Chapter5

• Outright purchase pay the stock price in cash, receive ownership of the stock.

• Fully leveraged purchase borrow the entire purchase price of the security.

  • Suppose borrow share price $S_0$,
  • agree to repay the borrowed amount at time $T$.
  • the continuously compounded interest rate is r,
  • at time T you would owe $e^{r^T}$ per dollar borrowed, or $S_0 e^{r^T}$.
    

• Prepaid forward contract pay for the stock today and receive the stock at an agreed-upon mature date.

  • you receive the stock at time T,
  • the price you pay is not necessarily the stock price.

• Forward contract both pay for the stock and receive it at time T, with the time T price specified at time 0.

prepaid forward contract:

no dividend:

$F^P_{~0,~T~}=S_0$

discounted present value:

• expected stock price at time $T$ is $E_0(S_T)$, $\alpha$ the expect return on the stock. then the prepaid forward price is $F^P_{~0,T~}=E_0(S_T)e^{- \alpha T}$

• $E_0(S_T)= S_0 e^{\alpha T}$

Arbitrage (free money)

happen when $F^P_{~0,~T~}>S_0$, should not take place in any situation.

With Dividends:

• Discrete dividends:

Quarterly continuously compounded rate is 2.5%, quarter dividend is 1.25$, stock cost 100$ today) 1 year prepaid forward contract for the stock costs:

• Continuous dividends: the annualized daily compounded dividend yield is daily dividend is

index is 125$ and the annualized daily compounded dividend yield is 3%. The daily dollar dividend is (0.03 + 365) x $125 = $0.01027

Forward contract

• No dividend: $F_{~0,~T~}=S_0 e^{rT}$, just future value

• Discrete dividends:

• Continuous dividends:

• Forward premium =$\frac{F_{0,T}}{S_0}$
  
  

Chapter 8

A swap is a contract calling for an exchange of payments over time.

A single payment today for a single delivery of oil in the future is a prepaid forward.
A single payment today to obtain multiple deliveries in the future is a prepaid swap.