International Journal of Analysis and Applications Volume 18, Number 1 (2020), 129-148 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-18-2020-129 ON COX-ROSS-RUBINSTEIN PRICING FORMULA FOR PRICING COMPOUND OPTION JAVED HUSSAIN∗, BAREERAH KHAN Sukkur IBA University, Pakistan ∗Corresponding author: javed.brohi@iba-suk.edu.pk Abstract. The fundamental objective of this paper is twofold. Firstly, to derive the Cox-Ross-Rubinstein type new formula for risk neutral pricing of European compound call option, where the underlying asset is also a European call option. Thirdly, to prove that our newly derived CRR risk neutral pricing formula for compound call option, converges in distribution to the well known, continuous time Black-Scholes formula for pricing the compound call option on call. 1. Introduction A compound option is an option that has further an option as the underlying asset. Compound options were first studied by Geske (1979, [9]), using a partial differential equation method and Fourier integrals. Afterward, several other approaches were introduced for pricing methods for compound options. For instance, Lajeri-Chaherli (2002, [15]) used the martingale method and by computing the expectation of truncated bivariate normal variables, priced the compound options. Agliardi (2003, [1]) priced a generalized time- dependent compound calls. Gukhal (2004, [12]) proposed a model for valuation of the compound option, in which the underlying option follows the log-normal jump-diffusion process. Fouque and Han (2005, [13]) employed a perturbation techniques to compute the prices of compound options. Chiarella and Kang (2011, [5]) and (2014, [4]) evaluated American style compound option with stochastic volatility. Griebsch (2013, [11]) Received 2019-10-25; accepted 2019-11-19; published 2020-01-02. 2010 Mathematics Subject Classification. 91G20. Key words and phrases. binomial pricing; compound options; Cox-Ross-Rubinstein framework; convergence in distribution. c©2020 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 129 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-129 Int. J. Anal. Appl. 18 (1) (2020) 130 used FFT (Fast Fourier Transform) to price the European style compound option prices, where volatility assumed to be stochastic. Since most studies have been done through continuous time approach so in this paper we emphasize on pricing the compound option through the discrete time Cox-Ross-Rubinstein (CRR) approach, we refer the reader to chapter 6 of [7]. They key novelty of the work is the detailed proof that discrete time CRR price of the compound option converges to the continuous time pricing formula of the compound option. We now give a brief description of all sections of this paper. Section 1 is running introduction. The section 2 gives A short introduction to mechanics of the compound option and its continuous time formula. Section 3 has been devoted to deriving, in a greater detail, the CRR formula for the risk-neutral pricing the European-style compound option on European call option. In Section 4, we have given the proof that the CRR price of compound option converges to the well-known continuous time pricing formula, in distribution. options. section 6 comprises of conclusion. 2. CRR Formula for Pricing Compound Options This section comprises of a brief introduction to compound option and our new result on the derivation of its CRR premium of European call option when the underlying asset is a European call option on stock. 2.1. Introduction to Compound Option. Here we briefly illustrate a compound op- tion which is a particular type of exotic option. To do that we need a brief time line here. T0 T1 T2 Today is time zero, T1 is the point time in future and T2 will be even later than that. If today at T = 0 we purchase a call option we are buying a right to exercise a price of K1 and exchange for that we will receive a call option. So in this case we are purchasing a call option on a call option. Mainly, there can be four possible variations of compound options. 1. Call with underlying Call. 2. Call with underlying Put. 3. Put with underlying Call. 4. Put with underlying Put. So today at T = 0 when we purchase a compound option it gives us a right to going forward to exercise that compound option. So let’s just say that stock increases little bit here at T = 1 and we exercise compound option. We pay the strike price of K1 and get the option not the stock. So we only do that if the value at T = 1 of call option is greater than the strike price C > K1. So after having exercise the compound, we now in fact own the more familiar plain vanilla call option on stock and we do need another strike price K2. The Int. J. Anal. Appl. 18 (1) (2020) 131 compound option has two strike prices, first only to purchase a compound option which is K1 the price we pay in order to exercise initially to purchase a call option and that call option has K2 its own strike price. If we go forward in time, say stock move in our favor such that at T1 , there is an intrinsic value in that call option and if S > K2 then we exercise the underlying option by paying the strike price K2 and receiving the underlying asset (in this case it’s a stock). When assumption of geometric Brownian motion is made, European style compound option can analyt- ically be valued in terms of integral of bivariate normal distribution. The value of European call on call option at T = 0 is C = SN ( a1,b1; √ T1 T2 ) −e−rT2K2N ( a2,b2; √ T1 T2 ) −e−rT1K1N (a2) (2.1a) where a1 = ln ( S0 S∗ ) + ( r − σ 2 2 ) T1 σ √ T1 ; a2 = a1 + σ √ T1 b1 = ln ( S0 K2 ) + ( r − σ 2 2 ) T2 σ √ T2 ; b2 = b1 + σ √ T2 N = cumulative bivariate normal distribution function. S∗ = critical stock price at T1 such that compound option is in the money. We refer to [14] for further details. 3. Derivation of CRR compound Option Formula Now we move towards the key objective of the paper i.e. pricing the compound option through binomial approach. We aim to construct a CRR formula for the compound options. Let me state the key theorem of the section. Theorem 3.1. In a viable Cox-Ross-Rubinstein model with parameters S0,T,r,u and d the fair price at time 0 of a European compound call option with expiry m and strike K1, contingent upon a European call option with expiry n and strike K2, can be given as: C = S0B (a,b; p ′; n,n−m) − (1 + r)−(n−m)K2B (a,b; p′,p; n,n−m) (3.1) −(1 + r)−mK1ψ (a; m,p) where p = r−u u−d, p ′ = pu (1+r) , a = [ ln S ∗ S0 −m ln d ln( ud ) ] + 1 and b = [ ln K2 S0 −n ln d ln( ud ) ] + 1, and B(·, · ; ·) is the complementary distribution function of bi-variate Binomial distribution. Int. J. Anal. Appl. 18 (1) (2020) 132 Proof. Consider the discrete time line, 0 m n where m,n ∈ N. We aim to price a compound call option where underlying asset is also a call option. The compound option will be exercised at m (i.e. after m steps/periods) and the underlying call option (whose value depends on underlying stock prices), will be exercised at n steps/periods. By assuming the absence of arbitrage opportunities in the binomial model, the price of the underlying call option either moves up by a factor u or down by a factor d in each period. The probability of an upward movement is p and the probability of a downward movement is 1 −p. In order to compute the fair premium of the compound option at inception, we will begin by employing Theorem 5.47 of [7]. The payoff of the compound option can be given as, payoff = max{Cstd(Sm,K2,n−m) −K1, 0} where E is the risk neutral expectation, K1 is the strike price for the compound option, and Cstd is the Black- Scholes price of underlying call option at the maturity of compound option i.e. m. Thus by the Theorem 5.46 of [7], the premium of the compound option can be given as, C = (1 + r)−mE (max{Cstd(Sm,K2,n−m) −K1, 0}) where Cstd(Sm,K2,n − m) is the price of the underlying option. The expectation can be written more explicitly as, C = (1 + r)−m m∑ i=0 max ( Cstd(Smu idm−i,K2,n−m) −K1, 0 ) m i  pi (1 −p)m−i . (3.2) The compound option will be exercised i.e. its payoff will be nonzero if Sm > S ∗, where S∗ denotes the worth of the stock such that the underlying option is in the money at time m i.e. S∗ solves the equation Cstd(Sm,K2,n−m) = K1. Assume that a is the smallest time period i such that. Sm > S ∗. Let us try to determine bound on a,. As S0u adm−a > S∗, it follows that, a > ln S ∗ S0 −m ln d ln ( u d ) . Int. J. Anal. Appl. 18 (1) (2020) 133 Let us return back to equation (3.2) and write the expectation more explicitly. C = (1 + r)−m m∑ i=a ( Cstd(Smu idm−i,K2,n−m)uidm−i −K1 ) m i  pi (1 −p)m−i = (1 + r)−m m∑ i=a Cstd(Smu idm−i,K2,n−m)   m i  pi (1 −p)m−i uidm−i −(1 + r)−m m∑ i=a K1   m i  pi (1 −p)m−i C = (1 + r)−m m∑ i=a Cstd(Smu idm−i,K2,n−m)   m i  pi (1 −p)m−i uidm−i −(1 + r)−mK1ψ (a,m,p) (3.3) where ψ is the complementary Binomial distribution function, ψ (a,m,p) = m∑ i=a   m i  pi (1 −p)m−i . We want to further explore the above first term of equation (3.3). Here Cstd(Sm,K2,n−m) represents the price of an underlying European option and the price is given as: Cstd(Smu idm−i,K2,n−m) = (1 + r)−(n−m)E ({max (Sm −K2, 0)}) Cstd(Smu idm−i,K2,n−m) = (1 + r)−(n−m) n∑ j=m max ( (Smu idm−i)ujdn−j −K2, 0 ) ×   n−m j  pj (1 −p)n−m−j . Assume that b < n in first instant such that b = min { a : Smu jdn−j } > K2. Let us try to find bound on b. By definition Smu jdn−j > K2, therefore it follows that b > ln K2 Sm −n ln d ln ( u d ) . Now Cstd(Smu idm−i,K2,n−m) = (1 + r)−(n−m) n∑ j=b Sm   n−m j  pj (1 −p)n−m−j ujdn−m−j −(1 + r)−(n−m)K2 n∑ j=b   n−m j  pj (1 −p)n−m−j . Int. J. Anal. Appl. 18 (1) (2020) 134 Substitute the above equation in equation (3.3), we infer that, C = (1 + r)−m m∑ i=a   (1 + r)−(n−m) n∑ j=b Sm   n−m j  pj (1 −p)n−m−j ujdn−m−j −(1 + r)−(n−m)K2 n∑ j=b   n−m j  pj (1 −p)n−m−j   ·   m i  pi (1 −p)m−i uidm−i − (1 + r)−mK1ψ (a,m,p) . C = (1 + r)−n m∑ i=a n∑ j=b Sm   m i     n−m j  pi+j (1 −p)n−(i+j) ui+jdn−(i+j) −(1 + r)−nK2 m∑ i=a n∑ j=b   m i     n−m j  pi+j (1 −p)n−(i+j) uidm−i −K1ψ ( a,m,p ′ ) = m∑ i=a n∑ j=b S0   m i     n−m j  ( pu (1 + r) )i+j ( (1 −p) d (1 + r) )n−(i+j) −(1 + r)−(n−m)K2 m∑ i=a n∑ j=b   m i     n−m j  ( pu (1 + r)−(n−m) )i ( (1 −p) d (1 + r)−(n−m) )m−i pj (1 −p)n−j − (1 + r)−(n−m)K1ψ (a,m,p) , where   m i   = m! i!(m−i)! and   n−m j   = (n−m)! j!(n−m−j)!. One can see that the first two terms in the last equation represent the cumulative distributive function of bivariate binomial distribution. Thus C = S0B (a,b; p ′; n,n−m) − (1 + r)−(n−m)K2B (a,b; p′,p; n,n−m) − (1 + r)−mK1ψ (a,m,p) (3.4) where p′ = pu (1+r) , a = [ ln S ∗ S0 −m ln d ln( ud ) ] + 1 and b = [ ln K2 S0 −n ln d ln( ud ) ] + 1, and ψ(·, ·; ·) is the complementary distribution function of bi-variate Binomial distribution. The formula above gives us the CRR premium of the compound European call on European call option. � 4. Convergence of Empirical Means and Volatilities of Log Returns to Actual Means and Volatilities Following is the key theorem that we intend to prove in this section. Theorem 4.1. Assume that we are in the framework of Theorem 3.1. Then following holds: i) If S∗ stock price at time T1 (i.e. maturity time for compound option) such that the compound option in Int. J. Anal. Appl. 18 (1) (2020) 135 the money then the empirical mean µ̂pm and variance of σ̂ 2 pm of log returns ln ( S∗ S0 ) converges, respectively, to µT1 and σ 2T1. ii) Let T be an instant between T1 and T2 and ST be the stock price at time T. Then the empirical mean µ̂p(n−m) and variance of σ̂2p(n−m) of log returns ln ( ST S0 ) converges, respectively, to µT2 and σ 2T2. Proof. of the compound option given by formula 2.1a. Let S∗ be the stock price at T1 such that. the price of compound option is in the money. So S∗ = S0U iDm−i ln ( S∗ S0 ) = ln (U) i + ln (D) m−i ln ( S∗ S0 ) = i ln U + (m− i) ln D = i ln ( U D ) + m ln D. (4.1) Here i is the binomial random variable with mean E [i] = mp and variance V ar [i] = mp (1 −p) . As m →∞ E [ ln ( S∗ S0 )] = E [ i ln ( U D )] + E [m ln D] . The expectation of constant remains same , So E [ ln ( S∗ S0 )] = E [i] [ ln ( U D )] + [m ln D] = mp [ ln ( U D )] + [m ln D] = [ p ln ( U D ) + ln D ] m. We suppose that the empirical mean value of ln ( S∗ S0 ) is µ̂pm and represent it as: µ̂pm = [ p ln ( U D ) + ln D ] m (4.2) and Var [ ln ( S∗ S0 )] = Var [ i ln ( U D )] + Var [m ln D] . The variance of (aX + b) is a2V arX, hence Var [ ln ( S∗ S0 )] = [ ln ( U D )]2 V ar [i] = mp(1 −p) [ ln ( U D )]2 . We suppose that the empirical variance value of ln ( S∗ S0 ) is σ̂2pm and represent it as σ̂2pm = p(1 −p) [ ln ( U D )]2 m. (4.3) Int. J. Anal. Appl. 18 (1) (2020) 136 If µT1 = E [ ln ( S∗ S0 )] and σT = Var [ ln ( S∗ S0 )] represent the values of actual mean and variance, we want µ̂pm converges to µT1 and σ̂ 2 pm approaches to σT1 as m →∞ : lim m→∞ mp ln ( U D ) + ln D = µT1, (4.4) and lim m→∞ mp(1 −p) [ ln ( U D )]2 = σ2T1. (4.5) Also keep in view the following relation between U and D, U = 1 D . (4.6) To have these convergence we need to choose the parameters U,D and p, which satisfies last three limiting equations. To do so let solve the system of nonlinear equations (4.4), (4.5) and (4.6 ), for U, D and p. Ignore the limits in the mentioned equations. Using equation (4.6) into (4.4), we may infer that p ln ( U2 ) + ln 1 U = µT1 m ln U = µT1 m(2p− 1) . (4.7) Using equation (4.6) into (4.5) we infer, p(1 −p) [ ln ( U2 )]2 = σT1 m [ln U] 2 = σT1 4p(1 −p)m . Substituting the equation (4.7) into last equation we get,( µT1 m(2p− 1) )2 = σT1 4p(1 −p)m . On solving the above quadratic equation for p we get, p = 1 2 + 1 2 µ σ √ T1 m . (4.8) Now substituting the value of p from (4.8) into getting the value of U by using equation (4.7) ln U = µT1 m · 1( 2 ( 1 2 + 1 2 µ σ √ T1 m ) − 1 ) ln U = σ √ T1 m U = e σ √ T1 m . (4.9) Int. J. Anal. Appl. 18 (1) (2020) 137 As D = 1 U , then the value of D is D = e −σ √ T1 m . (4.10) After plugging the calculated values of U,D and p in equations of µ̂pm and σ̂ 2 pm, we see how these empirical values approaches to the actual values of mean and variance. µ̂pm = [ p ln ( U D ) + ln D ] m. Note that ln ( U D ) = ln U − ln D = σ √ T1 m + σ √ T1 m = 2σ √ T1 m . Now µ̂pm = {[ 1 2 + 1 2 µ σ √ T1 m ][ 2σ √ T1 m ] −σ √ T1 m m } = [ σ √ T1 m + µ ( T1 m ) −σ √ T1 m ] m lim m→∞ µ̂pm = µT1. Next σ̂2pm = p(1 −p) [ ln ( U D )]2 m = [ 1 2 + 1 2 µ σ √ T1 m ][ 1 − { 1 2 + 1 2 µ σ √ T1 m }][ 2σ √ T1 m ]2 m = ( 1 4 − 1 4 µ2 σ2 T1 m ) 4σ2T1 lim m→∞ σ̂2pm = σ 2T1. (4.11) Now, at the second expiry of underlying stock: ST = S0U jD(n−m)−j ln ( ST S0 ) = ln (U) j + (D) (n−m)−j ln ( ST S0 ) = j ln U +(n−m)−j ln D = j ln ( U D ) + (n−m) ln D. Here j is the binomial random variable with mean E [j] = (n−m) p and variance V ar [j] = (n−m) (1 −p) p. As (n−m) →∞ E [ ln ( ST S0 )] = E [ j ln ( U D )] + E [(n−m) ln D] . Int. J. Anal. Appl. 18 (1) (2020) 138 The expectation of constant remains same , so E [ ln ( ST S0 )] = E [j] [ ln ( U D )] + [(n−m) ln D] = (n−m) p [ ln ( U D )] + [(n−m) ln D] = [ p ln ( U D ) + ln D ] (n−m) . We suppose that the empirical mean value of ln ( ST S0 ) is µ̂p(n−m) and represent it as: µ̂p(n−m) = [ p ln ( U D ) + ln D ] (n−m) and Var [ ln ( ST S0 )] = Var [ j ln ( U D )] + Var [(n−m) ln D] . Recall the fact that the V ar (aX + b) = a2V arX. By using this, Var [ ln ( ST S0 )] = [ ln ( U D )]2 V ar [j] = (n−m) p(1 −p) [ ln ( U D )]2 . We suppose that the empirical variance value of ln ( S∗ S0 ) is σ̂2p (n−m) and represent it as: σ̂2p (n−m) = p(1 −p) [ ln ( U D )]2 (n−m) . If µT2 = E [ ln ( ST S0 )] and σT2 = Var [ ln ( ST S0 )] represent the values of actual mean and variance, we want µ̂p(n−m) approaches to µT2 and σ̂2p (n−m) approaches to σT2 as (n−m) →∞ : p ln ( U D ) + ln D → µT2 (n−m) and p(1 −p) [ ln ( U D )]2 → σT2 (n−m) From the last two equations above, we have to calculate the values of parameters U,D and p subject to constraint U = 1 D due to the balance in the tree structure in the binomial model. Calculations are shown below: p = 1 2 + 1 2 µ σ √ T2 (n−m) . Using the relation U = 1 D Int. J. Anal. Appl. 18 (1) (2020) 139 p ln ( U D ) + ln D = µT2 (n−m) p ln (U) 2 + ln ( 1 U ) = µT2 (n−m) ln U = µT2 (n−m) . 1 (2p− 1) . Now plugging the value of p ln U = µT2 (n−m) · 1( 2 ( 1 2 + 1 2 µ σ √ T2 (n−m) ) − 1 ) ln U = µT2 (n−m) · 1( µ σ √ T2 (n−m) ) ln U = T2 (n−m) · σ(√ T2 (n−m) ). After solving the powers ln U = σ √ T2 (n−m) or U = e σ √ T2 (n−m) (4.12) As D = 1 U , then the value of D is D = e −σ √ T2 (n−m) . (4.13) After plugging the calculated values of U,D and p in the µ̂p(n − m) and σ̂2p (n−m) , we see how these empirical values approaches to the actual values of mean and variance. µ̂p(n−m) = [ p ln ( U D ) + ln D ] (n−m) . Note that ln ( U D ) = ln U − ln D = σ √ T2 (n−m) + σ √ T2 (n−m) = 2σ √ T2 (n−m). Now, µ̂p(n−m) = [ 1 2 + 1 2 µ σ √ T2 (n−m) ][ 2σ √ T2 (n−m) −σ √ T2 (n−m) ] (n−m) = [ σ √ T2 (n−m) + µ ( T2 (n−m) ) −σ √ T2 (n−m) ] (n−m) lim m→∞ µ̂p(n−m) = µT2. Int. J. Anal. Appl. 18 (1) (2020) 140 Next σ̂2p (n−m) = p(1 −p) [ ln ( U D )]2 (n−m) = [ 1 2 + 1 2 µ σ √ T2 (n−m) ][ 1 − { 1 2 + 1 2 µ σ √ T2 (n−m) }] × [ 2σ √ T2 (n−m) ]2 (n−m) = [ 1 2 + 1 2 µ σ √ T2 (n−m) ][ 1 2 − 1 2 µ σ √ T2 (n−m) ][ 4σ2T2 ] = ( 1 4 − 1 4 µ2 σ2 T2 (n−m) ) 4σ2T2 lim m→∞ σ̂2p (n−m) = σ 2T2. (4.14) � 5. Main Convergence Result In this section we prove the second most important result that the established general CRR formula for binomial pricing formula of a Compound call option, to the well-known continuous-time Black-Scholes formula for European Compound option. We state this assertion in the form of following Theorem. Theorem 5.1. Assume that we are in assumptions of Theorem 3.1 and Theorem 4.1. the fair price at time 0 of a European compound call option with expiry m and strike K1, contingent upon a European call option with expiry n and strike K2, can be given as: C = S0B (a,b; p ′; n,n−m) − (1 + r)−(n−m)K2B (a,b; p′,p; n,n−m) (5.1) −(1 + r)−mK1ψ (a; m,p) , converges in the distribution to the following continuous time price of European compound call option with expiry T1 and strike K1, contingent upon a European call option with expiry T2 and strike K2, S0N ( d1,d2; √ T2 T1 ) −e−rT2K2N ( d∗1,d ∗ 2; √ T2 T1 ) −e−rT1K1N(d∗1), where p = r−u u−d, p ′ = pu (1+r) , a = [ ln S ∗ S0 −m ln d ln( ud ) ] + 1 and b = [ ln K2 S0 −n ln d ln( ud ) ] + 1, and B(·, · ; ·) is the complementary distribution function of bi-variate Binomial distribution and d1 = ln S S∗ + ( ln(1 + r) − 1 2 σ2 ) T1 σ √ T1 ,d∗1 = d1 + σ √ T1 d2 = ln ( S K2 ) + ( ln(1 + r) − 1 2 σ2 ) T2 σ √ T2 ,d∗2 = d2 + σ √ T2. Int. J. Anal. Appl. 18 (1) (2020) 141 Proof. From the last section we know that our choice of parameter (1 + r) T1 m ensures that K1(1 + r) −m = Ke−rT1 and K2(1 +r) −(n−m) = Ke−rT2. Next, we are going to show that the CRR price of compound option converges to the continuous time Black-Scholes price of the compound option. To do let us begin with the following observation about the CDF of bivariate Binomial random variables i, j, where i and j denotes the number of times the price of underlying option and compound option, has went up, respectively. 1 −B (a,b; p′; n,n−m) = P (i ≤ a− 1,j ≤ b− 1) = P ( i−mp√ mp(1 −p) ≤ a− 1 −mp√ mp(1 −p) , j − (n−m) p√ (n−m) p(1 −p) ≤ b− 1 − (n−m) p√ (n−m) p(1 −p) ) (5.2) where a = min { i ∈ m; i ≥ ln S ∗ S0Dm ln U D } ∈ [ ln S ∗ S0Dm ln U D , ln K1 CDm ln U D + 1 ) b = min { j ∈ (n−m) ; j ≥ ln K2 SD(n−m) ln ( U D ) } ∈ [ ln K2 SD(n−m) ln ( U D ) , ln K2SD(n−m) ln ( U D ) + 1 ) . For the convergence to the continuous time, mean and variance of continuously compounded rate of returns of stocks with respect to P = (p, 1 −p) can be given as: µ̂pm = [ p ln ( U D ) + ln D ] m and σ̂2pm = mp(1 −p) [ ln ( U D )]2 (5.3) µ̂p(n−m) = [ p ln ( U D ) + ln D ] (n−m) andσ̂2pm = mp(1 −p) [ ln ( U D )]2 . (5.4) Recall that i represents the number of times the stock price goes up before the expiry of the compound option i.e. between 0 and m. Also recall the equation (4.1) i.e. ln ( S∗ S0 ) = i ln ( U D ) + m ln D. Deducing the value of i from this equation we may infer that i−mp√ mp(1 −p) = ln ( ST S0 ) −m ln D ln( UD ) −mp√ mp(1 −p) = ln ( ST S0 ) −m(ln D + p ln ( U D ) ) ln ( U D )√ mp(1 −p) . Using the equations (5.3) it follows that, i−mp√ mp(1 −p) = ln ( ST S0 ) − µ̂pm σ̂p √ m . (5.5) Let us fix ε ∈ ([0, 1]) . Now since a = min { i ∈ m; i ≥ ln S ∗ S0D m ln U D } ∈ [ ln S ∗ S0D m ln U D , ln K1 CDm ln U D + 1 ) therefore: a− 1 = ln S ∗ S0Dm ln ( U D ) −ε = ln S∗S0 −m ln D −ε ln (UD) ln U D Int. J. Anal. Appl. 18 (1) (2020) 142 and therefore, a− 1 −mp√ mp(1 −p) = ln S ∗ S0 −m ln D −ε ln U D −mp ln ( U D ) ln U D √ mp(1 −p) = ln S ∗ S0 −m ( p ln U D − ln D ) −ε ln ( U D ) ln U D √ mp(1 −p) = ln S ∗ S0 − µ̂pm−ε ln ( U D ) σ̂p √ m . (5.6) Next, let us turn towards the random variable j over (n−m) time steps, where j is the number of up moves of the stock price in the interval m to n. j − (n−m) p√ (n−m) (1 −p)p = ln ( ST S0 ) −(n−m) ln D ln( UD ) − (n−m) p√ (n−m) (1 −p)p = ln ( ST S0 ) − (n−m) ln D − ln ( U D ) (n−m) p√ (n−m) (1 −p)p . Using the equations (5.3) it follows that j − (n−m) p√ (n−m) (1 −p)p = ln ( ST S0 ) − µ̂p(n−m) σ̂p √ N . (5.7) Again since b = min { j ∈ (n−m) ; j ≥ ln K2 SD(n−m) ln( UD ) } ∈ [ ln K2 SD(n−m) ln( UD ) , ln K2 SD(n−m) ln( UD ) + 1 ) so it follows that b− 1 = ln K2 SD(n−m) ln U D −ε = ln ( K2 S ) − (n−m) ln D −ε ln ( U D ) ln U D and therefore, b− 1 − (n−m) p√ (n−m) (1 −p)p = ln ( K2 S ) − (n−m) ln D −ε ln ( U D ) − (n−m) p ln ( U D ) ln U D √ (n−m) p(1 −p) = ln ( K2 S ) − (n−m) ( p ln ( U D ) − ln D ) −ε ln ( U D ) ln U D √ (n−m) p(1 −p) = ln ( K2 S ) − µ̂p (n−m) −ε ln ( U D ) σ̂p √ (n−m) . (5.8) Using equations (5.5), (5.6), (5.7) and (5.8) into (5.2), it follows that, 1 −B (a,b; p′; n,n−m) = P   ln ( ST S0 ) −µ̂pm σ̂p √ m ≤ ln S ∗ S0 −µ̂pm−ε ln( UD ) σ̂p √ m , ln ( ST S0 ) −µ̂p(n−m) σ̂p √ n−m ≤ ln( K2 S )−µ̂p(n−m)−ε ln( U D ) σ̂p √ (n−m)   . (5.9) Int. J. Anal. Appl. 18 (1) (2020) 143 To proceed further we want to show that µ̂pm → ( ln(1 + r) − 1 2 σ2 ) T1 and µ̂p (n−m) →( ln(1 + r) − 1 2 σ2 ) T2 as m → ∞ and n → ∞, respectively. We will prove first the convergence, and the second will follow the analogue argument. Recall that between the time 0 to m, the value of m can be given as, p = 1 2 + 1 2 µ σ √ T1 m . In order to show that µ̂pm → ( ln(1 + r) − 1 2 σ2 ) T1 as m →∞, it is sufficient to show that, p = 1 2 + 1 2 µ σ √ T1 m → 1 2 + 1 2 ( ln(1 + r) − 1 2 σ2 ) σ √ T1 m as m →∞ or 2 √ m ( p− 1 2 ) → ( ln(1 + r) − 1 2 σ2 ) σ √ T1 as m →∞. Or it is sufficient to show that, lim m→∞ 2 √ m ( p− 1 2 ) = ( ln(1 + r) − 1 2 σ2 ) σ √ T1. Let us consider the term on left hand side of last equation. Using value of p from 2 √ m ( p− 1 2 ) = 2 √ m ( (1 + r) T1 m −D U −D − 1 2 ) = 2 √ m ( 2(1 + r) T1 m − 2D −U + D 2 (U −D) ) = √ m  2 (1+r) T1mD − UD − 1 U D − 1   . On substituting (1 + r) T1 m = e( T1 m ) ln(1+r), U = e σ √ T1 m and D = e −σ √ T1 m ( or U D = e 2σ √ T1 m ) into last equation 2 √ m ( p− 1 2 ) = √ m  2e( T1m ) ln(1+r)+σ √ T1 m −e2σ √ T1 m − 1 e 2σ √ T1 m − 1   . Taking limit m →∞, lim m→∞ 2 √ m ( p− 1 2 ) = lim m→∞ 2e( T1 m ) ln(1+r)+σ √ T1 m −e2σ √ T1 m − 1( e 2σ √ T1 m − 1 ) m− 1 2 , ( 0 0 form ) . Using L’Hospital rule we may infer that, lim m→∞ 2 √ m ( p− 1 2 ) = = lim m→∞ 2e( T1 m ) ln(1+r)+σ √ T1 m −e2σ √ T1 m − 1( e 2σ √ T1 m − 1 ) m− 1 2 = lim m→∞ 2e( T1 m ) ln(1+r)+σ √ T1 m (m−2T1 ln(1 + r) + 1 2 m− 3 2 σ √ T1) −e2σ √ T1 m m− 3 2 σ √ T1 e 2σ √ T1 m ( m−2T1 ln(1 + r) + 1 2 m− 3 2 ) − 1 2 m− 3 2 . Int. J. Anal. Appl. 18 (1) (2020) 144 On dividing the numerator and the denominator by m− 3 2 e 2σ √ T1 m , we get = lim m→∞ 2e( T1 m ) ln(1+r)+σ √ T1 m (m− 1 2 T1 ln(1 + r) + 1 2 σ √ T1) −σ √ T1 σ √ T1 m + 1 2 − 1 2 e 2σ √ T1 m , ( 0 0 form ) . Again using L’Hospital rule, we infer, = lim m→∞ 2e( T1 m ) ln(1+r)+σ √ T1 m (−m−2T1 ln(1 + r) + 12m −3 2 σ √ T1) ( m− 1 2 T1 ln(1 + r) + 1 2 σ √ T1 ) −1 2 m− 3 2 T1 ln(1 + r) −1 2 m− 3 2 σ √ T1 − 12m −3 2 e −2σ √ T1 m σ √ T1 . On multiplication of 2m 3 2 into the numerator and denominator and simplifying it follows that, = lim m→∞ 4e( T1 m ) ln(1+r)+σ √ T1 m (m−1 (T1 ln(1 + r)) 2 −m 1 2 T 3 2 1 σ ln(1 + r) + 1 4 T1σ − 12T1 ln(1 + r)) −σ √ T1 ( 1 + e −2σ √ T1 m ) = 4 ( 1 4 σ2 − 1 2 T1 ln(1 + r) ) −2σ √ T1 lim m→∞ 2 √ m ( p− 1 2 ) = ln(1 + r) − σ 2 2 σ √ T1. On returning the value of p, lim m→∞ p = lim m→∞ 1 2 + 1 2 µ σ √ T1 m = 1 2 + 1 2 ( ln(1 + r) − 1 2 σ2 ) σ √ T1 m . (5.10) On the same lines we can prove that, between m and n, lim m→∞ 1 2 + 1 2 µ σ √ T2 n−m = 1 2 + 1 2 ( ln(1 + r) − 1 2 σ2 ) σ √ T2 n−m . (5.11) Now recall from the equation (5.3), µ̂pm = [ p ln ( U D ) + ln D ] m. Substitute the values of U = e σ √ T1 m and D = e −σ √ T1 m ( or U D = e 2σ √ T1 m ) , we get, µ̂pm = [ p ln ( U D ) + ln D ] m µ̂pm = [ p ln ( e 2σ √ T1 m ) + ln e −σ √ T1 m ] m = [ 2σp √ T1 m −σ √ T1 m ] m = [2p− 1] mσ √ T1 m . Int. J. Anal. Appl. 18 (1) (2020) 145 Taking limit m →∞ and using equation (5.10), it follows that, lim m→∞ µ̂pm = lim m→∞ [ 2 ( 1 2 + 1 2 ( ln(1 + r) − 1 2 σ2 ) σ √ T1 m ) − 1 ] mσ √ T1 m = lim m→∞ [( ln(1 + r) − 1 2 σ2 ) σ √ T1 m ] mσ √ T1 m lim m→∞ µ̂pm = ( ln(1 + r) − 1 2 σ2 ) T1. (5.12) Now we turn towards the next interval i.e. m to n. Recall the equation (5.4), µ̂p(n−m) = [ p ln ( U D ) + ln D ] (n−m) . Substitute the values of U = e σ √ T2 n−m and D = e −σ √ T2 n−m ( or U D = e 2σ √ T2 n−m ) , we get, µ̂p(n−m) = [ p ln ( e 2σ √ T2 n−m ) + ln e −σ √ T2 n−m ] (n−m) = [ 2σp √ T2 n−m −σ √ T2 n−m ] (n−m) = [2p− 1] (n−m) σ √ T2 n−m . Taking limit (n−m) →∞ and using equation (5.11), it follows that, lim (n−m)→∞ µ̂p(n−m) = [ 2 ( 1 2 + 1 2 ( ln(1 + r) − 1 2 σ2 ) σ √ T2 n−m ) − 1 ] (n−m) σ √ T2 n−m = lim m→∞ [( ln(1 + r) − 1 2 σ2 ) σ √ T2 n−m ] (n−m) σ √ T2 n−m lim (n−m)→∞ µ̂p(n−m) = ( ln(1 + r) − 1 2 σ2 ) T2. (5.13) Now let us return to the equation (5.6). Take limit m → ∞ and ε → 0, and using the equations (5.12) and (4.11), we consider the following limit. lim m→∞ lim ε→0 ln S ∗ S0 − µ̂pm−ε ln ( U D ) σ̂p √ m = ln S ∗ S0 − ( ln(1 + r) − 1 2 σ2 ) T1 σ √ T1 = d1. (5.14) Similarly consider the equation (5.8). By application of limit (n−m) → ∞ and ε → 0, and using the equation (4.14) and (5.13), we consider the following limit, lim m→∞ lim ε→0 ln ( K2 S ) − µ̂p(n−m) −ε ln ( U D ) σ̂p √ (n−m) = ln ( K2 S ) − ( ln(1 + r) − 1 2 σ2 ) T2 σ √ T2 = d2. (5.15) Int. J. Anal. Appl. 18 (1) (2020) 146 Thus we are in a position to claim our key convergence, using bivariate Binomial convergence to bivariate binomial from [16] (page 9) and Theorem 4.1 we infer that 1 −B (a,b; p′; n,n−m) = P   ln ( ST S0 ) −µ̂pm σ̂p √ m ≤ ln S ∗ S0 −µ̂pm−ε ln( UD ) σ̂p √ m , ln ( ST S0 ) −µ̂p(n−m) σ̂p √ N ≤ ln( K2 S )−µ̂p(n−m)−ε ln( U D ) σ̂p √ (n−m)   → N ( −d1,−d2; √ T2 T1 ) as (n−m) , m →∞, where ln S ∗ S0 − ( ln(1 + r) − 1 2 σ2 ) T1 σ √ T1 : = −d1 ln ( K2 S ) − ( ln(1 + r) − 1 2 σ2 ) T2 σ √ T2 : = −d2. Since 1 −N ( −d1,−d2; √ T2 T1 ) = N (d1,d2) , hence B (a,b; p′; n,n−m) → N ( d1,d2; √ T2 T1 ) . where, ln S S∗ + ( ln(1 + r) − 1 2 σ2 ) T1 σ √ T1 : = d1 ln ( S K2 ) + ( ln(1 + r) − 1 2 σ2 ) T2 σ √ T2 : = d2. Precisely on the same line of argument we can show that B (a,b; p,p′; n,n−m) → N ( d∗1,d ∗ 2; √ T2 T1 ) where, d∗1 = d1 + σ √ T1 d∗2 = d2 + σ √ T2 and N ( ·, ·; √ T2 T1 ) is the bivariate standard normal CDF with correlation coefficient √ T2 T1 . Finally we will deal with the one-dimensional convergence i.e. ψ (i ≥ a,m,p) → N (d∗1) as m →∞. Again 1 −ψ (i ≥ a,m,p) = P (i ≤ a− 1) = P ( i−mp√ mp(1 −p) ≤ a− 1 −mp√ mp(1 −p) ) where a = min { i ∈ m; i ≥ ln S ∗ S0Dm ln U D } ∈ [ ln S ∗ S0Dm ln U D , ln K1 CDm ln U D + 1 ) . Int. J. Anal. Appl. 18 (1) (2020) 147 Using equation (5.5) and equation (5.6) into the last equation we infer that 1 −ψ (i ≥ a,m,p) = P  ln ( ST S0 ) − µ̂pm σ̂p √ m ≤ ln S ∗ S0 − µ̂pm−ε ln ( U D ) σ̂p √ m   . Using the convergence (5.14) and the one dimensional Central Limit Theorem, and arguing in same manner as we argued in the end of the section 4 , we may infer that 1 −ψ (i ≥ a,m,p) = P  ln ( ST S0 ) − µ̂pm σ̂p √ m ≤ ln S ∗ S0 − µ̂pm−ε ln ( U D ) σ̂p √ m   → N ( ln S ∗ S0 − ( ln(1 + r) − 1 2 σ2 ) T1 σ √ T1 −σ √ T1 ) as m →∞, where N (·) is the c.d.f of standard normal distribution. By the use of symmetry property of the standard normal c.d.f i.e. 1 −N (z) = N (−z), it follows that, ψ (i ≥ a,m,p) = N ( ln S S∗ + ( ln(1 + r) − 1 2 σ2 ) T1 σ √ T1 + σ √ T1 ) = N(d∗1). Hence, we are done with the conclusion that C = S0B (i ≥ a,j ≥ b; p′; n,n−m) − (1 + r)−(n−m)K2B (i ≥ a,j ≥ b; p′,p; n,n−m) −(1 + r)−mK1ψ (i ≥ a,m,p) → S0N ( d1,d2; √ T2 T1 ) −e−rT2K2N ( d∗1,d ∗ 2; √ T2 T1 ) −e−rT1K1N(d∗1), (5.16) where d1 = ln S S∗ + ( ln(1 + r) − 1 2 σ2 ) T1 σ √ T1 ,d∗1 = d1 + σ √ T1 d2 = ln ( S K2 ) + ( ln(1 + r) − 1 2 σ2 ) T2 σ √ T2 ,d∗2 = d2 + σ √ T2. Thus we have shown that our developed CRR formula converges to the standard well-known continuous time Black-Scholes price of the compound option. � 6. Conclusion The paper provides a comprehensive treatment of the binomial pricing of option of financial derivatives, in general, and options in particular. Following two new results have been proven. 1. 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