futures.knit.rmd

---
title: "Forwards and Futures Formula Sheet"
author: "Alex Dou"
date: "March 10, 2019"
output:
  pdf_document: default
---




### Hedging

$$ basis=S_t-F_t  $$

$$ h^* = \rho\frac{\sigma_S}{\sigma_F}  $$

$$ N^* = h^*\frac{V_A}{V_F}  $$

$$ N^* = \beta\frac{V_P}{V_F}  $$

$$ N^*=(\beta^*-\beta)\frac{V_P}{V_F}  $$

### Standard form

$$ F=Se^{rt}  $$

### With dividend

$$ F_0=S_0e^{(r-q)T}  $$


### Exchange Futures

$$ F_0=S_0e^{(r-r_f)T}  $$


### With storage cost

$$ F_0=(S_0+U)e^{rt}  $$

$$ F_0=S_0e^{(r+u)T}  $$

### With yield rate

$$  F_0=S_0e^{(r+u-y)T} $$

### With cost of carrying

$$ F_0=S_0e^{cT}  $$

$$ F_0=S_0e^{(c-y)T}  $$

### Interest Futures

$$ P=\frac{360}{n}(100-y)  $$

### Cheapest-to-deliver bond

$$ CTD = min(B_{quoted} -F_{quoted}CF)   $$


### Euro-Dollar Future

$$ F_{eurodollar} = 10000\times[100-0.25(100-Z)]  $$

$$ Value(1bp) = 25USD  $$

### Duration hedge
$$ N = -\frac{V_p D_p}{V_f D_f}$$

$$ N = -\frac{V_p D_p}{V_f D_f} \beta$$


### The value of forward

$$ V_{forward} = S_0 (F-K)e^{-rt}  $$