Mathematics for New Technologies in Finance

MNTF Mathematics for New Technologies in Finance

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professor : Josef Teichmann

author : walkerchi

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Approximation

Weierstrass

Weierstrass Approximation Theorem

$A$ is dense in $C(\mathcal X,\R^m )=\{f_i|f_i\in C_{pw}^0, f_i\in \mathcal X\to \R^m,\mathcal X\subset \R^n\}$ if

1. A contains all polynomial functions: $\mathcal P\subset A$
   1. $A$ is vector subspace of $C$ : $A\subset C(\mathcal X,\R^m)\quad $
      • $f_1(x)+f_2(x)=f_3(x)\quad \forall f_1,f_2\in A,\exists f_3\in A$
      • $cf_1(x)=f_2(x)\quad \forall c\in\R,\forall f_1\in A,\exists f_2\in A$
   2. $A$ is closed under multiplication : $f_1(x)f_2(x)=f_3(x)\quad \forall f_1,f_2\in A,\exists f_3\in A$
   3. $A$ contains constant function : $f(v) = c\quad \forall v\in \mathcal X,\exists f\in A$
2. points seperation : f(v)\neq f(w)\quad \forall v\neq w\and v,w\in\mathcal X

for shallow NN with ReLU

• [ ] contains all polynomial functions
  • [ ] vector space
  • [ ] closed under multiplication
  • [x] contains constant function
• [x] points seperation

$\Rightarrow$ NN with ReLU is dense in $C(\mathcal X,\R^m)$

Faber-Schauder

Faber-Schauder basis : $s_{n,k} = 2^{1+\frac{n}{2}}\int_0^t\psi_{n,k}(u)\text du\quad n,k\in\Z$

$$v = \sum_{n=0}^\infin \alpha_n b_n\quad \forall v\in \mathbb R,\exists\alpha_n\in\mathbb R,\exists b_n\in \{s_{*,*}\}$$

• equivalent to the linear combination of ReLU

Haar function : $\psi_{n,k}(t) =2^{n/2}\psi(2^nt-k)\quad n,k\in \Z\quad\psi(t) = \begin{cases}1&t\in[0, \frac{1}{2})\\-1&t\in[\frac{1}{2},1)\\ 0&\text{otherwise}\end{cases}$

• $\text{supp}(\psi_{n,k}) = [k2^{-n},(k+1)2^{-n})$
• $\int_\R \psi_{n,k}(t)\text dt = 0$
• $\Vert \psi_{n,k}\Vert_{L^2(\R)} = 1$
• $\int_\R \psi_{n_1,k_1}\psi_{n_2,k_2} \text d t=\delta_{n_1n_2}\delta_{k_1k_2}$

Banach

$A$ is Banach space if :

1. Cauchy sequence : $\Vert f_m-f_n\Vert\le \epsilon\quad \forall \epsilon>0,\exists N_\epsilon\in \N,\forall m,n>N_\epsilon$
2. completeness : $\Vert f-f_m\Vert\le \epsilon \quad \forall \epsilon>0, m\to\infin$

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Signature

for path/curve $X_t\in \R^d$, $X_t=\begin{bmatrix}X^1(t)&X^2(t)&\cdots&X^d(t)\end{bmatrix}^\top$ , signature could determine the curve in tree like equivalences

n-th level of signature : $S(X)_{a,b}^{i_1,i_2,\cdots,i_n}=\int_a^b\int_{a}^{t_{n-1}}\cdots \int_a^{t_2}\text dX_{t_1}^{i_1}\cdots \text dX_{t_n}^{i_n}\quad n \le d$

• $S(X)_{a,b}^{i_1,i_2,\cdots,i_n}\in \R^{n^{\otimes d}}$, it’s a $d$ dimension tensor with each dimension of span $n$

signature : $S(X)_{a,b} = (1,S(X)_{a,b}^{i_1},S(X)_{a,b}^{i_1,i_2},\cdots,S(X)_{a,b}^{i_1,i_2,\cdots,i_d})$

• the maximum length of $S(X)_{a,b}$ is $d^0+d^1+d^2 + \cdots + d^d = \frac{d^{d+1}-1}{d-1}$
• the length of depth $M$ is $d^0 + \cdots +d^{M} = \frac{d^{M+1} - 1}{d-1}$

normally we got : $X_a^i$ denotes $i$-th component at time $a$ of vector of function $X$

• $S(X)_{a,b}^i = \int_a^b dX = X_b^i-X_a^i$

• $S(X)_{a,b}^{i,j} = \int_a^b\int_a^{t_2}dX_{t_1}^{i}dX_{t_2}^{j} =\int_a^b (X_{t_2}^i-X_a^i)dX_{t_2}^j\overset{X=\alpha t+\beta}{=} \frac{1}{2}(X_b^{i}-X_a^{i})(X_b^{j}-X_a^{j})$

• $S(X)_{a,b}^{i,j,k} = \int_a^b\int_a^{t_3}\int_a^{t_2}dX_{t_1}^{i}dX_{t_2}^{j}dX_{t_3}^{k} \overset{X=\alpha t+\beta}{=} \frac{1}{6}(X_b^{i}-X_a^{i})(X_b^{j}-X_a^{j})(X_b^{k}-X_a^{k})$

• shuffle product rule : $S(X)_{a,b}^IS(X)_{a,b}^J = \underset{K=\text{shuff}([I_1,\cdots,J_1,\cdots])}{\sum}S(X)_{a,b}^K$

  • example : $S(X)^{1}_{a,b} S(X)^2_{a,b} = S(X)^{1,2}_{a,b}+S(X)^{2,1}_{a,b}$

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Financial Market

Notation

• $S_t^i$ : $i$-th asset prices at time $t$, $S\in\R^{N\times {d+1}}$, normally $S^0$ represent bank account
• $\phi_t^i$ : holdings/strategy in $i$-th assets at time $t$
• $V_t$ : value of portfolio at time $t$ , $V_t = \sum_i \phi_t^i S_t^i$

self-financing : $\text dV(t) = \sum_{i=1}^n \phi^i(t)\text dS^i(t)$

• $\sum_i \phi^i_{t+1}S_t^i = \sum_i\phi_t^i S_t^i\quad\forall t\in[0,N)$

value process : $V_{t+1}-V_t = \sum_i\phi_t^i(S_{t+1}^i-S_t^i)\quad \forall t\in[0,N)$

martingale : $\mathbb E[X_{n+1}|X_1,\dots,X_n] = X_n$

arbitrage : $\underbrace{P(V_t\ge 0 )=1}_{\text{no risk of losing money}}\land \underbrace{P(V_t\neq 0) >0}_{\text{portfolio value > 0}}\quad t\in (0,T),\underbrace{V_0=0}_{\text{requires no initial value}}$

Stochastic Differential Equation

Brownian motion/Wiener process : $W_{t+1} - W_t \sim \mathcal N(0,1)\quad W_0=0\rightarrow W_t$

Geometric Brownian motion : $dS_t = \mu ~S_t~ dt + \sigma~ S_t~ dW_t\Leftrightarrow S_t = S_0 e^{\left(\mu - \frac{\sigma^2}{2}\right)t+\sigma W_t}$

• $W_t$ is brownian motion/wiener process
• $\mu,\sigma$ is the expectation/variance for the GBM

Utility

utility function $u$ : the additional utility or satisfaction from consuming one more unit of a good decreases as more of the good is consumed.

• concave : $f''(x)<0$
• monotone increase : $f'(x)>0$

expected utility optimization problem : $\underset{\phi_t^i}{\text{argmax}}\mathbb~E[u(V_N)]$

Local Volatility Model : $\text d S_t = rS_t\text d t+\sigma(S_t,t)S_t\text dW_t$

• $S_t$ : underlying asset price at time $t$

• $r$ risk-free interesting rate

• $\sigma(S_t,t)$ : local volatility function

• $W_t$ is the Brownian motion/ Wiener process

Local Stochastic volatility model : $\begin{aligned}\text d S_t &= \mu S_t \text d t+\sqrt{\nu_t} S_t \text dW_t\\ \text d\nu_t &= \alpha_{t}(\nu)\text dt + \beta_{t}(\nu)\text d W'_t\end{aligned}$

• $\alpha_{t}(\nu),\beta_{t}(\nu)$ : functions based on $\nu$
• $W_t,W'_t$ : Wiener process with correlation factor $\rho$
• $\nu_t$ : model the variance of $S_t$, it relies on another stochastic process, so LSV is not a standard SDE

Heston model : $\text d\nu_t = \kappa(\theta-\nu_t)\text dt +\xi \sqrt{\nu_t}\text d W'_t$

• $\theta$ : long term variance

• $\kappa$ : rate of variance reverts toward it’s long term

• $\xi$ : volatility of volatility, the variance of $\nu_t$

• ambitious approach

  • modeling $\theta,\kappa,\xi,\rho,\mu$ where $\rho$ is the correlation between $W_t,W'_t$

• modest approach

  • modeling $\theta, \kappa,\xi$ , and $\rho,\mu$ from emperical

Ito’s lemma : $\text d f(S,t) = \left(\frac{\partial f}{\partial t} + \mu\frac{\partial f}{\partial S}+\frac{1}{2}\sigma^2\frac{\partial^2 f}{\partial S^2}\right)\text dt + \sigma\frac{\partial f}{\partial S}\text dW_t\quad \underbrace{\text dS(t)=\mu \text dt+\sigma \text d W_t}_{\text{Stochastic Differential Equation}}$

Black Scholes equation : $\frac{\partial C}{\partial t} + rK\frac{\partial C}{\partial K} +\frac{1}{2}\sigma^2 K^2\frac{\partial^2 C}{\partial K^2}-rC= 0$ : derive from Ito’s lemma

• $C(K,t)$ : European call option price, equivalent to value $V$
• $K$ : strike price, equivalent to assets/stock price $S$

Dupire’s formula : $-\frac{\partial C}{\partial T}-rK\frac{\partial C}{\partial K}+\frac{1}{2}\sigma^2K^2\frac{\partial^2 C}{\partial K^2}-\Delta C=0$

• when $r=0$ , $\sigma^2 = \frac{2\partial_T C}{K^2\partial^2_K C}$

Breeden-Litzenberger fromula : $\partial^2_K C(T,K)\text dK = p_T(K)\text d K$

• $p_T(K)\text dK$ is the risk neural probability, $p_T(K) = p(S_t\in[K,K+\text dK])$

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Deep portfolio optimization

$$\text d S_t = S_t\mu \text d t+S_t\sigma\text d W_t \\ \text dX_t = \alpha_t X_t\frac{\text dS_t}{S_t} + (1-\alpha) X_t r\text dt\quad \\ \underset{\alpha}{\text{max}}~\mathbb E[u(X_T)]$$

• $X_t$ is the money at time $t$
• $\alpha_t$ is strategy how much portion of money in the stock rather than in the bank at time $t$
• $S_t$ is the stocks prices, governed by parameter $\mu$ and $\sigma$ , with $W_t$ a brownian motion or wiener process
• $r$ is the interest rate saved in bank
• $u$ is the utility function ,normally $u(x)=\frac{x^\gamma -1}{\gamma}$

analytical solution : $\alpha^* = \frac{\mu-r}{\sigma^2(1-\gamma)}$

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Deep Hedging

$$\text d S_t = S_t\mu \text d t+S_t\sigma\text d W_t \\ \underset{H,\pi}{\text{min}}~\mathbb E\left[\left\Vert f(S_T) - \pi - \int_0^T H_t\text dS_t\right\Vert^2\right]$$

• $S_t$ is the risky stocks prices, governed by parameter $\mu$ and $\sigma$ , with $W_t$ a brownian motion or wiener process
• $f(S_t)$ is financial claim, the payoff is $f(S_T)=\text{max}(S_T-K,0)$ for European call, $K$ is the strike price
• $\pi$ the price of the option, the upfront payment you received
• $H_t$ is the hedge strategy at time $t$
• $T$ is the expire date

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Deep Calibration

Heston Calibration

$$\text dX_t = \left((q-r)-\frac{1}{2}Y_t\right)\text d t +\sqrt {Y_t}\text d W_t^1 \\ \text dY_t = (\theta-\kappa Y_t)\text d t +\sigma\sqrt{Y_t}\text dW^2_t \\ \underset{\theta,\kappa,\sigma}{\text{argmin}}\sum_{t=0}^T\Vert X_t-\text {log}(S_t)\Vert^2$$

• $r$ : interest rate
• $q$ : dividend
• $S_t$ : price of assets
• $X_t$ : predicted log price : $X_0 = \text{log}(S_0)$
• $Y_t$ : variance of Heston model : $Y_0 = \nu_0$

Utility Calibration

$$dS_t = S_t\alpha_t l(t,S_t)\text dW_t \\ \underset{l}{\text{argmin}} \left\Vert\mathbb E\left[\text{max}(S_T-K,0) - C(K,T)- \int_0^T H_t\text dS_t\right]\right\Vert^2$$

• $\alpha_t$ is exogenous process at time
• $l(t, S_t)$ is leverage function
• $S_t$ is the stocks prices, with $W_t$ a brownian motion or wiener process
• $H_t$ is the hedge strategy at time $t$
• $K$ is the strike price of European call
• $C$ is the European call option market price

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Deep Simulation

model controlled differential equation

$$\text d X_t = \sum_{i=0}^d \sigma(A_iX_t+b_i)\text du_i(t)$$

• $A_i, b_i$ are randomly generated matrices/vectors
• $u_i$ is control coefficient learned by network
• $\sigma$ is the sigmoid//tanh function

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Reinforcement Learning

• $a,s$ : action $a\in A$, state $s\in S$
• $V,V^*$ : value function, optimal value function, $V\in S\times T\to\R$
• $\pi(s)$ : policy , $\pi \in S\to A$
• $c(t,s,a)$ : cost function , $c\in T\times S\times A\to \R$
• $r,R(s,a)$ : reward, reward function , $r\in \R, R\in S\times A\to \R$
• $Q(s,a)$ ：Q/state action function, return the priority for each state and action, $Q\in S\times A\to \R$

[DPP] Dynamic programming principle : $V^*(t,s)=\underset{a}{\text{max}}\left\{\int_t^T c(\tau,s(\tau),a(\tau))\text d\tau+V*(T,s(T))\right\}$

• $V(s) = \underset{a}{\text{max}}\left(R(s,a)+\gamma\underset{s'\in S}{\sum}P(s'|s,a)V(s')\right)$

[HJB] Hamiton-Jacobi-Bellman equation : $\frac{\partial V(s,t)}{\partial t} + \underset{a}{\text{max}}\left(\frac{\partial V(s,t)}{\partial s}\cdot f(t,s,a)+c(t,s,a)\right)=0$

• $f(t,s,a)$ : system dynamics, how state change over time, $\frac{\text d s(t)}{\text dt} = f(t,s,a)$
• $V^*(s) = \underset{a\in A}{\text{max}} \left(R(s,a)+\gamma \underset{s'\in S}{\sum}P(s'|s,a)V^*(s')\right)$

Bellman equation : $Q(s,a) = r+\gamma~\underset{a'}{\text{max}}~Q(s',a')$

Value Iteration : $V^{(n+1)}=\underset{a}{\text{max}}\left\{R(s,a)+\gamma \sum_{s'}P(s'|s,a)V^{(n)}(s')\right\}$

Policy Iteration : $\begin{aligned}V^{\pi^{(n)}}(s) &= R(s,\pi(s))+\gamma\sum_{s'}P(s'|s,\pi(s))V^{\pi^{(n)}}(s')\\\pi^{(n+1)}&=\underset{\pi}{\text{argmax}}\left\{R(s,a)+\gamma\sum_{s'} P(s'|s,a)V^{\pi^{(n)}}(s')\right\}\end{aligned}$

Q learning(environment-known/model-based) : $Q(s,a)\gets R(s,a)+\sum_{s'} P(s'|s,a)\left[\gamma~\underset{a'}{\text{max}}~Q(s',a')\right]$

Q learning(environment-unknown/model-free) : $Q(s,a)\gets (1-\alpha)Q(s,a)+\alpha\left[r+\gamma~\underset{a'}{\text{max}}~Q(s',a')-Q(s,a)\right]$

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Optimization

inverse calibration : $\underset{\theta}{\text{argmin}}\Vert \textbf d - \mathcal {NN}_\theta\Vert^2$

• $\textbf d$ is the observed data
• $\mathcal {NN}_\theta\quad \theta\in\Theta$ is the pool of the model

optimization approach : $\underset{\theta}{\text{argmin}}\Vert \textbf d-\mathcal{NN}_\theta\Vert^2 + \lambda R_\theta$

• $\theta$ model parameters
• $R_\theta$ : regularization term ($|\cdot|$ : lasso(L1) or $\Vert\cdot\Vert^2$ : ridge(L2))

bayesian optimization :

$$P(M_i|\textbf d) = \frac{P(\textbf d|M_i)P(M_i)}{P(\textbf d)}\propto P(d|M_i)P(M_i)$$

• $P(M_i|\textbf d)$ posterior probability of model $M_i$ given data $\textbf d$
• $P(\textbf d|M_i)$ likelihood of data given model $M_i$
• $P(M_i)$ : prior probability of model $M_i$
• $P(\textbf d)$ : evidence likelihood

for linear model $Y\sim \mathcal N(\theta X,\sigma^2\textbf I), \theta\sim\mathcal N(0,\tau^2\textbf I)$, the maximizing posterior of $p(\theta|x,y)$ is ridge regression:

$$\begin{aligned} \underset{\theta}{\text{argmax}} ~p(\theta|x,y) &\propto \underset{\theta}{\text{argmax}}~p(\theta)p(y|x,\theta) \\ &\propto \underset{\theta}{\text{argmax}}~\text{exp}\left(-\theta^\top \textbf I\theta /\tau^2 \right)~\text{exp}\left(-(y-\theta x)^\top \textbf I(y-\theta x)/\sigma^2\right) \\ &\propto \underset{\theta}{\text{argmin}}~\frac{\sigma^2}{\tau^2}\Vert\theta\Vert^2 + \Vert y-\theta x\Vert^2 \end{aligned}$$

[SGLD] Stochastic Gradient Langevin Dynamics : gradient descent plus noise :

$$\text d\theta_t = \frac{1}{2}\nabla\text{log}~p(\theta_t|x_1,\dots,x_n)\text dt + \text dW_t$$

• escape from local minimal