Covered call (connection with put call parity)

[seidu]

Hi @David Harper CFA FRM

I am finding some difficulty understanding the relation of the put call parity to the covered call you made on page 141 on Hull, Options, Futures & Other Derivatives. See below:



I would appreciate if you could throw more light on it.

Thanks in advance.

 

[Arka Bose]

Hi there,
Covered call strategy means you have written a call (short call) and then purchased a stock to hedge against your short call position.
What the hull text says is that we an create a covered call through algebraic manipulation of the put call parity.
Here is how:
the PCP equation is S+P=C+K
Now, our target is to create a portfolio of S and '-C' (negative as we are short on the call option
So, to create the portfolio of S-C, just take the S-C in left hand side of the equation and put others on the right hand side.
Thus, it will stand out to be S-C= K-P
K-P can be though of a long zero coupon bond and a short position in a put option. Thus covered call, i.e S-C can be thought of a replicating portfolio of a ZCB and a short put.
Intuitively, we can create any replicating portfolio through this algebraic manipulation of the put call parity equation

 

[Dr. Jayanthi Sankaran]

Hi @seidu,

According to European put-call parity,

p + S0 = c + Ke-rT...................(1)

where p = price of the European put
S0 = stock price
c = price of European call
K = strike price of both call and put
r = risk-free interest rate
T = time to maturity of both the call and the put

Rearranging we get,

S0 - c = Ke-rT - p......................(2)

The left side of equation (2) is a portfolio consisting of a long position in a stock plus a short position in a European call. This is known as writing a covered call. The long stock position "covers" or protects the investor from the payoff on the short call that becomes necessary if there is a sharp rise in the stock price.

The right side of equation (2) is a portfolio of a short European put plus a certain amount of cash Ke-rT. Ke-rT is the present value of a zero-coupon bond (i.e. we are investing in a zero-coupon bond which has a principal of K at time T - when we invest, we are lending cash)

Hope that helps!

 

[seidu]

Thanks @arkabose and @Dr. Jayanthi Sankaran your comments are really helpful.

 

[brian.field]

I always find the signage confusing - some consider C - P = Soe^(-delta*t) - Ke^(-rt ) as a "long call and short put" equaling a "long stock and short risk free bond" but this seems unintuitive to me.

Perhaps more intuitive is the fact that you can also write the equation as:

-C + P = -Soe^(-delta*t) + Ke^(-rt ) and interpret that sign of the variable to be the direction of cashflows from your own perspective. So this epxression is more intuitive to me.

A cash outflow (-C) in the amount of the Call price would equate to a long call and a cash inflow (+P) in the amount of a Put price equates to a a short put, etc.

 

[David Harper CFA FRM]

That's interesting @brian.field, although I have a bit harder time with your interpretation (although of course I see why it's perfectly reasonable). Despite the study note extract above, my own personal preference is to start with:

S(0) + p = c + K*e^(-rt)

For two reasons. First, all terms are positives (+) so I tend to think of these as prices; i.e., cash outflows from my perspective as a buyer. Second, the first term is a protective put, so I just find it a little easier to remember as: protective put = call + cash, where all terms are payments (+, cash outflows). But it's just a matter of style i guess! To illustrate with an example, if S(0) = $9.00, K = $10.00,. Rf= 4.0%, σ = 25%, and T = 1.0, then:

• c = 0.65
• p = 1.26
• K*e^(-rT) = 9.61
• S = 9.00

So that:

• Brian' s: -c + p = -S + K*e(-rT) --> -0.65 + 1.26 = -9.00 + 9.61; i.e., cash outflow of $0.61
• Mine: S + p = c + K*e^(-rt) --> 9.00 + 1.26 = 0.65 + 9.61; same as my "price of 0.61"

And this can be re-arranged. Just to be whimsical for example, a straddle = long call + long put (with same strike price). The price of a purchase straddle, then, is given by:

• c+p = S + 2*p - K*e(-rT) = 9.00 + 2*1.26 - 9.61 = $1.91. Thanks!