update readme latex

qmcpy/README.md

@@ -11,13 +11,13 @@ The function to integrate.
 - Keister Function: $g(\boldsymbol{x}) = \pi^{d/2} \, \cos(||\boldsymbol{x}||_2)$
 - Custom Function
 - European Option
-    - stock price at time $jT/d$: $~~~~~~~~~$ $S(x_j)=S_0\exp\bigl((r-\sigma^2/2)(jT/d)+\sigma\mathcal{B}(t_j)\bigr)$
-    - discounted call payoff $= \max\left(S(x_d)-K\right),\: 0)  \,\exp(-rT)$
-    - discounted put payoff $= \max\left(K-S(x_d)\right),\: 0)\,\exp(-rT)$
+    - stock price at time $jT/d=\tau_j$: $~~~~~~~~~$ $S(\tau_j)=S_0\exp\bigl((r-\sigma^2/2)\tau_j+\sigma\mathcal{B}(\tau_j)\bigr)$
+    - discounted call payoff $= \max\left(S(\tau_d)-K\right),\: 0)  \,\exp(-rT)$
+    - discounted put payoff $= \max\left(K-S(\tau_d)\right),\: 0)\,\exp(-rT)$
 - Asian Call Option
-    - stock price at time $jT/d$: $~~~~~~~~~$ $S(x_j)=S_0\exp\bigl((r-\sigma^2/2)(jT/d)+\sigma\mathcal{B}(t_j)\bigr)$
-    - discounted call payoff $= \max\left(\frac{1}{d}\sum_{j=1}^{d} S(x_j)-K\right),\: 0)  \,\exp(-rT)$
-    - discounted put payoff $= \max\left(K-\frac{1}{d}\sum_{j=1}^{d} S(x_j)\right),\: 0)\,\exp(-rT)$
+    - stock price at time $jT/d=\tau_j$: $~~~~~~~~~$ $S(\tau_j)=S_0\exp\bigl((r-\sigma^2/2)\tau_j+\sigma\mathcal{B}(\tau_j)\bigr)$
+    - discounted call payoff airthmetic mean: $g_{\text{arith}}(\boldsymbol{t})= \max\left(\frac{1}{2d}\sum_{j=1}^d [S(\tau_{j-1})+S(\tau_j)]-K,\: 0\right)  \,\exp(-rT)$
+    - discounted call payoff geometric mean: $g_{\text{geo}}(\boldsymbol{t}) = \max\left(\biggl[\prod_{j=1}^d [S(\tau_{j-1})S(\tau_j)]\biggr]^{\frac{1}{2d}} -K,\: 0\right)\,\exp(-rT)$
 - Multilevel Call Options with Milstein Discretization 
 - Linear Function: $g(\boldsymbol{x}) = \sum_{j=1}^{d}x_{j}$