Binary Option

Allied Academies International Conference page 71

Proceedings of the Academy of Accounting and Financial Studies, Volume 9, Number 1 New Orleans, 2004

A STUDY ON THE BINARY OPTION MODELAND ITS PRICING

Bin Peng, Nanjing University of Science and TechnologyYuqi Han, Nanjing University of Science and Technology

[email protected]

ABSTRACT

In this paper, the valuation and applications of an exotic binary option are discussed thatincludes features of cash-or-nothing option and asset-or-nothing option. We established a pricingmodel for binary option and derived the analytical solutions of the model by using a conventionalBlack-Scholes option-pricing method. We further dissertated the application of a Binomial Treemethod on the binary option pricing and provided numerical experiments, which verify the validityof the Binomial Tree method. Therefore, we concluded that the Binomial Tree method is a goodestimator for the value of binary option.

Key word: Binary Option, Black-Scholes method, Binomial Tree method.

INTRODUCTION

A standard option is a contract that gives the holder the right to buy or sell an underlyingasset at a specified price on a specified date. The payoff depends on the underlying asset price. Thecall option gives the holder the right to buy an underlying asset at a strike price; the strike price istermed a specified price or exercise price. Therefore the higher the underlying asset price, the morevaluable the call option. If the underlying asset price falls below the strike price, the holder wouldnot exercise the option. Binary option is an exotic call option with discontinuous payoffs. The optionpays off a fixed, predetermined amount if the underlying asset price is beyond the strike price onits expiration date.. We discuss two types of binary options here: asset-or-nothing call option andcash-or-nothing call option. For the first type, the option pays off nothing if the underlying assetprice ends up below the strike price. It pays an amount equal to the underlying asset price if it endsup above the strike price. For the second type, the option pays off nothing if the underlying assetprice ends up below the strike price and pays a fixed amount Q if it ends up above the strike price.Note that for the binary option the underlying asset is the stock and the underlying asset price istermed the stock price.

Fischer Black and Myron Scholes (1973) created the Black-Scholes method of option pricingand Cox, J. C., S. Ross, and M. Rubinstein (Sept. 1979) proposed the Binomial Tree option pricingmethod, which laid the foundation on the new securities pricing. In recent years, more and morepeople pay much attention on the option pricing. Binomial Tree option pricing was found to be themost simple and powerful technique that can be used to solve many complex option-pricingproblems in contrast to the Black-Scholes method and other complex option-pricing methodsreferred by Wilmott, P., J. Dewynne, and S. Howison (May 1997). Single European and American

Allied Academies International Conference

New Orleans, 2004 Proceedings of the Academy of Accounting and Financial Studies, Volume 9, Number 1

standard option pricing formulae are published by Fischer Black and Myron Scholes (1973) andCox, J. C., S. Ross, M. Rubinstein (Sept. 1979) and Churchill, R. V. and J. w. Brown (1985.). Sincethe binary option are not combinations of these options, in the following section, we extend Black-Scholes method to derive valuation formula of the binary option of asset or nothing call option andcash or nothing call option. In the third section we shall study the Binomial Tree method applied tothe binary option pricing. At last, we provide simulated computation, which indicate the validity ofthe Binomial tree method compared to Black-Scholes method.

. BINARY OPTION VALUATION

Our objective is to establish the binary option pricing model and derive its analytic solutionin a Black-Scholes method. The asset price is assumed to follow the lognormal random walk, andthere are no transaction costs, The interest rate is taken to be continuous and constant over the optionlife, so the expected return is the risk free interest. The asset pays no dividends during the optionperiod. The valuation method is a risk-neutral valuation approach. Therefore according to John C.Hull (July 2002), we start the deviation from the Black-Scholes equation:

(1)rf

sfs

sfrs

tf =

++

2

222

21

Where f is the option value, s is the underlying asset price, t is the time, is the volatility, r is therisk-free interest rate. Considering the characteristics of the binary option, its pricing model is equalto the Black-Scholes equation (1).

A. ASSET-OR-NOTHING CALL OPTION

If the stock price never hits the strike price x at expiration, then the option is worthless, thus on thelines x and blow the lines x, the option value is zero. If surpass the price x, we let the finalTs

payment of the option be (stock price at maturity). If is the value of asset-or-nothing callTs ),(1 tsc Toption on its expiration date, then the final boundary condition of equation (1) is

for t=T (2).=),(1 tsc T 0

Tsxsxs

T

T

>

With the assumption that the expected return is the risk-free interest rate, we get

Such that ),(1)(

1 TsEcec TtTr = Tt 0

(3)( ) TTTtTr dssfTscec ){,1)(1 +

= Tx

TTtTr dssfse ){)( +=

Note that the limits on the integral go from x to +infinite. Since if the stock price was below x, theoption value at expiry would be zero and the function would use 0 instead.),(1 Tsc T

Recall that the stock price follows the lognormal distribution and it turns out that theprobability distribution function for isTs



With (4)

= 2

2

)(2)(ln

exp)(2

1)( tTus

stTsf T

T

T ))(2(ln2

tTrsu +=

Applying the relationship between s and , we let Ts

, ( ) ( )

+== tTrussssespss TTT 22

1lnexplnln

so equation can be expressed as( )3( ) ( ) ( )

( )( ) TTTx

T

dstT

usus

tTstTsc

= + 2

22

1 2ln

lnexp2

121exp

(5)( )( ) ( )

TT

xT

dstT

tTustTs

s

=

+22ln

21exp

21

By using the transformation , we can easily get( )[ ]2)(ln1 tTustTy T =

if =+infinite then y=+infinite, if then Ts xs T =( ) ( ) 122

1/ln1 dtTrxstT

y =

++=

So equation (5) can be written as

(6)( )( ) ( )1121 12exp2

11

dsNdNsdyyscd

==

= + B. CASH-OR-NOTHING CALL OPTION

In this option, if s never reaches strike price x, then the option is worthless, thus on the linex and below the line x, the option value is zero. If s exceed strike price x, the final valuation ofoption is equal to a fixed amount Q. If is the value of cash-or-nothing call option on its),(2 tsc Texpiration date, then the final boundary condition of equation (1) is

for t=T (7)( )

0

,2Q

ttscxsxs

t

t

>

By using the risk-neutral valuation approach, we can get,

=( )( )TscEec TtTr ,2)(2 =( ) TTTtTr dssfTsce ){,2)( +

( ) Tx

TtTr dssQfe += )(

(8)( ) Tx TtTr dssfQe += )( ( )xsPQe TtTr >= )(

dytTsds TT =



Note that the limits on the integral go from x to +infinite since if the stock price was below x, thevalue would be zero and function would use 0 instead. Using the assumption that stock),(2 Tsc Tprice follows the lognormal distribution

(9)

+ tTrssT ),2(ln~ln2

According to (9) we can obtain( ) ( )

( ) ( )

>

=

>=>

tT

tTrsx

tT

tTrssP

xsPxsP

T

TT

22

21lnln

21lnln

lnln

=

( ) ( )( ) ( )22

2

dNdN121/ln

1 ==

=tT

tTrsxN

So equation (9) becomes

where (10) ( ) ( )22 dNQec tTr = ( ) ( ) tTdtTrxsd =

+= 122 2

1/ln

III. BINOMIAL TREE METHOD TO THE BINARY OPTION PRICING

According to (6) and (10), we can derive the analytic solution of binary option in the Black-Scholes method. , which is the basis for determining how accurate the Binomial Tree method isIn the following section, we shall use the BTM to evaluate the binary option prices.

The Binomial Tree method uses the idea that asset price follows a multiplicative binomialprocess over discrete periods. Each small time steps, the asset price can either increase or decrease.In other words, there are two different possibilities at each point. , su is the new asset price if itincreases, and sd is the new asset price if it decreases, the probability that the asset increase is p,therefore the probability it decrease is 1-p.. This process is repeated for every small , until timetT is reached. Considering One assumption is that ud=1, so if the asset increase and then decreased,it would be back at the original starting asset price. The other assumption is that the expected returnfor an asset is the risk-free interest rate, therefore, we can derive the formulas for p, u and d are asfollow according to John, C, Hull (July 2002):

(11)( ) ( )dudep tr = / teu = ted = We assume the stock pays no dividends. The increasing rate of the asset is u, the decreasing

rate of the asset is d. And we have , ,. r is riskless interest rate and it is11 >+> ru rd +



values after two periods : , , ; at the end of three periods , the asset has fourt2 2su sud 2sd t3possible values: , , , . At the end of periods , the asset has possible3su dsu 2 2sud 3sd i ti 1+ivalues: ;at the expiration date , the asset has N+1 possible values( ) jij dsuis =+ 1 tNT =

. For j=0,1,2N. So, the asset prices are calculated using the following( ) jNj dsuNs =+ 1formula: (Cox, J. C., S. Ross, and M. Rubinstein. Sept. 1979)

. For j = 0, 1, 2, , i and i = 1, 2, , N (12)jij

ij dsus=

Basically, at time i+1, the asset changed i times, and could be any combination of increasing anddecreasing. It is important to pick N large enough. So is small enough to provide for a goodtsample of asset prices in the Binomial Tree.

Once the asset prices are calculated for the whole tree, the values are then computed bystarting at the last step in the tree and working backward. Since the value on the expiry date isknown by Rubinstein, M. and E. Reiner (October 1991). The formula for the binary option value atthe final time is as follows:

For the asset-or-nothing option,

j = 0, 1, ... , N (13)=

0,jnj

jNdsuf

xdsuxdsu

jNj

jNj

For the cash-or-nothing option,

j=0,1,..,N (14)= 0,Q

f jNxdsuxdsu

jNj

jNj

and moving back one step at a time, the remainder of the option values are calculated as follows:

, (15)( )[ ]1,1,1, 1 +=+ += jijitrji fppfef 10 Ni ij 0The logic behind this calculation is for option price , the probability it moves up is p, or thejif ,

probability in one time step and has value is p, then the probability it moves down and has1,1 ++ jif

value is 1-p, and then discount it back one time step for the present value of the option.1, +jif t

. NUMERICAL EXPERIMENT

We now give numerical experiment to determined how valid the Binomial Tree method forthe binary option pricing is in contrast to the Black-Scholes method. The parameters used in thenumerical experiment are: S=50, r=0.1, =0.3, the fixed amount Q=25. strike price was equal to50,51 and 52, the total time is 6 months. First using 256 total time steps and then increasing to 512and 1024 steps to improve accuracy ran the Binomial Tree. Table1 contains the numerical resultsfor the asset-or-nothing call option Table2 shows the numerical results for the cash-or-nothingoption. The numerical results for the analytical solution are computed using the Black-Scholesmethod .from formula (6) and (10). The values of binary option with finite time steps are computedusing the Binomial Tree method from formula (13), (14) and (15). It can be observed that the prices



of cash-or-nothing call option and asset-or-nothing call option are closer with the analytical solutionwith the increase of step times. All the numerical example verify the validity of the Binomial Treemethod.

Table1. Numerical results for asset-or-nothing call option (s=50,T=6months)

Strikeprice

Analyticalsolution(a)

N=256 N=512 N=1024

Value(b) %(b/a) Value(c) %(c/a) Value(d) %(d/a)505153

32.2029.9026.10

32.8530.5025.59

102.02102.0198.46

32.5129.1525.65

100.9697.4998.28

32.2729.9026.21

100.22100.00100.42

Table2. Numerical results for Cash-or-nothing call option (s=50,T=6months)

Strikeprice

Analyticalsolution(a)

N=256 N=512 N=1024

Value(b) %(b/a) Value(c) %(c/a) Value(d) %(d/a)505153

13.2012.2510.49

13.6912.5310.16

103.79101.4596.76

13.5311.8610.20

102.5096.8297.14

13.4112.2310.46

101.5999.0399.62

. CONCLUSION

The Black-Scholes method is an exact calculation of the option value for a predeterminedstream of stock prices. The analytical solution was the basis for determining how accurate theBinomial Tree method is. In this paper, we derived the analytical solution for the binary option inthe Black-Scholes method and numerical solution for the binary option in the Binomial Treemethod.. Moreover, we provided the example to verify the validity of the Binomial Tree methodfor the binary option pricing compared to the Black-Scholes method. It can be concluded thatBinomial Tree method is a strong predictor for the new type of the option-binary option pricing.

REFERENCES

Black, F and Scholes, M (1973). The pricing of option and corporate liabilities, Journal of Political Economy 81, 637-659.

Churchill, R, V and Brown, J, w (1985). Fourier Series and Boundary Value Problem, International Student Edition(McGraw Hill, New York).

Cox, J, C, Ross, S, and Rubinstein, M (Sept. 1979). Option Pricing: A Simplified Approach. Journal of FinancialEconomics 7, 229-263.

John, C, Hull (July 2002), Options, futures and other derivatives, Prentice Hall, 5th edition, New York.



Longstaff, F and Schwartz, E (2001). Valuing American options by simulation: a simple least-squares approach, Reviewof Financial Studies 14, 113-147.

Rubinstein, M and Reiner, E (October 1991). Unscrambling the binary code, Risk Magazine, 4(9), 75-83.

Rubinstein, M (1994), Implied binomial trees, Journal of Finance 49,771-818.

Wilmott, P, Dewynne, J and Howison, S (May 1997), Option Pricing: Mathematical Models and Computation, OxfordFinancial Press, Oxford.

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