[QF] 04-Stochastic Calculus

List of topics for quantitative finance with concise key points and useful resources. For Quant interview preparation and quantitative finance introduction, or could be cheatsheets. Motivated by Ran Ding.

resources

• <Stochastic Calculus for Finance II>
• <Introduction to the Mathematics of Financial Derivatives>

key points

• Derivative
  • notation
    • Leibniz's notation: $\cfrac{dy}{dx}, \cfrac{d^ny}{dx^n}$
    • Lagrange's notation: $f^{'}, f^{"}, f^{n}$
    • Newton's notation: $\dot{y}, \ddot{y}$
  • Rules for basic functions
    • $\frac{d}{dx}x^{a}=ax^{a-1}$
    • $\frac{d}{dx}e^{x}=e^{x}$
    • $\frac{d}{dx}a^{x}=a^{x}\ln(a),\ a>0$
    • $\frac{d}{dx}\ln(x)=\frac{1}{x},\ x>0$
    • $\frac{d}{dx}\log_{a}(x)=\frac{1}{x\ln(a)},\ x,a>0$
  • Rules for combined functions
    • Product rule: $(f g)^{\prime}=f^{\prime} g+f g^{\prime}$
    • Quotient rule: $\left(\frac{f}{g}\right)^{\prime}=\frac{f^{\prime} g-f g^{\prime}}{g^{2}}$
    • Chain rule: If $f(x)=h(g(x))$, then $f^{\prime}(x)=h^{\prime}(g(x)) \cdot g^{\prime}(x)$
• Differential
  • we call $dy$, $df$, or $f(x)dx$ be differential of a function (with respect to x)
• Taylor series
  • $f(x)=f(a)+\frac{f^{\prime}(a)}{1 !}(x-a)+\frac{f^{\prime \prime}(a)}{2 !}(x-a)^{2}+\frac{f^{\prime \prime \prime}(a)}{3 !}(x-a)^{3}+\cdots$
  • when $a=0$, also called a Maclaurin series
  • $f(x)=f(0)+\frac{f^{\prime}(0)}{1 !}x+\frac{f^{\prime \prime}(0)}{2 !}x^{2}+\frac{f^{\prime \prime \prime}(0)}{3 !}x^{3}+\cdots$
  • $e^{x}=\sum_{n=0}^{\infty} \frac{x^{n}}{n !}=1+x+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\cdots$
  • $\ln (1-x)=-\sum_{n=1}^{\infty} \frac{x^{n}}{n}$
• Moment generating function
  • $M_{X}(t)=\mathrm{E}[e^{t X}],\quad t\in\mathbb{R}$
  • $E(X^{n})=M_{X}^{(n)}(0)=\cfrac{d^{n}M_{X}}{d t^{n}}|_{t=0}$
  • prof:
    • $M_{X}(t)=\mathrm{E}\left(e^{t X}\right)=1+t \mathrm{E}(X)+\frac{t^{2} \mathrm{E}\left(X^{2}\right)}{2 !}+\frac{t^{3} \mathrm{E}\left(X^{3}\right)}{3 !}+\cdots+\frac{t^{n} \mathrm{E}\left(X^{n}\right)}{n !}+\cdots$
  • $\cfrac{d^{n}M_{X}}{d t^{n}}|_{t=0}=0+\cfrac{n!\mathrm{E}(X^n)}{n!}+0=\mathrm{E}(X^n)$
• p.d.f. of $y=g(x)$
  • we have p.d.f. of x be $f_x(x)$
  • $f_y(y)dy=f_x(x)dx$ (this is important)
  • $f_y(y)=f_x(x)|\cfrac{dx}{dy}|=f_x(g^{-1}(y))|\cfrac{d}{dy}g^{-1}(y)|$
• Martingale
  • $S_t$ is adapted to the filtration $\mathbb{F}_t$
  • $S_t$ is integrable: $E[|S_t|]<\infty$
  • $E_t[S_T]=S_t$
• Brownian Motion properties
  • $E[W_{t_{i+1}}-W_{t_{i}}]=0$
  • $\operatorname{Var[W_{t_{i+1}}-W_{t_{i}}]}=t_{i+1}-t_{i}$
  • $W_t$ are jointly normal distributed
    • If X and Y are normally distributed and independent, this implies they are "jointly normal distributed"
  • $\operatorname{Cov}(W_t,W_s)=\operatorname{min(s,t)}$
  • $E_t[W_s|\mathbb{F}_t]=W_t\ \forall\ s\geq t$
  • $dW_tdW_t=dt$
  • $dW_idW_j=0$
  • $dW_tdt=0$
• integration by parts (in Riemann-Stieltjes integral)
  • $\int^b_au(x)dv(x)=[u(x)v(x)]^b_a-\int^b_av(x)du(x)$
  • or compactly $\int udv=uv-\int vdu$
  • if $v(x)$ is $W_t$: $\int^t_0 f(s)dW_s=f(t)W_t-\int^t_0W_sf^{'}(s)ds$
• Itô's Lemma
  • core: $df(t,X_{(t)})=f_t(t,X_{(t)})dt+f_X(t,X_{(t)})dX_{(t)}+\frac{1}{2}f_{XX}(t,X_{(t)})dX^2_{(t)}$
  • product rule: $d(X_tY_t)=X_TdY_t+Y_tdX_t+dX_tdY_t$
  • stochastic integral by parts: $\int^t_0g^{'}(W_u)dW_u=[g(W_u)]^t_0-\frac{1}{2}\int^t_0g^{''}W_udu$
• Geometric Brownian Motion
  • stochastic process $S_{t}$
  • $d S_{t}=\mu S_{t} d t+\sigma S_{t} d W_{t}$
  • $S_{t}=S_{0} \exp \left(\left(\mu-\frac{\sigma^{2}}{2}\right) t+\sigma W_{t}\right)$
  • $\mathrm{E}\left(S_{t}\right)=S_{0} e^{\mu t}$
  • $\operatorname{Var}\left(S_{t}\right)=S_{0}^{2} e^{2 \mu t}\left(e^{\sigma^{2} t}-1\right)$
• Black-Scholes model
  • Black–Scholes equation
    • $\frac{\partial V}{\partial t}+\frac{1}{2} \sigma^{2} S^{2} \frac{\partial^{2} V}{\partial S^{2}}+r S \frac{\partial V}{\partial S}=rV$
    • prove it yourself once
  • Black–Scholes formula
    • $c=S_{0} N\left(d_{1}\right)-K e^{-r T} N\left(d_{2}\right)$
    • $p=-S_{0} N\left(-d_{1}\right)+K e^{-r T} N\left(-d_{2}\right)$
    • $d_{1}=\frac{-L n K+L n S_{0}+\left(r+1 / 2 \sigma^{2}\right) T}{\sigma \sqrt{T}}$
    • $d_{2}=\frac{-L n K+L n S_{0}+\left(r-1 / 2 \sigma^{2}\right) T}{\sigma \sqrt{T}}$
    • prove it yourself once
• Jump Process $N_t$
  • $P(N_t=n)=\cfrac{(ht)^n}{n!}e^{-ht}$
  • $h$ is the intensity of the poisson process (expected number in time 1)
  • compound poisson process
    • $Q_t=\sum^{N_t}_{i=1}y_i\ \text{where y be i.i.d. random variable with}\ E[y_i]=\beta$
    • $y$ and $N_t$ are independent
    • $E[Q_t]=\beta ht$
  • compensated compound poisson process
    • $Q_t-\beta ht$
    • martingale