option delta hedge – local anesthesia#RogerLee…

An intuitive explanation given by Roger, who pointed out that delta hedge insulates you from only small changes in stock (or another underlier).

Say, you use 50 shares to hedge an ATM option position with delta hedge ratio = 50%. Suppose your hedge is not dynamic so you don’t adjust the hedge, until price moves so much the option delta becomes close to 100%. Now a $1 move in your option is offset by $0.50 change in the stock position – insufficient hedge.

The other scenario Roger didn’t mention is, the option becomes deeply OTM, so delta becomes 1%. Now a $0.01 change in your option position is offset by $0.50 of the stock position – overhedged.


option math : thin -> thick -> thin

Many topics I may have to give up for lack of bandwidth. For the rest of the topics, let’s try to grow the “book” from thin to thick then to thin.



+ delta hedging

+ graphs of greeks

+ arbitrage constraints on prices of European calls, puts etc. Intuition to be developed

+ basic strategies like straddle                     


LG American options


LG binary options but ..

+ Roger’s summary on N(d1) and N(d2)

LG div

LG most of the stoch calc math but ..

LG vol surface models

+ some of the IV questions on martingale and BM


intuitive – dynamic delta hedging

Q: Say you are short put in IBM. As underlier falls substantially, should you Buy or Sell the stock to keep perfectly hedged?

As underlier drops, short Put is More like long stock, so your short put is now more “long” IBM, so you should Sell IBM.

Mathematically, your short put is now providing additional positive delta. You need negative delta for balance, so you Sell IBM. Balance means zero net delta or “delta-neutral”

Let’s try a similar example…

Q: Say you are short call. As underlier sinks, should you buy or sell to keep the hedge?
Your initial hedge is long stock.
Now the call position is Less like stock[1], so your short call is now less short, so you should drive towards more short — sell IBM when it sinks.

[1] Visualize the curved hockey stick of a Long call. You move towards the blade.

Hockey stick is one of the most fundamental things to Bear in mind.

compound options, basics

Based on P78 [[DerivativeFinancialProducts]].

Most popular Compound option is a call on a put. (I think vanilla Puts are the overall most popular option in Eq and FX, and embedded Calls are the most popular bond options.) Given a lot of buyers are interested in puts, there’s a natural demand for calls-on-puts.

In a Compound option there are 2 layers of fees (i.e. premiums) and 2 expiry dates —

The front fee is what you pay up front to receive the front option. If you exercise front option, you do so by buying the back option, paying the back fee as a 2nd premium to the dealer. Therefore the back fee is a conditional fee.

At end of front option’s lifespan, the back option’s protection period would start. (Remember in this case the back option is a protective put.)

It’s quite common that at end of the front window, the back option has become unnecessary, or back fee has become too expensive given the now reduced risk.

arb between 2 options with K1, K2

It’s not easy to get intuitive feel about arb inequality involving European put/calls of 2 strikes K1 < K2. No stoch or Black/Scholes required. Just use hockey stick diagram i.e. range-of-possibilities payoff diagram.

Essential rule #1 to internalize — if a (super-replicating) portfolio A has terminal value dominating B, then at any time before maturity, A dominates B. Proof? arbitrage. If at any time A is cheaper, we buy A and sell B and keep the profit. At maturity our long A will adequately cover our short B.

—-Q1: Exactly four assets are available: The bond Z with Z0 = $0.9; and three calls. The underlying is not available for you to trade. The calls have identical expiry T, strike K = 20; 22.5; 25, and time-0 price C0(K), where C0(20) = 6.40; C0(22.5) = 4.00; C0(25) = 1.00. Any arbitrage

Consider the portfolio B {long C(22.5) short C(25)} and A {25 Z – 22.5 Z} = {2.5 Z}

At maturity, A is worth $2.5 since the bond has maturity value $1 by definition.
At maturity, B < $2.5, obvious by hockey stick. This is the part to get intuitive with.

By Rule #1, any time before maturity, A should dominate B. In reality, A0 = 2.5 Z0 but B0 = $3 -> arb.

This trick question has some distractive information!

—-Q2: P141 [[xinfeng]] has a (simple) question — put {K=80} is worth $8 and put {K=90} is worth $9. No dividend. Covered on Slide 30 by Roger Lee. Need to remember this rule — 80/90*P(90) should dominates P(80) at time 0 or any time before expiry, otherwise arb.

Consider portfolio A {80/90 units of the K=90 calls} vs B {1 unit of the K=80 call}

If stock finishes above $80 then B = 0 so A >= B i.e. dominates
If stock finishes below $80, then B = 80-S. Hockey stick shows A = 80 – S*80/90 so A dominates.

Therefore at expiration A dominates and by Rule #1 A should be worth more at time 0. In reality, A0=B0.

implied vol vs forecast-realized-vol

In option pricing, we encounter realized vs implied vol (not to be elaborated here). In market risk (VaR etc), we encounter

past-realized-vol vs forecast-realized-vol. Therefore, we have 3 flavors of vol

PP) past realized vol, for a historical period, such as Year 2012

FF) forecast realized vol, for a start/end date range that's after the reference date or valuation date. This valuation date is

typically today.

II) implied vol, for a start/end date range that's after the reference date or valuation date. This valuation date is typically


PP has a straightforward definition, which is basis of FF/II.

Why FF? To assess VaR of a stock (I didn't say “stock option”) over the next 365 days, we need to estimate variation in the stock

price over that period.

FF calculation (whenever you see a FF number) is based on historical data (incidentally the same data underlying PP), whereas II

calculation (whenever you see a II number like 11%) is based on quotes on options whose remaining TTL is being estimated to show an

(annualized) vol of 11%.

See http://www.core.ucl.ac.be/econometrics/Giot/Papers/IMPLIED3_g.pdf compares FF and II.

premium adjusted delta – basic illustration

http://www.columbia.edu/~mh2078/FX_Quanto.pdf says “When computing your delta it is important to know what currency was used to pay the premium. Returning to the stock analogy, suppose you paid for an IBM call option in IBM stock that you borrowed in the stock-lending market. Then I would inherit a long delta position from the option and a short delta position position from the premium payment in stocks. My overall net delta position will still be long (why?), but less long than it would have been if I had paid for it in dollars.”

Suppose we bought an ATM call, so the option position itself gives us +50 delta and let us “control” 100 shares. Suppose premium costs 8 IBM shares (leverage of 12.5). Net delta would be 50-8=42. Our effective exposure is 42%

The long call gives us positive delta (or “positive exposure”) of 50 shares as underlier moves. However, the short stock position reduces that positive delta by 8 shares, so our portfolio is now slightly “less exposed” to IBM fluctuations.

2nd scenario. Say VOD ATM call costs 44 VOD shares. Net delta = 50 – 44 = 6. As underlier moves, we are pretty much insulated — only 6% exposure. Premium-adjusted delta is significantly reduced after the adjustment.

You may wonder why 2nd scenario’s ATM premium is so high. I guess
* either TTL(i.e. expiration) is too far,
* or implied vol is too high,
* or bid ask spread is too big, perhaps due to market domination/manipulation

y mkt-makers delta hedge — my take again

A1: earn bid/ask spread
A2: Well the same kind of problem as the unhedged long call — insufficient insulation.

Q: why do vol market makers take the trouble to maintain a dynamic hedge?

Say I am long call (IBM for eg). Tiny rises are welcome, but tiny drops hurt pnl. As a MM (i.e. market maker), i need to minimize any negative impact to my pnl, which threatens to destabilize me, and the SS (i.e. sell side) infrastructure.

Therefore on the SS we must immediately delta-hedge our long call, usually by short stock. Net delta = 0. This insulates us from both positive and negative impacts. (But then what’s our profit model? [Q1])

Most of the time during a single day IBM would move relatively slowly and by relatively small magnitude, so our initial delta hedge should suffice, but next day we should probably re-hedge or “dynamic hedging”. Key question — What if we don’t?

Say our initial call delta is +44, so initial hedge short 44 shares and achieve 0 delta. Next day our call delta becomes +55 [1], net delta is +11. (What’s the problem? [Q2]), so we short more stocks.

The nice thing (due to long gamma) is, we Sell IBM when it rises, and Buy when it drops. However, Bid/ask spread [2] and commission hurts the profit. Nevertheless, this is a standard SS strategy — long gamma and delta hedge dynamically.

Doesn’t work if we are short gamma.

[1] Over a longer period such as a day, underlier move is often too big for our delta hedge. (Big foot for small shoe).

[2] Note we PAY the bid/ask spread as market taker. Alternatively, it would be perfect if we hedge-sell/hedge-buy the stock using limit orders, thereby earning the bid/ask spread.

skew bump on a smile curve

Background — in volatility smile (rather than term structure) analysis, we often use a number (say, -2.1212) to measure the skewness of a given smile curve. Skew is one of several parameters in a calibrated formula that determines the exact shape of a given smile curve. Along with anchor volatility, skew is among the most important parameters in a parameterization scheme.

We often want to bump the skew value (say by 0.0001) and see how the smile changes.

A bump in skew would make the smile curve Steeper at the ATM point on the curve. You need to look at both the put (low strike) and call (high strike) sides of the smile curve. If the bump causes put side to move even higher and call side to move even lower, then skewness is further increased. In many cases, skew value of the entire curve is (approximately?) equal to the slope (first derivative) at the ATM point.

If you mistakenly look at only one side of the smile curve, say the put side, you might notice the curve flattening out when skew is bumped. Entire left half may become less steep, while the right half was rather flat to start with. So you may feel both halves become less steep when skew is bumped. That’s misleading!

Note skew is usually negative for equities. A bump is a bump in the magnitude.

ATM definitions in various asset classes

what's atmf exactly?

http://www.sdgm.com/Support/Glossary.aspx?term=ATM%20option says

An ATM is an option where the strike is the same as the current observed underlying which can be the current spot, forward rate or delta neutral straddle. You can have an:

ATMS (at-the-money spot) option, where the strike is the same as the current spot rate.
ATMF (at-the-money forward) option, where the strike is the same at the current forward rate.
ATM delta neutral straddle, where the strike gives a delta neutral straddle [1].

The default ATM in the:
FX market is the delta neutral straddle.
IR market is the forward rate.
CM (commodity) market is the forward rate.
EQ market is the spot price.

[1] net delta = 0. Therefore for small fluctuations in underlier, portfolio MV doesn't change.

flow-vol franchise lowers cost of a structured vol provider

A strong flow vol business Drastically reduces hedging cost of a structured vol deal maker.

A structured vol trader in a sell-side basically constructs structured deals for hedge fund, mutual fund, insurer and other institutional clients. Typically based on client request, but the sell side could presumably market a given instrument to a client on an unsolicited basis too. As a sell-side, this trader always, always need to hedge her exposure. Being a structured vol trader, the biggest exposure isn’t rates or credit but vol. The way to hedge away vol exposure is to enter into positions in options or var swaps — using Flow volatility instruments.

Since the vol exposure in a structured deal can be large in dollar terms, the hedge involves large flow vol trades. These trades tend to entail sizable amount of transaction cost, which is factored into the quote to client. The lower this hedging cost, the more competitive is our quote.

Best way to lower the hedging cost is a strong flow vol business. In the same way, a strong cash equity business reduces hedging cost of an equity volatility trader and makes his bid/ask more competitive.

For a commercial bank, it’s relatively easy to become a structured vol solution-provider. It’s harder to build flow vol business, the traditional strength of investment banks (and broker/dealers?).

In the retail space, there are parallels. Target sells lots of shoes and pays lots of rent, so it wants to make shoes and also buy (rather than rent) commercial property so as to drive down cost and offer more competitive prices to consumers.

long gamma + delta hedging

Long-gamma is one of the most important business models. I believe both buy-side and sell-side like it.

To keep delta-hedging you need to continually buy/sell underlier. There is a transaction cost per trade, so typically you adjust once a day, either buying or selling the udnerlier. Another way to reduce the transaction cost is via limit orders.

If you long a call,

* When stock rises, delta rises, you have too much delta. So you sell stock (high)

* When stock drops, delta drops, you have too little delta. So you buy stock (low)

If you long a put,

* When stock drops, your negative delta balloons, you have too much negative delta, so you Buy stock (low)

* when stock rises, your negative delta shrinks, you have too little negative delta, so you Sell stock (high)

Therefore, long-gamma would buy low and sell high the underlier. Someone said “the larger the underlier moves, the larger the profit”.

Long gamma has other costs — the initial premium, and theta decay. Someone said

“If realized vol loss

If realized vol > implied vol => profit”

I think the author means “implied vol at time of purchase”.

See also http://www.volcube.com/resources/options-articles/gamma-trading-and-option-time-decay/

vol surface needed for risk mgmt

How is vol surface used for risk management? Mostly for vega i.e. sensitivity to implied vol fluctuations.

There are 2 advanced concepts — term-structure vega vs vega maturity, but the basic one is just “vega”, defined as position MV change due to implied vol move.

For an European style position, you don’t need vol surface. Just simulate a bump in sigma and calculate the new MV using BS.

For any non-European style option or option strategy, we assume a parallel shift of entire vol surface.

A more fundamental need for vol surface is marking and PnL. Liquid option positions may not need the vol surface (can use market prices) but most other option positions do. Needless to day, realistic valuation is the basis of risk management.

Nomura FX option IV #Ldn video link

Q: what are the products traded on your desk
%%A: stock/ETF/index options + var swap for flow vol. For EFS, there are a lot – digital options, barrier options, Asian style options, structures with 2 underliers
Q: how do you monitor the barriers – discrete or continuous or …?
Q: what does a typical quant library API function looks like?
Q: You said c++ is more challenging, but what technical challenges do you see using c++ compared to java?
(Now I would say pointer, memory mgmt, STL, undefined behaviors…)
Q: how is a libor yield curve constructed?
Q: what’s the rationale for using swap rates to derive spot rates for a long tenor?
Q: what’s the rationale for using libor futures rates to derive spot rates? 
Q: for risk management, what features of the vol surface are measured/monitored?
%%A: skew bump, tail bump. Wings are known to be problematic.
Q: did you work on PnL explain?
Q: Our trades are often long gamma. What happens to their thetas?

IR exposure of a strucrtured vol dealer desk

I believe many (if not most) structured volatility contracts have some IR exposure. Forgive me for stating the obvious — exposure is felt by both the sell-side and the buy-side, though either side can hedge away that exposure. Let’s focus on the sell-side.

On a structured volatility dealer desk, I believe (confirmed by a veteran) the vol exposure outweighs any IR exposure, since the desk is experienced and set up to trade volatility, not IR. This means that interest rate fluctuations should have less PnL impact than volatility fluctuations.

However, the back office (including IT) must get the IR calc right, which is often non-trivial. Therefore to the back office employees, IR calc could be more complicated than the vol calc. For a given portfolio, I guess the business logic and processing complexity might theoretically be dominated by the IR component rather than the volatility component. This is suggested by anecdotal evidence.

Nomura FX option + java IV

Q: Does your system cover tenor based risk?

Q: Between Equity vol and FX vol systems, do you monitor the same greeks?
%%A: in eq, it’s about delta gamma vega and theta. In FX, I believe these are all important. But I guess rho is probably more important than in eq, since FX is very sensitive to interest rate and there are 2 interest rates in each currency pair.
A: delta/gama/vega/theta are the big 4. rho is insignificant in both eq and FX. See http://bigblog.tanbin.com/2011/06/rho-of-vanilla-fx-option.html

Q: You mentioned that you build thousands of Libor yield curves each day? How do you do that?
%%A: using deposit rates, futures rates, swap rates, and year-end turns

Q: You mentioned EOD marking that feeds into downstream PnL attribution. How do you do that?

What Unit test tools did you use?

What’s a good unit test?

In java how do you compare 2 strings?

Why is java string designed to be immutable ?

convertible bond – more like a call than cash stock or corp bond

Now I know that a vanilla convertible bond as an asset is more like a call [1] rather than a stock or a corporate bond. Some People said a CB is more like a cash stock than a bond, but I don’t think so. CBs are traded by the vol desk, not the cash equity desk or the fixed income desk.

[1] a call option (on the stock). I don’t there is a put-option like convertible bond

A CB has vega, a maturity, theta, delta.

In a typical contract, the indenture specifies a strike in the form of conversion-price or conversion-ratio. For example, a conversion ratio of 45:1 means one bond (with a $1,000 par value) can be exchanged for 45 shares of stock.

var swap daily mark-to-market

Imagine we buy a week-long var swap with variance notional $400K. Since there are 4 price relatives in a week and hence 4 daily realized var (DRVar) numbers, we divide the notional into 4 equal slices. Each day we record the DRVar and compute our “spread” over the strike level (say 30% annualized vol), before computing our gain or loss realized on that day.

If today’s DRVol annualized is -64.8029% our spread would be 0.648*0.648 – 0.09 = 0.33. We actually earn 0.33 * $100k on this day. It’s a realized gain. By the way, the minus sign is ignored !

If today’s DRVol annualized is 12.7507% our spread would be 0.1275*0.1275 – 0.09 = 0.0737. We actually lose 0.0737*$100k. This is realized loss.

This is like a cabby earning a daily wage. Each day’s earning is realized by end of the day. Next day starts on a clean slate. He could also make a daily loss due to rent, fuel, parking, or servicing.

Now imagine 400 slices. It’s possible to have a lot of small losses in the early phase, then some big profit, perhaps due to dividend anouncement. A shrewd market observer may have some insider/insight on these volatility swings — the volatility of volatility.

It’s wrong to project those early losses (or profits) into the remaining “slices”. This important principle deserves a special illustration. Supppose I strike a deal to sell you all the eggs my hen produces — one per day for 100 days, at a total price of $100. If you initially get a lot of small eggs (losses) you may complain and want to cancel at a negotiated price. However, the initial price is actually a reasonable price, because market participants expect bigger eggs later. After 20 days, the 80 upcoming eggs now have a fair market price of $88. First 20 eggs were perhaps priced from the onset at around $11.50-$12.70, though people couldn’t have predicted the exact value of first 20 eggs. Now if you do terminate, you get $88 + 20 eggs. Fair. Your 20 eggs’ value may exceed $12.

( Now imagine a long var swap position with 100 “slices” with 20 realized…..)   The eggs is an analogy of var swap daily market-to-market. It’s similar (slight messier than) to futures mark-to-market. Here “Market” refers to the market expectation of the remaining 80 daily price relatives (all remaining eggs). Remember each of the 80 slices would result in a daily realized PnL contribution. The market participants have some idea about the SUM[2] of these 80 forthcoming DRVariances — 80 eggs.

[2] variance is additive; volatility isn’t.

IV: local volatility by OCBC Bertrand

Q: tell me 1 or 2 low-latency challenges in your projects.
Q: what’s a variance swap?
Q: what’s your favorite messaging vendor product? (MQ is firm standard but too slow for this team.)
Q: Where do you get your implied vol numbers? From exchange or you do your own inversion? (I guess exchanges do publish quotes in vol.)
Q: how many tenors on your vol surface?
Q: how do you model term structure of vol?
Q: how do you perform interpolation when you query the vol surface?
Q: you mentioned various vol models (taylor, cubic etc) but how do you decide which model to use for each name?
Q: describe BS equation in simple language, and what are the inputs?
Q: FX vol is delta sticky. On X-axis you see 25 delta, 10 delta, atm vol etc. How about eq?
Q: any example of structured vol products?
Q: what’s dv01?
Q: which asset class are you most comfortable with?
Q: in your systems there are often multiple languages. How do you make them interoperable?
Q: what’s java generic?
Q: value type vs reference type in c#
%%Q: who will validate the prices and decide if it’s a reasonable or valid price?
A: traders. Even though they aren’t fully trained as quants are. Some managers also have pricing knowledge. Perhaps no real quants.
Q1: what’s fwd vol vs local vol?
Q: you said variance is additive? Can you elaborate?
Q: have you heard of total variance?
Q3: what’s put-call parity?
%%A: C + K = P + F. Each of the 4 can be synthesized using the other 3

Q3b: can you show me one synthetic portfolio, say a synthetic long call?
Q: Can you explain PCP in terms of deltas?

Q: you mentioned code bloat due to c++ templates, so how does c++ deal with that?
Q: have you heard of c++ strong exception guarantees vs basic exception guarantees?
A: http://en.wikipedia.org/wiki/Exception_guarantees

Q: how does java generic differ from c++ and c# — take a hashmap for example?
Q: c# hashmap of integer? Is it special?
Q: you mentioned java hashmap of integer involves autoboxing, so what happens during autoboxing?
Q: What smart pointers have you used?
Q: if i were to use a smart pointer with my legacy class, do I have to modify my class?

Q7: you mentioned java generic was designed for backward compatibility, but why not add a new type HashMap in addition to the old non-generic HashMap class?
%%A: old and new must be interoperable

Q7b: what do you mean by interoperable?

Q: tell me one project with low latency

— some answers revealed —
A1: local vol is designed to explain skew. During the diffusion, instantaneous vol is assumed to be deterministic, and a function of spot price at that “instant” and TTL.

I guess what he means is, after 88 discrete steps of diffusion, the underlier could be at any of 888888 levels (in a semi-continuous context). At each of those levels, the instantaneous vol for the next step is a function of 2 inputs — that level of underlier price and the TTL.

A3: easiest way is “call premium – put premium == fwd price” (cash amount to be added to LHS?)

PnL curve pushed towards hockey stick – 2 "pushers"

The vanilla European Call or Put’s valuation graph (against spot price) is a _smooth_ curve Above the kinky hockey stick. I like to remind myself the curve depicts a “range of possibilities”. The “original” curve is a Snapshot of a bunch of what-if possibilities such as “If underlier becomes $x my put’s theoretical value is $y”.

The curve drops towards the hockey stick as expiration approaches. This sentence is important but too complex too abstract. Let’s be concrete and say we are on the 2nd last day of an IBM put. Let’s say IBM is around $100 now.

To keep things simple, let’s say our option is ATM — Scenario 1. The range of possibilities remains the same because IBM can still become $1 or $1000 or any value in between. Theoretical value of our put in that range of scenarios would be different than the earlier days of the option. IBM = $50 -> put = $51… IBM=$100->put=$11… IBM=$200->put=$1.50. Note, in reality IBM will fluctuate around the spot rather than long-jumping, so we can erase all but the middle section of this new curve. Still a smooth curve. If you compare this smooth curve to the original smooth curve, new curve is physically closer to the hockey stick.

What if our 2nd put is deep OTM with K = $50 (Scenario 2)? Well the hockey stick is now a different hockey stick. (Actually it’s shifted to the left). But again as expiration approaches, the “smooth curve” moves down to the hockey stick.

This downward shift is known as option decay. There are rare exceptions, but vast majority of options lose value over time.

During the decay, the smooth curve presumably becomes more convex, as it morphs into the kinky hockey stick. This means gamma increses????

Now, there’s a 2nd way the smooth curve moves towards (or away from) the hockey stick. When IBM implied vol surface experiences a uniform drop, the smooth curve drops in a similar fashion. More specifically, if the implied vol smile curve for our maturity drops, the smooth curve drops towards the hockey stick. This is still too vague too abstract. Let’s be concrete and say smile curve drops to 0 for our maturity. Our put becomes identical to a short position in IBM if current price is below our strike, or worthless if above strike. That means, our PnL graph against IBM price is the hockey stick itself.

Therefore, drop in sigma_i has a similar effect as option decay. A surge in sigma_i will lift the smooth curve away from the hockey stick. Note all of the graphs we mentioned are plotted by BS — so-called theoretical values.

It’s all common sense after you internalize it.

#1 usage of volatility surface

I’d say end-of-day unrealized PnL is the most IMPORTANT usage. An integral part of it is mark-to-market. (However, For liquid option products with numerous “tight” market quotes, I don’t know if we really need the vol surface for PnL.)

A more “fundamental” need for vol surface is the valuation of non-liquid volatility contracts, including structured, exotic, tailor-made contracts with optionality features. I prefer the words “contract” or “deal” rather than “instrument”, “product”, “security” or “asset”. Contract means there are at 2 counter-parties. If they really do the deal, at contract termination each will end up with a realized PnL, potentially humongous. The estimate, risk-management and analysis of that realized PnL is often the biggest job in a trading desk.

In Equities and FX, vol surface is often the centerpiece (at least part thereof) of the valuation framework for such “contracts”. In a valuation framework, most other factors are simpler compared to the volatility factor.

In a real London structured eq vol desk, such a valuation requires a Monte Carlo simulation which queries one or more vol  surfaces repeatedly. However, i don’t think the valuation need to use the parameters (like skew, tail…). The surface is treated as a black box to query.

The vol surface must be constructed by taking into consideration a variety of observed market data. Therefore a good surface is consistent with a diverse variety of market data, including but not limited to
– dividend forecast,
– tax schedule on dividends,
– calendar convention,
– holiday schedules….

But the most important market data is the premium on the liquid instruments, which typically cover the first few years only. Long-dated instruments are much less liquid.

ATM ^ ATF European calls, briefly

Refer to my simplified BS formula in http://bigblog.tanbin.com/2011/06/my-simplified-form-of-bs.html.

Q: for a ATF European call, where K == S*exp(rt) i.e. struck slightly Above current spot, how would the BS formulas be simplified
d1 = -d2 = 0.5σt  =   \frac{\sigma\sqrt{t}}{2} …………………. (in more visual form)

C(S,t) = S * [ N(d1) – N(-d1) ] = S * [2N(d1)-1] and depends only on sigma scaled Up for 2.5 years (our t)

Q: how about an ATM European call, where S==K?
A: the ATM call (slightly Lower strike than ATF) has more moneyness than the ATF call , because stock will drift past K long before expiry. The diffusion of the stock prices is “centered” around the drift.

binomial trees — flexible and adaptable

Binomial Tree is widely used in investment banks such as GS and Barcap. One of the most popular numeric methods for option pricing. I think the only other common pricing methodology is PDE, but less popular.
Simpler pricers use formula directly. So I’d say binomial tree is the #1 most practical and wide-spread pricing methodology for non-trivial instruments.

All binomial trees I have seen are interlocked. After 54 intervals, there are exactly 55 possible prices. See http://bigblog.tanbin.com/2011/06/option-pricing-recombinant-binomial.html

“Level 55” refers collectively to the 55 tree nodes after 54 intervals. The word “Level” should NOT be used for price-level. “Level” means height of the tree as it grows. Rotate the tree to put the root down to visualize the 55 “Levels”. However, most btrees are drawn left to right because time (X-axis) grows rightward, and price values are naturally arranged along Y-axis.

The interlock/recombinant rule is simplifying, and all intervals are equal on a tree, but many other aspects of the tree are flexible/complex and need to be modeled. It’s these flexibilities that allow the tree to be so versatile and adaptable.

– the probability of an upswing at each node is unique and probably independent. On the 5 nodes at Level 5, there are 5 distinct up-probabilities, and 6 on the next level.
– the magnitude of the upswing at each node is independent. Ditto the downswing. However CRR btree don’t provide this flexibility.

Trinomial tree isn’t popular (perhaps never used) in the finance industry.

binary options, briefly

cash-or-nothing binary option and the asset-or-nothing binary option. CON is like a simple $5 bet on Worldcup Final.

I feel Binary options are important for
* interviews
* Often embedded in structured deals

Greeks (not that important) — Since a binary call is the first derivative (mathematical sense) of a vanilla call with respect to strike, the price of a binary call has the same shape as the delta of a vanilla call, and the delta of a binary call has the same shape as the gamma of a vanilla call

Usually European style

Exchange-traded, and also available OTC

Underlier can be FX, index, ETF or single stocks.

In FX, vanilla and Barrier options (knock…) are more popular than Binary options.

correlation and realized volatility — won’t stay constant

I read these findings sometime ago and now i feel more strongly that the asset correlation theory is rather impractical, unreliable, even misleading. It can give naive users a false sense of security, just like VaR does.

One problem i feel strongly about is that strength of correlation doesn’t last.

(There are many commonalities between vol and correlation as 2 statistical power-tools.) Just as observed volatility[1] changes over time[2], observed correlation between any 2 assets seldom stays stable. Intuitively, 2 assets can be highly correlated now and uncorrelated later[3]. This is an important fact to bear in mind when using correlation numbers to predict anything long-term.

FX rates often show strong correlation with one subset of “drivers” now, and then another subset of drivers later. If you follow the correlation with one particular driver, it rises and falls.

[2] Note vol is often assumed to have a lasting character particular to a given stock. Such an assumption in turn assumes history tends to repeat itself. What’s the pattern that repeats? For realized volatility of stocks/indices/currencies, there’s often an observed pattern that periods of high vol follow periods of low vol. I think a small number of symbols exhibit a consistently high volatility, often for obvious reasons.

The volatility of SPX (i.e. S&P 500) is reflected in the Volatility Index (VIX). You can see how VIX rises and ebbs

[1] using daily closing prices, but how about using hourly prices or monthly closing prices? Stdev/vol may look very different

[3]but probably won’t become anti-correlated. I guess 2 anti-correlated assets can show positive correlation in a credit crisis, when every security loses value relative to hard currencies or commodities.

reflex about option premium at different strikes

Example — Among spx calls, at lower strikes, premium should be ….? Higher. We need to develop reflex.

There's no greek or graph of premiums against strikes. This relation is too simple. But many option newbies lack quick confidence about this “simple” thing. When you reason about option spread strategies, you need this quick confidence.

Think of calls as shopping coupons “get a beer at $1 with this coupon”. The lower the “strike”, the more Valuable is the coupon.

Puts are actually more intuitive. Higher the strike, higher the premium, since buyer can “cash in” more.

misleading PnL graph – short option positions

PnL graph of short option position is misleading. Both call/put show a inverted hockey stick indicating huge potential loss when price move against option writer — when writer gets assigned (at or before expiration). Let's assume the writer wrote an OTM call/put, earning the premium.

Reality is, many option writers don't mind assignment.

A long term bull writes an OTM put, below spot, and doesn't mind buying at that slightly[1] low price, because she wants to buy it anyway and now she gets to buy it at a better/lower price! If you are long term bullish then you want to buy it anyway, so now you just need to hold it for a while. PnL graph shows a huge paper loss, but a long term bull won't liquidate and realize the paper loss. So the PnL graph is misleading.

[1] Usually an OTM put is slightly below spot. Most positions and trades are concentrated near ATM.

———— That's PUT. Now calls ———————

A long term bear writes an OTM call (like a “get a beer for $99 with this coupon”), above spot ($88), and doesn't mind selling $99, unless she is forced to buy it from open market at a sky high price. This unlucky scenario is uncommon because —
* PnL graph shows a huge loss assuming writer has to buy from open market, but many (if not most) OTM call writers are buy-writers, so they already hold the asset. If they end up selling at $99, they don't lose. Remember spot at the time of option sale is $88, so they sold a good price.
* stocks are less likely to rise sharply than drop sharply.

Also, a long term bear wants to sell the stock anyway. Now she gets to sell good price.

Lastly, don't forget the premium income.

how frequently can volatility change?

Remember r-vol is backward-looking while i-vol is forward-looking.

In a top 3 i-bank’s VaR engine, vol is assumed to change once a day. An expert mentioned vol “can change overnight”.

== i-vol based on mid-bid/ask can change by the minute.
== Realized-vol is OFTEN (usually?) measured based on closing prices, in which case it won’t change intra-day. http://en.wikipedia.org/wiki/Volatility_(finance)#Crude_volatility_estimation says “Suppose you notice that a market price index, which has a current value near 10,000, has moved about 100 points a day, on average, for many days. This would constitute a 1% daily movement, up or down.”

barrier (knock) option – key words

Cheaper — insurance than vanilla options
FX — barrier options are more common than in eq
Equities — barrier options are much less common than vanilla options

touch-and-go — is considered a “knock” (or “breach” of barrier). Once out, out forever; Once in, in forever.

knock-price — strike-price can be identical or different from Barrier-price aka knock-price

knock-IN — up-and-IN, down-and-IN
knock-OUT — up-and-OUT, down-and-OUT

in-out parity  — only works for European options without rebate.

valuation@various options: complexity imt…

“imt” = is more than

I only know bonds, FX, futures. I feel their valuation is simpler than options.

As stated in http://bigblog.tanbin.com/2011/01/fi-is-more-complex-due-to-time.html, all derivatives have the added complexity of time to maturity. In the same vein, many derivative positions are held “open” longer than cash positions. While open, sensitivity to a lot of variables must be monitored. This is the essence of risk management. Consequence is grave if you neglect these exposures. Valuation is key.

option valuation depends on many (fairly fast-moving) variables
– underlier price swing
– passage of time – theta
– volatility, which is unknown and must be inferred
** volatility itself is a random variable and has a volatility

Option traders not only monitor option valuation, they must monitor delta (#1 sensitivity) and delta’s sensitivity. Delta is affected by
– underlier price swing. If you have an open position, and its Delta value is 68%, it won’t be valid when underlier drops
– volatility, which is unknown and a guesstimate
– time to maturity. Your 68% will not stay the same tomorrow, even if underlier stays constant

I recently picked up a market maker’s brochure on variance swap. Designed for buy-side clients, not quants.
Even the entry-level concepts involve stdev. Since there’s a time element, all the vol values need adjustment. Even pnl involves sigma squared.

I was told fixed income is also complex in terms of math, but I feel the entry barrier is a bit higher in this space.

intuitive – quick reflex with option-WRITING, again

See also http://bigblog.tanbin.com/2011/04/get-intuitive-with-put-option.html. Imprecisely —
+ writing a call, I guarantee to “give” IBM when my counter-party “calls away” the asset.
+ writing a put, I guarantee to “take in” the dump, unloaded by the counter-party. Put holder has the right to “unload” the asset (IBM share) at a fixed price — a high price perhaps [1]

An in-out intuitive reflex —
– If I write a call, I must give OUT assets when option holder calls IN the asset;
– If I write a put, I must take IN when option holder “throws OUT” the junk

[1] in reality, put buyers usually buy puts at low strikes (OTM) therefore cheaper insurance.

H-Vol vs I-Vol – options #%%jargon

Volatility measures the dispersion / divergence / scatter / spread-out among snapshots of a fluctuating price, over a period. Most intuitive and simplest visualization of the spread-out is a histogram. Whenever I have problem understanding volatility, i go back to histogram.

– Frequency of observations can be high or low, usually daily.
– The fluctuating price can be a stock, Interest Rate, Forex, Index, ETF…
– But the period’s start/end date must be specified otherwise volatility is meaningless.

σh‘s start/end dates are always in the past. σi‘s start date is always today, and end date is typically 30 days later. In other words, σh is backward looking; σi is forward looking. Therefore only σi (not σh) can affect option pricing

σh‘s sample values are real snapshots. σi‘s “sample values” are unknowable. We predict that if we take snapshots over the next 30 days, stdev will be this σi value.

reflexes about the big 5 parameters in option valuation

Introducing the cast

+ val i.e. premium — option valuation, manifested in bid/ask
+ vol — implied vol assuming 30 days to maturity
+ delta — by far the most important sensitivity. All other greeks pale in significance, though vega and gamma are important.
+ strike
+ spot — underlier spot price
– one more … leverage – ratio of spot/val , typically 2 to 200. Measures how expensive an insurance this is. See http://www.tradingblock.com/Learn/public/ShowLearnContent.aspx?PageID=28 and my blog post http://bigblog.tanbin.com/2011/10/option-premium-should-be-low-cost.html Note delta is the first derivative of val against spot, comparable to the inverse of this ratio.

For a given option, these 5+1 variables move in tandem. They are intricately linked but not by a simple math formula. When an experienced trader sees a subset of these numbers, she has a reflex about the level of other numbers i.e. their actual values. This reflex is important. Val, Vol, Delta, Spot are actively monitored.

Unrealized PnL is derived from Val. VaR depends on Delta. Major decisions are made when these “risk” numbers become unacceptable.

In this list i have excluded many well-known parameters (omitted), and introduced a few unsung heroes. I feel this is a real list of important numbers we should master. http://www.ivolatility.com/calc/ lets us see how changes in some numbers come with corresponding changes in others.

Here are some Examples from P24 [[Option Vol Trading]] –

Val = $6.50
Spot = $77
Strike = $75 ITM
Delta ~= 60%
leverage = 77/6.5 = 12 times
Vol ~= 52% according to http://www.ivolatility.com/calc/ assuming 48 days to maturity

Val = $6.50
Spot = $24
Strike = $20 ITM
Delta ~= 75%
leverage = 24/6.5 = 4 times
Vol ~= 103% assuming 84 days to maturity

binary^European options – parity relation

I confirmed with my quant friend – the European call valuation formula by BS consists of 2 terms like “Term1 – Term2”

C = S*N(d1) – K*(e^-rt) N(d2)

Term1 is the valuation of an asset-or-nothing binary option.
Term2 is the valuation of a   cash-or-nothing binary option.

In other words, C = AON  –  CON

Now, before maturity, does the above equality hold?

N(d2) is the probability of the option finishing ITM, which is “easy to derive”

put spread – a frequently useful option strategy

(Based on a Barron's 6/20/2011 article) If you anticipate a stock (or ETF) depreciation in a few months, you can sell an OTM call, then use the /proceeds/ to finance a “put spread”.

Eg: iShares Trust MSCI EAFE Index Fund (Ticker EFA). In June, the EFT was 58.
– sell Jul 60 call for a premium
– buy Sep 56 put, which costs more than the other put, and need financing from the call's proceeds
– sell Sep 52 put

Note, as usual, all the calls and puts are OTM — in the spirit of term insurance, where premium should be low-cost to be efficient.

options, swaps and futures: 3 drv’s !! equally important to FX

options — eq, FX. Many bonds have embedded call/put options[1]
swaps — FI
futures/forwards — all

Why did the market evolve this way? Highly educational but I'm not knowledgeable.

(Commodities? slightly less “relevant” to an IT guy since there are few IT jobs in commodity trading. Partly because there's not much data or automation.)

[1] In fact an IT team was dedicated to refunding analysis i.e. how to price a proposal to an issuer to recall a bond and reissue at a lower coupon. A sizable IT team was tasked with creating an instrument of puttable floating notes. Tender Option Bond is another puttable bond.

buy OTM call, and sell OTM put

When GLD (Select sector Gold SPDR, an ETF on gold) was trading at $142.05, a veteran recommended buying June $150 calls and sell June $135 put to “position” GLD to exceed 150 by June.

This is known as Risk-Reversal. “It lowers the price of a bullish position by selling a bearish PUT.”

(As in all recommendations, the options to buy or sell are all OTM.)

It's relatively easy to plot the { pnl vs underlier terminal price }. I mean “pnl” — Generally I avoid { portfolio MV vs underlier expiration price } curve since it looks silly — a fairytale no-loss-always-profitable strategy.

Comparable to — a long call
Comparable to — a long underlier

Motivation and advantage of this trade — cash commitment (paid out of pocket) is lower than buying a call, thanks to PUT premium income.

3 types of ROI from owning a stock

“Some share holders of certain stocks (say XYZ) are now writing (i.e. selling) OTM call options to get cash income”. Classic buy-write (covered call) strategy, as illustrated on Barron’s.

These investors aren’t concerned about limiting potential upside. They want the cash income in the form of premium. If they are greedy, they could write slightly OTM calls.

The XYZ stock is probably paying insufficient dividend, if you ask such an investor.

This strategy is popular when implied volatility rises, i.e. extrinsic value rises (OTM valuation is purely extrinsic).

Now I see that buying a stock can provide 3 types of returns
1) capital gain
2) dividend
3) premium income by selling call options

buy a PUT to protect long position – simple examples

Suppose you bought IBM at $100 but it is $90. You fear it may drop below $80. You can buy a PUT option at $80. It's ITM so premium is at least (spot – strike), perhaps $12/share or $1200/contract. Any premium you pay is likely to go down the drain.

Now suppose you have a lot of USD. You may need a lot of SGD in x months. USD has dropped to SGD1.23, but you fear it may drop to $1.20. You may buy a PUT like the above. The put allows you to convert USD1 to SGD1.20.

what greek measures depth of moneyness@@

Q1: is there a greek that numerically measures moneyness that is, how deep ITM/OTM an option position is? 

There’s a key difference between the 1) probability of an option expiring ITM vs a 2) “weighted average”. The latter has many incarnations like
– expected PnL.
– area under the histogram weighted by PnL of each “column”
– area under the lognormal cuve waited by expiration payoff

Which of the 1) and 2) measurements do you want? I like 1.

A1: I feel delta is an approximate likelihood of an option expiring ITM. http://en.wikipedia.org/wiki/Moneyness confirms that delta is a good approximation of the probability of expiring ITM.

A1: deep ITM option would have a premium close to its intrinsicVal. A deep OTM option would have a low premium. TimeVal is close to 0 for deep ITM/OTM. Therefore,

A1b: {{ option price – intrinsicVal }} is another reasonable measurement of moneyness but not by the probability definition.

ceiling price of an option?

Ceiling price of an option? I don't think there's such a thing but there's an important floor price.

Q: any option (Am/Eu, call/put) has a floor/minimum value tied to the underlier spot price as the 2 prices move. But no max value tied to underlier spot price. Why?
A: any option *premium* quoted is always (intrinsicVal + timeVal). IntrinsicVal is tied to underlier spot price. Therefore option's minimum value is the intrinsicVal. TimeVal is a function of volatility and can be very high.

Note timeVal is each buyer's/seller's judgement, whereas intrinsicVal is converted, like temperature, from underlier spot price.

Q: how does that floor depend on ITM/OTM?
A: ITM has intrinsicVal. That “floor” is nothing but the instrinsicVal
A: OTM has $0 intrinsicVal, so “floor” = $0. The valuation and premium is pure timeVal

y no early exercise of American option (my2011writing

Suppose you have a microsoft American-style call, expiring end of next month, K = $20. You believe microsoft will drop to $19. Let me convince you that you should hold to expiration and never exercise the call.

(Assumptions: no dividend; short selling is nearly interest-free)

Strategy 1 (naive): exercise the call now and immediately sell the 100 shares for $2467. Hurry before it drops! Realized profit of $467. You get the intrinsic value and give up the time value.

Strategy 2 (slightly better): sell the call option immediately. You get (intrinsicValue + timeValue). This realized profit definitely exceeds intrinsicValue of $467. Why? Before expiration, a call is worth at least (underlyingSpotPrice – strikePrice) or (S – K) i.e. the intrinsic value. Remember in-the-money call option always sells slightly [1] above (S – K). Therefore it’s always always always better to sell the option rather than exercise it then sell the shares. But should you sell it *now* because you feel Microsoft will drop to $19 by the expiration date?

Strategy 4 (recommended): short sell 100 shares at $24.67, and keep the call to limit potential shortfall.
If indeed $19 on expiration day, short earns $567. call expires worthless. At expiration,
If S@T = $20.01, profit = $466 (short) + (call) $1 = $467 realized at T i.e. expiration.
if S@T = $24.67 same as today, then your short-sell breaks even, but the American call earns $467.

Now let’s try another perspective — forget about expiration scenarios. Look at price movements after your short sell. As of today, when spot = $24.67, unrealized profit = ($0 from the short + value of the call). intrinsicVal + timeVal is above intrinsicVal of (S – K = $467).

– Tomorrow, if MSFT edges above $20 (ie K), the short position has an unrealized profit just below $467, but together with the call, total unrealized profit will exceed $467.
short’s unrealized profit might be $466 (assuming S = 20.01)
intrinsicVal = $1
timeVal = some positive value
– Tomorrow, if MSFT is below $20,  the short alone would generate an unrealized profit exceeding $467.

==> Therefore, the short position + the call is better than $467 cash. Therefore, you should never exercise that American option (under above ASSUMPTIONS). Therefore the American call option and European version have identical values.

Strategy 3 (Reckless): sell the option and short the stock. You lose the protection from the option. If MSFT rises after your short sell, you would need the option to cover your losses. (Under those opening ASSUMPTIONS) Don’t exercise and don’t sell due to your view on the stock. I guess you should sell the option if you feel vol is going to drop.

[1] if volatility is assumed 0, then the gap (ie time value) would be 0

Update — I always feel compared to an European option, an American option has an element of  timing, surprise and flexibility — when the market condition is right, the owner should cash in. Now I feel there is indeed a time to sell — when implied volatility is higher than reasonable, then you should sell the option, but not exercise it. However this applies to European options too.

OTM put — on left or right of a graph against underlier

It’s well known that the implied volatility smile curve is skewed for stock options. The implied vol is Higher on the far Left than far right on a smile curve. The Left derive from OTM Put quotes from the market, whereas the Right is derived from OTM Calls.

In other words, on Low strikes implied sigmas are much higher, and derive from OTM Puts.

On a PnL graph however, you may see a different pattern. For example Look at a simple put PnL graph. OTM is on the Right — At sky high prices, our put is worthless and deep OTM.

So OTM put is on left or right?

Well, what’s the horizontal axis?
– it’s underlier prices on the PnL graph, where strike is a fixed part of the put contract we are analyzing.
– it’s strikes on the smile curve, where current spot level is (roughly) the lowest point on the smile. The current spot doesn’t move on this smile curve. The smile curve is a snapshot of the option quotes and implied sigmas.

Black-Scholes: flawless for volatility inversion

Inverting a given option premium to an implied vol by BS is uncontroversial and unaffected by volatility skew/smile. If risk-free rate, spot price, time to expiry (aka TTL) are all /unanimously/ observed, then according to BS equation there’s one-to-one mapping between option premium and implied vol. It’s like converting kilograms to pounds. Therefore exchanges often quote premiums in vol, forcing everyone to use BS to back out the dollar values.

That doesn’t mean we agree to all the BS assumptions, including the constant-vol assumption.

BS is perfectly fine for inversion + … perhaps … greek calculations….

We can safely use the original BS equation for inversion, and then completely discard the BS model after that. We can use a different model (usually related to BS) to describe or model the dataset of implied vol numbers. Some common models include

local-vol model
SABR model
(I guess both are stochastic vol models??)

intuitive – with PUT option, 1st look

When I read financial articles, I find PUT options harder to understand than most derivatives. Here’s my summary

–> You use a put as insurance, when you worry that underlying might fall.
–> A put insurance let’s you unload your worthless asset and cash in a reasonably high strike price

Here’s a longer version

–> You use a put insurance (on an underlying) at price $100 when you think that underlying might fall below $100. This insurance lets you “unload” your asset and cash in $100. Note most put or calls traded are OTM.

A good thing about this simplified intuitive definition is, current underlying price doesn’t matter.Specifically, it doesn’t matter whether current underlying price is below or above strike (ie in the money or out).

Q: both a short position (in the underlying) and a put holder benefits from the fall. Any difference?
A: I feel if listed puts are available, then they are preferable to holding a short position. Probably cheaper and don’t tie up lots of cash.

Q: how about the put writer?
A: (perhaps not part of the “intuitive” first lesson on puts)
A: sound byte — an “insurer” .
A: therefore they don’t want volatility.  They want underlying price to stay high or at least be stable

Simplest underlying is a stock.

fxoption vs eq-option popularity among non-retail traders

I spoke to a market data vendor’s presales. Let’s just say it’s a lady named AA.

Without referring to the Singapore market, she feels FXO is clearly more popular as a hedging tool than EqOptions. I feel that’s true among her clients (all institutional, no retail). She explains that only large equity funds would use eqo while virtually all importer/exporters would buy fxo (usually from banks). I asked “In that case how do the equity traders hedge their risk if Not with options?” She didn’t give a complete answer but cited eq index and futures.

I too feel import/export corporates outnumber equity trading houses (perhaps by a large margin), but I feel eqo is more liquid (thanks to exchanges) and more widespread than fxo. Also, eqo has retail demand.

Our conversation about fxo vs eqo was exclusively focused on the hedging usage. Eqo has other users including traders. I was told some hedge funds also trade fxo, but I feel it’s less popular due to exchanges and bid/ask spread.

She felt FXO must be on the books of every corporate (treasury). I asked why. In terms of FX risk hedging, she feels option is the true hedge, whereas fwd is a view on the market. I guess there’s some deeper meaning in her remark. Perhaps she means option is an insurance.

major valuation techniques for volatility instrument

1) formula-based
2) binary tree — fairly simple
3) solve partial differential equations — use numerical methods to “search” for a numeric solution(s) to the equations.

These are the major techniques used against familiar/everyday volatility instruments. For complex/exotic contracts, you need simulation.

Simulation may be justified even for RFQ valuation — Request/trade volume is low, so people spend hours to price a deal.

option valuation is never $0 (due to time-value and vol)

basic question: look at the option-payoff/asset-price curve of a call option. It’s flat till X then it rises linearly with asset price. But in-the-money and out-of-the-money options both have some fluctuating value. timeValue is alwyas positive, even with $0 intrinsicVal (out of the money). Why?

A: an out-of-the-money call has $0 valuation only on the expiry date or 0 volatility. Before expiry, it has a non-zero chance (due to vol) to become in-the-money. It has a non-zero valuation.

Imagine someone selling a deep out-of-the-money eurodollar call optoin (op on futures) with K=5000 bps/yr. K is too high, but what if the option premium is $0 ie free? I would “buy” it since there’s some chance eurodollar futures contract price (expressed in bps) may escalate to that level.

price of futures^options; demystifying MV^PnL

Commodity future is the simplest example.

If a 2022 futures contract is traded today at a price of $1k, it’s not a present market-value, and it’s not an amount to be transferred from the buyer to the seller TODAY. The $1k is the amount to be transferred in 2022 — scheduled payment.

Unlike stocks and bonds, Futures prices are not pinned to current market-value, but are actually forward prices, or scheduled-payments. However, the scheduled payment affects cash flow immediately – end of today. Introducing margin mark-to-market rule of the clearing house! This rule converts[1] the scheduled payment due 2022 into a cash transfer EOD _today_ between the long and the short accounts.

[1] using discount factor 1 !

Q: So what’s the current valuation of a long futures position?
%%A: wrong terminology
%%A: a security has a valuation. Based on the valuation, a position has an unrealized-PnL
%%A: PnL = current MV (ie mid-market price) – trade price.

Option current prices are pinned to current market value, which is not same as the payoff on expiry. If an option is traded at $1k, this price is computed from a formula using expected payoff.

Note a holder of a Call/Put option owns a never-negative valued security. A short seller of any option “owns” a security of always-negative value. If you understand that option is an insurance, then Insurance company can only pay (excluding the one-time premium) and the insured can only receive.

how bell+hockey stick move when vol drops or expiry nears

Q: how the bell and the hockey stick move when vol drops or expiration approaches?

Not complicated, but we need to Develop quick intuitions about these graphs. I feel these graphs are the keys to the mathematics.

Anyway, here are My Answers —

The Lognormal bell [1] Tightens as ttl [2] drops, assuming
– constant annualized vol therefore
– falling stdev (i.e. vol scaled-down for the shrinking TTL).

“Lognormal bell tightening” indicates lower stdev. stdev is basically the “scaled-down” vol.

Also, when ttl drops, the PnL graph drops towards the hockey stick. The hockey stick means 0 vol OR 0 ttl.

[1] the bell shape is skewed because lognormal isn’t symmetrical.
[2] TimeToLive, aka time to maturity or time to expiration.

what exactly is traded? the RIGHT to demand…cash or asset?

A tradable bond is a transferable “right to demand coupon+principal payments”. Was a paper certificate.
-} An MBS is a transferable “right to receive part of monthly installment by borrower”

A stock is the most intuitive tradable instrument as it represents part ownership of a company. Was a paper certificate.

A tradable call option is a transferable “right to buy the underlying”. However, when the underlying is a bond, concept is less intuitive.

A tradable put option is a transferable “right to cash in the underlying instrument at a preset sky-high price”. This is less intuitive. A right to sell IBM at $999,000 a share is valuable provided counter-party guarantees to buy it from us.

A basic forward contract is probably not tradable or transferable as it’s a bilateral agreement. Is a swap tradable? I don’t think so.

A copper futures contract is a transferable “right to demand delivery at the specified date, at the trade price[1]” but I think this is like a “call future”. How about a put …..? No need to go into that for the purpose of the post. Here we just want to get to the bottom of what’s traded in a copper futures contract. Now we know.

[1] which is NOT specified in the contract. A futures contract specifies everything but the price.

straddle – basics

* put/call simultaneously
* Buy only — for long straddle [1]
** max loss is the 2 premiums we pay

* payoff looks like a “V”
** losing if too quiet
** Profitable in both extremes
*** obviously long vega

Straddle is the most basic among all “strategies”. Quite common.

[1] short straddle is also common

option valuation graph – hovers above hockey stick

The vanilla European Call (or Put’s) valuation graph against spot price is a _smooth_ curve slightly above the kinky hockey stick. I like to remind myself the curve depicts a “range of possibilities”. The curve is a Snapshot of a bunch of what-if possibilities such as “If underlier becomes $S my call’s theoretical value is $C”.

Q: since S and C axes are both in dollars, is this graph drawn in scale? Can I take a point on the smooth curve and read actual (x=100,y=20) coordinate values and interpret them as S=$100, C=$20?
A: yes you can. In fact the slope of the hockey stick handle is  45 degrees (representing a delta of 100% when sigma is 0.)

Let’s assume strike is $100.

For OTM, C might be $7 when S is $50 (below strike). C is a small fraction of S. In other words S/C represents a high leverage ratio for an OTM call (or put). C is purely time-value, since intrinsic value is $0. Let me repeat, for OTM call (or put), leverage ratio is high because valuation is purely time-value.

The magnitude of the leverage ratio (and the valuation of the call/put) depends fundamentally on the contest between ln(S/K) vs σ√ t , as described elsewhere in this blog.

For ITM, leverage ratio is much lower and probably less meaningful. When S is $180, C is dominated by intrinsic value of S-K = $80. In other words, the smooth curve hovers slightly above hockey stick handle

option rule – delta converges to 50/50 with increasing vol

Better develop reflexes — Across all maturities, all ITM/OTM options’ delta would converge towards 50% when perceived and implied volatility intensifies. Option premium rises.

50 delta means ATM.

50 delta also means no-prediction about my option finishing ITM or OTM. When vol spikes, it becomes harder for “gamblers” to assess any given strike — will it finish ITM or OTM?

Let’s use a put for illustration. When underlier becomes very volatile,
– a previously deep OTM (hopeless) suddenly looks a useful insurance protection. eg – A ultra-low-strike put.
– a previously deep ITM (sure-win) suddenly looks “unsafe” — may finish worthless.

Rule) At expiry, underlier volatility doesn’t bother us and is treated as 0
Rule) At expiry, option delta == either 0 or 1/-1 never something else. Fully diverged
Rule) In general, 0 implied volatility means all options’ deltas == either 0 or 100%
Rule) Similarly low implied volatility means all options’ deltas are close to the 2 extremes.

vol skew and thick tail — bit of insight

(a beginner’s understanding.)

As observed on any equity vol surface, implied volatilities almost always increase with Decreasing strike – that is, OTM puts (dominating low strikes) trade at higher implied volatilities than OTM calls (dominating high strikes).

In theory, The naive constant-vol model [1] predicts the vol smile curve flattens to a flat line. Lognormal distribution assumes a stock is equally likely to gain 25% or drop 20% (see separate blog), but for a real stock, -20% is more likely. Real world stocks are more “panicky”.

[1] exact def? not sure. It could possibly mean “const local vol”. But i guess it means “at any moment in time, fair valuations (and bid/ask) of a chain of options should reflect the same implied volatility” — BS assumption.  When market players perceive a vol hike, it has to be a parallel shift across strikes, and all options on the chain should rise exactly those amounts to reflect the same but /heightened/ vol.

Skew steepens when markets decline — observation over the past decades (probably since 1987 crash). When markets crash, stocks can drop more than constant-vol suggests. Downside risk is higher than predicted. Option writers are insurers. Insurers know “downside” is bigger so they charge higher premiums. Buyers are actually willing to pay higher because they too know downside is more likely.

Therefore implied vol for low strike Puts are higher than ATM puts. Note bid/ask premium is obviously lower than near-the-money Puts — due to arbitrage.

If you plot log(daily close price relative) in a _histogram_, constant-vol predicts a bell histogram, but I believe a real histogram actually shows a thick tail.

options and futures – related

* options is about future prices. Closely related to futures contracts, futures pricing, futures trading.

* When CBOT was inventing (commodity) options, their options contracts were constructed around futures contracts

* In commodities biz, futures contracts have a right to exist and an economic value, for producers and manufacturers (buyers). options? questionable.

– A futures contract is a contract. One side must delivery; the other must take the delivery.
– An option is an obligation, a one-sided contract. The writer of the option has an obligation to provide (C option) or pay for (P option) the security upon request. Some say an option is not an obligation, but i disagree.

y delta overshadows other greeks

If your own money is at stake, which risk factor do you worry most? Which variable do you watch most? I bet #2 is theta, not vega, but #1 is delta/gamma i.e. how underlier moves.

One reason is, underlier changes much faster.
– Implied vol doesn’t change by the minute. I think people re-calibrate vol surfaces as frequently as every few minutes. However, imp vol is a guesstimate “soft” market data, with a lot of guessing.
– realized vol changes usually by the day.
– TTL doesn’t change by the hour. We typically treat it as changing by the day.

In contrast, Underlier spot price changes by the second, and is Directly observable. Therefore, among all of the input variables to option valuation models, spot price changes fastest.

Note spot isn’t always the most significant input in every pricing formula. Volatility and TTL (when expiring) are sometimes more significant.

Another reason — hedging. An option/swap often exists along with a position in underlier. In that case it’s paramount to analize the nett impact of an underlier move. No such thing with the other greeks.

pace of change — for TTL pace is constant, whereas pace of change in S can be very fast in a crash. Sigma can change fast, but not as fast as S.

I feel theta matters mostly in the last few months. “Decay” is a slow process until the dying days.

wrong way to look at option payoff graph

The at-expiration pnl graph looks stupid for a short call position i.e. when you write a naked call. The diagram shows limited reward but unlimited risk. Covered call is better but still limited-reward, almost-unlimited-risk. [1]

Why would anyone write a call? To answer that you had better compare the { PnL vs at-expiration underlier price} graph of short call vs a comparable strategy.
– Naked call-writing is comparable to a outright short position in the underlier.
– Covered call-writing is comparable to a outright long position in the underlier.

You will see that in many scenarios, the call writer profits more at expiration. If you have a view on the stock and its volatility, then you can take calculated risks writing a call.

[1] “Almost unlimited risk” basically means your position MV could drop to $0 meaning you lose all the money invested, which is a common downside. You get this “almost unlimited risk” whenever you buy a stock. What makes the graphs stupid is the “limited reward”.

In fact “unlimited risk” is common when you short anything.

equity drv, from an OTC options perspective

* listed options (standardized contracts) are guaranteed by the exchange. Exchange is the counterparty of every trade. You don’t need to know who the real counterparty is. Like futures, For every long position out there, there’s a short position for the same contract. Exchange takes no outstanding position. No default risk. OTC options have credit risk, so credit needs checking. Collateral is required.
* OTC option is custom made, rather than standardized. If you want a specific option (some strike price, expiry…) you ask the banks. They each spend some effort (won’t waste time if your order is small) to give you a quote.
* You might sell the option you bought. No restriction for listed. For OTC, your counterparty may disallow it. Remember it’s a bespoke bilateral contract.
* Are there more calls than puts? Possible.
** For every long call, there’s a short call.
** For every long put, there’s a short put.
* eq deriv instruments include exchange-listed options, OTC options (ie buyer and seller both come to a broker/dealer to trade????), index futures, futures on individual stocks, swaps, but not convertibles.
** eq cash and eq deriv are separate trading desks.
* OTC option trading has no low-latency requirement since there’s no exchange involved.
* Valuation (rocket science) is a shared module of option pricing and market risk.
* before an eq deriv trader places a trade, system lets her assess the aggregate risk exposure of the entire firm. Just like prime brokerage risk systems (GSS), EqDeriv Risk measures risk to the bank, not risk to the client. I think this means the bank is risking its own money, probably as a dealer not a broker.
* Futures contracts are obligations; options are one-sided obligations. How were financial futures invented? As hedging tools just like commodity futures. If your (short/long) position (in stocks or other assets) shows correlation to a futures, then the futures can reduce your risk, but also will limit profit potential. You can also use futures to lock in profit.

simplified ATM call/put valuation(mental arithmetic

Develop a mental valuation for ATM European call/put assuming 0 dividend/interest rate with 1 year TTL. (Note put and call have identical valuation in this context.)

if σai (annualized implied volatility) = 100%, then valuation/spot = 1/ =~ 0.4
if σai = 50%, then valuation/spot = 0.5 /   =~= 0.2. In other words, “leverage” is proportional to σai.

More generally, leverage is proportional to maturity-adjusted volatility. For example, 100 vol but 3 mth TTL, leverage is same as 50 vol 12 month.

Now the quick mental calc —
$1 spot = strike, 100% σai, 1 year, val = 0.4. This is the reference level.
50% σai, 1 year, val = reference * 50% = 0.2
100% σai, 3 month, val = reference * 3/12 = 0.2
80% σai, 3 month, val = reference * 0.8 * 3/12 = 0.16. (80% σai =~ 5% daily vol)

synthetic put – in real world Put valuation

http://25yearsofprogramming.com/blog/20070412.htm says

To use the Black-Scholes formula for put options you must incorporate a routine to calculate the price of an equivalent “synthetic put”, which involves

    selling the stock short and buying a call option on it.

Because this can be less expensive than buying a put option, the prices of real puts are driven DOWN to the values of their equivalent synthetic puts. Without the synthetic put calculation, the Black-Scholes formula produces a higher theoretical valuation and makes all traded puts appear undervalued.

long call butterfly – basics

* all-call — I don’t know other flies, but you can have an all-call fly or (equally popular) an all-put fly.
** in FXO, not so simple

* equidistant — among the 3 strikes.
** In FXO, equidistant in terms of delta. Eg 25 delta fly, where both “outside” options have delta values 0.25 away from the center

* size 1:2:1

* ATM – in the middle.

* wizard’s hat — the long call fly looks more like that than a butterfly.
Buy 1 ITM Call + Sell 2 ATM Call + Buy 1 OTM Call — long call butterfly — most common.
(sell the body buy the wings)

* less common in eqd — short call butterfly, short PUT butterfly, long PUT butterfly

Q: For a long call butterfly, the 4 premiums always add up to a net Cost. If underlier stays unchanged till expiry, is the payoff intuitive?
A: not intuitive. Only the low-strike long call makes money and more than covers your premium net cost. Other 3 expire worthless.

Actually the 2 “extreme” outcomes are more intuitive —
$ all 4 expires OTM — negative payoff == initial net cost
$ all 4 expires ITM — You pay initial premium, and leave the rest to clearinghouse, who gives the 2 lots you acquired (Wings) to your (Body) buyers.

In both cases there’s no more cash flow after the initial premium payment.

Q: How do you know if a butterfly strategy is long or short?
%%A: i don’t think you can know.

Q: in FXO, why is the standard strangle also called a fly?

compared to calls, put are more like insurance (barrier option

Compared to a call, a put is more like traditional insurance. (Barrier option is a cheaper insurance than vanilla options. Down-and-in European Put is a common barrier option.)

Having an ITM call is like a shopping _coupon_ ” buy a beer at $1 with this coupon “.
* you can resell the coupon or use as gift voucher

An OTM call (more common than ITM) is like an above-market airticket offer for those affluent travelers who need last minute booking.

For puts, Commodity puts are more intuitive than equity puts. FX options and swaptions are less intuitive. Here’s an illustration —

Suppose a farmer receives guarantee from Walmart to buy his 2019 produce at a pre-set fairly low price whenever he wants within the harvest season.

  • * OTM — pre-set price is low because Walmart is providing a “distress assistance” price
  • * farmer has to pay a “membership” for this guarantee
  • * membership can be traded
  • * put writer is typically a wholesaler like Walmart who needs to regularly source the product in bulk.
    • My friend Trevor Tang is also a put writer, who doesn’t mind receiving the underlyer

Commodity option is the most intuitive. When you write a put on wheat, you advertise to take IN wheat that’s PUT OUT by the option holder. Incidentally, the strike price is typically below the current underlier price (OTM), but the take-in-PUT-out actions happen when underlier drops below the strike.

ITM call/put are more intuitive, but OTM are more important in practice.

buy-write vs short put #my take

Put-call parity equates buy-write to selling puts (naked puts), but there are many differences glossed over.

Background — Buy-write means buying underlier and selling calls. Put-call parity shows the resulting expiration PnL graph resembles a short put position. All the trades reference the same amount of underlier asset, and identical strikes. Note the hockey stick PnL graph is an Expiration snapshot of a RangeOfPossibilities — see other blog posts.

Diff — many (at least for retail investors) BW happens on option exchanges, which are usually American style so no PCP. Notable exception — a few listed index options are European style.

Diff — cash outlay — usually 10 times or higher for BW, because you must pay (at least half) the full cash price for underlier. To short sell a put you probably can get away with lower margin. See p89 [[the math of option trading]] for real examples.

Diff — moneyness — typical BW uses an OTM calls (though ITM is common too). The corresponding (same strike) short puts are ITM.

Diff — recurring income — to the BW trader, but only if she’s consistently lucky that her calls sold always expires OTM.

Diff — BW can make you lose the best stocks in your portfolio if you do that habitually. No such side effect with short puts, because once you get burned you stop playing with fire.

Diff — after expiry — you end up with no position and no risk if using naked puts. With BW, you may still hold large risk, large profit potential and large exposure.

Diff — To execute a buy-write, you would need to overcome 2 bid-ask spreads.
Diff — To execute a buy-write, you would pay 2 commissions.

a view on a particular option instrument

When we see 2 quotes for the same IBM call option by 2 sellers (say UBS vs Citi), these are essentially 2 “views” on IBM volatility. The UBS view may assume a low volatility from now till expiration, but …

Q: does the UBS trader express a view on where underlying price will move? (Note “no-move” is also a view.)
My A: I don’t think so. I feel the entire view is expressed (and encapsulated) in the implied volatility.

Q: If the Citi quote is very, very high, then can we conclude either Citi trader feels IBM will rise, or Citi assumes a high volatility???

Q: From a slightly different angle, what if Citi trader feels IBM will rise over the next month? How does that affect his pricing?
%%A: I feel he can simply use a higher volatility estimate. or he can delta hedge.

Whenever discussing a VIEW like “what if traders feel …”, we can’t avoid comparing multiple strategies. Each strategy has a “risk/reward profile” (what I call RRP). In eq option world, a strategy is basically one or more positions you enter, whose combined portfolio MV is hopefully weather-proof. The layman’s obvious and simple strategy can be smart or naive.  (ignoring spread strategies,) Basic, common strategies include

– buy underlier
– short underlier – risk? unlimited
– buy call, ITM or OTM – risk? low
– buy put, ITM or OTM – risk? low
– [7] (naked call) write a call, OTM – risk? unlimited
– [7] (naked call) write a call, ITM – risk? unlimited and bigger
– (covered call) buy underlier + write call of identical qty
– [a] write an IMT put – risk unlimited (loosely)
– [8] write an OTM put – risk unlimited (loosely)
– [1] buy underlier using a limit order “buy when it drops to a value price of $0.01”
– [3] (protective put) – buy underlier + buy an OTM put of the same qty

[3] the long underlier position has big downside risk. The put protects it, like a low-cost insurance.
[7] naked call is related to a short in underlier. When I write a naked call, i agree to go short in underlier at, say $44.
[8] when I write a put, I agree to buy into a long position
[8] When I write a put at $999, someone may unload his rotten corn (something worthless) on me and I end up buying the corn at $999. I end up with a massive losing long position.
[8] is comparable to [1], more comparable to limit order BUY, but more sophisticated.
[a] I don’t know exactly why anyone would take this risk, but premium would be intrinsicVal + timeVal and a pretty high insurance premium

convert between i-volatility & option price (no real eg

In bond markets, bids/offers are often quoted in bps of yield. Example – Seller and buyer each convert 200bps yield into price and always get the same dollar price. It’s like converting between meter and inch, or between Celsius and Fahrenheit.

The conversion between hazard rate and market quotes is less clear.

Similarly, in many option markets, prices are quoted in implied volatility. Seller and buyer each convert 20%/year[1] to price, using Black-Scholes and always get the same dollar price.

Price/vol conversion is similar to price/yield conversion because …… in each case price is a poor parameter for relative value analysis, and investors found a superior apparatus to compare 2 securities — yield for bonds and i-volatility for options. Wikipedia says “Often, the implied volatility of an option is a more useful measure of the option’s relative value than its price.”

Price/vol conversion is similar to price/yield conversion because ….. from (mid-point of quoted) bid/ask prices, you get an implied-vol or implied-yield[3], over the remaining lifetime of the instrument. The vol and yield are both forecasts, verifiable at end of that “lifetime”. I-vol vs Historical-Vol. Implied Yield vs real inflation rate.

Price/vol conversion is similar to price/yield conversion because ….. everything else is held constant in the equation. Specifically,

** underlying price is held constant — Tricky. Even though underlying price changes every minute, investors each try to estimate the volatility of the underlying price over the _next_ month. If UBS assume stability, then UBS may use a 19%/year vol. As underlying price moves, UBS option pricing will move accordingly, but always lower than a hypothetical Citi trader (who assume instability).

When a buyer compares 2 competing quotes for the same option, she sees 2 i-vol estimates[2]. 19% by UBS, and 22% by Citi. As underlying price wiggles by the minute, both quotes move, but Citi quote always higher.

Next day, UBS uses a higher vol of 21.9%. As before, both quotes move along with underlying price, but now the spread between UBS and Citi quotes narrow.

At any underlier spot price, say underlying S = $388, one can convert the UBS quote to a lower implied vol and convert the Citi quote to a higher implied vol. During this conversion, underlying price is held constant.

** time to expiration is held constant — Tricky. This situation exists in price/yield conversion too.

[1] same dimension as yield? P276 [[complete guide]]
[3] Implied-Yield is not a common term but quite accurate.

FX option – collateral, briefly

A 2008 CFA textbook suggests that most fx options are tailor-made for a given client. Listed fx options trade volume is “fairly low” (2008)

According to Wikipedia, Most of the FX option volume is traded OTC , but a fraction is traded on exchanges like the International Securities Exchange (FX options), Philadelphia Stock Exchange (FX options), or the Chicago Mercantile Exchange (options on FX futures)

I guess one possible reason is – end users (large international organizations who need to convert currencies) don’t like the standardized terms on the exchanges??

Another possible reason is fee??

Most forex option trading is conducted via telephone. There are only a few forex brokers offering online forex option trading platforms.

The same fx option can be viewed as either a call or a put, but the writer is always the writer. Fx option writer (say DB) must put up collateral; fx option buyer must pay the writer an upfront premium. Intuitive for a fx call, but if you see it as a put, the same writer (DB) still has the same collateral/premium deal.

calendar spread – basics

Key point — sell time-value before it drops (to zero)

Take a PUT-calendar-spread for example.
– Sell (write) the near-date put
– buy the far-date put

An extreme case is easier to comprehend. The near-date instrument loses value (much) faster; while the far-date instrument barely loses value.

In a calendar spread portfolio, vega roll-up is meaningless. See http://bigblog.tanbin.com/2011/11/vega-roll-up-makes-no-sense.html

Key point — treat the near-date instrument as a –perishable-fruit– — Sell it! But to reduce delta exposure, buy the far-date instrument.