fwd price@beach^desert – intuitive

Not that easy to develop an intuitive understanding…

Q2.46 [[Heard on the street]]. Suppose the 2 properties both sell for $1m today. What about delivery in x months? Suppose the beach generates an expected (almost-guaranteed) steady income (rental or dividend) of $200k over this period. Suppose there’s negligible inflation over this (possibly short) period.

Paradox: you may feel after x months, the beach would have a spot price around $1m or higher, since everyone knows it can generate income.
%%A: there’s no assurance about it. It could be much lower. I feel this is counter-intuitive. There might be war, or bad weather, or big change in supply/demand over x months. Our calculation here is based solely on the spot price now and the dividend rate, not on any speculation over price movements.

I guess the fair “indifferent” price both sides would accept is $800k, i.e. in x months, this amount would change hand.
– If seller asks $900k forward, then buyer would prefer spot delivery at $1m, since after paying $1m, she could receive $200k dividends over x months, effectively paying $800k.
– If buyer bids $750k forward, then seller would prefer spot delivery.

What would increase fwd price?
* borrowing interest Cost. For a bond, this is not the interest earned on the bond
* storage Cost

What would decrease fwd price?
* interest accrual Income
* dividend Income

eq-fwd contract – delivery price K

Eg: me buying Blk 155 flat. In Oct we agreed on $615 delivery price. Cash-On-Delivery on the delivery date in Feb. “Logistics”… No exposure no mkt risk.

Eg from [[hull]] P104. $40.50 delivery price means $40.50 cash to change hand on the delivery date.

Simple rule — No cash flow on execution date – different from most other trades. Simple difference yet very confusing to some.

Simple rule — To understanding the “delay”, we can imagine sky high interest, carry and inflation rates.

Simple rule – Cash-On-Delivery

I feel the delayed cashflow is at the heart of the (simple) arb and math. If we aren’t absolutely clear about this delay, big messy confusions …

EE context — yes deliver price is an important factor to PnL, risk mgmt etc
QQ context – delivery price is the price quoted and negotiated

Q: In each fwd contract, are there 2 prices?
QQ context – 1 only
EE context – 2 indeed. Similar to simply buying then holding IBM shares. Both prices are relevant in the EE context —
• K is the negotiated “execution” price, implicitly written into the fwd contract
• there’s a live market price for the same fwd contract.

fwd px ^ px@off-mkt 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.

hockey stick – asymptote

(See also post on fwd price ^ PnL/MTM of a fwd position.)

Assume K = 100. As we get very very close to maturity, the “now-if” graph descends very very close to the linear hockey stick, i.e. the “range of (terminal) possibilities” graph.

10 years before maturity, the “range of (terminal) possibilities” graph is still the same hockey stick turning at 100, but the now-if graph is quite a bit higher than the hockey stick. The real asymptote at this time is the (off-market) fwd contract’s now-if graph. This is a straight line crossing X-axis at K * exp(-rT). See http://bigblog.tanbin.com/2013/11/fwd-contract-price-key-points.html

In other words, at time 0, call value >= S – K*exp(-rT)

As maturity nears, not only the now-if smooth curve but also the asymptote both descend to the kinked “terminal” hockey stick.

eq-forward – basic questions to internalize

See also post on equity forward. Better become very very comfortable answering these questions. They should be in your blood:)

Q: daily mark to market of an existing position, on some intermediate date “t” before maturity.

Q: market risk of an existing long position?
A: similar to a simple long spot position. When underlier appreciates, we have a positive  PnL. “Logistics”.

Q: delta of  such an existing fwd contract?

There are many relationships  among many variables –

K, T — part of the contract specification
Z0, S0, — observable today
F0 — defined in the EE context as the MTM value of a new position. Almost always $0
ZT := 1.0, STFT := ST – K
Zt, St, Ft,  — where t is an intermediate time between now and T. Since t is in the future, these values are unknown as of today.

An interviewer could ask you about the relationship among any 3 variables, or the relationship among any 4 variables.

Warning — I use F0 to denote today’s price of an off-mkt fwd contract with K and T. Some people use F0 to denote the fwd price of the stock S.

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.

predicted future px USUALLY exceeds current px – again

This is a pretty important concept …

Extreme example – Suppose home price is believed to be unpredictable or unstable for the next 3 months[1], and short term rental is impossible, and there are many equivalent houses on sale. You just bought one of these equivalent houses but need it in 3 months and you would have all the cash by then. Do you prefer to “settle” (i.e. pay cash and get key) now (A) or (B) in 3 months? You prefer B because A means you must start paying mortgage interest 3-month earlier.

Now suppose seller exploits your preference for B, and asks $1 more to do a fwd deal (B) instead of a spot deal (A), you would be wise to still prefer B because interest amount is likely to be thousands.

So $1 is too cheap. But what’s a fair price for the fwd deal? I think it’s exactly the spot price plus the mtg interest amount. For most securities, fwd price [2] is Higher than spot. (A few assets are exceptions and therefore important[3].) First suppose fwd price == spot price as of today, and ignore the positive/negative signs below —

* if interest_1 < rent_3, then seller gains. Some competing seller would sacrifice a bit of gain to sell at a lower fwd price. Fwd price is then driven down below spot price. This is the high-coupon case.
* if interest_1 > rent_3, then seller loses. She would simply reject the proposed trade. She would have to charge a Higher fwd price to compensate for her loss. This is the usual case, where rent_3 is $0 and there’s no repo or rent market for this asset.

I feel the fair theoretical fwd price is not affected by implied volatility, or by any kind of trend. A trend can continue or reverse. I feel the calculation of fwd price is based on assumption of constant asset price or random movement for the next x months.

[1] I think in most cases of fair pricing we assume the asset’s price has no up/down trend.
[2] If the spot contract doesn’t have a pair of start/end dates, i.e. a straightforward “cash-on-delivery” instrument, then I think in many cases “fwd price” means ” delayed settlement”.

[3] Their fwd price is Lower —
– High-coupon bonds such as treasury
– High-dividend stocks
– many currency pairs

Why the premium vs discount? There’s arbitrage mathematics at play. For most products, a fwd seller could 1) borrow cash to 2) buy the underlier today, 3) lend it out for the fwd term (say 90 days), and 4) deliver it on the fwd start date as promised. All deals are executed simultaneously today, so all prices fixed together, and a profit if any is locked in.

replicate eq-fwd contract, assuming a single dividend

See also
http://www.matthiasthul.com/joomla/attachments/article/121/ForwardReplication.pdf

Note replication portfolio is always purchased as a bundle, sometime (time t) before expiry (denoted time T).

First, let’s review how to replicate a forward contract in the absence of dividends. The replication portfolio is {long 1 share, short K discount bonds}. To verify, at T the portfolio payout is exactly like long forward. By arbitrage argument, any time before expiry the portfolio value must at all times equal the fwd contract’s price. I will spare you the math formula, since the real key behind the math is the replication and arbitrage.

Now, suppose there’s a percentage dividend D paid out at time Td before T. In this case, let’s assume the dividend rate D is announced in advance. To reduce the abstractness, let’s assume D=2%, K=$100, the stock is IBM. We are going to reinvest the dividend, not use it to offset the purchase price $100. (This strategy helps us price options on IBM.)

The initial replication portfolio now adjusts to –{ long 0.98 IBM, short 100 discount bonds}. At T, the portfolio is exactly like long 1 forward contract. Please verify!

(In practice, dividends are declared as fixed amount like $0.033 per share whatever the stock price, but presumably an analyst could forecast 2%.)

In simple quant models, there’s a further simplification i.e. continuous dividend yield q (like 2% annually). Therefore reinvesting over a period A (like 1Y), 1 share becomes exp(qA) shares, like exp(0.02*1) = 1.0202 shares.

Q: delta of such a fwd contract’s pre-maturity value? Math is simple given a good grip on fwd contract replication.
A: rep portfolio is { +1 S*exp(-qT),     -K bonds }.
A: key concept — the number of shares (not share price) in the portfolio “multiplies” (like rabbits)  at a continuous compound rate of q. Think of q = 0.02.
A: In other words

   F0 = S0*exp(-qT) – K*Z0

Differentiating wrt S0, delta = exp(-qT), which degenerates to 1 when q=0.

##basic steps in vanilla IRS valuation, again

* First build a yield curve using all available live rates. This “family photo” alone should be enough to evaluate any IRS
* Then write down all (eg 20) reset dates aka fixing date.
* Take first reset date and use the yield curve to infer the forward 3M Libor rate for that date.
* Find the difference between that fwd Libor rate and the contractual fixed rate (negotiated on this IRS contract). Could be +/-
* Compute the net cashflow to occur on that fixing/reset date.
* Discount that cashflow to PV. The discounting curve could be OIS or Libor based.
* Write down that amount.

Repeat for the next reset date, until we have an amount for each reset date. Notice all 20 numbers are inferred from the same “family photo”. Tomorrow under a new family photo, we will recalc/reval all 20 numbers.

Add up the 20 amounts to get the net PV in that position. Since the initial value of the position is $0, this net value is also the PnL.

FRA/ED-Futures – discount to fwd settlement date

–Example (from Jeff’s lecture notes)–
Assume on 12 Nov you buy (borrow) a 3×9 FRA struck at 5.5% (paying 5.5%) on 1M notional. On 12 Feb, 6M Libor turns out to be 5.74% , compensation due to you =

$1M x (0.0574-0.055) * 180/360 / (1 + 0.0574*180/360) = $1166.52
——–Notation ——-
Libor fixing date = 12 Feb

“accrual end date” (my terminology) = 12 Aug.

settlement could be either before or (occasionally) after the 6M loan tenor. This example uses (more common) fwd settlement.
disc factor from 12 Aug to 12 Feb = 1/ (1 + 0.0574 * 180/360)
————–
Note the “interest due date” is always end of the 6M accrual period. Since we choose fwd settlement, we discount that cashflow to the fixing date.

annualized interest Rate difference = 5.74 %- 5.5%
pro-rated  interest Rate difference = (0.0574-0.055) * 180/360
difference in interest amount (before discounting) = $1M x (0.0574-0.055) * 180/360. This would be the actual settlement amount if we were to settle after the 6M loan period. Since we choose fwd settlement …

discounting it from 12 Aug to 12 Feb = $1166.52
————
Now we come to the differences between FRA and ED Futures.
1) a simple difference is the accrual basis. ED futures always assumes 90/360 exactly. FRA is act/360.
2) Another simple difference is, ED Futures always uses 3M libor, so our example must be set on Mars where ED futures are 6M-Libor-based.

3) The bigger difference is the discounting to fwd settlement date or fixing date.
– EDF gets away without the PV discounting. It takes Libor rate as upfront interest rate like in Islamic banking. Since Libor turns out to be 5.74% but you “bought” at 5.5%, the difference in interest amount is, under EDF, due immediately, without discounting to present value.
– the payout, or price, is linear with the Libor rate L.
– this is essentially due to daily mark-to-market margin calculation
* FRA takes Libor rate as a traditional loan rate, where interest is due at end of loan period.
** under late settlement, the amount is settled AFTER the 6M, on the proper “interest due date”. (Linear with L)
** under fwd settlement, the amount is settled BEFORE the 6M, but PV-discounted. This leads to a non-linear relationship with libor rate and convexity adjustment.