__terminal payoff__analysis. Identify the 2

__replicating portfolios__, 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”.

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# Category: PCP

# dummy’s PCP intro: replicating portf@expiry=>pre-expiry

# fwd px ^ px@off-mkt eq-fwd

# PCP with dividend – intuitively

# option pricing – 5 essential rules n their assumptions

# eq-fwd contract pricing – internalize!

# intuitive – basic equivalence among option positions

# PCP synthetic positions – before expiration?

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 2 __replicating portfolios__, 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”.

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.

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.

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

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.

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.

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.

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.