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.

—–

+ 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.