Financial Engineering Lesson 2 Asian Options

Asian Options

Giampaolo Gabbi

Asian Options.Definition and Applications

3

Exotic (nonstandard) options

• Exotic options solve particular business problems that an ordinary option cannot

• They are constructed by tweaking ordinary options in minor ways

• Relevant questions:– How does the exotic payoff compare to ordinary option

payoff?

– Can the exotic option be approximated by a portfolio of other options?

– Is the exotic option cheap or expensive relative to standard options?

– What is the rationale for the use of the exotic option?

– How easily can the exotic option be hedged?

4

Asian options

• The payoff of an Asian option is based on the average price over some period of time path-dependent

• Situations when Asian options are useful:– When a business cares about the average exchange rate

over time– When a single price at a point in time might be subject to

manipulation– When price swings are frequent due to thin markets

• Example: – The exercise of the conversion option in convertible bonds

is based on the stock price over a 20-day period at the end of the bond’s life

• Asian options are less valuable than otherwise identical ordinary options

5

Asian options (cont.)

• There are eight (23) basic kinds of Asian options:– Put or call

– Geometric or arithmetic average

– Average asset price is used in place of underlying price or strike

• Arithmetic versus geometric average:

– Suppose we record the stock price every h periods from t=0 to t=T

– Arithmetic average: Geometric average:

A TN

Sih

i

N

( ) ==

∑1

1G T S S Sh h Nh

N( ) ( )

/= × × ×21

L

6


• Average used as the asset price: Average price option– Geometric average price call = max [0, G(T) – K]– Geometric average price put = max [0, K – G(T)]

• Average used as the strike price: Average strike option– Geometric average strike call = max [0, S

T– G(T)]

– Geometric average strike put = max [0, G(T) – ST]

• All four options above could also be computed using arithmetic average instead of geometric average

• Simple pricing formulas exist for geometric average options but not for arithmetic average options

7


• Example of use of average price option:– Receiving funds in a foreign currency every month for a

year, with translation of the cash flow into US Dollars taking place every month. At the end of the year, the firm is concerned with the average exchange rate obtained, i.e. does not want to have received funds at a rate below a specific exchange rate.

• Example of use of average strike option:

– Acquiring shares of a security S gradually by purchasing a fixed amount every month. Wanting to unload the full position at the end of the year, the firm needs an option that yields a payoff proportional to the difference between average security value and terminal value.

8


• Comparing Asian options:

9


• For Asian options that average the stock price, the averaging reduces the volatility of the value of the stock entered in the payoff.

• Hence the price of the option is less than an otherwise similar traditional option.

• For Asian options that average the strike price, more averaging reduces the correlation between the terminal value of the stock and the strike price (computed as average of past stock prices), hence increasing the value of the option.

10


• XYZ’s hedging problem– XYZ has monthly revenue of 100m, and costs in dollars– x is the dollar price of a euro– In one year, the converted amount in dollars is

– Ignoring interest what needs to be hedged is

• A solution for XYZ – An Asian put option that puts a floor K, on the average

rate received

10012 12

1

12

m × −

=

∑ x eir i

i

( ) /

x =x

i

i

ii12121

121

12

×

=

=∑∑

max K xi

i

01

12 1

12

, −

=

∑

11


• Alternative solutions for XYZ’s hedging problem

Asian Options.Example

13

Example: Asian Option

George Brickfield’s business is highly exposed to volatility in the cost

of electricity.

He has asked his investment banker, Lisa Siegel, to propose an option

whereby he can hedge himself against changes in the cost of a

kilowatt hour of electricity over the next twelve months.

14

Lisa thinks that an Asian option would work nicely for George’s

situation.

An Asian option is based on the average price of a kilowatt hour (or

other underlying commodity) over a specified time period.


15

In this case, Lisa wants to offer George a one year Asian option with

a target price of $0.059.

•If the average price per kilowatt hour over the next twelve months

is greater than this target price, then Lisa will pay George the

difference.

•If the average price per kilowatt hour over the next twelve months

is less than this target price, then George loses the price he paid for

the option (but he is happy, because he ends up buying relatively

cheap electricity).


16

What is a fair price for Lisa to charge for 1 million kwh worth of

these options?

Use the historical data provided and Monte Carlo simulation to

arrive at a fair price.


17

Month $/kwh Month $/kwh Month $/kwh

Jan-90 0.0510 May-93 0.0630 Sep-96 0.0580 Feb-90 0.0560 Jun-93 0.0710 Oct-96 0.0570 Mar-90 0.0540 Jul-93 0.0840 Nov-96 0.0550 Apr-90 0.0520 Aug-93 0.0770 Dec-96 0.0550

May-90 0.0520 Sep-93 0.0790 Jan-97 0.0520 Jun-90 0.0570 Oct-93 0.0660 Feb-97 0.0530 Jul-90 0.0670 Nov-93 0.0560 Mar-97 0.0500

Aug-90 0.0640 Dec-93 0.0690 Apr-97 0.0500 Sep-90 0.0640 Jan-94 0.0560 May-97 0.0530 Oct-90 0.0580 Feb-94 0.0540 Jun-97 0.0540 Nov-90 0.0540 Mar-94 0.0530 Jul-97 0.0540 Dec-90 0.0570 Apr-94 0.0560 Aug-97 0.0520 Jan-91 0.0590 May-94 0.0560 Sep-97 0.0500 Feb-91 0.0590 Jun-94 0.0580 Oct-97 0.0550 Mar-91 0.0560 Jul-94 0.0590 Nov-97 0.0520 Apr-91 0.0550 Aug-94 0.0530 Dec-97 0.0480





18

An important initial step is to study the historical behavior of electricity prices. Our model will be based not on the actual prices, but on monthly percent changes in price, so we add a column to our data set that calculates the percent change in price (or return):

1

2

3

4

5

6

7

A B C D E

Month $/kwh Return

Jan-90 0.0510

Feb-90 0.0560 0.09804

Mar-90 0.0540 -0.03571

Apr-90 0.0520 -0.03704

May-90 0.0520 0.00000

Jun-90 0.0570 0.09615

=(B3-B2)/B2


19

Now, we need to think about what sort of theoretical probability distribution would do a good job of approximating the empirical distribution in these data. A useful tool for studying distributions is the histogram:

Histogram of Electricity Returns

0

5

10

15

20

25

30

-0.200 -0.175 -0.150 -0.125 -0.100 -0.075 -0.050 -0.025 0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200 0.225 0.250 0.275 0.300

Monthly Price Change

Fre

qu

en

cy


20

It would appear that the percent price changes are approximately normally distributed, so we’ll use a normal distribution. We’ll use the sample mean and sample standard deviation from these data (0.001768 and 0.073462, respectively) as the mean and standard deviation of the input random variable for our model.


21

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

A B C D E F G H

Initial Electricity Price $0.05684

Target Price $0.05900

Mean monthly return 0.18%

Std dev monthly return 7.35%

# kwh per option 1,000,000

Month Return Price Average Price

Jan 1.13% 0.05748 0.06486

Feb -0.21% 0.05736

Mar 7.82% 0.06185 Payout

Apr -5.77% 0.05828 5,861.89$

May 6.54% 0.06209

Jun 8.31% 0.06725

Jul -12.05% 0.05915

Aug 9.71% 0.06489

Sep 1.56% 0.06590

Oct 6.84% 0.07041

Nov 9.39% 0.07703

Dec -0.50% 0.07664

=AVERAGE(C8:C26)

=C5*MAX(E8-C2,0)

=C15*(1+B16)


22

In B8:B19 we have 12 Crystal Ball assumption cells, normally distributed with the mean and standard deviation from our sample data (C3 and C4).

In C8:C19 we use the random percent returns to calculate monthly prices, which are averaged in E8 for the whole year.

E11 calculates the payout on the option (a Crystal Ball forecast cell).

The average value of E11 over many trials will be a reasonable estimate of the fair price for this option.


23

We’ll add a graph, to show the change in electricity prices over the course of each simulated year:

1

2

34

5

67

8

910

11

12

1314

15

1617

18

1920

21

22

23

A B C D E F G H I J K L

Initial Electricity Price $0.05684

Target Price $0.05900

Mean monthly return 0.18%Std dev monthly return 7.35%

# kwh per option 1,000,000

Month Return Price Average Price

Jan 1.13% 0.05748 0.06486

Feb -0.21% 0.05736Mar 7.82% 0.06185 Payout

Apr -5.77% 0.05828 5,861.89$

May 6.54% 0.06209

Jun 8.31% 0.06725Jul -12.05% 0.05915

Aug 9.71% 0.06489

Sep 1.56% 0.06590Oct 6.84% 0.07041

Nov 9.39% 0.07703

Dec -0.50% 0.07664

Price

0.04

0.05

0.06

0.07

0.08

0.09

0.10

0.11

0.12

Jan

Feb Mar Apr

May Ju

n

Jul

Aug Sep Oct

NovD

ec

Month

Pri

ce

($

/kw

h)


24


25


26

The frequency chart indicates that the option is frequently worthless (as evidenced by the tall bar at zero), but that the payout is occasionally $20,000 or more. Remember that the units here are millions of kilowatt hours.


27

To estimate a fair price, the most useful piece of the simulation output is the sample mean of approximately $3,138 per million kwh.


28

A 95% confidence interval is given by:

X n

s96.1±

$3,138.48 000,1

77.428,5$96.1±

( )67.171$96.1±

48.336$±

We are 95% confident that the true fair price is somewhere between $2,802.00 and $3,474.96, which seems like a very wide interval. We could narrow the interval around our estimate by running a longer simulation.


Asian Options. Pricing

30

Black-ScholesPoisson

(Cox and Ross)

Jump diffusion model

(Merton)

Dividends

Options on futures

(Black)

Multiple factors

Stochastic Volatility

(Hull and White)

Exchange one asset

for another (Margrabe)

Option on Max, Min Option on an average

(Asian options)

Path Dependent

31

Path Dependence:

Some derivatives depend on the path of the underlying asset.

Payoff: )0,max(0

1 ∫−T

tTTdtSS

A specific example is the Average strike option.

So, ),,(0

tdSSct

t ∫ ττ

We need to write an Ito equation for this.

The dependence on the entire path is a problem!

For example: An Asian option depends on the average price

of the underlying stock over a given time period.

32

The problem is the ∫t

dS0

ττterm

The general approach is to try to capture the path

dependence with another variable.

Now we can apply Ito’s lemma and continue in typical fashion...

Path Dependence:

Let’s see how this would work for an Asian option

∫=t

tdSI

0ττ

then dtSdItt

=Let’s assign:

),,(),,(0

tISctdSSctt

t

t=∫ ττSo where S and I are Ito processes

33

Asset price: dzdtS

dSσµ +=

Bond: rdtB

dB=

Path Dependence:

= ∫

t

tdSI

0ττdtSdI

tt=Path Dependence

Derivative:

dzc

Scdt

c

cSScScc

c

dcSSSISt

σσµ+

+++=

)(22

21

),,( tISc

34

[ ]λµ K1=Step 2: Apply

Path Dependence:

+

=

+++c

Sc

c

cSScScc

r

SSSISt

σσλλ

σµ

µ

0

1

1

1

)(10

22

21

rccSScrSccSSISt

=+++ 22

2

1 σ

σ

µσσµ )()(22

21 r

c

Scr

c

cSScSccSSSISt

−

+=

+++

35

Path Dependence:

rccSScrSccSSISt

=+++ 22

2

1 σ

For an average strike Asian option, the boundary condition is

)0,max(),,( ISTISc −= 0),,0( =tIc

We can pull the same sort of trick for other path dependence:

We can even handle ]),0[,max( tSMt

∈= ττ

Since it is the limit ofnt

n

nt

dSM

1

0lim

= ∫∞→

ττ

36

Path Dependence:

I won’t expand on this now since we will see another

approach to this later in the course...

We can even handle ]),0[,max( tSMt

∈= ττ

Since it is the limit ofnt

n

nt

dSM

1

0lim

= ∫∞→

ττ

You can derive that: MdtM

S

ndM

n

=

10

So, and option on a max solves the Black-Scholes equation, but

the boundary conditions are different.

37

Summary of Return Form Approach:

Three step algorithm:

(1) Derive factor models for returns of tradable assets.(often involves Ito’s lemma.)

(2) Apply absence of arbitrage condition.(m=[1 s]l)

(3) Apply appropriate boundary conditions and solve.(how to solve is your problem.)

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Financial Engineering Lesson 2 Asian Options