Modelling Crop Insurance Based on Weather Index Using The Homotopy Analysis for American Put Option

Article history: Received : 10-12-2020 Reviced : 02-02-2021 Accepted : 04-02-2021 The crop insurance in Indonesia (AUTP) is much focused on the area impacted by flood, drought, and pest attack. The complication of the procedure to claim the loss must follow several conditions. The different approaches in the insurance sector, using weather index can be taken into consideration to produce a variety of insurance products. This insurance product used the American put option with the primary asset is the rainfall and the cumulative rainfall to exercise the claim, considering the optimal execution limit. The homotopic analysis is used to determine the valuation of the American put option, which also becomes the insurance premium. The case study is focused on areas experiencing a drought so that insurance claims can be exercise when the rainfall index value is below a predetermined limit. Considering the normality of the rainfall data, the calculation of insurance premium was done for the first growing season. The insurance premium is varies based on the optimal execution limit, while the calculation of profit is based on the optimum limit exercise and the minimum rainfall for the growing season, and its different depended on insurance claim acceptance limits. Keyword: Homotopy; Crop Insurance; Weather Index; American Option This is an open access article under the CC BY-SA license. DOI: https://doi.org/10.30812/varian.v4i2.993 ———————————————————— A. INTRODUCTION The crop insurance in Indonesia (AUTP) is much focused on the area impacted by flood, drought, and pest attack. The status quo right now, that to join AUTP the farmers should pay the premium Rp 180000 per hectare which is subsidized for about 144000 by the government, where the farmers will receive Rp 600000 per Hectare if the farmers having crop failure minimum 75%, affected by flood, pest, and drought. it has premium insurance complication of the procedure to claim the loss must follow several conditions. According to Hidayat & Gunardi (2019) there are some tendency of the farmers is not interested to join the AUTP because the complexity of the claim process as well as the premium is still not reasonable. Thus, it’s important to adjust new types of insurance, which can give another option to farmer or group of farmers for insuring their lands. Insurance based on rainfall is becoming new trend in development country such as in China, Australia, and Iran (Adeyinka et al., 2016). Rainfall can be used for calculating the insurance premium because its dependency with the yield (Hidayat & Gunardi, 2019). There are several techniques that can be used to determine the premium of insurance premium of crop insurance based of cumulative rainfall such as copula (Bokusheva, 2018), weather index (Adeyinka et al., 2016) and American put option using Black Scholes Model (Putri et al., 2017). As we can see that option concept is common to be implemented for determining the premium of the insurance. 118| Jurnal Varian| Vol.4, No.2, April 2021, Hal. 117-124 American put option is an option that gives the right not obligation for the option holder to sell the asset at certain prices at any time, it is different with European put option which must be exercised at maturity time. The problems in the valuation of American put option price is related to the free limitation problems. To overcome the problems analytically, Cheng et al., (2010), Dyke & Liao (2012) used the homotopy analysis method to find the solution of pricing the American put option and it is also can be implemented for European option (Fadugba, 2020). Homotopy analysis method can be used to solve non-linear problems and it gives the freedom to choose the type of equation from linear problems and basic function from the solution. The implementation of homotopy method is common in sector such as science, finance, and engineering. The homotopy analysis is also can be used to do pricing under option using Levy process (Sakuma & Yamada, 2014) and stochastic volatility (Park & Kim, 2011). In this research we try to explore more about the implementation of homotopy analysis method which also have been introduced by Zhu (2006) to analyse the solution of the price of American put option. We try to assess the price of American put option for calculating crop insurance premium which used the cumulative rainfall as the basic assets. The result is expected able to give another spectrum of implementation crop insurance in Indonesia, hence there is various type of crop insurance that can be choose by the farmers in the future to insure their assets. B. LITERATURE REVIEW 1. American put Option American put option gives the right not obligation to the option holder to sell the asset at certain time from the beginning up to the maturity time (Higham, 2004). The calculation of American option price using the Black-Scholes model involving the log normal return of the assets. Rt = ln ( St St−1 ) (1) where Rt is return assets at time t, St is stock price at time t, and St−1 is the stock price at time t − 1. For the implementation for this research, the term of log normal return of stock is modified with the log normal ratio of cumulative rainfall. According to Hull (2009) the volatility of stock, σ, is the uncertainty from the return of the stock. There are two types of volatility which are historical volatility and implied volatility. Historical volatility of the stock is estimated from the historical data of the stock prices. 2. Black Scholes formula Assume that the asset prices follow the formula of asset price movement model, St = S0 exp (μ − 1 2 σ) t + σ√tZ (2) where St is the price of assets at time t, S0 is the asset price at time 0, μ is the mean of asset return, σ is the volatility of the asset price, both μ and σ constant, and Z~N(0,1). The Black-Scholes option pricing formula for European type put option with strike price X, maturity time T, and risk-free interest rate r is, VE = X exp(−rT) N(−d2) − S0N(−d1) (3) given d1 = ln( S0 X )+(r+ 1 2 σ2)T σ√T , d2 = d1 − σ√T, and N(x) is cumulative distribution function from standard normal. 3. Decomposition of American Put Option In American put option, if Pt denoted the price of the option at time t ∈ [0, T, for each time t ∈ [0, T] there exist an optimal execution limit, Bt , which is optimal to exercise the option when S is figure 1 or above Bt . Agus Sofian Eka Hidayat, Modelling Crop Insurance...119 Figure 1. Domain Function of American Put Option Graduate & Myneni (1992) explained that in continues area C, the price of American put option, P0, can be decomposed from European put option, p0, and the premium for early exercises, e0: P0 = p0 + e0 (4) where e0 = rX ∫ exp(−rt) N ( ln( Bt S0 )−e2t σ√t ) T 0 dt and e2 = r − σ2 2 . 4. Homotopy Analysis Method Homotopy between two continues function f(x) and g(x) from topology space X to topology space Y is defined as continues function H: X × [0, 1] → Y from multiplication of space X with interval [0,1] to Y such that if x ∈ X then H(x; 0) = f(x) and H(x; 1) = g(x). In topology f(x) and g(x) is called homotopic, H: f(x)~g(x). From the set of real function C[a, b] if f ∈ C[a, b] continually deformed into g ∈ C[a, b] then we can perform homotopy H: f(x)~g(x) with H(x; q) = (1 − q)f(x) + qg(x), x ∈ [a, b]. The parameter q ∈ [0,1] from homotopy is called homotopy parameter (Dyke & Liao, 2012). Furthermore, for each homotopy, H(x; q) = (1 − q)f(x) + qg(x), x ∈ [a, b] we can define the first derivative of homotopy, ∂H(x; q) ∂q = g(x) − f(x), q ∈ [0, 1] (5) Which describe the ratio or rate continue deformation from f(x) to g(x). Dyke & Liao (2012) explained that the nonlinear function ε1 which has at least one solution u(z, t) with z and t are independent spatial and temporal variable, respectively. Given, q ∈ [0,1] denote the homotopy parameter and ε̅(q) is the null deformation equation which is connected the original ε1 and ε0 with approximation u0(z, t). Assume that the null deformation of ε̅(q) performed well which make the solution φ(z, t; q) exists, for q = 0. Hence the null deformation equation is φ(z, t; 0) = u0(z, t) while for q = 1, ε̅(q) equivalent with ε1 thus φ(z, t; 1) = u0(z, t). The Maclaurin sequence from φ(z, t; q) is φ(z, t; q)~u0(z, t) + ∑ un +∞