Ch 10. Arithmetic Average Options and Asian Opitons


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1 Ch 10. Arithmetic Average Options an Asian Opitons I. Asian Option an the Analytic Pricing Formula II. Binomial Tree Moel to Price Average Options III. Combination of Arithmetic Average an Reset Options Asian options are path epenent erivatives whose payoffs epen on the average of the unerlying asset prices uring the option life. They were originally issue in 1987 by Bankers Trust Tokyo on crue oil contracts an hence with the name Asian option. The features or avantages of Asian options are as follows. 1. Asian options are appropriate to meet the heging nees of users of commoities, energies, or foreign currencies who will be expose to the risk of average prices uring a future perio. 2. Since the volatility for the average of the unerlying asset prices is lower than the volatility for the unerling asset prices, Asian options are less expensive than corresponing vanilla options an are therefore more attractive for some investors. 3. Asian options are also useful in thinlytrae markets to prevent the manipulation of the unerlying asset price. In this chapter, for the teaching purpose, average options an Asian options are classifie epenent on either the price of the unerlying asset at maturity or the strike price being replace by the average price. average price call: max(save K, 0) Average options average price put: max(k Save, 0) average strike call: max(s T Save, 0) Asian option average strike put: max(save S T, 0) 101
2 I. Asian Option an Its Analytic Pricing Formula If Save is efine as the geometric average of stock prices, since the prouct of lognormally istribute ranom variables also follows the lognormal istribution, Save is lognormally istribute. In the riskneutral worl, the process of Save over a certain perio T is with the expecte continuously compouning growth rate 2 1 σ2 (r q 6 )T (i.e., E[S ave] = S 0 e 1 σ2 (r q )T 2 6 ) an the volatility σ T / 3. For geometric average options, because the role of Save is the same of S T in the payoff function, base on the lognormal istribution of Save an the BlackScholes formula, the price formula for geometric average option can be erive straightforwar. For a geometric average call, option value = S 0 e (a r)t N( 1 ) Ke rt N( 2 ) = e rt [S 0 e at N( 1 ) KN( 2 )] = e rt [E[geometric average until T ]N( 1 ) KN( 2 )] 1 = 2 = a = σ A = σ 3 ln(s0/k)+(a+ 1 2 σ2 A)T σ A T 1 σ A T 1 σ2 2 (r q 6 ) = ln(s0eat /K)+( 1 2 σ2 A)T σ A T Kemna an Vorst (1990), A Pricing Metho for Option Base on Average Asset Values, Journal of Banking & Finance 14, pp
3 If Save is efine as the arithmetic average of stock prices, it is more ifficult to price the arithmetic average option. An approximation metho is escribe as follows. First, calculate the first an the secon moments of Save uring the option life T. M 1 = e(r q)t 1 (r q)t S 0 = E[arithmetic average until T ] M 2 = 2e (2r 2q+σ2 )T S 2 0 (r q+σ 2 )(2r 2q+σ 2 )T 2 + 2S2 0 (r q)t 2 ( 1 2(r q)+σ 2 e(r q)t r q+σ 2 ) Secon, assume that Save is lognormally istribute with the first an secon moments mentione above. Finally, base on the BlackScholes formula for futures options, the value of an arithmetic average call can be approximate as follows. c = e rt [F 0 N( 1 ) KN( 2 )] 1 = ln(f0/k)+σ2 T/2 σ T 2 = 1 σ T where F 0 = M 1, σ 2 = 1 T M2 ln( ) M1 2 (In the original formula for futures option, F 0 is the current futures price mature at T, an σ 2 is the corresponing variance of the futures price.) Turnbull an Wakeman (1991), A Quick Algorithm for Pricing European Average Option, Journal of Financial an Quantitative Analysis 26, pp
4 II. Binomial Tree Moel to Price Average Options The naive pricing metho base on the treebase moel, which tracks all possible arithmetic average prices reaching each noe, is able to erive exact option values for both arithmetic an geometric average options. The naive pricing metho only works for geometric average options. It is intractable to price arithmetic average options ue to the exponential growth of the number of possible arithmetic average prices with respect to the number of time steps, n. Instea of keeping track of all possible arithmetic average prices, Hull an White (1993) introuce representative average prices to be (logarithmically) equallyspace place between the maximum an minimum arithmetic average prices for each noe. In aition, the piecewise linear interpolation is employe to approximate the corresponing option values for nonexistent average prices uring the backwar inuction. The algorithm of Hull an White (1993): (1) For any noe(i, j), the maximum arithmetic average price is contribute by a price path starting with i j consecutive up movements followe by j consecutive own movements, an the minimum arithmetic average price can be calculate from a price path starting with j consecutive own movements followe by i j consecutive up movements. Figure 101 S 0 A noe(1,0) S 0 u noe(0,0) noe(1,1) S 0 max ( i, j) A min ( i, j) noe(i, j) Su 0 i j j i j up movements j own movements { }} { { }} { A max (i, j) = S 0 (1 + u + u u i j + u i j + u i j u i j j )/ (i + 1) = (S 0 1 u i j+1 1 u + S 0 u i j 1 j )/(i + 1) 1 j own movements i j up movements { }} { { }} { A min (i, j) = S 0 ( j + j u + j u j u i j )/(i + 1) = (S 0 1 j S 0 j u 1 ui j )/(i + 1) 1 u 104
5 (2) For each noe, representative average prices are arraye (logarithmically) equallyspace from the maximum to the minimum arithmetic average prices for each noe via the following formula. A(i, j, k) = M k M A max(i, j) + k M A min(i, j), for k = 0,..., M. ( ( M k A(i, j, k) = exp M ln(a max(i, j)) + k ) ) M ln(a min(i, j)), for k = 0,..., M. (3) For each terminal noe(n, j), ecie the payoff for each representative average price A(n, j, k). Figure 102 M k k Amax ( i, j) Amin ( i, j), for k 0,1,2,..., M M M noe( n, j) Su 0 n j j M+1 representative average prices An (, j,0) A ( n, j) max An (, jk, ) An (, jm, ) A ( n, j) min max( An (, j,0) K,0) max( An (, jk, ) K,0) max( An (, jm, ) K,0) 105
6 (4) Backwar inuction Figure 103 noe( i1, j) S u 0 i1 j j Ai ( 1, j,0) A ( i1, j) max A( i 1, j, k 1) u Ci ( 1, j,0) C( i 1, j, k 1) u A i 1, j, k ) C i 1, j, k ) ( u ( u noe( i, j) S u 0 i j j A u Ai ( 1, jm, ) A ( i1, j) min Ci ( 1, jm, ) Ai (, j,0) A (, i j) max Ci (, j,0) A( i, j, k) C( i, j, k) noe( i1, j1) Ai (, jm, ) A (, i j) Ci (, jm, ) min S u 0 i1( j1) j1 A Ai ( 1, j1,0) A ( i1, j1) max A( i 1, j 1, k 1) Ci ( 1, j1,0) C( i 1, j 1, k 1) A i 1, j 1, k ) C i 1, j 1, k ) ( ( Ai ( 1, j1, M) A ( i1, j1) min Ci ( 1, j1, M) For A(i, j, k), 0 j i n, an k= 0, 1,..., M, A u = (i+1)a(i,j,k)+s0ui+1 j j i+2 Suppose A u is insie the range [A(i + 1, j, k u ), A(i + 1, j, k u 1)]. The corresponing option value C u for A u can be approximate by the linear interpolation, i.e., C u = w u C(i + 1, j, k u ) + (1 w u )C(i + 1, j, k u 1), where w u = A(i + 1, j, k u 1) A u A(i + 1, j, k u 1) A(i + 1, j, k u ). A = (i+1)a(i,j,k)+s0ui+1 (j+1) (j+1) i+2 Similarly, if A is insie the range [A(i+1, j +1, k ), A(i+1, j +1, k 1)]. The corresponing option value C for A can be approximate by the linear interpolation following the same logic as above. 106
7 C(i, j, k) = (P C u + (1 P ) C ) e r t If American arithmetic average options are consiere, the option value C(i, j, k) = max(a(i, j, k) K, (P C u + (1 P ) C ) e r t ). As a consequence, the interpolation error emerges an pricing results might not converge to exact option values unless the number of representative average prices for each noe, M, is sufficiently large an well collocate with the number of time steps, n, in the tree moel. Generally speaking, with the increase of the number of time steps in the tree moel, more representative average prices are neee for each noe to erive convergent results. 107
8 III. Combination of Arithmetic Average an Reset Options This section introuces a financial innovation to combine two attrative features, the Arithmetic Average an Reset Options, to form a new options. The pricing moel of this new option is first propose by Kim, Chang, an Byun (2003), Valuation of Arithmetic Average Reset Options, Journal of Derivatives 11, pp The payoff of a stanar reset call: max(s T K T, 0). Since the strike price is reset ownwar for calls, K T = min(k 0, S t1, S t2,, S ti ), where t 1, t 2,..., t I are reset ates. Arithmetic average reset calls: the same payoff function as that for stanar reset calls, except that K T = min(k 0, A t1, A t2,, A ti ). The avantages of the arithmetic average reset options: Avoi manipulation on (or near) the reset ate. The arithmetic average feature can reuce the option premium. Figure 104 Suppose t T / n, an the reset ates t nt. i i n 0 n 1...n i1 n i m n i+1... n I1 n I n I+1 t 0 t 1... t i1 t i t i+1... t I1 t I t I+1 = = 0 T At 1 A A ti t i 1 A ti = ( S S S )/( n n ) ( ni11) t ( ni12) t nit i i1 for n m n i i1 Am t( S( n 1) ( 2) )/( ) i ts ni t Sm t mni Km t min( K0, At, A,, ) 1 t A 2 ti 108
9 The evolution rule of state variables (K t, A t ): (i) For the root an the reset time points, the state variables at the next time point is (K t+ t, A t+ t ) = (K t, S t+ t ) (A t+ t = S t+ t inicates the start (or restart) of calculating the arithmetic average price at the next time point). (ii) For time points just before the reset time points, i.e., (n i 1) t, the state variables at the next time point is (K t+ t, A t+ t ) = (min(k t, G(A t, S t+ t )), G(A t, S t+ t )), where G(A t, S t+ t ) is an upating function for the arithmetic average price, which returns A t+ t given A t an S t+ t. (iii) For time points other than those in (i) an (ii), only upate the arithmetic average price such that the state variables at the next time point is (K t+ t, A t+ t ) = (K t, G(A t, S t+ t )). The ata structure of each noe: Representative values for A (an K) are logarithmically equallyspace place with the ifference h between the maximum an minimum arithmetic average prices (an the maximum an minimum strike prices) for each noe. Figure 105 S(m+1,j+1) K(m,j,k)=K max (m,j) exp(k h) K min A min K max A(m,j,l)=A max (m,j) exp(l h) S(m,j) A u (m,j,l) ln( Amax ) ln( Amin ) 1 h K min A min A max K max ln( Kmax ) ln( Kmin ) 1 h A (m,j,l) A max K min A min S(m+1,j) K max A max 109
10 The upating function for the arithmetic average price, G(A t, S t+ t ): For A(m, j, l) an n i < m < n i+1 A u (m, j, l) = [(m n i )A(m, j, l) + S(m + 1, j + 1)]/(m n i + 1) A (m, j, l) = [(m n i )A(m, j, l) + S(m + 1, j)]/(m n i + 1) Backwar inuction (i) Decie the payoff for each pair of (K, A) on terminal noes. The payoff is max(s T K T, 0), which is inepenent of the average variable A, so for each column with the same representative values of K, the payoff is the same (see Figure 106). Figure 106 (i) S(n I+1,n I+1 ) S(n I+1,j) S(n I+1,0) n I n I+1 (ii) A min A max K min K(n I+1,j,k) K max max(s(ni+1,j)kmin, 0) max(s(ni+1,j)k(ni+1,j,k), 0) max(s(ni+1,j)kmax, 0) (ii) For m = n I, n I + 1, n I + 2,..., n I+1 1, V (m, j, K(m, j, k), A(m, j, l)) = [P u V (m + 1, j + 1; K(m, j, k), A(m, j, l))+ P V (m + 1, j; K(m, j, k), A(m, j, l)]e r t (For the time perio between (n I + 1) t an (n I+1 1) t, the strike price K will not be reset, an the arithmetic average A will not change either at the next time point. Therefore, it is only necessary to fin option values at the next time point with state variable (K, A) ientical to the values of K(m, j, k) an A(m, j, l).) 1010
11 (iii) If m t is one of the reset ates for m = n 1, n 2,..., n I 1, V reset (m, j; K(m, j, k), A(m, j, l)) = [P u V (m + 1, j + 1; K(m, j, k), S(m + 1, j + 1))+ P V (m + 1, j; K(m, j, k), S(m + 1, j))]e r t (Since K(m, j, k) represents the strike price after the reset, the strike price K will not change at the next time point. Therefore, fin option values with the state variable K which is ientical to the value of K(m, j, k). As to the average state variable A, because the calculation of the arithmetic average price will restart at the next time point, fin option values with the state variable A which is equal to the stock prices of the following chil noes.) (iv) If m is the time point just before the reset ate, V (m, j; K(m, j, k), A(m, j, l)) =[P u V reset (m + 1, j + 1; min(k(m, j, k), A u (m, j, l)), A u (m, j, l)) + P V reset (m + 1, j; min(k(m, j, k), A (m, j, l)), A (m, j, l))]e r t (First, the arithmetic average price will be upate to be A u (m, j, l) for the upper chil noe an A (m, j, l) for the lower chil noe. Secon, the both strike prices are reset to be the minimums between K(m, j, k) an A u (m, j, l) for the upper chil noe an K(m, j, k) an A (m, j, l) for the lower chil noe.) (v) For values of m other than those in cases (i), (ii), (iii), an (iv), V (m, j; K(m, j, k), A(m, j, l)) = [P u V (m + 1, j + 1; K(m, j, k), A u (m, j, l)) +P V (m + 1, j; K(m, j, k), A (m, j, l))]e r t (Since the strike price will not be reset at the next time point, it is only necessary to take the upate of the arithmetic average price into account. So, fin option values with the state variable (K, A) to be (K(m, j, k), A u (m, j, l)) for the upper chil noe an (K(m, j, k), A (m, j, l)) for the lower chil noe.) During the backwar inuction process, if there are no matche representative arithmetic average price an strike price, fin the ajacent representative arithmetic average prices an ajacent representative strike prices to contain the target arithmetic average price an strike price. Then apply the twoimensional linear interpolation to erive the corresponing option price. In aition to the above algorithm of the backwar inuction, it is also important to ecie K min, K max, A min, an A max for each noe. In fact, it is necessary to erive A min an A max for each noe first, then to etermine K min an K max for the noes at the time points just before the reset ates, an finally to erive K min an K max for other noes following a backwar inheritance process
12 For n i + 1 m n i+1, an i = 0, 1,..., I 1, [S(m, j) + S(m 1, j) + + S(n i + 1, j)]/(m n i ) if j n i + 1 A max (m, j) = {[S(m, j) + S(m 1, j) + + S(j, j)]+ [S(j 1, j 1) + S(j 2, j 2) + + S(n i + 1, n i + 1)]} /(m n i ) if j > n i + 1 (For the upper case, trace the upper parent noe backwar until m = n i + 1. For the lower case, trace the upper parent noe backwar first. Once reaching the uppermost noe of the tree, trace the lower parent noe backwar until m = n i + 1.) Figure 107 n i+1 n i n i +1 m A min (m, j) = [S(m, j) + S(m 1, j 1) + + S(n i + 1, j m + n i + 1)]/(m n i ) if j m n i 1 {[S(m, j) + S(m 1, j 1) + + S(m j, 0)]+ [S(m j 1, 0) + + S(n i + 1, 0)]} /(m n i ) if j < m n i 1 (For the upper case, trace the lower parent noe backwar until m = n i + 1. For the lower case, trace the lower parent noe backwar first. Once reaching the lowermost noe of the tree, trace the upper parent noe backwar until m = n i + 1.) Figure 108 n i+1 n i n i +1 m 1012
13 For m = n i+1 1, an i = 1, 2,..., I, K max (m, j) = min(a max (n i, min(j, n i )), K 0 ), where the outsie minimum operator is to ensure the possible strike price after resets must be smaller than K 0. min(a min (n q 1, 0), K 0 ) if q < i + 1 K min (m, j) =, min(a min (n q, j (m n q )), K 0 ) otherwise where q is chosen to satisfy n q 1 m j < n q. Figure 109 n4 1 The path with the highest strike price noe( m, j) The path with the lowest strike price n1 n2 n3 n 4 For the time points n i+1 2, n i+1 3,..., n i, the K min an K max for each noe at these time points can be etermine backwar given the K min an K max for each noe at the time point of n i+1 1: { Kmin (m, j) = K min (m + 1, j + 1) (inherit from the upper chil noe) K max (m, j) = K max (m + 1, j) (inherit from the lower chil noe) The metho propose by Kim, Chang, an Byun (2003) to etermine K min an K max for each noe is complicate. In fact, K min an K max for each noe can be set to be 0 an K 0, respectively. Because the strike price is reset ownwar, the maximum value for K max of all noes must be K 0. In aition, since the stock price cannot be negative, it is impossible that the minimum value for K min becomes negative, an thus we can set K min for each noe to be 0. The above alternative by setting K min an K max to be globally minimum an maximum for each noe is much simpler. However, the larger ifference between K min an K max will increase the number of representative strike prices for each noe an in turn cause the heavier usage of the memory space an the CPU power to calculate the option value
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