Tuesday, February 12, 2008

PCP, intuitively, without Fwd contract

The notations C, P, F are very ambiguous. F means what? MV of an existing fwd position?

Tip -- either think in terms of terminal value of pre-expiry. Don't mix them
Tip -- think of owning the underlier (or a fully-paid fwd position). The standard Fwd position's MV and PnL are ... non-intuitive and confusing to me.Tip -- think in terms of change in MV. MV itself is poorly defined. See post on MktVal confusion. PnL is also confusing due to the fee paid upfront...

Tip -- don't bother with the innocent-looking details of some "fwd struck at K". It's not so simple, after you deal with it for 5 years, and the complexity is unneeded, when we can use a fully-paid fwd position.
Tip -- either think of premium paid (and locked in) or think of pre-expiry live PnL of existing positions. Don't mix them.
Tip -- don't bother with the premium paid in the past. That amount is just a can of worm that gives nothing but confusion. The only premium thing relevant here is the value of an advertized option.

PCP assumes 0 bid/ask and 0 commission. Let's consider the most liquid family -- Eur/Usd or SPX index. The option bid/ask is large.

For a beginner, this non-trivial concept is simplified by substituting a fixed amount of cash for the bond.
http://www.theoptionsguide.com/understanding-put-call-parity.aspx has a nice diagram to illustrate the call portfolio vs the put portfolio. When studying any payoff diagram, #1 key point to bear in mind is that the X-axis is the possible underlying asset prices at_expiration. X-axis is not time. Time is irrelevant in the PCP analysis if we use cash to substitute the bond. Now let's focus on an intuitive illustration of the call-put parity.

(In this first exmple, let's assume an option contract has just 1 share, rather than the 100 shares in real contracts.)

A portfolio of 1 share of IBM + 1 put at $100 (Protective put -- original motivation of PUT contracts and its simplest usage.)

Suppose our PUT is deep in the money, and IBM is trading at $77. Portfolio value is simply $100, since we have the right to cash in our share at $100. When IBM trades at $111, portfolio is worth $111.This payoff is equivalent to a call (K=100)+ $100 cash. In terms of terminal MV,

   K (i.e. cash) + C = P + S (i.e. underlier, paid in full)

In this illustration, we completely avoid the (confusing) fwd or futures contract. Instead, we deal with options and underlier asset only, so the MV concept is clear.

This relation holds at expiry. By arbitrage argument, it also holds any time before maturity.
------ There's a simpler version involving just 3 trades
* Suppose I buy an $100 ATM Put for $9 and also buy 100 IBM at the current price of $100/share.(A protective put?)
* My friend buys the $100 ATM Call also at $9. (No surprise the ATM call and put have identical valuations.)
* We both start with $10900 cash.

After the purchase, my portfolio has $0 cash + the put + the stock; my friend has $10000 cash + the call.

At expiration, Our 2 portfolios are identical. In this version, you don't see the bond position!

------ Now let's try to make sense of PCP with a fwd but no bond. Let's try to simplify the fwd contract. There are many standard ways to define it. As a holder, an "K-based fwd" [5] requires me to pay a fixed price $K (eg $100) at expiry to receive the asset. No upfront payment.
pnl at expiry = S_T - K, which could be negative
pnl at a time before expiry = S - K, which could be negative
MV is ... defined as same as pnl. It's non-intuitive.


   C = P + F(K)

where C and P mean the strictly positive MV of the call/put, ignoring the premium paid, and F(K) means the +/- MV of the K-based fwd contract.

The expiration range-of-possibility graph would show the RHS is the same hockey stick as the call... same terminal payoff.

This relation holds at maturity. By arbitrage argument, it also holds any time before maturity.

[5] "K-based fwd", denoted F(K) means an off-market fwd contract struck at K. The on-market fwd has a strike price equal to the fair fwd price. In FX, the on-market fwd contract is struck at the fwd rate.

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