Introduction to exotic and path-dependent derivatives

European-style options
With the exception of some American put options (end of Lecture 10),
every derivative we have valued has had European-style payoffs.
– Specifically, the derivative payoff depends only on the price of the
underlying asset at expiry time T (ST).
– The path travelled by share price leading up to time T is irrelevant.
This meant we only needed to be concerned with the distribution of
share prices at the expiry date of the derivative (time T).
– The Binomial tree models how many different share prices are possible at
time T,
– Pascal’s triangle tells us how many paths lead to each time-T price, and
– We calculate the risk-neutral path probability for each path.
– We calculate the expected payoff to the derivative.
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Naturally, it is useful (and necessary) to spend a lot of time studying
European-style derivatives:
– We learned new ideas and approaches to valuation (replication, delta
hedging, risk neutral).
– We saw both analytic formulae to value options (Black-Scholes, put-call
parity) and numerical approximation techniques (Binomial trees).
– On the whole, we have developed good intuition for derivative valuation
and some very useful techniques.
However, the derivatives that trade in the real world rarely have such
simple features.
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Path-dependent derivatives
This lecture considers path-dependent derivatives.
The payoff to a path-dependent derivative depends, in some way, on
the path the underlying asset travels over the life of the derivative.
This path-dependency significantly complicates any attempt to
analytically value the derivative.
Thankfully, numerical methods like the Binomial approach still work
fine.
4
Overview of this lecture
In this lecture, we will:
– Introduce a handful of exotic and path-dependent derivatives.
– Demonstrate how their payoff depends on the path travelled by the
underlying asset.
– Given that we don’t have the mathematical background to value these
derivatives analytically,
– We will explore how path-dependent derivatives can be valued using the
Binomial approach.
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Lookback options
Lookback options have a payoff that depends in some way on either
the minimum or maximum share price attained over the option’s life.
Floating lookback options:
– Vanilla options have a fixed strike price (X). The payoff to a call option is
max(0, ST – X). The payoff to a put option is max(0, X – ST).
– With a floating lookback option, the fixed strike price (X) is replaced with
either the minimum or maximum share price over the option life.
Floating lookback call has a payoff equal to the time-T share price
less the minimum share price over option life: max (0, ST – Smin).
Floating lookback put has a payoff equal to the maximum share price
over option life less the time-T share price: max (0, Smax – ST).
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Lookback options
Fixed lookback options:
– A fixed lookback option has a fixed strike price (just like a vanilla option).
– However, the payoff depends on the minimum or maximum share price
over the option life (not the time-T price ST).
Fixed lookback call has a payoff equal to the maximum share price
over the life of the option, less the fixed strike price: max (0, Smax – X).
Fixed lookback put has a payoff equal to the fixed strike price (X) less
the minimum share price over the life of the option: max (0, X – Smin).
Clearly, these lookback options are “path dependent”, because Smax
and Smin depend on how stock price moves between now and expiry.
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Valuing a lookback option
Valuing a derivative with a payoff structure like this is not easy:
– With a plain-vanilla European option, all we need to worry about is the
range of possible share prices at expiry time T.
– With lookback options, we need to be considering the range of possible
share prices at every point in time (so that we can model the minimum or
maximum share price likely over the option life).
Amazingly, mathematicians can do this and have derived analytic
formulae to value lookback options. Section 26.11 of Hull presents
these formulae.
Alternatively, we can use a simple Binomial tree, in conjunction with
risk-neutral valuation, to get a numerical approximation of the value of
lookback options.
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Floating lookback call
9
Floating lookback put
10
Asian options
The payoff to an Asian option depends on the average price of the
underlying asset during some part of the option’s life.
Rather than the payoff depending on the time-T expiry share price
(ST), the payoff depends on an average price.
– an Asian call option has a payoff: max(0, Savg- X)
– an Asian put option has a payoff: max(0, X – Savg)
The period over which the share price is averaged is pre-determined
and written into the option contract.
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Asian options
Consider, for example, an Asian call option with strike price X and one
year to expiry. The payoff is max(0, Savg- X).
– One possibility is that Savg might be the average of the share price on the
last trading day of each of the 12 months.
– Or perhaps Savg might be the average of the daily share price over the last
30 trading days leading up to expiry.
Hull Section 26.13 presents very-complicated analytic formulae to
value Asian options.
Or, we can use a simple Binomial tree, in conjunction with risk-neutral
valuation, to get a numerical approximation of the value of an Asian
option. 12
Clearly, Asian options are “path dependent”,
because the payoff depends on how stock
prices moves between now and expiry
Ratchet (or reset) options
A ratchet (or reset) option has a strike price that changes throughout
the life of the option.
An example: a reset call option has one year to expiry. It is an
American-style option so it can be exercised at any point in time.
– The strike price is $10 if exercised during months 1-4,
– The strike price is $12 if exercised during months 5-8, and
– The strike is $14 if exercised during months 9-12.
These are not too difficult to value – they are a bit like a combination
of three options each with different strike prices.
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Chooser options
These are sometimes called “as you like it” options.
Essentially, you purchase the security. It is not a call and not a put.
However, at a certain point in time, the holder gets to choose whether
it is a call or a put.
For example:
– A chooser option has a strike price of X and expires in one year.
– On a pre-determined date during it’s life (e.g., after 3 months), the holder
of a chooser option must make a decision whether it is a call or a put.
Chooser options are “path dependent”. Naturally, if share price falls
over the first three months, we will probably choose it to be a put
option. If share prices rises, we elect to make it a call option.
We know how to value a call and we know how to value a put, but
how to value something that may end up being a call or a put? 14
Compound options
A vanilla option gives you the right (with no obligation) to buy or sell
the underlying asset. For example, with a call option, you can pay the
strike and exercise your right to buy the underlying asset (usually a
share).
A compound option is an option on an option. The asset underlying
the option is itself an option!
There are four types of compound options:
– A call on a call a right to buy a right to buy the underlying
– A put on a call a right to sell the right to buy the underlying
– A call on a put a right to buy a right to sell the underlying
– A put on a put a right to sell a right to sell the underlying
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Compound options
A compound option will have two strike prices and two expiry dates.
Example: a call on a put:
– You enter a call on a put.
– In 6 months time, you can exercise your right to buy something at the
strike price of X1.
– The “something” that you buy under the call is a put option!
– If you exercise the call (by paying the strike X1), you become the owner of
a put option.
– You now have the right (with no obligation) to sell some underlying asset
at the strike price X2 at some future expiry date.
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Barrier options
The most common path-dependent options are barrier options.
A barrier option is like a normal call or put option, except …
whether or not you are allowed to exercise the option depends on
whether the underlying asset price hits a pre-determined barrier
during the option’s life.
There are two main types of barrier options:
– Knock-in barrier option: starts off “dead” but comes to life if the barrier is
hit.
– Knock-out barrier option: starts off “alive” but is knocked out if the barrier
is hit.
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Knock-in barrier options
A knock-in option only comes to life if share price hits a predetermined
barrier (the option is knocked-in to action).
Where is the barrier? The barrier (H) could be below the starting
share price or above it:
– if the barrier H is set below the initial share price, option will be knocked-in
if share price drifts low enough: a down-and-in option.
– if the barrier H is set above the initial stock price, the option is knocked-in
if share price drifts high enough: an up-and-in option.
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Call was never activated
strike is X
barrier is H
Time
Price of underlying
S0 > H
Down-and-in call option
expiry
Zero payoff,
despite finishing
in the money
19
But call expires
out of the money
Call is activated
Time
Down-and-in call option
Price of underlying
strike is X
barrier is H
S0 > H
expiry
20
Call is activated
and it finishes
in the money
Time
Down-and-in call option
Price of underlying
strike is X
barrier is H
S0 > H
expiry
21
Time
Call is activated
but it expires
out of the
money
Up-and-in call option
Price of underlying
strike is X
barrier is H
S0 < H expiry 22 Call was never activated Time Up-and-in call option Price of underlying strike is X barrier is H S0 < H expiry Zero payoff, despite finishing in the money 23 Call is activated and it finishes in the money Time Up-and-in call option Price of underlying strike is X barrier is H S0 < H expiry 24 Down-and-in barrier call option Despite the complexity of a barrier option, it is possible to derive an analytic formula to value a down-and-in barrier call option:   Ny T  S N y Xe H S C S H rT DI                       2 2 0 2 0 0 T T S X ln H y r                0 2 2 2 1 Or, just use a Binomial tree to get a numerical approximation 25 Knock-out barrier options A knock-out option starts off as a vanilla option, but is knocked out (killed) if the underlying asset price hits a pre-determined barrier. The barrier (H) could be set below the starting share price or above it: – if the barrier H is set below the initial share price, option will be knocked out if share price drifts low enough: a down-and-out option. – if the barrier H is set above the initial stock price, the option is knocked out if share price drifts high enough: an up-and-out option. 26 Call is knocked out Time Down-and-out call option barrier is H Price of underlying strike is X S0 > H
expiry
Zero payoff,
despite finishing
in the money
27
but call expires
out of the money
Call is never knocked out
Time
Down-and-out call option
barrier is H
Price of underlying
strike is X
S0 > H
expiry
28
Call is never knocked out
and call finishes
in the money
Time
Down-and-out call option
barrier is H
Price of underlying
strike is X
S0 > H
expiry
29
Call is knocked out
Time
Up-and-out call option
barrier is H
Price of underlying
strike is X
S0 < H
expiry
30
but call finishes
out of the money
Time
Call is never knocked out
Up-and-out call option
barrier is H
Price of underlying
strike is X
S0 < H
expiry
31
and call finishes
in the money
Time
Call is never knocked out
Up-and-out call option
barrier is H
Price of underlying
strike is X
S0 < H
expiry 32
Down-and-out barrier call option
Here is a very ugly formula to value a down-and-out barrier call
option:
   
   



 


  


     
 

  
 

 N y T
S
Xe N x T H
N y
S
C S N x S H
rT
DO
 


1
2 2
0
1
1
2
0
0 1 0
T
T
S
ln H
y
T
T
H
ln S
x





 

 








0
1
0
1
Or, just use a Binomial
tree to get a numerical
approximation
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Do a PhD in mathematics?
No – just value them numerically!
These analytic formulae are getting pretty messy.
However, if we are comfortable with the Binomial approach (and
especially if we are able to write some computer code to implement a
binomial tree),
then we can price path-dependent derivatives using a Binomial
approximation in conjunction with risk-neutral methods.
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Example: Down-and-out barrier call option
S0 = $20; r = 6%; t = 0.25; u = 1.25; d = 0.80
A down-and-out barrier call option has:
– Strike price X = $15
– Barrier H = $14
– T = 1 year to expiry
Use a 4-step Binomial tree to calculate a numerical approximation to
the price of this down-and-out barrier call option.
This implies that
σ = 44.63%
35
48.83
39.06
31.25 31.25
25 25
20 20 20
16 16
12.80 12.80
10.24
8.19
Node Time:
0.0000 0.2500 0.5000 0.7500 1.0000
With a 4-step tree:
5 ending “nodes”
24 = 16 different paths
For a down-out barrier option:
any path that goes down
below $14 is knocked out
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X = $15
H = $14
48.83
39.06
31.25 31.25
25 25
20 20 20
16 16
12.80 12.80
10.24
8.19
Node Time:
0.0000 0.2500 0.5000 0.7500 1.0000
With a 4-step tree:
5 ending “nodes”
24 = 16 different paths
For a down-out barrier option:
any path that goes down
below $14 is knocked out
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X = $15
H = $14
H=14
Pascal’s Triangle
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Calculate Risk-Neutral Probability
   
  
 
   
0.06 0.25
*
*
0.80 0.4780
1.25 0.80
1 1 0.4780 0.5220.
er t d e p
u d
p
39
No. of
Paths
(1)
Stock Price
Moves
Finishing
Stock
Price
Option
Payoff
(2)
Path
Probability
(3)
(1)(2)×(3)
1 uuuu $48.83 $33.83 0.47804 1.7660
4
uuud
uudu
uduu
duuu
$31.25
$16.25
0.47803  0.52201
3.7057
6
5
uudd
udud
uddu
duud
dudu
dduu
$20.00
$5.00
0.47802  0.52202
1.5565
4
uddd
dudd
ddud
dddu
$12.80
nil
0
1 dddd $8.19 nil 0
$7.0281
Discount: e-0.06×1 = 0.94176
Option Value $6.62
These 5 paths all finish out of the money (below
$15 strike). And they get knocked out anyway.
If we plugged into the
analytic formula, price =
$6.10
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The “dduu” path travels through $12.80 therefore hits the $14 barrier
Example: Floating lookback call option
Assume S0 = $10,  = 0.40, r = 5%, t = 3/12
Floating lookback call option: T = 1, payoff = max(0, ST – Smin)
0.05 0.25
*
*
0.8187 0.4814
1.2214 0.8187
1 1 0.4814 0.5186
er t d e p
u d
p
   
  
 
   
0.8187
1.2214
1 1
0.40 3 /12 1.2214
  
  
u
d
u e t e
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With a 4-step tree, there are 24 = 16 different paths the stock can
travel over the next 12 months.
No easy way to handle the lookback call:
– Just trace each path one by one
– Take note of the smallest stock price (Smin) on that path, and
– Plug it into the payoff function.
In calculating Smin on each path, we will disregard the starting price S0
= $10.
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22.26
18.22
14.92 14.92
12.21 12.21
10 10.00 10.00
8.19 8.19
6.70 6.70
5.49
4.49
Node Time:
0.0000 0.2500 0.5000 0.7500 1.0000
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payoff = max(0, ST – Smin)
Stock
Price
Moves
Finishing
Stock
Price
Lowest
Price
(Smin)
Option
Payoff
(1)
Path
Probability
(2)
(1)(2)
uuuu 22.26 12.21 10.05 0.48144 0.5397
uuud
uudu
uduu
duuu
14.92
12.21
12.21
10.00
8.19
2.71
2.71
4.92
6.73
0.48143  0.51861
0.1568
0.1568
0.2847
0.3894
uudd
udud
uddu
duud
dudu
dduu
10
10.00
10.00
8.19
8.19
8.19
6.70
0
0
1.81
1.81
1.81
3.30
0.48142  0.51862
0
0
0.1128
0.1128
0.1128
0.2057
uddd
dudd
ddud
dddu
6.70
6.70
6.70
6.70
5.49
0
0
0
1.21
0.48141  0.51863
0
0
0
0.0812
dddd 4.49 4.49 0 0.51864 0
Expected Payoff $2.1527
Discount: e-0.05×1 0.95123
Option Value $2.05
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payoff = max(0, ST – Smin)
Key takeaways from this lecture
We have spent most of our time this semester on “vanilla” calls and
puts. However, the options that trade in markets tend to be much
more exotic.
Derivatives can have very exotic features and path-dependent
payoffs.
This makes it very difficult (and often impossible) to derive an analytic
formula to price them.
Fortunately, we can rely on numerical approaches like a Binomial tree
with risk-neutral valuation.
These only give an approximate value, but if we use a large number
of branches in the tree, it becomes increasingly accurate.
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