options pricing - fuzzy term paper

8/7/2019 Options Pricing - Fuzzy Term Paper

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MAL 717

Fuzzy Sets and Applications

Paper overviewA Multiperiod Binomial Model for Pricing Options in a

Vague Worldby

Silvia Muzzioli and Costanza Torricelli

Submitted by

Achal Premi (2005MT50558)Sameer Jain (2005MT50445)Department of Mathematics

IIT Delhi


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Contents

S. No. Titles Page Nos.

1. Introduction 31.1

1.2

Options

Options Pricing

3

4

2 Fuzzy Binomial Option Pricing Model

(FBOPM)

4

2.1 Fuzzy Binomial Tree 4

2.2 Risk Neutral Probability Intervals 6

2.3 Pricing of an option in a vague world SinglePeriod Model 11

2.4 Multi Period Model 13

3 Defuzzification 15

4 Proposed Extension 15 4.1

4.2

4.3

Integration of fuzzy BOPM into fuzzy logic

Justification for values of control parameter f

Pattern Recognition & Similarity Measures

15

17

17

5 Conclusion 18

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Prepared by: Achal Premi & Sameer Jain


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1. IntroductionThis paper gives an overview of the Binomial Option Pricing Model in a

vague world published by Silvia Muzzioli, Costanza Torricelli in the

Journal of Economic Dynamics & Control 28 (2004)and provides an

extension to the model proposed by integrating the fuzzy pricing model

with fuzzy logic rules and pattern recognition to capture the information

about different states of an economy in calculating the price of an option.

1.1. Optionsan option is a contract that gives someone the right to buy or sell an

asset for a specified time at a specified price, but unlike a forward or a

future contract, the buyer of the option is not under any obligation to

exercise the option. The asset can be a real asset such as real estate,

agricultural products or natural resources, or it can be a financial asset

such as stock, bond, stock index, foreign currency, etc. There are two

types of options, namely, call and put options

Call Option:The buyer of a call option has the right but not theobligation to buy an agreed quantity of an asset (the underlying security)

from the seller (writer) of the option at a certain time (the expirationdate) for a certain price (Strike price). The seller is obligated to sell the

asset if the buyer wishes to exercise his right. Initially, the buyer of the

option has to pay a certain amount to the seller for purchasing the option

which is called the premium or the call price. Given below are the payoff

and profit curves for the buyer and seller of a call option.

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FROM BUYER'S PERSPECTIVE FROM SELLER'S

PERSPECTIVE

Put Option:The buyer of a put option has the right but not theobligation to sell an agreed quantity of an asset (the underlying security)to the seller (writer) of the option at a certain time (the expiration date)

at a certain price (Strike price). The seller is obligated to buy the asset if

the buyer wishes to exercise his right. Given below are the payoff and

profit curves for the buyer and seller of a put option

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FROM BUYER'S PERSPECTIVE FROM SELLER'S

PERSPECTIVE

1.2. Option Pricing

Option pricing is an attempt to determine the fair market value of the

premium that should be paid by the buyer to the seller of the option.

(Here the term fair refers to a desirable condition that both buyer and

seller should have equal expected returns from the option contract).

The aim is to develop a model that takes into account as much market

information as possible and come up with a fair value for the option

price (call price or put price).

The different models differ in their ability to cover different aspects of the

market. Thus, a model can be considered to be good if it can take account

of the future movements of underlying security with some precision and

hence determine a fair premium value payable at the time of buying

the option.

2. Fuzzy Binomial Options pricing model

(FBOPM)2.1. Fuzzy Binomial Treelet us consider a one period model where t {0, 1} is the time. Let P0 be

the price of the stock at time t = 0 and P1 be the price at time t = 1, u &

d be the up and down jump factors with the probabilities p and (1-p); p

[0, 1].

In the standard binomial option pricing model (BOPM), we assume u and d

to be crisp values so that the stock value at time t = 1 is either

P1= uP0 with probability p OR

P1= dP0 with probability 1-p.

But this assumption of the BOPM neglects the vagueness involved in the

stock price movements in the real world and hence is not expected to

yield a fair option price value. To counter this limitation the authors

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have assumed u and d to be fuzzy numbers. For the sake of

computational simplicity they are assumed to be triangular fuzzy

numbers (TFNs).

Fig 1: The two possible jump factors

But How to get the jump factors u & d?

Standard BOPM Analogously for the

fuzzy BOPMWork done by Cox et al. (1979)

leads to

et

u

=

ud e

t1==

, where

- volatility of the underlying

asset,

t - length of the time period.

Triangular fuzzy numbers u = (u1,

u2, u3) & d=(d1, d2, d3) are givenby

et

u

=1

1

, et

u

=2

2

,

et

u

=3

3

( 321


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Where

PT defuzzified option price calculated from

the modelPM actual market price of the optionn number of observations from the past

datar continuously compounded interest rate

The second condition in the NLPP takes care of the no arbitragecondition (which says that no transaction or portfolio can make a profit

without risk). This condition is also intuitively appealing because

if etr > e

t1then

o no investor would wish to invest in the risky assets and

would confine only to the risk free investments and

if etr < e

t 1

o no investor would wish to make any risk free investment

2.2. Risk neutral probability intervals

We introduced p and 1-p as the probabilities of the up and down jump

factors, but how do we get these probabilities. As for the pricing

methodology, a risk neutral (fair) valuation approach is used therefore

these probabilities should be such that they incorporate the fair (risk

neutral) pricing strategy.

The aim of this section is to derive these risk neutral probabilities in order

to price a call written on a stock. Let us consider a one period model

where t {0, 1} is the time

Let pu & dp be the up and down jump probabilities. Then

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up + dp

= 1


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and from the idea ofrisk neutral valuation, we require

1(000 rPpuPpdP ud +=+

LHS = expected value of P1 (expected value of stock at t =1).RHS = the value of a risk free investment of amount P0 (invested at time

t=0) at time t=1.

Since u & d are fuzzy numbers, writing them in terms of cut we get

(1)Solving these equations for a particular value of d

[d

1 ,d

3 ] and a

particular value of u

[u

1 ,u

3 ] we get crispdp

andup

values. On

varying d & u values on these intervals we get range of values forup

anddp

.

We get maximum possible range ofup

&dp

values at a given

cut

by solving the following two systems of equations

Solving system (2) yields

Solving system (3) yields

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The two solutions thus give the bounds on the probability intervals:

It is easy to check that the following duality relations hold:

up + dp

=1d

p + up

=1Hence for a given -cut we get intervals for the possible values of up &

dp . These intervals have a maximum spread at = 0 and reduce to crisp

values at = 1. This is justified because it can be easily seen that

derivative w.r.t is positive for both the left bounds and negative for

both the right bounds of the probabilities.

Also it is easy to verify the following claimSecond derivative

positive condition

Second derivative

negative condition

Second derivative

zero condition

up u3-u2 > d3-d2 u3-u2 < d3-d2 u3-u2 = d3-d2

up u2-u1 > d2-d1 u2-u1 < d2-d1 u2-u1 = d2-d1

dp u2-u1 < d2-d1 u2-u1 > d2-d1 u2-u1 = d2-d1

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dp u3-u2 < d3-d2 u3-u2 > d3-d2 u3-u2 = d3-d2

Taking use of the information in the above table we can have different

shapes for curves of up and dp depending upon the relative positionsof u1, u2, u3 and d1, d2, d3 which are illustrated in tables below

TABLE 2

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Table3

These tables do not depict the cases in which the second derivative of up ,

up , dp , dp are zero when up & dp are TFNs.

By inspection of the tables 2 & 3, it is clear that the shape of the risk

neutral probabilities depends on the relative positioning of u & d. In

particular if we fix the two peaks, d2 and u2 of the two jump factors, from

table 2 we can see that when the distribution of u is closer to that of d i.e.

u & d are less distinct, then both up & dp are closer to a crisp number.

Conversely, when u & d are more distinct, up & dp are more vague

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2.3. Pricing of an option in vague world

single period model

Having obtained the risk neutral probabilities up & dp in the previous

section we now use them to price an option on a stock. We only considerthe one period model in this section.

At the time of expiry a call option has a positive value (payoff) if the priceof the underlying stock at that time is greater than the strike price and iszero otherwise. Now in our fuzzy model at time t=1 the stock price iseither P0d or P0u, which are triangular fuzzy numbers (because u & dare triangular fuzzy numbers.)

To consider an interesting contract let us assume that the strike price (X)

is between the highest value of stock in the down state and the lowest

value in the up state i.e.

Let us denote

C (u) =the payoff of call in the up state

C (d) = the payoff in the down state.

Then clearly

C (u) = (P0u-X) = (P0u1-X, P0u2-X, P0u3-X)

C (d) = 0

Since u is a TFNC (u) is also a TFN which is given in the -cut

representation as

Now by the risk neutral valuation strategy we can determine the call price

C0 at t=0 by

Where

E refers to the expectation under the risk neutral probabilitiesC1is the call payoff at time t=1

This equation signifies that

The discounted value of expected call payoff at time t=1 equals

the value of call at time t=0.

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condition that the expected returns of the writer and the buyer

should be same at the time of expiry which is precisely the ideology

behind risk neutral valuation.

Since Cd = 0 in our case therefore

The rules of multiplication between fuzzy numbers then lead to

It is easy to prove that as increases this interval size of call option

price shrinks.

In particular for =1 the interval reduces to a single crisp value

which is the same result as in standard BOPM (crisp case) with u2 and d2

as up and down jump factors as expected.

Also it is easy to verify the following claims

Second derivative

positive condition

Second

derivative

negative

condition

Second

derivative zero

condition

0

C (u3-d3)(p0u2-X) >

(u2-d2)(p0u1-X)

(u3-d3)(p0u2-X)

(u1-d1)(p0u2-X)

(u2-d2)(p0u3-X) 1

Neutral f 1

Depression f < 1


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These f values may be decided from the expert knowledge of the

investors, after carrying out extensive research on the historical data.

4.2. Justification for values of Control

Parameter f

As an example consider the expression for C0 interval in the one period

fuzzy BOPM

This can also be considered as

C0 =

+

+ )

11(),

11(

rr

where , , , are some constants for a given value of and R =

1+r.

Clearly to increase C0 in the booming economy by using R* instead of

R we need to have R* > R which in turn implies f >1. Similar observations

can be made for the other economy states.

But the question which arises now is how you would identify the state of

economy at any particular point of time. This can be achieved through the

concept of fuzzy pattern recognition which in turn uses similarity

measures.

4.3. Pattern Recognition & Similarity

Measures

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Let at any point in time the jump factors as interpreted by an investor

be u & d (TFNs). Now we wish to know the state of the economy to

which these jump factors resemble the most. For this we need the

concept of similarity measures.

Let B = (u,d) be the new fuzzy input (consisting of two fuzzy numbers:

u and d)

E1, E2, E3 be the three fuzzy classes each having two features Ei1 & Ei2,

where

Ei1 refers to up jump factor in Ei.

Ei2 refers to down jump factor in Ei.

Define the similarity measure between B & Ei as

[B,Ei] = (u,Ei1) + (d,Ei2) , where

(u,Ei1) =2

1(u 1iE + 1iEu )

u 1iE = xmax min { )(),(

1xx

Eiu } = inner fuzzy product

between u & 1iE

u 1iE = xmin max { )(),(

1xx

Eiu } = outer fuzzy product

between u & 1iE

Let max [B, Ei] = [B, Er] 1i3then we say that the current market state resembles Er state the most

Now once we know the current market state by the above pattern

recognition procedure, we take the corresponding control parameter value f

and use the modified risk free rate of interest R* in the fuzzy BOPM to

calculate the option price which is expected to capture a broader knowledge

of the market as compared to the earlier fuzzy BOPM.

5. Conclusion

This paper tells about the basic aim and strategy involved in developing

any Option Pricing model and provides an overview of the Risk Neutral

valuation strategy for pricing options in a vague market published by

Silvia Muzzioli and Costanza Torricelli in 2004.

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But this existing model doesnt take into account the role of overallmarket sentiments in an investors decisions at any point of time. Thisincapability is partly due to the use of risk free rate of interest directlyinto the model, whatever may be the state of the market. Thisassumption is a valid one only when the expected returns from riskyinvestments are comparable to those from risk free investments whichis not the case in skewed economy states like when the economy isbooming or in depression.

Thus this paper review tries to overcome this incapability by integratingthe existing model with fuzzy logic and fuzzy pattern recognition toidentify the state of economy at any point in time and consequently usea modified risk free rate of interest (R* = Rxf) while calculating the

premium value. This approach is intuitively more appealing and is thusexpected to give better accuracy than the existing model.

A further possible extension could be to use these steps on past dataand validating these results

P 21 f 21

options pricing - fuzzy term paper

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