Department of Mathematics Weather Derivatives Pricing Using Regime Switching Models Emanuel Evarest, Fredrik Berntsson, Martin Singull and Xiangfeng Yang LiTH-MAT-R--2017/04--SE Department of Mathematics Link¨oping University S-581 83 Link¨oping Weather Derivatives Pricing Using Regime Switching Models Emanuel Evarest1,2, Fredrik Berntsson1, Martin Singull1 and Xiangfeng Yang1 1Department of Mathematics Linköping University SE-581 83 Linköping, Sweden [email protected], [email protected], [email protected], [email protected] 2Department of Mathematics University of Dar Es Salaam P.O.Box 35062, Dar Es Salaam, Tanzania Abstract In this study we discuss the pricing of weather derivatives whose underlying weather variable is temperature. The dynamics of temperature in this study follows a two state regime switching model with a heteroskedastic mean reverting process as the base regime and a shifted regime defined by Brownian motion with mean different from zero. We develop the mathematical formulas for pricing futures contract on heating degree days (HDDs), cooling degree days (CDDs) and cumulative average temperature (CAT) indices. We also present the mathematical expressions for pricing the corresponding options on futures contracts for the same temperature indices. The local volatility nature of the model in the base regime captures very well the dynamics of the underlying process, thus leading to a better pricing processes for temperature derivatives contracts written on various index variables. We provide the description of Montecarlo simulation method for pricing weather derivatives under this model and use it to price a few weather derivatives call option contracts. Keywords:Weather derivatives, Arbitrage-free pricing, Regime switching, Monte Carlo simulation, Option pricing 1 2 1 Introduction During the 20th century, weather derivatives emerged in the financial market as an in- strument for hedging various weather related risks in the energy sector and other closely related sectors. Weather derivatives were officially introduced in the major financial mar- kets like Chicago Mercantile Exchange (CME) in 1999 [9, 8]. Over the recent years, weather Derivatives have gained popularity where apart from energy and power indus- tries who were the first industries in the business, other industries and sectors joined the business. These industries include agricultural, tourism, insurance and reinsurance companies, brokers as well as retail business, which are now among the many partici- pants in the weather derivatives markets [18]. Currently, CME offers weather derivatives futures contracts and options based on daily average temperature indices in over 46 dif- ferent cities around the world, see [19, 25]. The key factor in using weather derivatives as an instrument for hedging risk is a reliable pricing method that could accelerate the development of weather derivatives which seems to be impended by lack of standard pricing method [16]. The situation has been contributed by its nature, where the under- lying weather variable is not a tradable asset. Also there is little liquidity in the market, implying that the weather derivatives market is an incomplete market. Due to the na- ture of incomplete market pricing models in acknowledging the presence of both hedge- able and unhedgeable risks, they are considered as more appropriate for pricing weather derivatives [6]. Weather derivatives market comprises of derivatives written on various weather variables like temperature, precipitation, wind and Hurricanes. Temperature derivatives contract are written based on heating degree days (HDDs), cooling degree days (CDDs), cumulative average temperature (CAT) and Pacific Rim (PRIM) indices [3, 18]. For a daily average temperature process Td(t), these temperature indices are defined as AccHDDs t2 T T˜ t dt, AccCDDs t2 T˜ t T dt. = t1 max ref − d( ) = t1 max d( ) − ref Also, CAT tR2 T˜ t dt and PRIM 1 t2 T˜ t dt,R where T is the reference = t1 d( ) = t2 t1 t1 d( ) ref temperature forR regulating domestic heating− andR cooling demands. For weather deriva- tives based on other weather variables like precipitation and wind see [18, 25]. Generally, a standardized weather derivative contract comprises of a weather measurement station, a weather variable (s), a contract period, a payoff function, and a premium depending on the nature of contract. The variables defining the payoff function also varies according to the type of contract that might be swaps, options, collars, straddles, strangles and binaries, see [23, 18]. In this study we focus on temperature derivatives pricing because the largest proportion ( about 98%) of weather derivatives currently traded in the market are based on temper- ature [14]. Specifically, we focus on pricing these derivatives whose underlying tem- 3 perature process is governed by a regime switching model with a heteroskedastic base regime. The model allows the volatility of the underlying process in the base regime to vary with changes in temperature process. In general regime switching models allow the underlying process to switch between states of the process over time. That means there are periods where the stochastic process is under the base regime and periods when it is under the shifted regime. The switching process is caused by changes in volatility of the underlying financial variables like interest rate, energy prices, weather variables etc. As a result, regime switching models have attracted many applications in various problems in economics and finance particularly in the area of option valuation, see [5, 4, 10, 20, 24]. For temperature dynamics, factors like deforestation, urbanization, changes of weather station and clear skies contribute to the discrete shifts of temperature stochastic process from one state to another [15]. The regime switching models represent temperature dynamics relatively better than single regime models due to its ability to capture the discrete shifts in the process. Then pric- ing of weather derivatives based on regime switching models will also provide relatively good pricing method since the underlying temperature models captures most of the nec- essary features of the underlying variables. Most of the research on weather derivatives pricing has been done on single regime models for different noise driving processes like Brownian motion, fractional Brownian and Levy process see [7, 1, 2, 21, 22]. The model for pricing weather derivatives used in this study is adopted from our previous article [13]. The model allows the volatility of temperature process to vary locally under its base regime temperature process. Thus, the price process of weather derivatives contracts un- der the equivalent measure will have the mean and variance as functions of the underlying variable dynamics. The contribution of this study is as follows: We develop mathematical expressions for pricing temperature derivatives contracts written on CAT, HDDs and CDDs futures. Then we formulate the dynamics of these futures prices under the equivalent probability mea- sure. Also pricing formulas for European call option price written on futures contracts are developed. Due to the complicated nature of the pricing formulas developed for weather derivatives contracts, they dont give explicit formulas for the expected payoff of the HDDs, CAT and CDDs call option contracts. Therefore, we describe the Monte Carlo simulation approach for the underlying temperature dynamics model and then use it to price the call option contracts. The remaining part of the article is organized as follows: In Section 2, we present a regime switching model for temperature derivatives pricing, and transformed to an equiv- alent probability measure Q by introducing the market price of risk. Also the closed form 4 solution of the regime switching model under the Q measure is presented. In Section 3 we provide closed form expressions for arbitrage free pricing of CAT, HDDs and CDDs fu- tures contracts together with the dynamics of these futures with respect to the underlying regime switching process. In Section 4 we describe a Monte Carlo simulations method for pricing temperature derivatives for European call options and then compute the expected payoff of these contracts under the market price of risk. Finally, in Section 5 we give concluding remarks and possible future work. 2 Regime Switching Model for Temperature Derivatives In this section, we consider the regime switching model for temperature dynamics for pricing weather derivatives contracts. The model allows us to reflect different states of weather dynamics contributed by various factors. We adopt the model from [13], defining temperature dynamics by two states regime switching model, comprising of mean revert- ing heteroskedastic process as base regime and Brownian motion as the shifted regime. The model represent the dynamics of the deseasonalized temperature T˜d(t), after remov- ing the seasonality Sd(t) from the daily average temperature Td(t). The model is given as T˜t,1 : dT˜t,1 = (µ1 − βT˜t,1)dt + σ1T˜t,1dWt, with probability q1, T˜d(t)= (2.1) T˜t,2 : dT˜t,2 = µ2dt + σ2dWt, with probability q2, µ1 where β is the mean reversion speed, β is the long-term mean and σ1 is the volatility of the mean-reverting heteroskedastic proces in base regime while µ2 and σ2 are the mean and volatility of the shifted regime process respectively. The seasonality is defined by 2π S (t)= A sin (t − A ) + A t + A , (2.2) d 1 365 2 3 4 where A1 is the amplitude, A2 is the phase angle, A3 and A4 are constants defining the linear trend. The movement of the regime switching process from one state to another is driven by the transition probabilities given by Pij = P (St = j|St 1 = i) for i, j =1, 2. (2.3) − Using the Ito formula, an integral representation to (2.1) can be derived and written as µ1 µ1 βt t β(t s) T˜t,1 : T˜t,1 = + (T˜0,1 − )e− + σ1T˜s,1e− − dWs T˜ (t)= β β 0 (2.4) d ˜ ˜ ˜ t R Tt,2 : Tt,2 = T0,2 + µ2t + 0 σ2dWs.