# Utilizing Quantum Monte Carlo Simulation for Call Option Pricing Problem
*National Yang Ming Chiao Tung University Dep. Applied Methematics
Advisor: Su Cheng-Fang
Student & Editor: Wang Xuan-Wei
Time: 03. Jan. 2024
Contact: [e-mail]*

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## Contents
### 1. Abstract & Poster
### 2. Call Option Pricing Problem
### 3. Black-Scholes Model
### 4. Main algorithm Framework and Quantum Circuit Construction
### 5. Conclusion
### 6. Reference Papers

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## 1. Abstract and Poster
The option pricing problem poses a fsumormidable challenge for financial experts worldwide, as they aspire to forecast the price distributions of all assets, enabling them to make more informed investment decisions.

In this semester's project, we elucidate the significance and primary challenges associated with the call option pricing problem. Specifically, we delve into the intricate aspects of resolving high-dimensional Black-Scholes partial differential equations (PDEs) and outline the fundamental structure of the algorithm.

[Poster link](https://drive.google.com/file/d/1_npuh0J-jaQyTjUAKn73J77rO2wH83bu/view?usp=sharing)

## 2.Call Option Pricing Problem

**Option :**
An option is a financial *'contract'* that grants its owner the right to buy or sell a specific quantity of an underlying asset, such as stocks, indexes, and exchange-traded funds. Notably, an option holder has the discretion to either exercise or not exercise the option.

> **Key Takeaways**:
> * Options are financial derivatives that give buyers the right, but not the obligation, to buy or sell an underlying asset at an agreed-upon price and date.
> * Call options and put options form the basis for a wide range of option strategies designed for hedging, income, or speculation.
> * Options trading can be used for both hedging and speculation, with strategies ranging from simple to complex.
> * Although there are many opportunities to profit with options, investors should carefully weigh the risks.

In this context, we focus on the 'buy' opportunity through a call option. Call options allow the holder to buy the asset at a stated price within a specific timeframe. Each option contract has a precisely defined expiration date. As the expiration date approaches, the value of the contract tends to decrease and becomes more variable. Therefore, it is crucial to calculate the time effects on the option."

**Refers to:** [What are options? Types, Spreads, Example and Risk Metries](https://www.investopedia.com/terms/o/option.asp)

## 3. Black-Scholes Model

**BS Model :**
It is a mathematics model for the dynamics of a financial market containing derivative investments, using various underlying assumptions. It allows to evaluate the value of options which only depend on the underlying single security.

>However, in the multidimensional setting of the Black-Scholes model, there are no explicit expression for the solution of the corresponding PDE, and hence numerical methods are necessary to approximately aolve these high-dimensional PDEs. [1]

Background assumptions of BS Model:
1. European option: the call(or put) action will only take place on the date of option maturity.
2. Random walk: the instantaneous log-return of the stock price follows a geometric Brownian motion.
3. The stock does not pay a dividend.
4. No arbitrage opportunity.
5. Ability to borrow and lend any amount, even fractional, of cash at the riskless rate.
6. Ability to buy and sell any amount, even fractional, of the stock.
7. The above transactions do not incur any fees or costs.

Then we consider the multiple-asset BS PDE:$$\frac{\partial u}{\partial t}+\frac{1}{2}\sum^{d}_{i,j=1}C_{ij}x_ix_j\frac{\partial^2 u}{\partial x_i\partial x_j}+\sum^{d}_{i=1}rx_i\frac{\partial u}{\partial x_i} - ru=0,\ \ \text{in}\ [0,T)\times\mathbb{R}^d_+$$
* $r\in(0,\infty)$ : risk-free interest
* $T\in(0,\infty)$ : finite time horizon determine the maturity
* $d\in\mathbb{N}$ : the number of the assets

Then subject to a terminal condition that $u(T,\cdot)=h(\cdot)$, where $h:\mathbb{R}^d_+\rightarrow\mathbb{R}$ represents the payoff function, and $u(t,x)$ represents the option price at time $t$ with spot price $x$.

And we forcus on the option price $u(t,x)$:$$u(t,x)=e^{-r(T-t)}\mathbb{E}[h(S_T)|S_t=x]=e^{-r(T-t)}\int_{\mathbb{R}^d_+}h(y)p(y,T;x,t)dy$$

The $d$ stocks are modeled by a multidimensional GBM:$$dS^i_t=S^i_t(rdt+\sum^d_{j=1}\sigma_{ij}dW^i_t),\ \ \text{for}\ i=1,...,d$$
* $(\Omega, \mathcal{F},\mathbb{P})$ : a prob. space
* $W=(W^1,...,W^d):[0,T]\times\Omega\rightarrow\mathbb{R}^d$ : a standard $d$-dimensional brownian motion
* $S_t=(S^1_t,...,S^d_t):[0,T]\times\Omega\rightarrow\mathbb{R}^d_+$ : the stock price process for time $t\in [0,T]$

From the stocks above, the joint probability density funciton is given by:$$p(y,T;x,t)=\frac{exp(\frac{1}{2}(log(y)-\mu)^\top\Sigma^{-1}(log(y)-\mu)}{(2\pi)^{d/2}(det\Sigma)^{1/2}\Pi^{d}_{i=1}y_i}$$
* $x=(x_1,...,x_d)\in\mathbb{R}^d_+$ : spot prices of all assets
* $\mu=(\mu_1,...,\mu_d)\in\mathbb{R}^d,\ \mu_i=ln(x_i)+(r-\frac{1}{2}\sigma_{ij}^2)(T-t)$ for $i=1,...,d$ : log-mean of the log-normal distribution
* $\Sigma\in\mathbb{R}^{d\times d},\ \Sigma=(T-t)C$ : log-covariance matrix of the log-normal distribution

We then consider the payoff functions restricted to the class of continuous piecewice affine (CPWA) functions as the following form:$$h(x)\sum^K_{k=1}\xi_k\ max\{a_{k,l}\cdot x+b_{k,l}: l=1,...,I_K\}$$
* $K,I_K\in\mathbb{N}$
* $\xi_k\in\{-1,1\}$
* $a_{k,l}\in\mathbb{R}^d,\ b_{k,l}\in\mathbb{R}\ \text{for}\ k=1,...,K\ \text{and}\ l=1,...,I_K$

## 4. Main algorithm Framework and Quantum Circuit Construction
### Main steps:
1. Upload the joint transition probability density function $p(\cdot,T;x,t)$
2. Upload the CPWA payoff function $h:\mathbb{R}^d_+\rightarrow\mathbb{R}$
3. Apply the modified IQAE algorithm to obtain an estimated amplitude $\hat{a}\in[0,1]$
4. Rescale the estimated amplitude $\hat{a}$ to output $\widetilde{U}_{t,x}$

### Algorithm framwork
:::info
***Intput:***
* $\epsilon\in(0,1)$
* $\alpha\in(0,1)$
* $d\in\mathbb{N}$
* $r,T\in(0,\infty)$
* $(t,x)\in[0,T)\times\mathbb{R}^d_+$
* Covariance matrix $C_d\in\mathbb{R}^{d\times d}$
* CPWA function $\mathbb{R}^d_+\ni x\mapsto h(x)=\sum^{K}_{k=1}\xi_k\ max\{a_{k,l}\cdot x+b_{k,l}:l=1,...,I_K\}$

***Output:***
* $\widetilde{U}_{t,x}\in\mathbb{R}$
:::
:::success
***Steps:***
1. Set constants $C_1,C_2,C_3$
2. Set $n_1:=\lceil n_{1,d,\epsilon}\rceil,\ n_2:=1+\lceil log_2(C_2)\rceil,\ m_1:=\lceil m_{1,d,\epsilon}\rceil,\ m_2:=\lceil m_{2,d,\epsilon}\rceil$
3. Set $N,\gamma, \mathfrak{s}$ and $\tilde{a}_{n_2,m_2,k,l}, \tilde{b}_{n_2,m_2,k,l}$ for $k=1,...,K, l=1,...,I_K$
4. Construct probability distribution quantum circuit $\mathcal{P}=\mathcal{P}_{d,\epsilon}$
5. Construct CPWA payoff with rotation quantum circuit $\mathcal{R}_h$
6. Construct the quantum circuit $\mathcal{A}=\mathcal{R}_h(\mathcal{P}\otimes I_2^{\otimes(N-d(n_1+m_1))})$
7. Set $\hat{a}=\frac{a_u-a_l}{2}$ using the output $[a_l,a_u]$ from the modified iterative quantum algorithm
8. Return $\widetilde{U}_{t,x}:=\mathfrak{s}^{-1}\gamma e^{-r(T-t)}(2\hat{a}-1)$
:::

### Step 1-3
***Step 1. Set constants $C_1,C_2,C_3$***

>**Assumption 1.**
>There is a constant $C_1\in[1,\infty)$ not depending on the dimension $d\in\mathbb{N}$, such that the covariance matrix $C\equiv C_d=((C_d)_{i,j})_{i,j=1}^d\in\mathbb{R}^{d\times d}$ satisfies for every $i,j=1,...,d$ that$$|(C_d)_{i,j}|\leq C_1$$ $\diamond$

>**Assumption 2.**
>There is a constant $C_2\in[1,\infty)$ not depending on the dimension $d\in\mathbb{N}$ such that the CPWA function $h:\mathbb{R}^d_+\rightarrow\mathbb{R}$ satisfies both $$max\{||a_{k,l}||_\infty,|b_{k,l}|: k=1,...,K,\ l=1,...,I_K \}\leq C_2$$ and $$K\cdot max\{I_1,...,I_K\}\leq C_2d$$ $\diamond$

>**Assumption 3.**
>Let $m,n\in\mathbb{N}$ and $T>0$. For every $d\in\mathbb{N}$ and $(t,x)\in[0,T)\times\mathbb{R}^d_+$, let $p_d(\cdot,T;x,t):\mathbb{R}^d_+\rightarrow\mathbb{R}$ be th log-normal transition probability density function. Then we assume that there exists a constant $C_3\in[1,\infty)$ such that for every $d\in\mathbb{N}$ and $\epsilon >0$ there exists a quantum circuit $\mathcal{P}_{d,\epsilon}$ on $d(n+m)$ qubits such that the number of elementary gates used to construct $\mathcal{P}_{d,\epsilon}$ is at most $$C_3d^{C_3}(n+m)^{C_3}(log(\epsilon)^{-1})^{C_3}$$...$\diamond$

***Step 2. Set $n_1:=\lceil n_{1,d,\epsilon}\rceil,\ n_2:=1+\lceil log_2(C_2)\rceil,\ m_1:=\lceil m_{1,d,\epsilon}\rceil,\ m_2:=\lceil m_{2,d,\epsilon}\rceil$***

>**Proposition 4.**
>Let $\epsilon\in(0,1),\ d\in\mathbb{N},\ r,T\in(0,\infty),\ (t,x)\in(0,T]\times\mathbb{R}^d_+$. Let $u(t,x)$ be the option price. Ler $h:\mathbb{R}^d_+\rightarrow\mathbb{R}$ be the CPWA function. Let Assumption 1. and Assumption 2. hold with respective constants $C_1,C_2\in[1,\infty)$ and let $$n_2:=1+\lceil log_2(C_2) \rceil$$ For every $\eta\in(0,1)$, let $M_{d,\eta}\in[1,\infty)$ satisfy$$M_{d,\epsilon}=2C_2^2d^{5/2}\epsilon^{-1}e^{4C_1^2T^2}e^{2rT}\ max_{i=1,...,d}\{1,x_i^2\}$$ Let $n_{1,d,\epsilon},\ m_{1,d,\epsilon},\ m_{2,d,\epsilon}\in(0,\infty)$ be defined by $$n_{1,d,\epsilon}:=1+log_2\Big(M_{d,\frac{\epsilon}{6}}\Big)$$ $$m_{1,d,\epsilon}:=log_2\Big(C_2^2d^2(\frac{\epsilon}{6})^{-1}\Big)$$ $$m_{2,d,\epsilon}:=n_{1,d,\epsilon}+m_{1,d,\epsilon}$$ ...$\diamond$

***Step 3. Set $N,\gamma, \mathfrak{s}$ and $\tilde{a}_{n_2,m_2,k,l}, \tilde{b}_{n_2,m_2,k,l}$ for $k=1,...,K, l=1,...,I_K$***

> $$N:=d(n_1+m_1)+\sum^{K}_{k=1}q_k+2K(n+m+d+5)$$ where $q_k:=I_K(n+m+d+p)+(I_k-2)(n+m+d)+4(I_k-1)$ for $k=1,...,K$. $\diamond$

> $$\gamma=\sum_{j\in\mathbb{K}^d_{n_1,m_1,+}}p_{j,m_1}\in(0,1)$$ where $p_{j,m_1}:=\int_{Q_{j,m_1}}p(y,T;x,t)dy$ for $Q_j:=[j_1,j_1+2^{-m_1})\times\cdots\times[j_d,j_d+2^{-m_1})$. $\diamond$

> $$\mathfrak{s}\equiv\mathfrak{s}_{d,\frac{\epsilon}{6}}:=\sqrt{\frac{\epsilon/6}{(C_2^2d^22^{n_1+1})^3}}$$ where $\epsilon\in(0,1)$. $\diamond$

> **Propositon 5.**
> Let $d]in\mathbb{N},\ r,T\in(0,\infty),\ (t,x)\in[0,T)\times\mathbb{R}^d_+\ n_1\in\mathbb{N}\cap{2,...,3},\ m_1\in\mathbb{N}_0$. Let $h:\mathbb{R}^d_+\rightarrow\mathbb{R}$ be the CPWA function, i.e.$$h(x)\sum^K_{k=1}\xi_k max\{a_{k,l}\cdot x+b_{k,l}:\ l=1,...,I_K\}$$ For every $m_2\in\mathbb{N}_0$ define the function $[\cdot]_{m_2}:\mathbb{R}\rightarrow 2^{-m_2}\mathbb{Z}$ by $[x]_{m_2}:=\frac{\lfloor 2^{m_2}x\rfloor}{2^{m_2}},\ x\in\mathbb{R}$. For every $m_2\in\mathbb{N}_0,\ k=1,...,K,\ l=1,...,I_K$, we define $\tilde{a}_{n_2,m_2,k,l}=(\tilde{a}_{n_2,m_2,k,l,1}\cdots\tilde{a}_{n_2,m_2,k,l,d})\in\mathbb{R}^{d}$ and $\tilde{b}_{n_2,m_2,k,l}\in\mathbb{R}$ by $$\tilde{a}_{n_2,m_2,k,l}:=[\big(a_{k,l}\big)_i]_{m_2}$$ $$\tilde{b}_{n_2,m_2,k,l}:=[\big(b_{k,l}\big)]_{m_2}$$ ...$\diamond$

### Step 4-6
***4. Construct probability distribution quantum circuit $\mathcal{P}=\mathcal{P}_{d,\epsilon}$***

>**Assumption 3. (Complete)**
Let $m,n\in\mathbb{N}$ and $T>0$. For every $d\in\mathbb{N}$ and $(t,x)\in[0,T)\times\mathbb{R}^d_+$, let $p_d(\cdot,T;x,t):\mathbb{R}^d_+\rightarrow\mathbb{R}$ be th log-normal transition probability density function. Then we assume that there exists a constant $C_3\in[1,\infty)$ such that for every $d\in\mathbb{N}$ and $\epsilon >0$ there exists a quantum circuit $\mathcal{P}_{d,\epsilon}$ on $d(n+m)$ qubits such that the number of elementary gates used to construct $\mathcal{P}_{d,\epsilon}$ is at most $$C_3d^{C_3}(n+m)^{C_3}(log(\epsilon)^{-1})^{C_3}$$ And that $\mathcal{P}$ satisfies $$\mathcal{P}_{d,\epsilon}|0\rangle_{d(n_1+m_1)}=\sum_{i=(i_1,...,i_d)\in\mathbb{F}^d_{n_1,m_1,+}}\sqrt{\tilde{p}_i}|i_1\rangle_{n_1+m_1}\cdots|i_d\rangle_{n_1+m_1}$$ where the cofficients $\tilde{p}_i\in[0,1]$ satisfying $$\sum_{i\in\mathbb{F}^d_{n_1,m_1,+}}\tilde{p}_i=1$$ and $$\sum_{i\in\mathbb{F}^d_{n_1,m_1,+}}|\tilde{p}_i-\gamma^{-1}p_{i,m_1}|\leq\epsilon$$ Here $$p_{j,m_1}:=\int_{Q_{j,m_1}}p(y,T;x,t)dy$$ for $Q_j:=[j_1,j_1+2^{-m_1})\times\cdots\times[j_d,j_d+2^{-m_1})$, $D_{n,m}:\mathbb{F}_{n,m}\rightarrow\mathbb{K}_{n,m}$ Moreover, $$\gamma=\sum_{j\in\mathbb{K}^d_{n_1,m_1,+}}p_{j,m_1}\in(0,1)$$ is a normalized constant.

***5. Construct CPWA payoff with rotation quantum circuit $\mathcal{R}_h$***

> **Proposition 6.**
> Let $d\in\mathbb{N},\ n_1,n_2,m_1,m_2\in\mathbb{N},\ \xi_1,...,\xi_k\in\{-1,1\}, s\leftarrow\mathfrak{s}\in(0,1)$. Define $n:=n_1+n_2,\ m=m_1+m_2$, and $p:=d(2m+3)+m+(d-1)(n+m)$.
> Let $\{a_{k,l,j}\}_{k=1,...,K;l=1,...,I_K;j=1,...,d},\ \{b_{k,l}\}_{k=1,...,K;l=1,...,I_K}\subset\mathbb{F}_{n_2,m_2}$ and $a_{k,l,j}\leftarrow E_{n_2,m_2}(\tilde{a}_{n_2,m_2,k,l,j}),\ b_{k,l}\leftarrow E_{n_2,m_2}(\tilde{b}_{n_2,m_2,k,l})$
>
> Let $h_{k,l}:\mathbb{F}^d_{n_1,m_2}\rightarrow\mathbb{F}_{n+d,m}$ be functions defined by $$h_{k,l}(i):=\bigoplus^d_{j=1}(a_{k,l,j}\odot i_j)\oplus b_{k,l},\ \ \forall i(i_1,...,i_d)\in\mathbb{F}^d_{n_1,m_1}$$ where $\oplus:\mathbb{F}_{n_1,m_1}\times\mathbb{F}_{m_1,m_2}\rightarrow\mathbb{F}_{max\{n_1,n_2\}+1,max\{m_1,m_2\}}$, and $\odot:\mathbb{F}_{n_1,m_1}\times\mathbb{F}_{m_1,m_2}\rightarrow\mathbb{F}_{n_1+n_2,m_1+m_2}$
> For $k=1,...,I_K$, let $\bar{\xi_k}\in\mathbb{F}_{2,0}$ by $$\bar{\xi_k}:=E_{2,0}(\xi_k)$$ where $E_{n,m}:\mathbb{K}_{n,m}\rightarrow\mathbb{F}_{n,m}$
>
> Let $M_{I_K,n+d,m}:\mathbb{F}^{I_k}_{n+d,m}\rightarrow\mathbb{F}_{n+d,m}$ be defined by $$M_{I_k,n+d,m}(i_1,...,i_{I_k}):=E_{n+d,m}(max\{D_{n+d,m}(i_1),...,D_{n+d,m}(i_{I_k})\})\ \ \forall (i_1,...,i_k)\in\mathbb{F}^{I_k}_{n+d,m}$$
>
> And let $h_k:\mathbb{F}^d_{n_1,m_1}\rightarrow\mathbb{F}_{n+d,m}$ be defined by $$h_k(i):=M_{I_K,n+d,m}(h_{k,1}(i),...,h_{k,I_k}(i))\ \ \forall i=(i_1,...,i_d)\in\mathbb{F}^d_{n_1,m_1}.$$
>
> Let $h:\mathbb{F}^d_{n_1,m_1}\rightarrow\mathbb{F}_{n+d+K+1},m$ be defined by $$h(i):=\bigoplus^{K}_{k=1}(\bar{\xi}_k\odot h_k(i)),\ \ \forall i=(i_1,...,i_d)\in\mathbb{F}^d_{n_1,m_1}$$
>
> Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function defined by $f(x)=sx+\frac{\pi}{2}$, and define $\bar{f}:\mathbb{F}_{n+d+K+1,m}\rightarrow\mathbb{R}$ by $$\bar{f}(i):=\circ D_{n+d+K+1}(i),\ \ \forall i\in\mathbb{F}_{n+d+K+1,m}$$
>
> Then, there is a quantum circuit $\mathcal{R}_h$ on $N$ qubits (as step 3.) such that for any $i=(i_1,...,i_d)\in\mathbb{F}^d_{n_1,m_1}$, $$\begin{align}\mathcal{R}_h&:\ \ \ |i_1\rangle_{n_1+m_1}\cdots|i_d\rangle_{n_1+m_1}|0\rangle_{q_1+...+1_K}|0\rangle_{2K(n+m+d+5)}\\&\rightarrow |i_1\rangle_{n_1+m_1}\cdots|i_d\rangle_{n_1+m_1}|anc\rangle_{q_1+...+1_K+2K(n+m+d+5)-1}[cos(\frac{\bar{f}(h(i))}{2})|0\rangle+sin(\frac{\bar{f}(h(i))}{2})|1\rangle]\end{align}$$ $\diamond$

***6. Construct the quantum circuit $\mathcal{A}=\mathcal{R}_h(\mathcal{P}\otimes I_2^{\otimes(N-d(n_1+m_1))})$***

> $$\begin{align}\mathcal{R}_h&:\ \ \ |i_1\rangle_{n_1+m_1}\cdots|i_d\rangle_{n_1+m_1}|0\rangle_{q_1+...+1_K}|0\rangle_{2K(n+m+d+5)}\\&\rightarrow |i_1\rangle_{n_1+m_1}\cdots|i_d\rangle_{n_1+m_1}|anc\rangle_{q_1+...+1_K+2K(n+m+d+5)-1}[cos(\frac{\bar{f}(h(i))}{2})|0\rangle+sin(\frac{\bar{f}(h(i))}{2})|1\rangle]\end{align}$$

> $$\mathcal{P}_{d,\epsilon}|0\rangle_{d(n_1+m_1)}=\sum_{i=(i_1,...,i_d)\in\mathbb{F}^d_{n_1,m_1,+}}\sqrt{\tilde{p}_i}|i_1\rangle_{n_1+m_1}\cdots|i_d\rangle_{n_1+m_1}$$

> Then, we combine the two quantum circuits:$$\mathcal{A}=\mathcal{R}_h(\mathcal{P}\otimes I_2^{\otimes(N-d(n_1+m_1))})$$

### Step 7-8

> **Modified IQAE Algorithm Introduction**
> QAE algorithm appears as a subroutine in many application for quantum computers. The optimal query complexity achievable by a quantum for QAE is $O\big(\frac{1}{\epsilon}log(\frac{1}{\alpha})\big)$ queries, providing a speedup of a factor of $\frac{1}{\epsilon}$ over any other classical algorithm for the same problem
> Given a unitary operator $\mathcal{A}$ that acts on $(n+1)$ qubits such that $$\mathcal{A}|0\rangle_{n}|0\rangle=\sqrt{1-a}|\psi_0\rangle_n|0\rangle+\sqrt{a}|\psi_1\rangle_n|1\rangle$$ where $a\in[0,1]$ is unknown and $|\psi_0\rangle,\ |\psi_1\rangle$ are two normalized states. Given this setting, the objective is **to find a good approximation for $a$ with as few applications of $\mathcal{A}$ as possible.** Brassard et al. presented the problem as a generalization of Grover's algorithm and demostrate that techniques used by Grover could be used to solve QAE.

> Brassard et al.'s algorithm enables a quadratic speedup for many applications that are solved classically using Monte Carlo simulation, which itself proves useful in finance for applications such as risk analysis, or option pricing, as well as other general tasks like numerical integration.
> Futhermore, amplitude estimation is a widely used subroutine in many quantum algorithms, making it an algorithm of high interest to implement in the future.

***7. Set $\hat{a}=\frac{a_u-a_l}{2}$ using the output $[a_l,a_u]$ from the modified iterative quantum algorithm***

> Back to our quantum circuits for option pricing problem. Let $\alpha,\epsilon\leftarrow\frac{\epsilon\mathfrak{s}}{12}\in(0,1)$, and let $\mathcal{A}$ be $(n+1)$-qubit quantum circuit satisfying $$\mathcal{A}|0\rangle_{n}|0\rangle=\sqrt{1-a}|\psi_0\rangle_n|0\rangle+\sqrt{a}|\psi_1\rangle_n|1\rangle$$where $a\in[0,1]$ is unknown and $|\psi_0\rangle,\ |\psi_1\rangle$ are two normalized states, and $\mathcal{A}$ can be constructed with $N_{\mathcal{A}}\in\mathbb{N}$ number of elementary gates.

>Then, the following holds:
>The modified IQAE algorithm outputs a confidence interval $[a_l,a_u]$ that satisfies $$a\notin[a_l,a_u]$$ with probability at most $\alpha$, where $0\leq a_u-a_l\leq 2\epsilon$.
>In particular, the estimator $\hat{a}:=\frac{(a_u-a_l)}{2}$ of a satisfies $$|a-\hat{a}|<\epsilon$$ where $a\in[a_l,a_u]$ with probility $(1-\alpha)$.
>...$\diamond$

***8. Return $\widetilde{U}_{t,x}:=\mathfrak{s}^{-1}\gamma e^{-r(T-t)}(2\hat{a}-1)$***

> **Continue from Proposition 4.**
> Moreover, for every $n_1,m_1,m_2\in\mathbb{N}$ satisfying $n_1\geq n_{1,d,\epsilon},\ m_1\geq m_{1,d,\epsilon},\ m_2\geq m_{2,d,\epsilon}$. Let $\{\tilde{p}_{j,\epsilon/6}:j\in\mathbb{K}^d_{n_1,m_1,+}\}\subset[0,1]$ satisfy $$\sum_{j\in\mathbb{K}^d_{n_1,m_1,+}}\tilde{p}_{j,\epsilon/6}=1$$ $$\sum_{j\in\mathbb{K}^d_{n_1,m_1,+}}|\tilde{p}_{j,\epsilon/6}-\gamma^{-1}p_{j,m_1}|\leq\frac{\epsilon}{6C^2_2d^22^{n_1+1}}$$, where for all $j=(j_1,...,j_d)\in\mathbb{K}^d_{n_1,m_1,+}$, $$p_{j,m_1}:=\int_{Q_{j,m_1}}p(y,T;x,t)dy,\ \ Q_{j,m_1}=[j_1,j_1+2^{-m_1})\times\cdots\times[j_1,j_1+2^{-m_1})$$ $$\gamma:=\sum_{j\in\mathbb{K}^d_{n_1,m_1,+}}p_{j,m_1} \in(0,1)$$

> Let $\mathfrak{s}\equiv\mathfrak{s}_{\epsilon/6}\in(0,\infty)$ be defined by $$\mathfrak{s}\equiv\mathfrak{s}_{d,\epsilon/6}:=\sqrt{\frac{\epsilon/6}{(C^2_2d^22^{n_1+1})^3}}$$ Let $\tilde{h}_{n_2,m_2}:\mathbb{R}^d\rightarrow\mathbb{R}$ be given by $$\tilde{h}_{n_2,m_2}(x):=\sum^K_{k=1}\xi_k max\{\tilde{a}_{n_2,m_2,k,l}\cdot x+\tilde{b}_{n_2,m_2,k,l}: l=1,...,I_K\}$$

> Let $a\equiv a_{n_1,m_1,n_2,m_2,\mathfrak{s},\epsilon/6}\in[0,1]$ be the amplitude given by $$a_{n_1,m_1,n_2,m_2,\mathfrak{s},\epsilon/6}:=\sum_{j\in\mathbb{K}^d_{n_1,m_1,+}}\tilde{p}_{j,\epsilon/6}sin^2\Big(\frac{\mathfrak{s}\tilde{h}_{n_2,m_2}(j)}{2}+\frac{\pi}{4}\Big)$$ Let $\hat{a}\in[0,1]$ satisfy $$|a-\hat{a}|\leq\frac{\epsilon\mathfrak{s}}{12}$$ And let $\tilde{U}_{t,x}$ be the approximated solution given by $$\tilde{U}_{t,x}:=\mathfrak{s}^{-1}\gamma e^{-r(T-t)}(2\hat{a}-1)$$ Then, $$|u(t,x)-\tilde{U}_{t,x}|\leq\epsilon$$

Then we complete the whole algorithm. :heart:

## 5. Conclusion
Through this semester's project, we have acquired fundamental knowledge related to option pricing problems, particularly focusing on a crucial model known as the Black-Scholes (BS) Model. Towards the conclusion, we introduce a quantum circuit framework for addressing the problem. Nevertheless, constructing the entire system through computer programming remains challenging. Consequently, our aim is to physically build the circuit and conduct experiments.

## 6. Reference Papers
1. [Yongming Li, Ariel Neufeld, *“Quantum Monte Carlo algorithm for solving Black-Scholes PDEs for high-dimensional option pricing in finance and its proof of overcoming the curse of dimensionality”*, March 14,2023](https://arxiv.org/abs/2301.09241)
2. [Shion Fukuzawa, Christopher Ho, Sandy Irani, and Jasen Zion *“Modified Iterative Quantum Amplitude Estimation is Asymptotically Optimal”*, arXiv:2208.14612 (2022).](https://arxiv.org/abs/2208.14612)