• We are pleased to announce that the winner of our Feedback Prize Draw for the Winter 2024-25 session and winning £150 of gift vouchers is Zhao Liang Tay. Congratulations to Zhao Liang. If you fancy winning £150 worth of gift vouchers (from a major UK store) for the Summer 2025 exam sitting for just a few minutes of your time throughout the session, please see our website at https://www.acted.co.uk/further-info.html?pat=feedback#feedback-prize for more information on how you can make sure your name is included in the draw at the end of the session.
  • Please be advised that the SP1, SP5 and SP7 X1 deadline is the 14th July and not the 17th June as first stated. Please accept out apologies for any confusion caused.

Call-put parity

S

Son Isfa

Member
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
 
Last edited by a moderator:
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.
 
Back
Top