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Introduction 12;9G0 60G22. 91G80; 91G20; . 1 mcnign nunderly- on contingent im ueia results numerical d nieyt rnlt the translate to unlikely hs oesassume models these e edbeoptions. tendible atnwinformation new tant nmto with motion an rat ihexercise with arrants s,PO o 700, Box P.O. asa, hr ar u the out carry chers o.Tecompound The ion. swoematurities whose ns e ncroaefi- corporate in sed dtevlainof valuation the ed nhle,adthis and holder, on drterisk the nder o ihjmsto jumps with ion nteastprice asset the on e-omsolution sed-form hr fcommon of share eeo e of set a develop hyaewidely are They ik underlying risky ta 3 and [3] al et n inbigthe being tion s pound 2 SHOKROLLAHI match the abnormal fluctuation of financial asset price, which was introduced into derivation of the option pricing model. Based on this theory, Dias and Rocha [8] considered the problem of pricing extendible options under petroleum concessions in the presence of jumps. Kou [17] and Cont and Tankov [7] also considered the problem of pricing options under a jump diffusion environment in a larger setting. Moreover, Gukhal [13] derived a pricing model for extendible options when the asset dynamics were driven by jump diffusion process. Hence, the analysis of compound and extendible options by applying jump process is a significant issue and provides the motivation for this paper. All this research above assumes that the logarithmic returns of the exchange rate are independent identically distributed normal random variables. However, the empirical studies demonstrated that the distributions of the logarithmic returns in the asset market generally reveal excess kurtosis with more probability mass around the origin and in the tails and less in the flanks than what would occur for normally distributed data [7]. It can be said that the properties of financial return series are nonnormal, nonindependent, and nonlinear, self-similar, with heavy tails, in both autocorrelations and cross-correlations, and volatility clustering [14, 4, 15, 16, 9]. Since fractional Brownian motion (F BM) has two substantial features such as self- similarity and long-range dependence, thus using it is more applicable to capture behavior from financial asset [22, 5, 27, 26, 28]. Unfortunately, due to F BM is neither a Markov process nor a semimartingale, we are unable to apply the classical stochastic calculus to analyze it [2]. To get around this problem and to take into account the long memory property, it is reasonable to use the mixed fractional Brownian motion (MFBM) to capture fluctuations of the financial asset [6, 10]. The MFBM is a linear combination of Brownian motion and F BM processes. Cheridito [6] proved that, for H ∈ (3/4, 1), the mixed model with dependent Brownian motion and F BM was equivalent to one with Brownian motion, and 1 hence it is arbitrage-free. For H ∈ ( 2 , 1), Mishura and Valkeila [21] proved that, the mixed model is arbitrage-free. In this paper, to capture the long range property, to exclude the arbitrage in the environment of F BM and to get the jump or discontinuous component of asset prices, we consider the problem of compound option in a jump mixed fractional Brownian motion (JMFBM) environment. We then exert the result to value ex- tendible options. We also provide representative numerical results. The JMFBM is based on the assumption that the underlying asset price is generated by a two- part stochastic process: (1) small, continuous price movements are generated by a MFBM process, and (2) large, infrequent price jumps are generated by a Poisson process. This two-part process is intuitively appealing, as it is consistent with an efficient market in which major information arrives infrequently and randomly. The rest of this paper is as follows. In Section 2, we briefly state some definitions related to MFBM that will be used in forthcoming sections. In Section 3, we analyze the problem of pricing compound option whose values follow a JMFBM process and present an explicit pricing formula for compound options. In Section 4, we derive an analytical valuation formula for pricing extendible option by compound option approach with only one extendible maturity under risk neutral measure, then extend this result to the valuation of an option with N extendible maturity. Section 5 deals with the simulation studies for our pricing formula. Moreover, the comparison of our JMFBM model and traditional models is undertaken in this Section. Section 6 is assigned to conclusion. EXTENDIBLEANDCOMPOUNDOPTIONS 3
2. Auxiliary facts
In this section we recall some definitions and results which we need for the rest of paper [21, 10, 28]. Definition 2.1: A MFBM of parameters ǫ, α and H is a linear combination of F BM and Brownian motion, under probability space (Ω,F,P ) for any t ∈ R+ by:
H H (2.1) Mt = ǫBt + αBt , H where Bt is a Brownian motion , Bt is an independent F BM with Hurst param- eter H ∈ (0, 1), ǫ and α are two real invariant such that (ǫ, α) 6= (0, 0).
Consider a frictionless continuous time economy where information arrives both continuously and discontinuously. This is modeled as a continuous component and as a discontinuous component in the price process. Assume that the asset does not pay any dividends. The price process can hence be specified as a superposition of these two components and can be represented as follows:
dSt = St(µ − λκ)dt + σStdBt H (2.2) + σStdBt + (J − 1)StdNt, 0 H where µ, σ, λ are constant, Bt is a standard Brownian motion, Bt is a indepen- dent F BM and with Hurst parameter H , Nt is a Poisson process with rate λ, J−1 1 2 2 is the proportional change due to the jump and k ∼ N(µJ = ln(1 + k) − 2 σJ ,σJ ). H The Brownian motion Bt , the F BM , Bt , the poisson process Nt and the jump amplitude J are independent. Using Ito Lemma [18], the solution for stochastic differential equation (2.2) is 1 1 (2.3) S = S exp (r − λk)t + σB + σBH − σ2t − σ2t2H J(n)N(t). t 0 h t t 2 2 i n where J(n) = i=1 Ji for n ≥ 1, Jt is independently and identically distributed Q St and J0 = 1; n is the Poisson distributed with parameter λt. Let xt = ln S0 . From Eq. (2.3) easily get 1 2 2 2H−1 H (2.4) dxt = r − λk − σ − Hσ t dt + σdBt + σdB + ln(J)dNt. 2 t Consider a European call option with maturity T and the strike price K written on the stock whose price process evolves as in Eq. (2.2). The value of this call option is known from [25] and is given by C(S0,K,T − T0) ∞ ′ e−λ (T −T0)(λ′(T − T ))n (2.5) = 0 S Φ(d ) − Ker(T −T0)Φ(d ), X n! 0 1 2 n=0 4 SHOKROLLAHI where S0 1 2 2 2H 2H 2 ln K + rn(T − T0)+ 2 [σ (T − T0)+ σ (T − T0 )+ nσJ )] d1 = , 2 2 2H 2H 2 qσ (T − T0)+ σ (T − T0 )+ nσJ 2 2 2H 2H 2 d2 = d1 − qσ (T − T0)+ σ (T − T0 )+ nσJ , ′ n ln(1+k) λ = λ(1 + k), rn = r − λk + T −T0 and Φ(.) is the cumulative normal distri- bution. 3. Compound options In order to derive a compound option pricing formula in a jump mixed fractional market, we make the following assumptions. (i) There are no transaction costs or taxes and all securities are perfectly divisible; (ii) security trading is continuous; (iii) there are no riskless arbitrage opportunities; (iv) the short-term interest rate r is known and constant through time; (v) the underlying asset price St is governed by the following stochastic dif- ferential equation Consider a compound call option written on the European call C(K, T2) with expi- ration date T1 and exercise price K1 , where T1 < T2 . Assume CC [C(K, T2), K1, T1] denotes this compound option. This compound option is exercised at time T1 when the value of the underlying asset, C(S1,K,T1, T2), exceeds the strike price K1 . When C(S1,K,T1, T2) < K1 , it is not optimal to exercise the compound option and hence expires worthless. The asset price at which one is indifferent between exercising and not exercising is specified by the following relation: (3.1) C(S1,K,T1, T2)= K1. ∗ Let, S1 shows the price of indifference which can be obtained as the numerical solution of the Eq. (3.1). When it is optimal to exercise the compound option at time T1 , the option holder pays K1 and receives the European call C(K, T1, T2). This European call can in turn be exercised at time T2 when ST exceeds K and expires worthless otherwise. Hence, the cashflows to the compound option are an ∗ outflow of K1 at time T1 when S1 >S1 , a net cashflow at time T2 of ST −K when ∗ S1 > S1 and ST > K , and none in the other states. The value of the compound option is the expected present value of these cashflows as follows: CC [C(K, T2), K1, T0, T1] − 2− 0 − 1− 0 = E e r(T T )(S − K)1 + E e r(T T )(−K )1 ∗ T0 h T ST >K i T0 h 1 S1>S1 i − 1− 0 − 2− 1 = E e r(T T )E e r(T T )(S − K)1 1 ∗ T0 h T1 h T ST >Ki S1>S1 i − 1− 0 −E e r(T T )K 1 ∗ T0 h 1 S1>S1 i − 2− 0 − 1− 0 (3.2) = E e r(T T )C(S ,K,T , T )1 ∗ − E e r(T T )K 1 ∗ T0 h 1 1 2 S1>S1 i T0 h 1 S1>S1 i where C(S1,K,T1, T2) is given in Eq. (2.5). EXTENDIBLEANDCOMPOUNDOPTIONS 5 Let, the number of jumps in the intervals [T0, T1) and [T1, T2] denoted by n1 and n2 , respectively and m = n1 + n2 shows the number of jumps in the interval [T0, T2]. Then, use the Poisson probabilities, we have − 2− 0 E e r(T T )C(S ,K,T , T )1 ∗ T0 h 1 1 2 S1>S1 i − 1− 0 − 2− 1 = E e r(T T )E e r(T T )(S − K)1 1 ∗ T0 h T1 h T ST >K i S1>S1 i ∞ ∞ ′ ′ e−λ (T1−T0)(λ′(T − T ))n1 e−λ (T2−T1)(λ′(T − T ))n2 = 1 0 2 1 X X n1! n2! n1=0 n2=0 − 1− 0 − 2− 1 ×E e r(T T )E e r(T T )(S − K)1 1 ∗ |n ,n T0 h T1 h T ST >Ki S1>S1 1 2i ∞ ∞ ′ ′ e−λ (T1−T0)(λ′(T − T ))n1 e−λ (T2−T1)(λ′(T − T ))n2 = 1 0 1 X X n1! n2! n1=0 n2=0 − 2− 0 ×E e r(T T )(S − K)1 1 ∗ |n ,n T0 h T ST >K S1>S1 1 2i The evaluation of this expectation requires the joint density of two Poisson weighted sums of correlated normal. From this point, we work with the loga- St rithmic return, xt = ln S0 , rather than the stock price. It is important to know that the correlation between the logarithmic return xT1 and xT2 depend on the number of jumps in the intervals [T0, T1) and [T1, T2]. Conditioning on the number of jumps n1 and n2 , xT1 has a normal distribution with mean 1 µ = (r − λk)(T − T ) − σ2(T − T ) JT1−T0 1 0 2 1 0 1 1 − σ2(T 2H − T 2H )+ n [ln(1 + k) − σ2 ] 2 1 0 1 2 J σ2 = σ2(T − T )+ σ2(T 2H − T 2H )+ n σ2 , JT1−T0 1 0 1 0 1 J 2 and xT2 ∼ N(µJ − ,σ ) where T2 T0 JT2−T0 1 µ = (r − λk)(T − T ) − σ2(T − T ) JT2−T0 2 0 2 2 0 1 1 − σ2(T 2H − T 2H )+ m[ln(1 + k) − σ2 ] 2 2 0 2 J σ2 = σ2(T − T )+ σ2(T 2H − T 2H )+ mσ2 . JT2−T0 2 0 2 0 J The correlation coefficient between xT2 and xT1 is as follows cov(x ,x ) ρ = T1 T2 . var(x ) × var(x ) p T1 T2 Evaluation the first expectation in Eq. (3.2) gives − 2− 0 E e r(T T )C(S ,K,T , T )1 ∗ T0 h 1 1 S1>S1 i ∞ ∞ ′ ′ e−λ (T1−T0)(λ′(T − T ))n1 e−λ (T2−T1)(λ′(T − T ))n2 = 1 0 2 1 X X n1! n2! n1=0 n2=0 −r(T2−T0) (3.3) ×hS0Φ2(a1, b1, ρ) − Ke Φ2(a2, b2, ρ)i 6 SHOKROLLAHI where S0 2 ln ∗ + µJ − + σ S1 T1 T0 JT1−T0 2 a1 = , a2 = a1 − σ q JT1−T0 σ2 q JT1−T0 S0 2 ln + µJ − + σ K T2 T0 JT2−T0 2 b1 = , b2 = b1 − σ q JT2−T0 σ2 q JT2−T0 Φ(x) is the standard univariate cumulative normal distribution function and Φ2(x, y, ρ) is the standard bivariate cumulative normal distribution function with correlation coefficient ρ. The second expectation in Eq. (3.2) can be evaluate to give − 1− 0 E e r(T T )K 1 ∗ T0 h 1 S1>S1 i ′ ∞ −λ (T1−T0) ′ n1 e (λ (T1 − T0)) − 1− 0 = E e r(T T )K 1 ∗ |n T0 h 1 S1>S1 1i X n1! n1=0 ′ ∞ −λ (T1−T0) ′ n1 e (λ (T1 − T0)) −r(T1−T0) (3.4) = K1e Φ(a2), X n1! n1=0 where a2 is defined above. Then, the following result for a compound call option is obtained. Theorem 3.1. The value of a compound call option with maturity T1 and strike price K1 written on a call option, with maturity T2 , strike K , and whose under- lying asset follows the process in Eq. (2.2), is given by CC [C(K, T2), K1, T0, T1] ′ ′ ∞ ∞ −λ (T1−T0) ′ n1 −λ (T2−T1) ′ n2 e (λ (T1 − T0)) e (λ (T2 − T1)) = n X X n1! n2! n1=0 n2=0 −r(T2−T0) ×hS0Φ2(a1, b1, ρ) − Ke Φ2(a2, b2, ρ)io ′ ∞ −λ (T1−T0) ′ n1 e (λ (T1 − T0)) −r(T1−T0) − K1e Φ(a2) X n1! n1=0 where a1, a2, b1, b2, and ρ are as defined previously. For a compound option with dividend payment rate q , the result is similar with Theorem 4.1, only r replaces with r − q . 4. Extendible option pricing formulae Based on the assumptions in the last Section, let EC be the value of an ex- tendible call option with time to expiration of T1 . At the time to expiration T1 , the holder of the extendible call can (1) let the call expire worthless if ST1 If at expiration time T1 the option is worth more than the extendible value with a new strike price of K2 for a fee of A for extending the expiration time T1 to T2 , then it is best to exercise; that is, ST1 − K1 ≥ C(ST1 , K2, T2 − T1) − A. Otherwise, it is best to extend the expiration time of the option to T2 and exercise when it is worth more than zero; that is, C(ST1 , K2, T2 − T1) − A> 0. Moreover, the holder of the option should be impartial between extending and not exercising at value L and impartial between exercising and extending at value M . Therefore, the critical values L and M are unique solutions of M − K1 = C(M, K2, T2 − T1) − A and M − K1 = C(L, K2, T2 − T1) − A = 0. See Longstaff [19] and Gukhal [13] for an analysis of the conditions. The value of a call option, C at time T1 with a time to expiration extended to T2 , as the discounted conditional expected payoff is given by EC(S , K , T , K , T , A) = E e−r(T1−T0)(S − K )1 0 1 1 2 2 T0 h T1 1 ST1 >M i + E e−r(T1−T0) C(S , K , T − T ) − A 1 T0 h T1 2 2 1 L≤ST1 ≤M i = E e−r(T1−T0)(S − K )1 T0 h T1 1 ST1 >M i −r(T1−T0) + ET0 he C(ST1 , K2, T2 − T1) − A (4.1) × 1 − 1 . ST1 ≥L ST1 ≥M i Then, by the same way of the call compound option, we have E e−r(T1−T0)(S − K )1 T0 h T1 1 ST1 >M i ∞ ′ e−λ (T1−T0)(λ′(T − T ))n1 (4.2) = 1 0 E e−r(T1−T0)(S − K )1 |n , T0 h T1 1 ST1 >M 1i X n1! n1=0 E e−r(T1−T0) C(S , K , T − T ) − A 1 − 1 T0 h T1 2 2 1 ST1 ≥L ST1 ≥M i = E e−r(T1−T0)E e−r(T2−T1)(S − K )1 1 − 1 T0 h T1 T2 2 ST2 >K2 ST1 ≥L ST1 ≥M i − E e−r(T1−T0)A 1 − 1 T0 h ST1 ≥L ST1 ≥M i ′ ′ ∞ ∞ −λ (T1−T0) ′ n1 −λ (T2−T1) ′ n2 e (λ (T1 − T0)) e (λ (T2 − T1)) = n X X n1! n2! n1=0 n2=0 −r(T2−T0) × ET0 e (ST2 − K )1S >K2 1S >L|n ,n 2 T2 T1 1 2o ′ ′ ∞ ∞ −λ (T1−T0) ′ n1 −λ (T2−T1) ′ n2 e (λ (T1 − T0)) e (λ (T2 − T1)) − n X X n1! n2! n1=0 n2=0 −r(T2−T0) × ET0 e (ST2 − K )1S >K2 1S >M |n ,n 2 T2 T1 1 2o ∞ ′ e−λ (T1−T0)(λ′(T − T ))n1 (4.3)− 1 0 E e−r(T1−T0)A(1 |n − 1 |n ) . n T0 ST1 >L 1 ST1 >M 1 o X n1! n1=0 8 SHOKROLLAHI Now, we assume that the asset price satisfies in Eq. (2.2). Then, by calculating the expectations in Eqs. (4.2) and (4.3), the following result is derived. Theorem 4.1. The price of an extendible call option with time to expiration T1 and strike price K1 , whose expiration time can extended to T2 with a new strike price K2 by the payment of an additional premium A, is given by EC(St, K1, T1, K2, T2, A) ∞ ′ e−λ (T1−T0)(λ′(T − T ))n1 = 1 0 S Φ(a ) − K e−r(T1−T0)Φ(a ) X n1! h 0 1 1 2 i n1=0 ′ ′ ∞ ∞ −λ (T1−T0) ′ n1 −λ (T2−T1) ′ n2 e (λ (T1 − T0)) e (λ (T2 − T1)) +n X X n1! n2! n1=0 n2=0 −r(T2−T0) ×hS0Φ2(b1, c1, ρ) − K2e Φ(b2, c2, ρ)io ′ ′ ∞ ∞ −λ (T1−T0) ′ n1 −λ (T2−T1) ′ n2 e (λ (T1 − T0)) e (λ (T2 − T1)) −n X X n1! n2! n1=0 n2=0 −r(T2−T0) −hS0Φ2(a1, c1, ρ) − K2e Φ(a2, c2, ρ)io ′ ∞ −λ (T1−T0) ′ n1 e (λ (T1 − T0)) −r(T1−T0) −n S0Ae X n1! n1=0 (4.4) ×hΦ(b2) − Φ(a2)io, where S0 2 ln + µJ − + σ M T1 T0 JT1−T0 2 a1 = , a2 = a1 − σ q JT1−T0 σ2 q JT1−T0 S0 2 ln + µJT −T + σJ − L 1 0 T1 T0 2 b1 = , b2 = b1 − σ q JT1−T0 σ2 q JT1−T0 S0 2 ln + µJ − + σ K2 T2 T0 JT2−T0 2 c1 = , c2 = c1 − σ q JT2−T0 σ2 q JT2−T0 Φ(x) is the standard univariate cumulative normal distribution function and Φ2(x, y, ρ) is the standard bivariate cumulative normal distribution function with correlation coefficient ρ. 1 Corollary 4.1. If H = 2 , the asset price satisfies the Merton jump diffusion equation (4.5) dSt = St(µ − λκ)dt + σStdBt + (J − 1)StdNt, 0 When λ = 0 , the asset price follows the MFBM model shown below H (4.6) dSt = Strdt + σStdBt + σStdBt . and the formula (3.3) reduces to the diffusion case. The result is in the following. EXTENDIBLEANDCOMPOUNDOPTIONS 9 Corollary 4.2. The price of an extendible call option with time to expiration T1 and strike price K1 , whose expiration time can extended to T2 with a new strike price K2 by the payment of an additional premium A and written on an asset following Eq. (4.6) is EC(St, K1, T1, K2, T2, A) −r(T1−T0) = S0Φ(a1) − K1e Φ(a2) −r(T2−T0) +S0Φ2(b1, c1, ρ) − K2e Φ(b2, c2, ρ) −r(T2−T0) −hS0Φ2(a1, c1, ρ) − K2e Φ(a2, c2, ρ)i −r(T1−T0) (4.7) −Ae hΦ(b2) − Φ(a2)i, where 2 2 S0 σ σ 2H 2H ln M + r(T1 − T0)+ 2 (T1 − T0)+ 2 (T1 − T0 ) a1 = , 2 2 2H 2H qσ (T1 − T0)+ σ (T1 − T0 ) 2H 2H a2 = a1 − σqT1 − T0 + T1 − T0 2 2 S0 σ σ 2H 2H ln L + r(T1 − T0)+ 2 (T1 − T0)+ 2 (T1 − T0 ) b1 = , 2 2 2H 2H qσ (T1 − T0)+ σ (T1 − T0 ) 2H 2H b2 = b1 − σqT1 − T0 + T1 − T0 2 2 S0 σ σ 2H 2H ln K2 + r(T2 − T0)+ 2 (T2 − T0)+ 2 (T2 − T0 ) c1 = . 2 2 2H 2H qσ (T2 − T0)+ σ (T2 − T0 ) 2H 2H c2 = c1 − σqT2 − T0 + T2 − T0. Let us consider an extendible option with N extended maturity times, the result is presented in the following corollary. Corollary 4.3. The value of the extendible call expiring at time T1 , written on an asset whose price is governed by equation (2.2) and whose maturity extend to T2 < T3 <, ..., < TN+1 with new strike of K2, K3, ..., KN+1 by the payment of corresponding premium of A1, A2, ..., AN+1 , is given by N+1 EC (S , K , T , T ) = S Φ (a∗ ,R∗) − K er(Tj −t)Φ(a∗ ,R∗) N 0 1 0 1 X nh 0 j 1j j j 2j j i j=1 ∗ ∗ r(Tj −t) ∗ ∗ − hS0Φj(c1j,Rj ) − Kje Φ(c2j,Rj )i r(Tj −t) ∗ ∗ ∗ ∗ (4.8) − Aje hΦ(b2j,R−1j) − Φ(a2j,R−1j )io ∗ ∗ where A0 = 0, Φj(a1j,Rj ) is the j -dimensional multivariate normal integral with ∗ upper limits of integration given by the j -dimensional vector a1j and correlation ∗ ∗ matrix Rj and define a1j = a1(M1, T1 − t), −a1(M2, T2 − t), ..., −a1(Mj, Tj − t) . ∗ ∗ ∗ ∗ The same as Φj(c1j ,Rj ) and Φj(b2j,Rj ) and define 10 SHOKROLLAHI ∗ c = b (L , T − t), a (M , T − t), ..., b (Lj− , Tj− − t), a (Mj, Tj − t) 1j 1 1 1 1 2 2 1 1 1 1 ∗ b = b (L , T − t), b (M , T − t), ..., b (Lj, Tj − t) 2j 2 1 1 2 2 2 2 ∗ ∗ ∗ and Φ1(c1j,Rj ). Rj is a j × j diagonal matrix with correlated coefficient ρp−1,p as the pth diagonal element, 0 and negative correlated coefficient ρj−1,j , respectively, as the first and the last diagonal element, and correlated coefficient ρp−1,s(s = p + 1, ..., j). As to the rest of the elements, we note that ρp−1,s is equal to negative correlated coefficient ρpj when s = j and ρp−1,s is equal to zero when p = 1,s = 0, ..., p−1, the term Tj and Mj,Lj respectively represents the j th “time instant” and the critical price as defined previously. As N increases to infinity the exercise opportunities become continuous and hence the value of the approximate option will converge in the limit to the value of the extendible option. Thus, the values EC1,EC2,EC3, ... form a converg- ing sequence and the limit of this sequence is the value of the extendible, i.e. limN→∞ ECN (S0, K1, T0, T1) = EC(S0, K1, T0, T1). To minimize the impact of this computational complexity, we use the Richardson extrapolation method [12] with two points. This technique uses the first two values of a sequence of a sequence to obtain the limit of the sequence and leads to the following equation, (4.9) EC2 = 2EC1 − EC0, where EC2 stands for the extrapolated limit using EC1 and EC0 . 5. Numerical studies Table 1 provides numerical results for extendible call options when the underlying asset pays no dividends. Column (3) displays the value obtained using the Merton model and column (4) shows the results using the Gukhal [13] method. Column (5) indicates the results by the JMFBM model and values using the Richardson extrapolation technique for EC1 and EC0 are shown in column (6). By comparing columns Merton, Gukhal, JMFBM and Richardson in Table 1 for the low- and high-maturity cases, we conclude that the call option prices obtained by these valuation methods are close to each other. Table 1. Results by different pricing models. Here, r = 0.1,σ = 0.1,L = 5, M = 15, A = 0.05,H = 0.8,S = 12,σJ = 0.3, k = −0.004. T1 K Merton Gukhal JMFBM Richardson 1 10 0.1127 0.11143 0.1228 0.1330 1 11 0.0960 0.0997 0.1075 0.1190 1 12 0.0812 0.0852 0.0922 0.1031 1 13 0.0687 0.0707 0.0768 0.0850 1 14 0.0587 0.0561 0.0615 0.0566 0.5 10 1.0347 0.7521 0.7799 0.5250 0.5 11 0.8387 0.6541 0.6783 0.5180 0.5 12 0.6662 0.5560 0.5768 0.4875 0.5 13 0.5412 0.4579 0.4753 0.4094 0.5 14 0.4598 0.3598 0.3738 0.2871 EXTENDIBLEANDCOMPOUNDOPTIONS 11 Fig 1 displays the price of extendible call option difference by the Merton, Guukhal and JMFBM models, according to the primary exercise date T1 and strike price K1 . JMFBM versus Merton model JMFBM versus Gukhal model 1 0.8 0.6 0.4 0.2 Difference 0 −0.2 −0.4 1 0.9 0.8 1 0.9 0.7 0.8 0.6 0.7 0.5 0.6 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 K 1 T (Year) 1 Figure 1. The relative difference between our JMFBM , Guukhal and Merton models. 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