Pricing Formulae of Power Binary and Normal Distribution Standard Options and Applications
aa r X i v : . [ q -f i n . P R ] M a r Pricing Formulae of Power Binary and Normal DistributionStandard Options and Applications Hyong-Chol O, Dae-Sung Choe , Faculty of Mathematics,
Kim Il Sung
University, Pyongyang , D P R Ke-mail: [email protected] Abstract
In this paper the Buchen’s pricing formulae of (higher order) asset and bond binaryoptions are incorporated into the pricing formula of power binary options and a pricingformula of ”the normal distribution standard options” with the maturity payoff relatedto a power function and the density function of normal distribution is derived. And astheir applications, pricing formulae of savings plans that provide a choice of indexingand discrete geometric average Asian options are derived and the fact that the price ofdiscrete geometric average Asian option converges to the price of continuous geometricaverage Asian option when the largest distance between neighboring monitoring timesgoes to zero is proved.
Keywords power binary option; normal distribution standard option; savings plansthat provide a choice of indexing; discrete geometric average Asian option
Option pricing problem is one of the main questions in financial mathematics. Thestudying method expressing complex exotics or corporate bonds as a portfolio of binariesbecomes a widely used way in solving financial practical problem [2, 3, 5, 7, 8, 10–15].In this paper, solution representations of Black-Scholes PDE with power functions or cu-mulative distribution functions of normal distribution as a maturity payoff are provided andusing them savings plans that provide a choice of indexing and discrete geometric averageAsian options are studied. This becomes binary option methods and their applications.The basic idea of expressing the payoffs of complex options in terms of binary optionswas initiated in Rubinstein and Reiner(1991) [7], where they considered the relationship ofbarrier options and binaries. Ingersoll(2000) [5] extended the idea by expressing complexderivatives in terms of event- driven binaries. Event ε driven binary option pays one unitof underlying asset if and only if the event ε occurs, otherwise it pays nothing.Buchen(2004) introduced the concepts of first and second order asset and bond binaryoptions and provided the pricing formula for them using expectation method. And thenhe applied them to pricing some dual expiry exotic options including compound option,chooser option, one time extendable option, one time shout option, American call optionwith one time dividend and partial barrier options.Skipper(2009) [15] introduced a general concept of multi-period, multi-asset M-binarywhich contains asset and bond binaries of Buchen [2] and provided the pricing formula byexpectation method. They noted that portfolios of M-binaries can be used to staticallyreplicate many European exotics and applied their results to pricing a simple model of anexecutive stock option (ESO) which has been used in practice to value corporate remuner-ation packages. Hyong-chol O and Dae-sung Choe
In [8], the authors extended the concepts of Buchen’s first and second order asset andbond binary options to a concept of higher order binary option, deduced the pricing formulaeby PDE method and use it to get the pricing of some multiple-expiry exotic options suchas Bermudan option and multi time extendable option and etc.In [11] and [14] the authors obtained some pricing formulae of corporate bonds withcredit risk including the discrete coupon bonds using the method of higher order binaryoptions. In [12] the authors extend higher order binary options into the case with timedependent coefficients and using it studied the discrete coupon bonds.Some banks have a service of savings account with choice item of interest rates, theholder of which has the right to select one among domestic and foreign interest rates and [1]calculated the price of such savings plans that provide a choice of indexing using expectationmethod. In [9] the authors studied the price of such savings plans that provide a choice ofindexing using PDE method. In such savings plans, the foreign currency price of the expirypaypff is related to the the inverse of underlying asset price [9].The geometric average Asian option is one of well studied exotics. In [6] they studiedcontinuous geometric average Asian options using PDE method and provided the pricingformulae, call-put parities, binomial tree models and explicit difference scheme on charac-teristic lines. In [3] they studied continuous models of geometric average Asian options withzero dividend rates using expectation and PDE methods and provided the pricing formulaeof discrete geometric average Asian options with expectation method. They showed thatthe prices of discrete geometric average Asian options converge to the price of the corre-sponding continuous geometric average Asian options when the monitoring time intervalgoes to zero. The expiry payoffs of discrete geometric average Asian options are related tothe n-th root of the price of underlying asset.The purpose of this article is to study the pricing model of the option whose terminalpayoff is related to the power of the price of the underlying asset in the viewpoint of PDEand apply them into pricing problems of some financial contracts.First we find the pricing formula of so called ”power binary option”, the expiry payoffof which is the power of underlying asset price, by using the partial differential equationmethod. Our formula includes the pricing formulae of [2, 8] for the (higher order) asset andbond binary options. We apply it to calculate the price of savings plans that provide achoice of indexing.Then we derive the solution representation of the terminal value problem of the Black-Scholes partial differential equation whose terminal value is the product of a power functionand a normal distribution function and using it we obtain the pricing formula of discretegeometric average Asian option. And we prove that the price of discrete geometric averageAsian option converges to the price of the continuous geometric average Asian option asthe interval between neighboring monitoring times goes to zero. It is very interesting in theview of PDE that the limit of the solutions to connecing problems of initial value problemsof Black-Scholes partial differential equations on several time intervals becomes the solutionto the initial value problem of another partial differential equation. This shows that usingthe backgrounds of problems a solving problem of a complicated differential equation canbe reduced to solving problems of simple differential equations.The rest of this paper is organized as follow. In section 2 we provide the pricing formulaof an European option (power binary option and normal distribution vanilla option) whosepayoff is related to the power of the underlying asset price and density function of the ricing Formulae of Power Binary and Normal Distribution Standard...
Definition 1
Let r, q, σ be interest rate, the dividend rate and the volatility of the under-lying asset respectively. Consider the terminal-value problem of the following Black-Scholesequation. ∂V∂t + 12 σ x ∂V∂x + ( r − q ) x ∂V∂x − rV = 0 , (0 < t < T, x >
0) (1) V ( x, T ) = x α (2)According to preposition 1 of [8], the solution of this equation is given as following: V ( x, t ) = e − r ( T − t ) σ p π ( T − t ) Z ∞ z e − σ T − t ) (cid:16) ln xz + (cid:16) r − q − σ (cid:17) ( T − t ) (cid:17) z α dz (3)This V ( x, t ) is called a ”standard power option” or ”standard α -power option” with expiry payoff x α . We denote the price of standard α -power option as M α ( x, t ).The pricing formula of standard power option is provided as following. Theorem 2.1
The price of α -power standard option, a solution of the Black-Scholes equa-tion (1) , (2) is given as M α ( x, t ) = e µ ( T − t ) x α (4)Here µ ( r, q, σ, α ) = ( α − r − αq + σ α − α ) (5) Proof
Using the transformation y = (cid:20) ln xz + (cid:18) r − q − σ ασ (cid:19) ( T − t ) (cid:21) (cid:16) σ p ( T − t ) (cid:17) − Then the equation (3) is changed into the following. V ( x, t ) = e µ ( T − t ) x α √ π Z ∞−∞ e − y dy = e µ ( T − t ) x α . (Q.E.D) Hyong-chol O and Dae-sung Choe
Definition 2
The binary contract based on the standard α -power option is called ” α -power binary option”. i.e the price of a power binary option is a solution of (1) satisfyingthe terminal value condition V ( x, t ) = x α · sx > sξ ) (6)Here s(+ or -) is called ”sign indicator” of upper or lower binary options and we denote theprice of a α -power option as ( M α ) sξ ( x, t ) .Since( M α ) + ξ ( x, T ) + ( M α ) − ξ ( x, T ) = M α ( x, T )we have the following symmetric relation between the α -power standard option priceand the corresponding upper α -power binary option and lower α -power binary option.( M α ) + ξ ( x, t ) + ( M α ) − ξ ( x, t ) = M α ( x, t ) (7) Theorem 2.2
The price of α -power binary option (solution of (1) , (5) ) is given as follow-ing. ( M α ) sξ ( x, t ) = e µ ( T − t ) x α N ( sd ) (8) Here d = d ( xξ , r, q, σ, α, T − t )= (cid:20) ln xξ + (cid:18) r − q − σ ασ (cid:19) ( T − t ) (cid:21) ( σ √ T − t ) − (9) N ( d ) = 1 √ π Z d −∞ e − y dy Proof
From the preposition 1 of [8], we have( M α ) sξ ( x, t ) = e − r ( T − t ) σ p π ( T − t ) Z ∞ z e σ T − t ) (cid:20) ln xz + (cid:16) r − q − σ (cid:17) ( T − t ) (cid:21) z α sz > sξ ) dz Let’s use the variable transformation y = (cid:20) ln xz + (cid:18) r − q + σ ασ (cid:19) ( T − t ) (cid:21) ( σ p ( T − t )) − Then ln z = (cid:20) ln x + (cid:18) r − q − σ ασ (cid:19) ( T − t ) (cid:21) − y ( σ p ( T − t ))and consider 1( sz > sξ ) = 1( s ln z > s ln ξ ) = 1( sy < sd )Then we have ( M α ) + ξ ( x, t ) = e µ ( T − t ) x α √ π Z + ∞−∞ e − − y sy < sd ) dy = I ricing Formulae of Power Binary and Normal Distribution Standard... y ′ = sy, dy ′ = sdy we have I = e µ ( T − t ) x α √ π s Z + s ∞− s ∞ e − − y ′ y ′ < sd ) dy ′ = e µ ( T − t ) x α √ π Z sd −∞ e − − y ′ dy ′ == e µ ( T − t ) x α N ( sd )(QED) Remark 1:
Consider the case when the binary condition of the expiry payoff functionhas more general form. i.e V ( x, t ) = x α · sx β > sξ ) (10)We have 1( sx β > sξ ) = 1( s · sgn ( β ) x > s · sgn ( β ) ξ β ) = 1( tx > tζ )So if we t = s · sgn ( β ) , ζ = ξ β , the solution of (1),(10) is given as ( M α ) tζ ( x, t ). Remark 2:
When α = 0 , the α -power binary option is the cash binary option andwhen α = 1 , it is the asset binary option. Theorem (2.2) includes the results for the firstorder binary options of [2, 3, 6] as a special case. And as shown in section 3, by using the α -power binary options we can represent the prices of financial contracts which are not ableto be represented in terms of cash or asset binary options. Definition 3
The solution of (1) with the expiry payoff V ( X, T ) = X β N (cid:20) δ (cid:18) X i K , τ , τ ′ , α (cid:19)(cid:21) (11) δ (cid:18) X i K , τ , τ ′ , α (cid:19) = (cid:20) ln X i K + (cid:18) r − q − σ ασ (cid:19) τ (cid:21) ( σ p τ ′ ) − is called a ”price of power normal distribution standard options”.Denote τ = T − t . Theorem 2.3
The solution of (1) , (11) is provided as follows V ( X, τ ; τ , τ ′ ) = X β e µ ( β ) τ N ( d ) (12) where d = (cid:20) ln X i K + (cid:18) r − q − σ iβτ + ατ iτ + τ σ (cid:19) ( iτ + τ ) (cid:21) ( σ p i τ + τ ′ ) − == δ (cid:18) X i K , iτ + τ , i τ + τ ′ , iβτ + ατ iτ + τ (cid:19) (13) and µ ( β ) = ( β − r − βq + σ β − β ) can be given by (5) Hyong-chol O and Dae-sung Choe
Proof
From proposition 1 of [8], the solution of (1),(11) is expressed as follows: V ( X, τ, τ , τ ′ ) == e − rτ σ √ πτ Z + ∞ z β z √ π Z δ ( ziK ,τ ,τ ′ ,α ) −∞ e − y dy e − σ τ (cid:16) ln Xz +( r − q − σ ) τ (cid:17) dz = I (14)Here δ ( z i K , τ , τ ′ , α ) = (cid:20) ln z i K + ( r − q − σ ασ ) τ (cid:21) ( σ p τ ′ ) − In (14) take the variable transformation: y = y p τ ′ p τ ′ + i τ + i (cid:20) ln Xz + ( r − q − σ βσ ) τ (cid:21) ( σ p τ ′ + i τ ) − (15) y = (cid:20) ln Xz + ( r − q − σ βσ ) τ (cid:21) Then we have (cid:12)(cid:12)(cid:12)(cid:12) ∂ ( y , y ) ∂ ( y, z ) (cid:12)(cid:12)(cid:12)(cid:12) = 1 zσ √ τ p τ ′ p τ ′ + i τ And z > ⇔ y < + ∞ and y < δ ( z i K , τ , τ ′ , α ) ⇔⇔ y < σ p τ ′ + i τ (cid:20) ln X i K + ( r − q − σ τ + iτ ) + ( ατ + iβτ ) σ = (cid:21) = δ (cid:18) ln X i K , τ + iτ, τ ′ + i τ, ατ + iβττ + iτ (cid:19) = d In the exponent of the last exponential function in the integrand of (14), we consider exp (cid:18) β ln Xz (cid:19) = X β z β e − rτ · e ( rβ − qβ + σ ( β − β ) ) τ y = p τ ′ + i τ p τ ′ y − y i √ τ p τ ′ Then we can rewrite equation (14) by follows: I = X β e µ ( βτ ) π p τ ′ + i τ p τ ′ Z d −∞ Z + ∞−∞ e √ τ ′ i τ √ τ ′ y − y i √ τ √ τ ′ y dy dy (16) ricing Formulae of Power Binary and Normal Distribution Standard... p τ ′ + i τ p τ ′ y − y i √ τ p τ ′ ! + y = p τ ′ + i τ p τ ′ y − y i √ τ p τ ′ ! + y Since √ τ ′ + i τ √ τ ′ R + ∞−∞ e √ τ ′ i τ √ τ ′ y − y i √ τ √ τ ′ ! dy = √ π , we can rewrite equation (16) as fol-lows: I = X β e µ ( β ) τ τ √ π Z d −∞ e − y dy = X β e µ ( β ) τ N ( d ) . ( QED ) In the savings plans with choice item of interest rates, the holder has the right to select oneamong the domestic rate and foreign rate. Thus this savings contract is an interest rateexchange option [1, 9].Let r d denote the domestic (risk free) interest rate, r f the foreign (risk free) interest rate, X ( t ) domestic currency / foreign currency exchange rate, Y ( t ) = X ( t ) − : foreign currency/ domestic currency exchange rate. Assumption
1) The invested quantity of money to the savings plan is 1 unit of (domestic) currency, r d , r f are both constants.2) The maturity payoff is as follows: V f = V d · X − ( T ) = max e rd T X − (0) e r f T ( unit of f oreign currency )3)The exchange rate satisfies the model of Garman-Kohlhagen [4]: dX ( t ) X ( t ) = λ · dt + σdW ( t ) .
4) The foreign price of the option is given by a deterministic function V f = V f ( X, t ). Underthe assumptions 1)-4), V f = V f ( X, t ) , the foreign price function of the Savings Plans thatprovide a choice of indexing is the solution of the following initial value problem of PDE ∂V f ∂t + 12 X ∂ V f ∂X + (cid:0) σ + r d − r f (cid:1) X ∂V f ∂X − r f V f = 0 (17) V f ( X, t ) = max { e r d T X − ( T ) , X − (0) e r f T } . (18)In fact, by It formula and assumption 4), d (cid:18) X ( t ) (cid:19) = ( − µX − ( t ) + σ X − ( t )) dt − σX − ( t ) dW ( t ) Hyong-chol O and Dae-sung Choe
By ∆-hedging, construct a portfolio ΠΠ = V f ( X, t ) − ∆ · X − (This portfolio consists of a sheet of option, ∆ units of domestic currency and its price iscalculated in foreign currency.) We select ∆ so that d Π = r f dt . By It formula, we have dV f = (cid:18) ∂V f ∂t + 12 σ X ∂ V f ∂X + µX ∂V f ∂X (cid:19) dt + σX ∂V f ∂X dW. Then d Π = dV f − ∆ (cid:18) X (cid:19) − ∆ · r d X − dt == (cid:18) ∂V f ∂t + 12 σ X ∂ V f ∂X + µX ∂V f ∂X (cid:19) + σX ∂V f ∂X dW − ∆ · ( − µX − + σ X − ) · dt −− ∆ σX − dW − ∆ r d X − dt = (cid:18) ∂V f ∂t + 12 σ X ∂ V f ∂X + µX ∂V f ∂X − ∆ · ( − µX − + σ X − ) − ∆ r d X − (cid:19) dt ++ (cid:18) σX ∂V f ∂X − ∆ σX − (cid:19) dW = r f Π dt = r f ( V f − ∆ · X − ) · dt Thus (cid:18) ∂V f ∂t + 12 σ X ∂ V f ∂X + µX ∂V f ∂X − ∆ · ( − µX − + σ X − ) − ∆ r d X − (cid:19) dt ++ (cid:18) σX ∂V f ∂X − ∆ σX − (cid:19) dW = 0If we select ∆ = X ∂V f ∂X , then we have ∂V f ∂t + 12 σ X ∂ V f ∂X + µX ∂V f ∂X − X ∂V f ∂X ( µX − − σ X − − r d X − + r f X − ) − r f V f = 0Thus we have the PDE model of option price. ∂V f ∂t + 12 σ X ∂ V f ∂X + ( σ + r d − r f ) X ∂V∂X − r f V f = 0(17) is a BS-PDE with rate r f , dividend 2 r f − r d − σ and volatility σ . We can representthe solution of the problem (17),(18) in terms of the power binary options. x = X (0) isthe known quantity and the maturity payoff function (18) is rewritten as follows. V f ( X, T ) = max { e r d T X − , x − e r f T } == e r d T X − · (cid:16) e r d T X −
1) = − r d , µ (0) = − r f ) Theorem 3.1
Under the assumptions 1 4, the foreign price of the savings plans thatprovide a choice of indexing is the solution of the problem (17) , (18) and given by V f ( x, t ) = e r d t X − N ( − d ) + x − e r f t N ( d ) (19) Here d = ln x − e r f t e r d tX − − σ T − t ) σ √ T − t , d = ln x − e r f t e r d tX − + σ T − t ) σ √ T − t (20) Remark.
The financial meaning of this formula is clear. e r d T X − is the foreign priceof the present value of the savings account when 1 unit (domestic currency) is saved withthe domestic rate and x − e r f t is the present value (foreign currency) of the savings accountwhen 1 unit (domestic currency) is saved with foreign rate. N ( − d ) and N ( d ) are theproportion of these two quantities in the option price. This proportion depends on which isbigger among them and especially at expiry date one is 0 and the other is 1. The formula(19),(20) is equal to that of [9] when the inflation rate is 1. The geometric average Asian option is an option whose expiry payoff depends on notonly the terminal price of the underlying asset but also the geometrical average of theunderlying asset’s prices on its life time [6]. The price of this option at time t depends onthe underlying asset’s price at time t as well as the geometrical average J t of the underlyingasset’s prices on the time interval [0 , t ].Let 0 = T < T < · · · < T m = T be discrete monitoring times and X n be the price ofthe underlying asset. Then the discrete geometric average of the underlying asset’s pricestill the time T n is expressed by J n = J T n = n Y i =1 X i ! n = e n P ni =1 ln X i The option with expiry payoff( J T − K ) + = ( m p X · · · X m − K ) + is called an discrete geometrical average Asian option .Similarly the geometrical average of the underlying asset’s price to time [0 , t ] is expressedas J t = e t R t ln S τ dτ Hyong-chol O and Dae-sung Choe
The option with expiry payoff( J T − K ) + = (cid:16) e t R t ln S τ dτ − K (cid:17) + is called a continuous geometric average Asian (call) option(with fixed exerciseprice) . The price of the continuous geometric average Asian (call) option (with fixedexercise price) is the solution of the following problem ∂V∂t + J ln X − ln Jt ∂∂J + σ X σ V∂X + ( r − q ) X ∂V∂X − rV = 0 (21) V ( X, J, T ) = ( J T − K ) + on the domain { ≤ X < ∞ , ≤ J < ∞ , ≤ t ≤ T } . Lemma 3.1 [6] The price of the geometric average Asian call option with fixed exerciseprice is given by V ( X, J, t ) = e − r ( T − t ) { [ J t S T − t ] T e (cid:18) r ∗ + ( σ ∗ )22 (cid:19) ( T − t ) N ( d ∗ ) − KN ( d ∗ ) } (22)Here d ∗ = 1 T ln J t X T − t K T + [ r ∗ + ( σ ∗ ) ]( T − t ) σ ∗ √ T − t , d ∗ = d ∗ − σ ∗ √ T − tr ∗ = ( r − q − σ T − t T , σ ∗ = σ ( T − t ) √ T We now establish the differential equation model of the price of the discrete geometricaverage Asian option. let 0 = T < · · · < T n = T be fixed monitoring times and X , · · · , X n be the price of the asset at time T , · · · , T n respectively. The price of discrete geometricaverage Asian option with fixed price with n monitoring times is denoted by V n ( X, t ) . Bydefinition the expiry payoff is equal to V n ( X n , T n ) = (cid:0) n √ X · · · X n − K (cid:1) + . The price of theoption on the time interval T i < t ≤ T i + 1 is denoted by V in ( X, t ) , i = 1 , · , n − T n − < t ≤ T n the price of asset X , · · · , X n − is known quantities, theexpiry payoff can be written as V n − n ( X, T n ) = (cid:16) n p X · · · X n − X n − K (cid:17) (23)On the interval T n − < t ≤ T n , this can be seen as a normal vanilla option with expirypayoff (23) and thus in this interval, the price of discrete geometrical average Asian optionis the solution to the terminal value problem of Black-Scholes equation LV n − n = ∂V n − n ∂t + 12 σ X ∂ V n − n ∂X +( r − q ) X ∂V n − n ∂X − rV n − n = 0 , ( X > , T n − < t < T n ) ,V n − n ( X, T n ) = (cid:16) n p X · · · X n − X n − K (cid:17) + (24) ricing Formulae of Power Binary and Normal Distribution Standard... V n − n ( X, t ) depends
X, t as well as X , · · · , X n − and so V n − n can be writtenas V n − n ( X, t ; X , · · · X n − ). Since X = X n − at time T n − , especially the option price is V n − n ( X, T n − X , · · · , X n − , X )On the interval T n − < t ≤ T n − , the prices X , · · · , X n − , at the monitoring time T , · · · , T n − at T n − < t ≤ T n − is the known quantities and thus at this interval theoption becomes a vanilla option (the solution of BS pde) whose payoff is V n − n ( X, T n − X , · · · , X n − , X ) . The price depends on X, t as well as X , · · · , X n − . By repeatingthis process we get LV in = 0 ( X > , T i < t < T i +1 ) (25) V in ( X, T i ) = V i +1 n ( X, T i ; X · · · , X i , X ) , i = ¯1 , n − Theorem 3.2
The price of discrete geometric average Asian options with fixed exercisestrike price with n monitoring times is expressed by V n − kn ( X, t ) = n p X · · · X n − k X kn e θ k ( t ) N ( d n − k ( t ) + 1 n ∆ k ( t )) − Ke − r ( T n − t ) N ( d n − k ( t )) T n − k ≤ t < T n − k +1 , ( k = ¯1 , n −
1) (26)
Here ∆ k ( t ) = σ vuut k ( T n − k +1 − t ) + k − X i =1 i ( T n − i +1 − T i )) d n − k ( X, t ) = ∆ − n − k { ln X · · · X n − k X k K n + (cid:18) r − q − σ (cid:19) [ k ( T n − k +1 − t )+ k − X i =1 i ( T n − i +1 − T n − i )] } θ k ( t ) = µ ( kn )( T n − k +1 − t ) + k − X i =1 µ (cid:18) in ( T n − i +1 − T i ) (cid:19) = µ ( 1 n )( T n − T n − ) + µ ( 2 n )( T n − − T n − ) + · · · + µ ( kn )( T n − k +1 − t ) Especially at time T = 0 we have V n ( X,
0) = Xe θ n − ( T ) N ( d + 1 n Σ) − Ke − r ( T n − t ) N ( d ) Here
Σ = ∆ n − (0) σ vuut n − X i =1 ( i ( T n − k +1 − T n − i ))2 Hyong-chol O and Dae-sung Choe d = d ( X, T ) = Σ − " ln X n K n + ( r − q − σ n − i X i =1 i ( T n − i +1 − T n − i ) θ n − ( T ) = n − X i =1 µ ( in )( T n − i +1 − T n − i ) Proof
First we find a solution of (24).The expiry payoff of (24) can be written as V n − n ( X, T n ) = ( n p X · · · X n − X n − K ) + = ( n p X · · · X n − X n − K ) · n p X · · · X n − X n > K )= ( n p X · · · X n − X n − K ) · (cid:18) X > K n X · · · X n − (cid:19) . This is expressed by the combination of the prices of power binary options and thus fromTheorem 2.2 for T n − < t < T n we have V n − n ( X, t ; T n ) = n p X · · · X n − (cid:16) M n (cid:17) + KnX ··· Xn − ( X, t ; T n ) − K ( M ) + KnX ··· Xn − ( T, t ; T n )= n p X · · · X n − e µ ( n )( T n − t ) N ( d n ( X, t )) − Ke − r ( T n − t ) N ( d n ( X, t )) (27)Here d n − ( X, t ) = (cid:20) ln X · · · X n − K n + (cid:18) r − q − σ (cid:19) ( T n − t ) (cid:21) ( σ p T n − t ) − d n − ( X, t ) = (cid:20) ln X · · · X n − K n + (cid:18) r − q − σ σ n (cid:19) ( T n − t ) (cid:21) ( σ p T n − t ) − = d n − ( X, t ) + 1 n σ p T n − t = d n − ( X, t ) + 1 n ∆ ( t )Thus (26) has been proved for T n − < t < T n Now we assume that (26) holds on the interval T n − k < t < T n − k +1 and find the priceformula on the interval T n − k − < t < T n − k . On the interval T n − k − < t < T n − k , X , · · · , X n − k − is known quantities and especially X = X n − k at time T n − k . Thus attime T n − k , we have V n − kn ( X, T n − k ; T n − k +1 , T n − , T n ) == n p X · · · X n − k − X ( k +1) /n e θ k ( n − k ) N ( d n − k ( X, T n − k )) − ke − r ( T n − t ) N ( d n − k ( X, T n − k ))= f ( X ) − g ( X ) ricing Formulae of Power Binary and Normal Distribution Standard... d n − k ( X, T n − k ) = ( ln X · · · X n − k − X k +1 K n + ( r − q − σ " k X i =1 i ( T n − k +1 − T n − i ) · σ vuut k X i =1 i ( T n − i +1 − T n − i ) = δ X · · · X n − k − X k +1 K n , k X i =1 i ( T n − k +1 − T n − i ) , , k X i =1 i ( T n − i +1 − T n − i ) ! d n − k ( X, T n − k ) = d n − k ( X, T n − k ) + 1 n ∆ k ( T n − k )= (cid:8) ln X · · · X n − k − X k +1 K n + ( r − q − σ " k X i =1 i ( T n − k +1 − T n − i ) + σ n k X i =1 i ( T n − i +1 − T n − i ) (cid:9) · σ vuut k X i =1 i ( T n − i +1 − T n − i ) = δ X · · · X n − k − X k +1 K n , k X i =1 i ( T n − k +1 − T n − i ) , n P ki =1 i ( T n − i +1 − T n − i ) P ki =1 i ( T n − k +1 − T n − i ) , k X i =1 i ( T n − i +1 − T n − i ) ! where δ is defined by (11). And from (25), V n − k − n ( X, t ) is the solution of (1) withexpiry payoff f ( X ) − g ( X ). Now in order to find the solution V f ( X ) of (1) with the expirypayoff f ( X ) ,we let τ = k X i =1 i ( T n − i +1 − T n − i ) , τ = T n − k − t, τ ′ = k X i =1 i ( T n − i +1 − T n − i ) α = n P ki =1 i ( T n − i +1 − T n − i ) P ki =1 i ( T n − k +1 − T n − i ) , β = k + 1 n , i = k + 1and apply Theorem 2.3. Thus we have V f ( X, t ) = n p X · · · X n − k − X ( k +1) /n e θ k +1 ( t ) N ( d n − k − )Here d n − k − ( X, t ) = (cid:26) ln X · · · X n − k − X k +1 K n + ( r − q − σ " ( k + 1)( T n − k − t ) + k X i =1 i ( T n − i +1 − T n − i ) ++ σ n " ( k + 1) ( T n − k − t ) + k X i =1 i ( T n − i +1 − T n − i ) · σ vuut ( k + 1) ( T n − k − t ) + k X i =1 i ( T n − i +1 − T n − i ) − Hyong-chol O and Dae-sung Choe θ k +1 ( t ) = µ ( k +1 n )( T n − k +1 − t ) + P ki =1 µ ( in )( T n − i +1 − T n − i )In order to find the solution V g ( X, t ) of (1) with expiry payoff g ( X ) ,we let τ = k X i =1 i ( T n − i +1 − T n − i ) , τ = T n − k − t, τ ′ = k X i =1 i ( T n − i +1 − T n − i ) α = 0 , β = 0 , i = k + 1and apply Theorem 2.3. Thus we have V g ( X, t ) = Ke − r ( T n − t ) N ( d n − k − )Here d n − k − ( X, t ) = (cid:26) ln X · · · X n − k − X k +1 K n + ( r − q − σ " ( k + 1)( T n − k − t ) + k X i =1 i ( T n − i +1 − T n − i ) · σ vuut ( k + 1) ( T n − k − t ) + k X i =1 i ( T n − i +1 − T n − i ) − So we have V n − k − n ( X, t ) = V f ( X, t ) − V g ( X, t )= n p X · · · X n − k − X ( k +1) /n e θ k +1 ( t ) N ( d n − k − ) − Ke − r ( T n − t ) N ( d n − k − )Thus we have proved (26).(QED)And now let’s find the limit of the prices of discrete geometric averageAsian options asdiscrete monitoring interval goes to zero. Let V n ( X, t ) be the price of discrete geometricaverage Asian options with fixed exercise price and n monitoring times. Then we have ∀ t ∈ [0 , T ) , ∃ k ∈ { , · · · , n − } , T n − k ≤ t < T n − k +1 : V n ( X, t ) = V n − kn ( X, t )And let V ( X, J, t ) be the price of continuous geometric average Asian call option with fixedexercise price and J ( t ) = e t R t ln X τ dτ .Then we have the following covergence theorem. Theorem 3.3
As the maximum length of the subintervals of the partition goes to zero andlet n go to infinity, then the price of (26) of discrete geometric average Asian call optionsconverges to the price of (22) of continuous geometric average Asian options. That is, wehave lim n →∞ V n ( X, t ) = V ( X, J, t ) , ∀ t ∈ [0 , T ) Proof
For simplicity of discussion, fix time t (0 < t < T ) and assume that { T , T , · · · , T n = T } is the partition of [0 , T ] wiht (n-1) subintervals with same length. (In the case with anypartition, the convergence can be proved in the same way.) Then ∃ k, T n − k ≤ t < T n − k +1 and we have V n ( X, t ) = V n − kn ( X, t ; T n − k +1 , · · · , T n − , T n ) == n p X · · · X n − k X k/n e θ k N ( d n − k ) + Ke − r ( T n − t ) N ( d n − k ) ricing Formulae of Power Binary and Normal Distribution Standard... d n − k ( X, t ) and d n − k ( X, t ) can be rewritten as following. d n − k ( X, t ) = (cid:26) ln X · · · X n − k X k K n + ( r − q − σ (cid:20) k ( T n − k +1 − t ) + k − X i =1 i ( T n − i +1 − T n − i ) (cid:21) ++ σ n " k ( T n − k +1 − t ) + k − X i =1 i ( T n − i +1 − T n − i ) ·· σ p k ( T n − k + 1) − t + k − X i =1 i ( T n − i +1 − T n − i ) ! − = (cid:26) nk ln n p X · · · X n − k X k/n K n + ( r − q − σ (cid:20) ( T n − k +1 − t ) + 1 k k − X i =1 i ( T n − i +1 − T n − i ) (cid:21) ++ kσ n " ( T n − k +1 − t ) + 1 k k − X i =1 i ( T n − i +1 − T n − i ) ·· σ p ( T n − k + 1) − t + 1 k k − X i =1 i ( T n − i +1 − T n − i ) ! − d n − k ( X, t ) = (cid:26) nk ln n p X · · · X n − k X k/n K n + ( r − q − σ (cid:20) ( T n − k +1 − t ) + 1 k k − X i =1 i ( T n − i +1 − T n − i ) (cid:21)(cid:27) ·· σ p ( T n − k + 1) − t + 1 k k − X i =1 i ( T n − i +1 − T n − i ) ! − Then we have T n − k ≤ t < T n − k +1 ↔ n − k − n − T ≤ < t < n − kn − T ↔ k − n − < T − tT ≤ kn − n →∞ kn = T − tT (28)Using (28) we haveln (cid:16) J ( t ) tT (cid:17) = tT · t Z t ln X τ dτ = 1 T lim n →∞ n − k X i =1 (cid:18) Tn − X i (cid:19) == lim n →∞ n n − k X i =1 ln X i = lim n →∞ ln n p X · · · X n − k == ln lim n →∞ n p X · · · X n − k So we have J ( t ) tT = lim n →∞ n p X · · · X n − k (29)6 Hyong-chol O and Dae-sung Choe ( T n − k +1 − t ) + 1 k k − X i =1 i ( T n − i +1 − T n − i ) = ( T n − k +1 − t ) + Tn − k k − X i =1 i = ( T n − k +1 − t ) + Tn − k ( k − k − k == ( T n − k +1 − t ) + k − n − (cid:18) − k (cid:19) T Using (28) and lim n →∞ ( T n − k +1 − t ) = 0, we havelim n →∞ ( T n − k +1 − t ) + 1 k k − X i =1 i ( T n − i +1 − T n − i ) = 13 ( T − t ) (30)Using k ln X ··· X n − k X k K n = nk ln n √ X ··· X n − k X kn K and (28) and (29),we havelim n →∞ nk ln n p X · · · X n − k X k/n K = TT − t ln J ( t ) tT X T − tT K (31)Using k P k − i =1 i ( T n − i +1 − T n − i ) + ( T n − k +1 − t ) = Tn − k − + ( T n − k +1 − t ) and (28) wehave lim n →∞ k k − X i =1 i ( T n − i +1 − T n − i ) + ( T n − k +1 − t ) = T T − tT = T − t d n − k ( X, t ) = (cid:26) ln X · · · X n − k X k K n + ( r − q − σ (cid:20) k ( T n − k +1 − t ) + k − X i =1 i ( T n − i +1 − T n − i ) (cid:21) ++ σ n " k ( T n − k +1 − t ) + k − X i =1 i ( T n − i +1 − T n − i ) ·· σ p k ( T n − k + 1) − t + k − X i =1 i ( T n − i +1 − T n − i ) ! − = (cid:26) nk ln n p X · · · X n − k X k/n K n + ( r − q − σ (cid:20) ( T n − k +1 − t ) + 1 k k − X i =1 i ( T n − i +1 − T n − i ) (cid:21) ++ kσ n " ( T n − k +1 − t ) + 1 k k − X i =1 i ( T n − i +1 − T n − i ) ·· σ p ( T n − k + 1) − t + 1 k k − X i =1 i ( T n − i +1 − T n − i ) ! − ricing Formulae of Power Binary and Normal Distribution Standard... n →∞ d n − k ( X, t ) = TT − t ln J ( t ) tT X T − tT K + ( r − q − σ ) T − t + T − tT T − tT T − t σ σ q T − t == T ln J ( t ) t X T − t K T + (cid:16) ( r − q − σ ) T − t T + (cid:0) T − tT (cid:1) σ (cid:17) ( T − t ) σ T − tT √ T − t √ == T ln J ( t ) t X T − t K T + [ r ∗ + ( σ ∗ ) ]( T − t ) σ ∗ √ T − t = d ∗ (33)Similarly rewrite as d n − k ( X, t ) = (cid:26) nk ln n p X · · · X n − k X k/n K n + ( r − q − σ (cid:20) ( T n − k +1 − t ) + 1 k k − X i =1 i ( T n − i +1 − T n − i ) (cid:21)(cid:27) ·· σ p ( T n − k + 1) − t + 1 k k − X i =1 i ( T n − i +1 − T n − i ) ! − From (30),(31),(32) we havelim n →∞ d n − k ( X, t ) = TT − t ln J ( t ) tT X T − tT K + ( r − q − σ ) T − t σ q T − t == T ln J ( t ) t X T − t K T + (cid:16) ( r − q − σ ) T − t T (cid:17) ( T − t ) σ T − tT √ T − t √ == T ln J ( t ) t X T − t K T + r ∗ ( T − t ) σ ∗ √ T − t = d ∗ (34)where r ∗ = ( r − q − σ T − t T , σ ∗ = T − t √ T Hyong-chol O and Dae-sung Choe
Consider θ k ( t ) = µ ( 1 n )( T n − T n − ) + µ ( 2 n )( T n − − T n − ) + · · · + µ ( kn )( T n − k +1 − t ) == Tn − X i =1 k − in −
1) + ( kn − T n − k +1 − t ) ! r − Tn − k − X i =1 + kn ( T n − k +1 − t ) ! q −− σ Tn − X i =1 k − i n − in ) + ( k n − kn )( T n − k +1 − t ) ! == (cid:18) Tn − (cid:18) k ( k − n − k (cid:19) ( kn − T n − k +1 − t ) (cid:19) r − (cid:18) Tn − k ( k + 1)2 n + kn ( T n − k +1 − t ) (cid:19) −− σ (cid:18) Tn − (cid:18) k ( k − k − n + k ( k − n (cid:19) + (cid:18) k n − kn (cid:19) ( T n − k +1 − t ) (cid:19) From (30),(31),(32) we havelim n →∞ θ k ( t ) = (cid:18) ( T − t ) T − ( T − t ) (cid:19) r − ( T − t ) T q − (cid:18) ( T − t ) T + ( T − t ) T (cid:19) σ r − q − σ T − t ) T + ( T − t ) T σ − r ( T − t ) == ( r ∗ + ( σ ∗ ) T − t ) − r ( T − t )) (35)From (33),(34),(35) we have the required result (QED).We can obtain the result of the geometric Asian options with floating price in the sameway as in Theorem 3.2 and Theorem 3.3.Let 0 = T < · · · < T n = T be the fixed monitring times and X , · · · , X n the assetprices at the monitoring times, respectively. Denote the price of discrete geometric Asianoption with floating exercise price by V n ( X, t ) . From the definition, the expriy payoff is V n ( X n , T n ) = (cid:16) X − n p X · · · X n (cid:17) + Denote the option’s price on the interval T i < t ≤ X n by V in ( X, t ) , i = 1 , · · · n − T n − < t < ≤ T n , the assets prices X , · · · , X n − are known quantities, wecan rewrite as. V n − n ( X, T n ) = (cid:16) X − n p X · · · X n − X /n (cid:17) + (36)Thus in this interval the discrete geometric Asian options with floating exercise price is avanilla option with the expiry payoff (36) and the pricing model is given by LV n − n = 0 ( X > , T n − < t < T n ) V n − n ( X, T n ) = (cid:16) X − n p X · · · X n − X /n (cid:17) + (37) V n − n ( X, t ) depends on
X, t and X , · · · , X n − and so V n − n ( X, t ) can be rewritten by V n − n ( X, t ; X , · · · , X n − ) . Especially X = X n − at time T n − , the price V n − n ( X, T n − ricing Formulae of Power Binary and Normal Distribution Standard... V n − n ( X, t ; X , · · · , X n − , X ). On the interval T n − < t ≤ T n − the assetprices X , · · · , X n − at the monitoring times T , · · · , T n − are known quantities and thus inthis interval the option is a vanilla option with the expiry payoff V n − n ( X, t ; X , · · · , X n − , X ).Again V n − n ( X, t ) depends on
X, t as well as X , · · · , X n − . Repeating these process, weget LV in = 0 ( X > , T i < t < T i +1 ) V in ( X, T n ) = V i +1 n ( X, T i ; X · · · X i , X ) , i = ¯1 , n − . (38)The pricing model of discrete geometric Asian options with floating exercise price andn monitoring times is (37) and (38). Theorem 3.4
The price of discrete geometric Asian options with floating price and n mon-itoring times (the solution to (37) , (38) ) is given by V n − kn ( X, t ; T n − k +1 , · · · , T n − , T n ) = Xe − q ( T n − t ) N ( d n − k ) − n p X · · · X n − k X k/n e θ k ( t ) N ( d n − k )(39) T n − k ≤ t < T n − k +1 , ( k = ¯1 , n − Here d n − k = (cid:26) ln X n − kn − n − p X · · · X n − k + ( r − q − σ n − kn − T n − k +1 − t ) + k − X j =1 n − jn − T n − j + 1 − T n − j ) ++ σ k ( n − k ) n ( n −
1) ( T n − k +1 − t ) + k − X j =1 j ( n − j ) n ( n −
1) ( T n − j +1 − T n − j ) (cid:27) ·· σ vuut(cid:18) n − kn − (cid:19) ( T n − k +1 − t ) + k − X j =1 (cid:18) n − jn − (cid:19) ( T n − j +1 − T n − j ) − d n − k = (cid:26) ln X n − kn − n − p X · · · X n − k + ( r − q − σ n − kn − T n − k +1 − t ) + k − X j =1 n − jn − T n − j + 1 − T n − j ) ·· σ vuut(cid:18) n − kn − (cid:19) ( T n − k +1 − t ) + k − X j =1 (cid:18) n − jn − (cid:19) ( T n − j +1 − T n − j ) − θ k ( t ) = µ ( kn )( T n − k +1 − t ) + k − X j =1 µ ( jn )( T n − j +1 − T n − j ) µ ( β ) = ( β − r − βq + σ β − β )The proof is omitted as it is similar with the proof of Theorem 3.2.0 Hyong-chol O and Dae-sung Choe
Theorem 3.5
In the case with floating exercise price, if we increase the number n ofmonitoring times, the price of discrete geometric Asian options converges to the price of V ( X, J, t ) continuous geometric Asian options with floating exercise price. That is, we have lim n →∞ V n ( X, t ) = V ( X, J, t ) Here V ( X, J, t ) is given by V ( X, J, t ) = e − q ( T − t ) XN ( d ∗ ) − J tT X T − tT e θ ∗ N ( d ∗ ) where d ∗ = √ σ √ T − t (cid:20) t ln XJ + ( r − q + σ T − t − σ T − t T (cid:21) d ∗ = √ σ √ T − t (cid:20) t ln XJ + ( r − q + σ T − t (cid:21) θ ∗ = − q ( T − t ) − ( r − q + σ T − t T + σ ( T − t )6 T The proof is omitted as it is similar with the proof of Theorem 3.3.
Remark : Remark: This formula is slightly different from that of [6]. But it is equal tothe formula of [3] in the case that q=0.
Definition 4 A second order α -power binary option is defined as the binary contractwith expiry date T underlying on an α -power binary option with expiry date T . In otherwords, a second order α -power binary option’s price is the solution of (1) with time T payoff of the following form V ( x, T ) = ( M α ) s ξ ( x, T ; T )1( s x > s ξ )Here ( M α ) s ξ ( x, T ; T ) is the price at time T of the α -power binary option with expirydate T . The price of the second α -power binary option is denoted by ( M α ) s s ξ ξ ( x, t ; T , T )and this is called the second order α -power binary option with expiry dates T , T . Corre-spondingly, α - power binary option is called the first order α -power binary option.Since ( M α ) s s ξ ξ ( x, T ; T , T ) = ( M α ) s ξ ( x, T ; T )1( s x > s ξ )we have ( M α ) + s ξ ξ ( x, T ; T , T ) + ( M α ) − s ξ ξ ( x, T ; T , T ) = ( M α ) s ξ ( x, T ; T )Thus we have a parity relation between the price of the first order α -power binary optionand the prices of the corresponding second order α -power binary options.( M α ) + s ξ ξ ( x, t ) + ( M α ) − s ξ ξ ( x, t ) = ( M α ) s ξ ( x, t ) , t < T ricing Formulae of Power Binary and Normal Distribution Standard... Theorem 4.1
The price of the second order power binary option is given by ( M α ) s s ξ ξ ( x, t ; T , T ) = e µ ( T − t ) x α N ( s d , s d ; s s ρ ) (40) Here d i = (cid:18) ln xξ i + (cid:18) r − q − σ ασ (cid:19) ( T i − t ) (cid:19) ( σ p T i − t ) − , ρ = r T − tT − tN ( d , d ; ρ ) = 12 π p − ρ Z d −∞ Z d −∞ e − y − ρy y y − ρ dy dy Proof
By the definition and Preposition 1 of [8] we have( M α ) s s ξ ξ ( x, t ) == e − r ( T − t ) σ p π ( T − t ) Z ∞−∞ z e − σ T − t ) (cid:16) ln xz + (cid:16) r − q − σ (cid:17) ( T − t ) (cid:17) s z > s ξ )( M α ) s ξ ( z , T ; T ) dz == e − r ( T − t ) Z + ∞−∞ e − σ T − t ) (cid:16) ln xz + (cid:16) r − q − σ (cid:17) ( T − t ) (cid:17) z σ p π ( T − t ) 1( s z > s ξ ) e µ ( T − T ) z α Z + ∞−∞ e − σ T − T (cid:16) ln xz + (cid:16) r − q − σ + ασ (cid:17) ( T − T ) (cid:17) z σ p π ( T − T ) 1( s z > s ξ ) dz dz = x α e µ ( T − t ) Z + ∞−∞ Z + ∞−∞ e − σ T − t ) (cid:16) ln xz + (cid:16) r − q − σ + ασ (cid:17) ( T − t ) (cid:17) e − σ T − T (cid:16) ln xz + (cid:16) r − q − σ + ασ (cid:17) ( T − T ) (cid:17) πσ z z p ( T − t )( T − T ) s z > s ξ )1( s z > s ξ )In this integral, we use the change of variables y i = (cid:18) ln xz i + (cid:18) r − q − σ ασ (cid:19) ( T i − t ) (cid:19) ( σ p T i − t ) − , i = 0 , dy i = ( σ √ T i − t ) − − dz i z i . Considering − y − ( y σ √ T − t − y σ √ T − t ) σ ( T − T ) = y − ρy y + y − ρ ) , ρ = r T − tT − t we have( M α ) s s ξ ξ ( x, t ) == x α e µ ( T − t ) π p − ρ Z + ∞−∞ Z + ∞−∞ e y − ρy y y − ρ s y < s d )1( s y < s d ) dy dy = I Hyong-chol O and Dae-sung Choe
We use the following change of variables again. y ′ = s y , y ′ = s y The Jacobians of this change is (cid:12)(cid:12)(cid:12)(cid:12) det (cid:18) s s (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = 1so we have( M α ) s s ξ ξ ( x, t ) == x α e µ ( T − t ) π p − ρ Z + ∞−∞ Z + ∞−∞ e y ′ − s s ρy ′ y ′ y ′ − ρ s y ′ < s d )1( s y ′ < s d ) dy ′ dy ′ = x α e µ ( T − t ) π p − ρ N ( s d , s d ; s s ρ )(QED). Definition 5 An n-th α -power binary option is inductively defined as the binary con-tract with the expiry date T underlying on an (n-1)-th order α -power binary option. Thatis, an n-th order α -power binary option ’s price is the solution of (1) with the time T payoff of the following form V ( x, T ) = ( M α ) s ··· s n − ξ ··· ξ n − ( x, T ; T , · · · , T n − )1( s x > s ξ ) (41)Here ( M α ) s ··· s n − ξ ··· ξ n − ( x, T ; T , · · · , T n − is the price of (n-1)-th order α -power binary optionwith expiry dates T , · · · , T n − .The price of this n-th order α -power binary option is denotedby ( M α ) s ,s ··· s n − ξ ,ξ ··· ξ n − ( x, t ; T , T , · · · , T n − ) and this option is called an n-th order α -powerbinary option with n expiry dates T , T , · · · , T n − .Between the prices of n -th order α -power binary options and the corresponding price of( n − α -power binary option, there is a following parity relation( M α ) + ,s ··· s n − ξ ,ξ ··· ξ n − ( x, t ) + ( M α ) − ,s ··· s n − ξ ,ξ ··· ξ n − ( x, t ) = ( M α ) s ··· s n − ξ ,ξ ··· ξ n − ( x, t ) , t < T Theorem 4.2
The price of n-th α -power binary option (the solution to the problem ( (1) , (41) )is given as following. ( M α ) s ,s ··· s n − ξ ,ξ ··· ξ n − ( x, t ; T , T , · · · , T n − ) = x α e µ ( T n − − t ) N n ( s d , · · · , s n − d n − ; A n ( s · · · s n − )) Here d i = (cid:18) ln xξ i + (cid:18) r − q − σ ασ (cid:19) ( T i − t ) (cid:19) ( σ p T i − t ) − N n ( d , · · · , d n − ; A ) = (cid:18) π (cid:19) n | detA | Z d −∞ Z d n − −∞ e − y T A − y dy · · · dy n − ricing Formulae of Power Binary and Normal Distribution Standard... A ( s s · · · s n − ) = ( s i s j a ij ) n − i,j =0 , det [ A ( s s · · · s n − )] = det ( A ) And A = ( a ij ) i,j =0 , ··· ,n − is n-th matrix which is defined as following like [8]. a = ( T − t ) / ( T − T ) ,a n − ,n − = ( T n − − t ) / ( T n − − T n − ) ,a ii = ( T i − t ) / ( T i − T i − ) + ( T i − t ) / ( T i +1 − T i ) a i,i +1 = a i +1 ,i = − p ( T i − t )( T i +1 − t ) / ( T i +1 − T i ) , ≤ i ≤ n − And for other indices, a ij = 0 . Proof
The cases of n = 1 and n = 2 were proved by Theorem 2.2 and Theorem 4.1. Inthe case of n > n −
1. From the definition 4.2,( M α ) s ,s ··· s n − ξ ,ξ ··· ξ n − ( x, t ; T , T , · · · , T n − ) satisfies (40) and V ( x, T ) = ( M α ) s ··· s n − ξ ··· ξ n − ( x, T ; T , · · · , T n − ) · s x > s ξ )Here ( M α ) s ··· s n − ξ ··· ξ n − ( x, T T , · · · , T n − ) is the price of the (n-1)-th power binary option.Therefore by preposition 1 of [8]( M α ) s s ··· s n − ξ ξ ··· ξ n − ( x, t ; T , · · · , T n − ) == e − r ( T − t ) Z + ∞−∞ s z > s ξ ) σ p π ( T − t ) 1 z e − (cid:18) ln xz + (cid:18) r − q − σ (cid:19) ( Ti − t ) (cid:19) σ T − t ) ( M α ) s ··· s n − ξ ··· ξ n − ( z, T ) dz By induction-assumption, the result of Theorem 4.2 holds for ( M α ) s ··· s n − ξ ··· ξ n − ( z, T ). Thus wehave ( M α ) s ··· s n − ξ ··· ξ n − ( z, T ) = x α e µ ( T n − − T ) N n ( s d , · · · , s n − d n − ; A n ( s · · · s n − ))Substitute this equality into the above integral representation and calculate the integral,then we have the result of Theorem 4.2 for the case n > References [1] Benninga, S., Bjrk, T. and Wiener, Z. On the use of numeraires in option pricing. TheJournal of Derivatives. Winter 2002. 10(2): 1-16.[2] P. Buchen., The Pricing of dual expiry exotics, Quantitative Finance, 4, 2004, 101-108.[3] P. Buchen (2012) An Introduction to Exotic Option Pricing, CRC Press, 107-124.[4] M. Garman, S. Kohlhagen, ”Foreign currency option values” , Journal of internationalmoney and Finance, 2(1983), 231-237.[5] Ingersoll, J. E., Digital contract: simple tools for pricing complex derivatives, J. Busi-ness, 73 (2000), 67-884