Transcript Option dynamic replication and RNV

Option Dynamic Replication
References:
See course outline
1
Option Replication
BOPM based on the idea that, since we
have two traded assets (underlying stock
and risk-free bond) and there are only
two states of the world, we should be
able to replicate the option payoffs
 That is, we should be able to form a
self-financing risk-less portfolio made
up of the stock, the option and the riskfree asset.
 How?

The Binomial Model (from previous
lecture example)


A stock price is currently $20
In three months it will be either $22 or $18
Stock Price = $22
Stock price = $20
Stock Price = $18
The Call Option

Option tree:
Stock Price = $22
Option Price = $1
Stock price = $20
Option Price=0.633
Stock Price = $18
Option Price = $0
Setting Up a Risk-Less Portfolio

Consider the Portfolio:
long D shares
short 1 call option
22D – 1
18D

Portfolio is riskless when 22D – 1 = 18D or
D = 0.25
Valuing the Portfolio
(Risk-Free Rate is 12%)

The riskless portfolio is:
long 0.25 shares
short 1 call option

The value of the portfolio in 3 months is
22 Δ -1 ( = 220.25 – 1) = 4.50
Or,
18 Δ ( = 180.25) = 4.50

The value of the portfolio today must be the PV of 4.50 at the
risk-free rate, i.e.
4.50e – 0.120.25 = 4.3670
Hedging and Valuation

So the risk-less portfolio is worth 4.367.

This portfolio is
long 0.25 shares
short 1 option

The value of the shares is
5.000 (= 0.2520 )

By LOOP, the value of the option is therefore
0.633 (= 5.000 – 4.367 )
Self-financing risk-less portfolio

Because it is riskless, the portfolio can be made self-financing if
we add 4.3670 of borrowings or short a 3-month zero-coupon
bond with face value 4.50 = 4.3670e0.120.25.

The value of the resulting self-financing replication portfolio is
zero by construction and, if LOOP holds, its value remains
constant, i.e. zero, regardless of the state of the world at the end
of the period.
◦
The debt contracted at time 0 to fund the purchase of  units of stock is repaid,
with interest, by selling the shares and using the proceeds from having invested at
the risk-free rate the premium received at time t.
Generalization

Consider the portfolio that is long D shares and short 1
derivative
S0uD – ƒu
ΔS0 – f
S0dD – ƒd

The portfolio is riskless when S0uD – fu = S0d D – fd or
fu  f d
D
S 0u  S 0 d
Delta-Hedging



Delta (D) is the ratio of the change
in the price of an option to the
change in the price of the
underlying asset
More in-the-money, more delta..
and vice versa
Delta is a sort of RN probability of
exercise
Delta

For a call:
Option
price
Slope = D
B
A
Stock price
Generalization
(continued)

Value of the (delta-)hedged portfolio at time T is
S0u D – fu

Value of the portfolio today is
(S0u D – fu )e–rT

Another expression for the portfolio value today is
S0D – f

Hence, by LOOP,
S0D – f = (S0u D – fu )e–rT
Delta Hedging and RNV

Substituting for D,
f  S0 D  ( S0uD  fu )e  rT



  rT
fu  f d
fu  f d
 S0
  S0u
 fu  e
S 0u  S 0 d 
S 0u  S 0 d


D
D


erT  d
 rT
  fu q  f d (1  q) e
q
ud
(see algebraic steps of proof next slide)
…Proof…



  rT
fu  f d
fu  f d
f  S0
  S 0u
 fu  e
S0u  S0 d 
S 0u  S 0 d


D
D


f  f d  uf u  uf d

 u

 fu  e  rT
ud  ud

f u  f d  uf u  uf d  f u u  f u d

ud 
ud
f  f d  uf d  f u d   rT
 u

e
u  d  u  d 

  rT
e

 f u e rT  f u d  f d e rT  uf d   rT

e 
u

d


  e rT  d 
 e rT  u    rT
  fu 
  fd 
 e
 u  d 
  ud 
  f u q  f d (1  q )  e
 rT
e rT  d
q
ud
e rT  u
1 q 
ud
RN Probabilities

Notice that, in order to interpret the
q and 1 – q thus obtained as RN
probabilities, they must be positive.

That is, we can derive them using
LOOP alone but, to use them to
price other derivatives written on
the same underlying, we must
assume NA.
Multi-period



The value of D varies from node to
node
More in-the-money, more delta..
and vice versa
Need to replicate the option at each
node of multi-period tree
…Multi-period

In every one-period step, we can form
risk-less one-period portfolios.

As D changes, we rebalance.

….
…Multi-period

The sequence of one-period risk-less
portfolios results in a dynamically
rebalanced multi-period portfolio, that is
kept LOCALLY riskless by dynamic
hedging of the option component.
◦
If we think of the initial one-period portfolios as selffinancing, the dynamically rebalanced portfolio must be, by
LOOP, self-financing too. The value of the underlying asset
hedge required to ‘keep the portfolio hedged’ changes with
Δ and so, as the latter changes, funding must change too.
But, under LOOP, the value of the dynamically rebalanced
portfolio must remain equal to zero from start to end.
The Two-Step Example (K = 21)
D
22
20
1.2823
A
B
2.0257
18
24.2
3.2
E
19.8
0.0
C
0.0
F
16.2
0.0
Dynamic Replication and Pricing

In multi-period setting, delta-hedging leads to
dynamic replication.

This nails multi-period option prices down to
NA values, by enforcing the LOOP alone.

Conversely, given the price of the underlying
asset with which the option is dynamically
replicated, the NA option price can be obtained
using RN valuation of multi-period payoffs.
◦ That is, we can take the RN expectation of the final
payoffs and discount at the risk-free rate, but first we
need to specify how one-period distributions
integrate to multi-period ones (e.g., i.i.d. assumption
of typical BOPM).
Problems with Dynamic Replication


LOCALLY (i.e., only between adjacent nodes)
risk-less does not mean GLOBALLY risk-less
What could go wrong?
◦ Hedge cannot be adjusted fast enough (underlying
asset moves too fast, e.g. price jumps) or ‘cheaply’
enough (when liquidity “dries out”)

There are more risk-factors than we are
modeling
◦ e.g., interest rates are stochastic, volatility as well as
returns is stochastic, etc.


The consequence is a possibly poor replication
and hence poor pricing and hedging
Interesting and important topic but to be left to
more advanced courses
Delta and Delta Hedging in B&S

The delta of a European call on a stock paying
dividends at rate d is N(d 1)e– dT

The delta of a European put is e– dT [N (d 1) – 1]

Just like in BOPM, a position in Δ units of the
underlying locally replicates the option and
delta-hedging involves maintaining a delta
neutral portfolio
Using Futures for Delta Hedging

The delta of a futures contract is e(r-q)T
times the delta of a spot contract

The position required in futures for delta
hedging is therefore e-(r-q)T times the
position required in the corresponding
spot contract
RN vs Real World Parameters
N(d2) is the risk neutral probability of exercise
 The implied volatility of an option is the risk
neutral volatility for which the Black-Scholes
price equals the market price

◦ If Black-Scholes held true, this should be also the
market expectation of future volatility (under real
world probabilities)
◦ The is a one-to-one correspondence between prices
and implied volatilities
◦ Traders and brokers often quote implied volatilities
rather than dollar prices
Other ‘Greeks’


Consider option with price f
Besides delta, there are at least two other
key sensitivity parameters are
◦ Gamma (G), the rate of change of delta (D) with
respect to the price of the underlying asset
◦ Theta (Q), the rate of change of f with respect
to the passage of time
◦ But also vega, rho, etc.
Price
Time value
TV = price - IV
Intrinsic value
IV = Max(S – K, 0)
Gamma
Gamma (G) is the rate of change of delta (D) with
respect to the price of the underlying asset
 Curvature  the hedge position must be
dynamically rebalanced
 By no-arbitrage:

◦ Delta hedging a written option must involve a “buy
high, sell low” trading rule
◦ Delta hedging a long option must involves a “buy low,
sell high” trading rule
Gamma Addresses Delta Hedging Errors
Caused By Curvature
Call
price
C’’
C’
C
Stock price
S
S’
Theta

Theta (Q) of a is the rate of change of the value
with respect to the passage of time,
Q  dS/dt
Interpretation of Gamma

Neglecting funding costsgains and second and higher order
terms in dt, for a delta neutral portfolio we must have
dP » Q dt + ½GdS2
dP
dP
dS
dS
Positive Gamma
Negative Gamma
Vega


But volatility is not constant
Vega (n) is the rate of change of the value
of a derivatives portfolio with respect to
volatility
Implied Volatility
N(d2) is the risk neutral probability of exercise
 The implied volatility of an option is the risk
neutral volatility for which the Black-Scholes
price equals the market price
 The is a one-to-one correspondence between
prices and implied volatilities
 Traders and brokers often quote implied
volatilities rather than dollar prices

Causes of Volatility Changes

Volatility is usually much greater when the
market is open (i.e. the asset is trading)
than when it is closed
e.g. French & Roll (1986)

For this reason time is usually measured in
“trading days” not calendar days when
options are valued
Rho


Rho is the rate of change of the
value of a derivative with respect
to the interest rate
For currency options there are 2
rho’s
Relationship Between Greeks

With ‘plain vanilla’ options:
◦ Positions that are long delta are also long rho
(long call, short put).
◦ Positions that are long gamma are also long
vega and short theta (long call, put).
◦ Why?
Managing Delta, Gamma, & Vega


D can be changed by taking a position in
the underlying
To adjust G and n it is necessary to take a
position in an option or other derivative
Hedging in Practice



Traders usually ensure that their portfolios
are delta-neutral at least once a day
Whenever the opportunity arises, they
improve gamma and vega
As portfolio becomes larger hedging
becomes less expensive
Hedging vs Creation of an Option
Synthetically

When we are hedging we take
positions that offset D, G, n, etc.
 When we create an option
synthetically we take positions that
match D, G, & n
Portfolio Insurance


In October of 1987 many portfolio managers
attempted to create a put option on a
portfolio synthetically
This involves initially selling enough of the
portfolio (or of index futures) to match the D
of the put option
Portfolio Insurance
continued


As the value of the portfolio increases, the
D of the put becomes less negative and
some of the original portfolio is
repurchased
As the value of the portfolio decreases, the
D of the put becomes more negative and
more of the portfolio must be sold
Portfolio Insurance
continued
Powerful tool...
but the strategy did not work well on
October 19, 1987
See notes on portfolio insurance