⋯𝜔_k) = S _k−1(𝜔₀⋯𝜔_k−1)(1 + 𝜎𝜔_k), and compute $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \hspace{-3.0pt}{\rm\Delta}_{k}({\omega}_{0}\cdots {\omega}_{k})=(1+r)^{-(N-k+1)}\frac{1}{N-k}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad \mathop{\sum }_{j=0}^{N-k}\frac{(N-k)!}{(N-k-j)!j!}p^{j}(1-p)^{N-k-j}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad \times f(S_{k}({\omega}_{0}\cdots {\omega}_{k})(1+{\sigma})^{j}(1-{\sigma})^{N-k-j})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad \frac{{\mathcal{W}}_{k}({\omega}_{0}\cdots {\omega}_{k})+j\frac{q}{\sqrt{pq}}+(N-k-j)\frac{-p}{\sqrt{pq}}}{2{\sigma}\sqrt{pq}S_{k}({\omega}_{0}\cdots {\omega}_{k})}\nonumber\end{eqnarray}$

end for

_________________________________________________

We show a numerical example for the scheme for the case N = 2, S ₀ = 100, r = 0, 𝜎 = u = −d = 0.2, $f(x)=\max \{x-100,0\}$ . In the setting, the process of the delta is given by Δ₁(𝜔₀𝜔₁) = 0.9166667 (𝜔₁ = 1), Δ₁(𝜔₀𝜔₁) = 0.0 (𝜔₁ = −1) and Δ₀ = 0.55. We remark that the value of the delta of the Black–Scholes model, the corresponding continuous time model, at time 0 (with the maturity T = 2, S ₀ = 100, r = 0, 𝜎 = 0.2 and the same payoff function) is 0.5562315, which indicates that our method gives an approximation for the Bismut formula in continuous time diffusion model.
6. Concluding remarks

In the paper, we showed the conditional integration by parts formula after we provided a simple and elementary proof for discrete Clark–Ocone formula. The discrete Bismut formula is introduced for asymmetric random walk model. Finally, for the discrete-time market, we provided the discrete Bismut formula for the delta hedging process as the extension of the Ocone–Karatzas formula where the conditional integration by parts formula plays an important role. The algorithm for computing delta-hedging process with the discrete Bismut formula is shown with a numerical example and a comparison analysis.

We should further study whether the discrete Bismut formula can be applied to the problems on expansion or/and discretization of (non-smooth) functionals of stochastic processes as in [18–22] as a next topic.

Footnotes

Acknowledgements

We thank an anonymous referee for valuable comments. We also thank Prof. Koichiro Takaoka (Chuo University) for useful advices for the method of this paper. This work is supported by JSPS KAKENHI (grant no. 19K13736), MEXT, Japan and JST PRESTO (grant no. JPMJPR2029), Japan.

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