Investment and Financial Markets

1.1 Introduction to Derivatives
1.1.0 Introduction
1.1.1 Derivative Basics

• What is a derivative?
• Why use derivatives?
  • To manage risk
  • To speculate
  • To reduce transaction costs
  • To minimize taxes/avoid regulatory issues
• Who uses derivatives?
  • End-users
  • Market-makers
  • Economic observers
• Underlying Assets
  • Stocks
  • Indices
  • Commodities
  • Currencies

1.1.2 Buying and Selling Assets

• Market-Makers
• Commission
• Bid-Ask Price
  • Bid-Ask Spread = Ask Price - Bid Price Bid Ask Market-Maker Buys Low Sells High Investor Sells Low Buys High
• Ways to Buy or Sell

1.1.3 Accounting Profits vs. Economic Profits

• Accounting profit = Revenue - Cost
• Economic profit = Revenue - Explicit Cost - Implicit Cost

1.1.4 Short Selling Assets

• Long vs. Short
• What is Short-Selling?
  • Borrow shares of stock now
  • Immediately sell the borrowed stock
  • Buy the shares back
• More Details on Short-Selling
  • The Short-Sale Proceeds
  • A Haircut
  • Interest
    • short rebate
    • repo rate
  • Dividends
    • lease rate

1.1.5 Payoff and Profit

• Profit = Accumulated Value of cash flows at the risk-free rate
• Payoff & Profit for a Nondividend-Paying Stock
  • Long Position
  • Short Position
• Payoff & Profit for Zero-coupon Bond
  \(\text{Profit}_\text{bond}=V+\left(-Ve^{-rT}\right)e^{rT}=0\)

1.2.0 Introduction
1.2.1 Forward Contract Basics

• Motivation
• Forward Contract: Example

1.2.2 Payoff and Profit

• Payoff of a Forward
  \(\text{Payoff}_\text{long forward} = \text{Spot price at expiration}\,– \text{Forward price}\) \(\text{Payoff}_\text{short forward} = \text{Forward price}\,– \text{Spot price at expiration}\)

• Profit of a Forward \(\begin{align} \text{Profit}_\text{long forward} &= \text{AV(Cash flows)} \\ &= \text{Payoff}_\text{long forward}+ \text{AV(Cash flows at time 0)} \\ &= \text{Payoff}_\text{long forward}\,+ 0 \end{align}\) \(\begin{align} \text{Profit}_\text{short forward} &= \text{AV(Cash flows)} \\ &= \text{Payoff}_\text{short forward}+ \text{AV(Cash flows at time 0)} \\ &= \text{Payoff}_\text{short forward}\,+ 0 \end{align}\)

1.2.3 Four Ways of Buying a Stock

1. Outright purchase: Pay for the stock at time 0 and receive it at time 0
2. Forward Contract: Pay for the stock at time T and receive it at time T
3. Prepaid forward contract: Pay for the stock at time 0 and receive it at time T
4. Fully leveraged purchase: Receive the stock at time 0 and pay for it at time T

1.2.4 Pricing a Forward & Prepaid Forward

• Pricing a Prepaid Forward Contract
  • Forward Contract
  • Prepaid Forward Contract
    \(F_{0,T}=F_{0,T}^{P}\cdot e^{rT}\)

• Nondividend-Paying Stock
  \(F_{0,T}^{P}= S(0)\)

• Dividend-Paying Stock
  • CASE 1: DISCRETE DIVIDENDS
    \(F_{0,T}^{P}=S(0)-\text{PV(Divs)}\)
  • CASE 2: CONTINUOUS DIVIDENDS
    \(F_{0,T}^{P}= S(0)\cdot e^{-\delta T}\)

1.2.5 Synthetic Forwards

1.2.6 Exploiting Arbitrage

• cash-and-carry
• reverse cash-and-carry

1.2.7 Summary

• FORWARD CONTRACT BASICS
• PAYOFF AND PROFIT
• FOUR WAYS OF BUYING A STOCK
• PRICING A FORWARD & PREPAID FORWARD
• EXPLOITING ARBITRAGE

3.1.1 Option Pricing: Replicating Portfolio

• Replicating Portfolio
• Formulas for Replicating Portfolios
  \(\begin{align} (\Delta e^{\delta h})(S_0u)+Be^{rh}=V_u \\ (\Delta e^{\delta h})( S_0 d)+Be^{rh}=V_d \end{align}\)
  • number of shares to purchase in order to replicate an option
    \(\Delta = e^{-\delta h} \cdot \dfrac{V_u-V_d }{S_{0}\left(u-d \right)}\)
    \(B= e^{-rh}\left(\dfrac{V_du -V_ud}{u-d} \right)\) \(V_0=\Delta S_0 + B\)

3.1.3 Constructing a Binomial Tree

• General Method
  \(\begin{align} u &= e^{(r-\delta)h+{\sigma} \sqrt{h}} \\ d &= e^{(r-\delta)h-{\sigma} \sqrt{h}} \end{align}\)

• Standard Binomial Tree
  \(\begin{align} p^{*} &=\dfrac{e^{(r-\delta)h}-d}{u-d}\\&=\dfrac{e^{(r-\delta)h}-e^{(r-\delta)h-\sigma\sqrt{h}}}{e^{(r-\delta)h+\sigma\sqrt{h}}-e^{(r-\delta)h-\sigma\sqrt{h}}} \\ &=\dfrac{1-e^{-\sigma\sqrt{h}}}{e^{\sigma\sqrt{h}}-e^{-\sigma\sqrt{h}}} \\ &= \dfrac{1-e^{-\sigma\sqrt{h}}}{(1-e^{-\sigma\sqrt{h}}) (1+e^{\sigma\sqrt{h}})} \\ &= \dfrac{1}{1+e^{\sigma\sqrt{h}}} \end{align}\)

3.1.4 Multiple-Period Binomial Option Pricing