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.

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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.

—–

+ PCP

+ 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 GBM

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, Put is More like stock, so your short put is now more “short” IBM, so you should Buy IBM.

Mathematically, your short put is now providing additional negative delta. You need positive delta for balance, so you Buy 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?
Call 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

today.

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

why market 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.