Call put parity

[marchaiseng]

Good Evening,

I'm reading John Hull and I don't understand very well "call put parity".

Why c + Ke-rT = p + So ?

And a last question : why So = Ke-rT ?

Thanks,

PS : c = call
p = put
K = strike - exercise price
r = risk-free interest rate
So = underlying
(e = exponential)

 

[MrGecko]

.

 

[Martinghoul]

What's the question exactly?

 

[marchaiseng]

Why reasons for c + Ke-rT = p + So ?

You have two portfolio :
- A: a call and liquidity about Ke-rT
- B: a put and the underlying.

Thanks ,

 

[Martinghoul]

It's just a no-arbitrage argument. The two portfolios have identical payoffs under all scenarios.

 

[marchaiseng]

it's so simple ? Thanks, in fact it's just a basic condition.

 

[BJ21]

The relationship between the price of a call and a put both with same maturity and strike.

Like you said 2 portfolios :
Port 1 : 1 call and K bonds (or discounted value of the strike).
Port 2 : 1 put and 1 unit of asset.

These are the payoff's in equations,
At T (time)
Port 1 : Max (S-K,0) + K.B(t,T) = Max(S,K)
[ K if S<=K call = 0 bonds = K ]
[ S if S>=K call = S-K bonds = k]

Port 2 : Max (K-S,0) +S = Max(S,K)
[ K if S<=K put = K-S Shares = S]
[ S if S>=K put = 0 Shares = S]

Both have the same value "[Max(S,K)]" at Maturity so both are worth the same ...

Max (S-K,0) + K.B(t,T) = Max (K-S,0) +S

Subtitute in C and P as Call and Put

C + K.B(t,T) = P + S

Bond interest is constant compound so :

C + Ke-rt = P + S
only holds for american C + PV(x) = P+S holding a call is the same as holding a put and 1 unit of underlying ect.......

 

[marchaiseng]

OK, thanks a lot !!!!

You explain very well. Thanks again.

 

[meanreversion]

If you purchase a call and sell a put with the same strike and same expiry, at expiry one of two things can happen

1. Spot is above the strike, thus you exercise your call and buy at price K (= strike)

2. Spot is below the strike, thus the put you sold is exercised on you, and you buy at price K

Either way, you buy at K. Thus C-P = underlying long position.

Now this relationship is useful because (for example) the vega (dp/dvol) of an underlying position is zero. Thus the vega of the call and the put are IDENTICAL.

In fact, the volatility assigned to the strike is the same regardless of whether it's a call or a put.

Or to put it another way, there is no arbitrage between buying the call/selling the put/hedging with the underlying. In the early days of options there was ... low prem calls were too high and high prem puts were too low, so simply sell the call/buy the put/hedge 100 pct of underlying and you're guaranteed to make money.

 

[Martinghoul]

Yep, meanie is 100% right... One thing to note is that all this applies to European options and doesn't apply to American ones.

 

[marchaiseng]

Yes, this explication is useful.
The value of the both portfolio is always K.

I knew that it's good for European options and not to american options, because an american option is "exercisable" during the full time (I don't know whether I am well expressed or not)