Introduction to Derivative Pricing

“Prevent arbitrage opportunities”

Arbitrage opportunity

1. no cost
2. no possibility of loss
3. possibility of profit

Given two portfolios \(A\) and \(B\), according to Arbitrage Pricing Principle I (APP I):

• \(a_T<b_T\quad\rightarrow\quad a_t<b_t\)
• \(a_T=b_T\quad\rightarrow\quad a_t=b_t\)

Forward price

\[F_0=S_0(1+i)^T=S_0e^{rT}\] where the interest rates, \(i\) (annual) and \(r\) (continuous), are risk-free.

Put-Call parity for European options

\[c_t+e^{-r(T-t)}K=p_t+S_t\] where \(c_t\) and \(p_t\) are respectively prices of call option and put option at time \(t\).

Upper and lower bounds on European option prices

Call option

\[(S_t-Ke^{-r(T-t)})_+\leq c_t\leq S_t\]

Put option

\[(Ke^{-r(T-t)}-S_t)_+\leq p_t\leq Ke^{-r(T-t)}\]

Upper and lower bounds on American option prices

Call option

\[(S_t-Ke^{-r(T-t)})_+\leq C_t\leq S_t\]

Put option

\[\max(p_t,(K-S_t)_+)\leq P_t\leq K\]