Call-put parity

We consider an European call (denoted C) and a European put (denoted P) with the same underlying asset S, the same maturity T and the same strike K. We denote by B(t,T) the time-t price of a zero-coupon bond with T. Give the call-put parity relation when the underlying asset pays dividends \[ d_1, ..., d_p\[ at dates t_1, ..., t_p such that t < t_1 < ...<t_p <T

 

Before proceeding lets define some notations first

\(S_0\): Price of asset S at time \(t=0\)
\(S_T\): Price of asset S at time \(t=T\)
\(p\) : Price of European put with underlying asset S
\(c\) : Price of European call with underlying asset S

Now lets consider two portfolios:
Portfolio A: A call option with strike \(K\) and the investment in \(p+1\) zero coupon bonds \(d_1B(0,t_1),d_2B(0,t_2), \ldots d_pB(0,t_p), KB(0,T) \)

Portfolio B: A put option with strike \(K\) and 1 unit of investment in asset S

Now lets see the payoffs of Portfolio A and Portfolio B at time \(t_j\) for \(1\le j\le p\)
Portfolio A: The investment of \(d_jB(0,t_j\) gives a payoff of \(d_j\)

Portfolio B: A dividend worth \(d_j\) is received

Now lets see what happens at time \(T\)

There are two cases to consider,

\(S_T \ge K\)
Portfolio A is worth \((S_T-K)+K = S_T\) as call option is exercised
Portfolio B is worth \((0+S_T = S_T\) as put option isn't exercised

\(S_T < K\)
Portfolio A is worth \((0)+K = K\) as call option isn't exercised
Portfolio B is worth \((K-S_T)+S_T = K\) as put option is exercised

What do we see here? Portfolio A and Portfolio B both pays same payoffs and has same worth at time \(T\) therefore by no arbitrage the cost of portfolios must be the same i.e.

\begin{equation}\boxed{ c+\sum_{j=1}^{p}d_jB(0,t_j)+KB(0,T) = p + S_0 }\end{equation}

We established the put call parity relation.