Introduction to Derivative Pricing

“Prevent arbitrage opportunities”

Arbitrage opportunity

• no cost
• no possibility of loss
• 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

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