dummy’s PCP intro: replicating portf@expiry=}pre-expiry #4IV

To use PCP in interview problem solving, we need to remember this important rule.

If you don’t want to analyze terminal values, and instead decide to analyze pre-expiry valuations, you may have difficulty.

The right way to derive and internalize PCP is to start with terminal payoff analysis. Identify the replicating portfolio pair, and apply the basic principle that

“If 2 portfolios have equal values at expiry, then any time before expiry, they must have equal value, otherwise arbitrage“.

Even though this is the simplest intro to a simple option pricing theory, it is not so straightforward!


fwd px ^ px@off-market eq-fwd

fwd price ^ price of an existing eq-fwd position. Simple rule to remember —
QQ) not $0 — fwd price is well above $0. Usually close to the current price of the asset.
EE) nearly $0 — current “MTM value” (i.e. PnL) of an existing fwd contract is usually close to +-$0. In fact, at creation the contract has $0 value. This well-known statement assumes both parties negotiated the price based on arb pricing.

Q: With IBM fwd/futures contracts, is there something 2D like the IBM vol surface?

2 contexts, confusing to me (but not to everyone else since no one points them out) —

EE) After a fwd is sold, the contract has a delivery price “K” and also a fluctuating PnL/mark-to-market valuation “f” [1]. Like a stock position (how about a IRS?) the PnL can be positive/negative. At end of day 31/10/2015, the trading venue won’t report on the MTM prices of an “existing” contract (too many), but the 2 counter-parties would, for daily PnL report and VaR.

If I’m a large dealer, I may be long/short a lot of IBM forward contracts with various strikes and tenors — yes a 2D matrix…

[1] notation from P 109 [[hull]], also denoted F_t.

QQ) When a dealer quotes a price on an IBM forward contract for a given maturity, there’s a single price – the proposed delivery price. Trading venues publish these live quotes. Immediately after the proposed price is executed, the MTM value = $0, always

The “single” price quoted is in stark contrast to option market, where a dealer quotes on a 2D matrix of IBM options. Therefore the 2D matrix is more intrinsic (and well-documented) in option pricing than in fwd contract pricing.

In most contexts in my blog, “fwd price” refers to the QQ case. However, in PCP the fwd contract is the EE type, i.e. an existing fwd contract.

In the QQ context, the mid-quote is the fwd price.

Mathematically the QQ case fwd price is a function of spot price, interest rate and tenor. There’s a simple formula.

There’s also a simple formula defining the MTM valuation in EE context. Its formula is related to the QQ fwd quote formula.

Both pricing formulas derived from arbitrage/replication analysis.

EE is about existing fwd contracts. QQ is about current live quotes.

At valuation time (typically today), we can observe on the live market a ” fwd price”. Both prices evolve with time, and both follow underlier’s price S_t. Therefore, both prices are bivariate functions of (t,S). In fact, we can write down both functions —

QQ: F_t = S_t / Z_t ….. (“Logistics”) where Z_t is the discount factor i.e. the T-maturity discount bond’s price observed@ t
EE: p@f = S_t – K*Z_t

( Here I use p@f to mean price of a fwd contract. In literature, people use F to denote either of them!)

To get an intuitive feel for the formulas, we must become very familiar with fwd contract, since fwd price is defined based on it.

Fwd price is a number, like 102% of current underlier price. There exists only one fair fwd price. Even under other numeraires or other probability measures, we will never derive a different number.

In a quiz, Z0 or S0 may not be given to you, but in reality, these are the current, observed market prices. Even with these values unknown, F_t = S_t / Z_t formula still holds.

Black’s model – uses fwd price as underlie, or as a proxy of the real underlier (futures price)

Vanilla call’s hockey stick diagram has a fwd contract’s payoff curve as an asymptote. But this “fwd contract’s payoff curve” is not the same thing as current p@f, which is a single number.

PCP with dividend – intuitively

See also posts on PCP.
See also post on replicating fwd contract.

I feel PCP is the most intuitive, fundamental and useful “rule of thumb” in option pricing. Dividend makes things a tiny bit less straightforward.

C, P := call and put prices today
F := forward contract price today, on the same strike. Note this is NOT the fwd price of the stock.

We assume bid/ask spread is 0.

    C = P + F

The above formula isn’t affected by dividend — see the very first question of our final exam. It depends only on replication and arbitrage. Replication is based on portfolio of traded securities. (Temperature – non-tradable.) But a dividend-paying stock is technically non-tradable!

* One strategy – replicate with European call, European put and fwd contract. All tradable.

* One strategy – replicate with European call, European put, bond and dividend-paying stock, but no fwd contract. Using reinvestment and adjusting the initial number of shares, replication can still work. No need to worry about the notion that the stock is “non-tradable”.

Hockey stick, i.e. range-of-possibility graphs of expiration scenarios? Not very simple.

What if I must express F in terms of S and K*exp(-rT)? (where S := stock price any time before maturity.)

  F = S – D – K*exp(-rT) … where D := present value of the dividend stream.

option pricing – 5 essential rules n their assumptions

PCP — arb + extremely tight bid/ask spread + European vanilla option only. GBM Not assumed. Any numeraire fine.

Same drift as the numeraire — tradeable + arb + numeraire must be bond or a fixed-interest bank account.

no-drift — tradeable + arb + using the numeraire

Ito — BM or GBM in the dW term. tradable not assumed. Arb allowed.

BS — tradable + arb + GBM + constant vol

eq-fwd contract pricing – internalize

Even if not actively traded, the equity forward contract is fundamental to arbitrage pricing, risk-neutral pricing, and derivative pricing. We need to get very familiar with the math, which is not complicated but many people aren’t proficient.

At every turn on my option pricing learning journey, we encounter our friend the fwd contract. Its many simple properties are not always intuitive. (See P 110 [[Hull]])

* a fwd contract (like a call contract) has a contractual strike and a contractual maturity date.Upon maturity, the contract’s value is frozen and stops “floating”. The PnL gets realized and the 2 counter-parties settle.
* a fwd contract’s terminal value is stipulated (ST – K), positive or negative. This is a function of ST, i.e. terminal value of underlier. There’s even a “range of possibilities” graph, in the same spirit of the call/put’s hockey sticks.
* (like a call contract) an existing fwd contract’s pre-maturity MTM value reacts to 1) passage of time and 2) current underlier price. This is another curve but the horizontal axis is current underlier price not terminal underlier price. I call it a “now-if” graph, not a  “range of possibilities” graph. The curve depicts

    pre-maturity contract price denoted F(St, t) = St                    – K exp(-r (T-t)  ) ……… [1]
    pre-maturity contract price denoted F(St, t) = St exp(-q(T-t)) -K exp(-r(T-t)) .. [1b] continuous div

This formula [1b] is not some theorem but a direct result of the simplest replication. Major Assumption — a constant IR r.

Removing the assumption, we get a more general formula
              F(St, t) = St exp(-q(T-t)) – K Zt
where Zt is today’s price of a $1 notional zero-bond with maturity T.

Now I feel replication is at the heart of everything fwd. You could try but won’t get comfortable with the many essential results [2] unless you internalize the replication.

[2] PCP, fwd price, Black model, BS formula …

Notice [1] is a function of 2 independent variables (cf call).  When (T – now) becomes 0, this formula degenerates to (ST – K). In other words, as we approach maturity, the now-if graph morphs into the “range of possibilities” graph.

The now-if graph is a straight line at 45-degrees, crossing the x-axis at    K*exp(-r  (T-t)  )

Since Ft is a multivariate function of t and St , this thing has delta, theta —

delta = 1.0, just like the stock itself
theta = – r K exp(-r  (T-t)  ) …… negative!

(Assuming exp(-q(T-t)) = 0.98 and
To internalize [1b], recall that a “bundle” of something like 0.98 shares now (at time t) continuously generates dividend converting to additional shares, so the 0.98 shares grows exponentially to 1.0 share at T. So the bundle’s value grows from 0.98St to ST , while the bond holding grows from K*Zt to K. Bundle + bond replicates the fwd contract.

 —————Ft / St is usually (above or below) close to 0 when K is close to S.  For example if K = $100 and stock is trading $102, then the fwd contract would be cheap with a positive (or negative) value.
** most fwd contracts are constructed with very low initial value.
* note the exp() is applied on the K. When is it applied on the S? [1]
* compare 2 fwd contracts of different strikes?
* fwd contract’s value has delta = 1

[1] A few cases. ATMF options are struck at the fwd price.

intuitive – basic equivalence among option positions

To a person unfamiliar with options, there’s confusion over the interchange-ability among

Long call
Long put
Short put
Short call

(Focus on regular options and ignore barrier option for now.)

Simple rule – in FX options,
– call-buying===buying-put;
– call-writing===writing-put. Not equities, index or commodity options though.

There’s never any equivalence between long vs short option positions. A short position (call or put) represents unlimited downside in FX. An experienced FX option trader once said option-writing is abhorrent.

After we get comfortable with these equivalence rules, then we can throw in PCP (put-call parity). Mixing PCP and the equivalence rule too early leads to brain-damage.

PCP synthetic positions – before expiration?

In most of the books I have seen, all the synthetic option positions (buy-write vs short put for eg) are analyzed at expiration, using the hockey stick PnL graphs. But what about valuations before-expiration — is the synthetic also a substitute?
Let me give you a long answer. Bear in mind all the synthetics are based on PCP, applicable to European (E) options only.( For Europeans, real valuation happens only at expiration.)

By arbitrage analysis, and assuming 0 bid/ask spread 0 commission European style, I believe we can prove that n days (eg 5) before expiration, BW valuation must match the naked short put.

5 days before expiration, even with very low spot level, even with very high implied volatility, the option holder can’t exercise and must wait till expiration. At expiration, the 2 portfolios have identical payoff. Therefore the 2 are equivalent any time before expiration.

first lessons on option delta

I feel delta is the #1 most important greek for a new guy trying to understand option valuation sensitivities.

If you are long or short any security, then you want to monitor your sensitivity to a few key variables. For fixed income positions, you want to monitor sensitivity to IR and credit rating change, among others. For FX positions, you want to monitor sensitivity to IR of multiple currencies…For option positions, you monitor
* (vega) volatility changes. The underlier can exhibit very different volatility from Day 1 to Day 2.
* (delta) underlier price
* (theta) speed of decay
* since delta is such a important thing to watch, you also want to monitor gamma, i.e. how fast your delta changes in response to underlier appreciations.

For a typical option’s valuation, sensitivity to underlier is the biggest sensitivity. To trade volatility, you first need to insulate yourself from directional changes. Call it direction-neutral or direction-indifferent. I was told in most cases 0 delta won’t “happen to us”, so we need to calculate or design our trades so portfolio has 0 delta. Note you get zero delta only at a particular spot price of the underlier. When underlier moves, your portfolio delta won’t stay zero.

To learn basic option trading, First a student needs a good grasp over option payoff at expiration. Actually non-trivial. Even for a basic call option, there is a payoff graph like a hockey stick. We need to understand the payoff graph of all 4 basic positions + payoff graph of basic strategies like protective put. Also PCP.

A more realistic graph is portfolio pnl at expiration. “Portfolio pnl” includes the cash you paid/received (realized) in addition to unrealized PnL. In this case, hockey stick crosses x-axis, which is completely realistic. It means your portfolio pnl can be positive or negative depending on the at-expiration price of underlier. Premium cost is a very practical consideration, so it’s naive to ignore it in the payoff diagram. Prefer portfolio-PnL.

Next graph or curve[1] is {{ option valuation vs spot price }} i.e. option valuation/premium relative to underlier spot price. Obviously option premium is priced by each trader’s pricing engine taking inputs of strike price, time to expiration, vol etc, but here we need to hold all other parameters constant and focus on spot price’s effect on option MV. This is how to get the curve. Within this simplified context, we need to
– compare all the basic strategies
– know the difference of ITM vs OTM
– know how a basic call’s curve depends on vol. We can plot the curve for different vol values

Lastly, Delta is treated like a soft market data on option MV. The slope of the curve in [1] is Gamma. Charm is also a derivative of delta.

Practical usage of delta? Delta is used in delta hedging and delta-neutral trading.

For a typical option’s price, sensitivity to underlier is the biggest sensitivity. Therefore delta overshadows vega, theta and rho. But Delta is definitely not the only important greek. In fact, vol is probably the most important factor in option pricing, so vega is rather important.

delta of a call vs put (ex-div

Simplified put-call parity stipulates {{ call =~= put + underlier }}. PCP holds true whether underlier (say MSFT) spot is $20, $22.2 or whatever. Remember delta is always measured with tiny changes around a particular value of underlier SPOT price, so let’s assume MSFT spot price is now $20 ,

delta(call) =~= 1 + delta (put)

(You derive this by taking the first derivative of each item in the equation.)

Since delta(put) is always negative, you can see that given identical strike prices, the _magnitude_ of delta(call) + delta(put) is roughly 100%. — worth remembering.

There’s a bit of tricky fine print for the undaunted. In reality, the 2 magnitudes sometimes add up to exceed 100% for American options. I was told the key reason is early exercise [2]. Since the 2 options have identical strikes, exactly one of them is ITM. Just before dividend, the ITM option (either the put or the call) would have a higher delta thanks to the dividend. If 99% of the players in the market agree this ITM should be early exercised due to the dividend, then IMHO 1 lot of this ITM option feels like equivalent to 100 shares of MSFT, either long or short. Therefore delta of the ITM is similar to 100%.

Q: on the day before ex-dividend day, does the ITM option’s delta approach 100%?
A: I was told yes.

[2] there are other reasons like interest rate.

PCP – how to internalize, using PnL

(PCP under continuous dividend model? See http://bigblog.tanbin.com/2013/11/equity-fwd-contract-pricing-internalize.html)

key PCP concept — the equivalence of values of 2 portfolios ANY time after you buy them, not just at expiration. However, the valuation (plotted against S) at any time before expiration is non-intuitive and hard to grapple.

Suppose A buys a long European call and B buys [a long European put + a futures]. All instruments were bought at a fair price, so PnL both start $0. (MV is irrelevant at this stage.) N days later, when market rates (spot, implied vol…) have moved a bit, we would expect both portfolios to show small but identical PnL[1]. Therefore, looking at PnL rather than MV, the cash component disappears from the equation, since cash will (almost) always have zero PnL.

Note MV is a poorly defined (non-intuitive) concept for futures and a lot of derivatives. See post on MktVal clarified.

Similar to the PnL view, the delta view involves only 3 positions — call/put/forward, not  the cash.

[1] exact PnL amount is hard to visualize as it involves BS.

(A: European options have no assignment before maturity.)

Now let’s look at MV or valuations. Valuations are more important in practice and relate to observed market prices. At t=0 portfolio MV are equal only if we started at t= -1 with an equal amount of seed capital. At t=0, MV becomes (assuming zero interest rate)

  Premium_c + $K cash = Premium_p + f

N days later, or at expiration, MV becomes

  MV_c + $K cash = MV_p + MV_f

Some people (like my boss Saurabh) say a long call + a short put == a forward [2], but I find it less intuitive. LHS is the difference between the 2 premiums, which could be 0 or negative.

Update – Now I agree C = P + F is the best way to remember it, once you recognize that you must get down to details with the fwd contract and build intimate knowledge thereof.

I guess the statement in [2] assumes a long position in the forward contract can become either an asset or liability any time before expiration. If I must translate [2] into English, i would say a long call combining a short put has identical PnL to a forward contract (assuming European options). Suppose both portfolios start with just the positions + no cash. At T=0, all 3 securities are bought at fair values, so Portfolio A has PnL=0, so does B. Based on the delta rule above, a 1 cent change in underlier would result in identical changes in the 2 portfolio’s valuations, so the 2 portfolios always have identical valuations, either positive or negative.

What if your short put gets assigned? Answer is hidden somewhere in this blog. If you indeed lose the short position, the delta rule will stop working.

Is PCP compromised by any of the “unrealistic/simplistic” assumptions of BS? No. PCP is model-independent.

Is PCP affected by the vol skew or the vol term structure ? I don’t think so.