constructing bear spreads

hi,

i'm trying to understand the construction of bear spreads using puts or calls.

the definition is pretty simple:

a) short call with strike K1 and long call with strike K2, where K2 > K1

or

b) short put with strike K1 and long put with strike K2, where K2 > K1

(these come from Hull).

Now, if i work out the profits at expiry to a) (ignoring discounting), I get:

profit = max(S_T - K2, 0) - max(S_T - K1, 0) +C_K1 - C_K2

where C_K1 - C_K2 is the money received for shorting the call less the money paid for buying the second call. This will be positive, since lower strike price calls cost more.

We can be more specific about the profit:

if: S_T > K2, profit = -(K2 - K1) + C_K1 - C_K2
K1 < S_T < K2, profit = -(S_T - K1) + C_K1 - C_K2
S_T < K1, profit = C_K1 - C_K2

this is all fine and agrees with Hull.

Now, I have issues with b). The profit this time is:

profit = -max(K1 - S_T,0) + max(K2 - S_T,0) - (P_K2 - P_K1)

where P_K2 - P_K1 is the initial investment due to buying a put with a higher strike price, than the one shorted.

This will give profits as:

if: S_T > K2, profit = -(P_K2 - P_K1)
K1 < S_T < K2, profit = K2 - S_T - (P_K2 - P_K1)
S_T < K1, profit = K_2 - K_1 - (P_K2 - P_K1)

The profit is less clear cut here, as it's not simply the difference of option premiums.

So let's compare the potential profit from each bear spread, in the case share prices fall.

a) gives: C_K1 - C_K2
b) gives: K_2 - K_1 - (P_K2 - P_K1)

Are these two quantities going to be similar? Or will one tend to provide a better payoff? (on the basis you are a speculator who predicts share prices will fall, and want to maximise your profit in this scenario).

 

major component of profits is the premiums and in (ii) the strikes.
I would have thought thus put call parity would shed light on these relativities

or is that inconsistent with problem of assuming no discounting?

If c(k1) – c(k2) = (p(k1) + s – pv (k1) ) – ( p(k2) + s – pv( k2) )

Note c(k1), c(k2) , p (k1) p (k2) are option premiums at k1/k2 and pv(k1) means present value of strike price….etc and s is current share price

Using put call parity for c – p = s – pv (exercise price)

So above is
C(k1) – c(k2) = ( Pv(k2) – pv(k1) )+ ( p(k1) – p(k2))

Now assume equal maturities so pv factor is same for both options and can be factorised:

Pv ( k2 – k1) - ( p(k2) – p(k1) )

But isn’t this less than

(k2 – k1 ) - ( p(k2) – p(k1) )

showing the call strategy less profitable than the put strategy

there must be an economic/option type argument somewhere for this...

 

you are right, once you allow for discounting, the put-call parity makes these two the same...doh!

 

Agreed Gareth.

Examstudent if you factor in discounting to the original payoff setup you don't encounter your (seemed) paradox.

 

Did either of you start thinking you may have have hit upon a hitherto unknown arbitrage?

 

thanks guys for pointing this out
i see both maturity payoffs revaluing everyhing in terms of time zero prices...
ollie - you make reference to the term arbitrage, and seeing these identical values at zero, in a no arbitrage world why would one prefer one strategy over another?
but in the real world i suppose there is a preference, i take it gareth your initial question is hinting at that?

ummmm..... lol