# A Dynamic Programming Approach for Pricing Weather Derivatives under Issuer Default Risk

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## Abstract

**:**

## 1. Introduction

## 2. Dynamic Pricing Model for Weather Derivatives

#### 2.1. Assumptions and Notation

- Assets. There are WDs on S weather indices (at different geographical sites and/or on different weather events) that are priced at times $t=0,1,\dots ,T-1.$ At T the payoff of each WD is determined and the cash settlement takes place. The non-negative price of the WD on underlying s in time t is denoted as ${W}_{t,s}$ where $s=1,\dots ,S$ and $t=1,\dots ,T-1$. The final value ${W}_{T,s}$ corresponds to the non-negative payoff of the sth WD. We denote the vector of prices at t as ${W}_{t}={({W}_{t,1},{W}_{t,2},\dots ,{W}_{t,S})}^{\mathrm{\top}}$. Besides the WDs, a risk free asset ${B}_{t}$ with a constant per period return r is available. Trading with ${B}_{t}$ is not restricted in any way, that is, unlimited borrowing and lending at the interest r in each t is allowed. We assume there is no transaction costs on the asset market. No capital addition or withdrawals are possible throughout the investment horizon, such that the agents are exposed to self-financing constraints.
- Agents. There are $J+1$ heterogeneous market participants, indexed by i, with the risk preferences described by the exponential utility function of the form ${U}_{i}\left(x\right)=-exp(-{a}_{i}x)$, where ${a}_{i}>0$ is the risk aversion of agent i. All agents have the same multi-period investment horizon of length T. They invest at $t=0$ and they consume their terminal wealth at $t=T$. At $t=1,\dots ,T-1$ agents rebalance their weather portfolios and renegotiate the prices for WDs. All agents are endowed with an initial wealth of zero monetary units. We distinguish between J buyers, indicated by subscript j, $j=1,\dots ,J$, who hedge weather exposure of their random income ${I}_{j}$, and a purely financial investor, indicated by subscript m, who issues WDs. Each buyer holds a basket of WDs on the relevant weather indices to hedge weather caused fluctuations in her profits. The issuer holds positions in all S WDs. A portfolio of agent i includes ${\alpha}_{i,t}={({\alpha}_{i,t,1},\dots ,{\alpha}_{i,t,S})}^{\mathrm{\top}}$ shares of the corresponding WDs and ${\beta}_{i,t}$ shares of the asset ${B}_{t}$. Both ${\alpha}_{\xb7,\xb7}$ and ${\beta}_{\xb7,\xb7}$ are real valued, that is, all assets are perfectly divisible and short sales are allowed. We denote the value of ith agent’s portfolio at time t as ${V}_{i,t}$, where ${V}_{i,t}={\alpha}_{i,t}^{\mathrm{\top}}{W}_{t}+{\beta}_{i,t}{B}_{t}$. In each period t of the investment horizon, the agents maximise their expected utility of the terminal wealth with the available WDs and attain their demand and supply for the WDs. That is, in each period $t<T$ every agent i determines her self-financing trading strategy ${({\alpha}_{i,t+1},{\beta}_{i,t+1})}_{t=0,1,\dots ,T}^{\mathrm{\top}}$, in particular, she constructs the optimal hedging portfolio given the state of the system at time t. Partial market clearing with respect to WDs determines the equilibrium prices for the WDs.
- State. The observable state of the system at time t, denoted by ${\mathcal{W}}_{t}$, contains values of the underlying weather indices at t and the variables regarding default risk at t. The random state ${\mathcal{W}}_{t+1}$ is characterised by the conditional distribution function ${\mathsf{\Phi}}_{t}(w\prime ,w)=Pr({\mathcal{W}}_{t+1}\le w\prime |{\mathcal{W}}_{t}=w)$. We assume that this transition function $\mathsf{\Phi}$ satisfies the Feller property Stokey et al. (1989). Expectation taken with respect to ${\mathsf{\Phi}}_{t}(\xb7)$ is denoted by ${\mathrm{E}}_{t}(\xb7)$.Each agent $j\in J$ is faced with the following discrete time stochastic control system:$${V}_{j,t+1}={g}_{j,t}\{{V}_{j,t},{\left({\alpha}_{j,t,s}\right)}_{s\in {S}_{j}},{\mathcal{W}}_{j,t}\},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}t=0,1,\dots ,T$$

#### 2.2. Pricing WDs without Default Risk

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

#### 2.3. Default Risk

#### 2.4. Alternative Investment

- 1a
- Assets. Let assumption 1. hold. Let ${F}_{t}$ be a quoted price of an exchange traded financial asset at time t. While ${F}_{t}$ is given, ${F}_{t+1}$ is random, bounded, and predictable at t. Trading with ${F}_{t}$ is not restricted in any way, that is, short and long positions in the asset in each t are possible. We assume there is no transaction costs on the asset market. As before, no capital addition or withdrawals are possible throughout the investment horizon, such that the agents are exposed to self-financing constraints.

- 2a
- Agents. Let assumption 2. hold. Now, issuer m holds additionally ${f}_{m,t}$ shares of the exchange traded financial asset with exogenous price ${F}_{t}$. Also, ${f}_{\xb7,\xb7}$ is real valued, that is, all assets are perfectly divisible and short sales are allowed. The value of the issuer’s portfolio at time t becomes ${V}_{m,t}={\alpha}_{m,t}^{\mathrm{\top}}{W}_{t}-{f}_{m,t}{F}_{t}+{\beta}_{m,t}{B}_{t}$.

- 3a
- State. Let assumption 3. hold. The observable state of the system at time t, denoted by ${\mathcal{W}}_{t}$, contains additionally the quoted price ${F}_{t}$. The random state ${\mathcal{W}}_{t+1}$ is characterised by the conditional distribution function ${\mathsf{\Phi}}_{t}$. Expectation taken with respect to ${\mathsf{\Phi}}_{t}$ is denoted by ${\mathrm{E}}_{t}(\xb7)$.

## 3. Pricing Weather Derivatives Using Weather Data

#### 3.1. Pricing Chinese Rain

#### 3.1.1. Setup

#### 3.1.2. Generation of Dependent Rainfall Paths on a Daily Basis

#### 3.1.3. Results

## 4. Summary

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

**Proof**

**of**

**Proposition**

**1.**

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1. | Station numbers given by the World Meteorological Organisation are 57662 for Changde and 57447 for Enshi. |

**Figure 1.**Shifts of the demand curves by increasing default risk (

**top**) and capital costs (

**bottom**).

**Top**: the default risk probability ${p}_{t}=p$ changes from 0 (dashed) to 0.01, 0.05, 0.10 (solid lines, thicker with increasing p).

**Bottom**: capital cost p.a. r changes from 1% (dashed) to 5%, 10%, 15% (solid lines, thicker with increasing r) for ${\sigma}_{F}=0.1$. Both plots: the x-axis shows ${\alpha}_{j,1,1}$, the buyer’s position in the put option on the rainfall in Changde during May ($C{R}_{May,1}$) with strike $K=100$ and investment horizon $T=1$; the y-axis shows ${W}_{0,1}\left({\alpha}_{j,1}\right)$, the price our representative buyer is willing to pay for such an option, where ${\alpha}_{j,1}={({\alpha}_{j,1,1},{\alpha}_{j,1,2})}^{\mathrm{\top}}$ and while ${\alpha}_{j,1,2}$, the position in the other put option on the rainfall in Enshi during May ($C{R}_{May,2}$) is kept constant.

**Figure 2.**Shifts of the demand curves by increasing investment horizon T. Buyer’s demand with investment horizon $T=1$ (dashed) and $T=2$ (solid). Thinner lines correspond to the case capital costs $r=5\%$ and probability of issuer’s default ${p}_{t}=p=0$; thicker lines correspond to capital costs $r=5\%$ and probability of issuer’s default ${p}_{t}=p=0.05$ for ${\sigma}_{F}=0.1$. The x-axis shows ${\alpha}_{j,1,1}$, the buyer’s position in the put option on the rainfall in Changde during May ($C{R}_{May,1}$) with strike $K=100$; the y-axis shows ${W}_{0,1}\left({\alpha}_{j,1}\right)$, the price our representative buyer is willing to pay for such an option, where ${\alpha}_{j,1}={({\alpha}_{j,1,1},{\alpha}_{j,1,2})}^{\mathrm{\top}}$ and while ${\alpha}_{j,1,2}$, the position in the other put option on the rainfall in Enshi during May ($C{R}_{May,2}$) is kept constant.

**Figure 3.**Shifts of the supply curves by increasing volatility of the alternative investment. Investor’s supply with investment horizon $T=1$ (dashed) and $T=2$ (solid). Thinner lines correspond to the case low volatility of the alternative investment; thicker lines correspond to high volatility. The x-axis shows ${\alpha}_{m,1,1}$, the investor’s position in the put option on the rainfall in Changde during May ($C{R}_{May,1}$) with strike $K=100$; the y-axis shows ${W}_{0,1}\left({\alpha}_{m,1}\right)$, the price for which the investor is willing to sell such an option, where ${\alpha}_{m,1}={({\alpha}_{m,1,1},{\alpha}_{m,1,2})}^{\mathrm{\top}}$ and while ${\alpha}_{m,1,2}$, the position in the other put option on the rainfall in Enshi during April and May ($C{R}_{May,2}$) is kept constant.

Station | Number | Latitude | Longitude | Start Date | End Date |
---|---|---|---|---|---|

Changde | 57,662 | 29.05 | 111.68 | 1951/01/01 | 2009/11/30 |

Enshi | 57,447 | 30.28 | 109.47 | 1951/08/01 | 2009/11/30 |

Order/BIC | Changde | Enshi |
---|---|---|

0 | 70.83 | 60.02 |

1 | 53.21 | 43.21 |

2 | 53.47 | 44.69 |

3 | 65.64 | 59.72 |

**Table 3.**Maximum Likelihood estimator for the mixture of two exponential distributions for the rainfall amounts.

Parameter | Changde | Enshi |
---|---|---|

${\gamma}_{\xb7,t\in [{\tau}_{1},{\tau}_{2}]}$ | 0.78 | 0.58 |

${\beta}_{1,\xb7,t\in [{\tau}_{1},{\tau}_{2}]}$ | 15.90 | 23.14 |

${\beta}_{2,\xb7,t\in [{\tau}_{1},{\tau}_{2}]}$ | 0.62 | 1.86 |

**Table 4.**Put option prices on cumulative rainfall in Changde and Enshi under different market scenarios.

Scenarios | Put on ${\mathit{C}\mathit{R}}_{\mathit{M}\mathit{a}\mathit{y},1}$ | Put on ${\mathit{C}\mathit{R}}_{\mathit{M}\mathit{a}\mathit{y},2}$ | ||||
---|---|---|---|---|---|---|

$\mathit{T}=1$ | $\mathit{T}=2$ | $\mathit{T}=1$ | $\mathit{T}=2$ | |||

${\sigma}_{F}=0.1$ | ||||||

$r=1\%$ | $p=0$ | 100.00 | 100.30 | 100.00 | 96.68 | |

$r=5\%$ | $p=0$ | 99.67 | 99.95 | 98.81 | 96.35 | |

$r=1\%$ | $p=0.05$ | 91.22 | 93.95 | 86.74 | 87.45 | |

$r=5\%$ | $p=0.05$ | 90.87 | 93.62 | 85.95 | 87.12 | |

${\sigma}_{F}=0.25$ | ||||||

$r=1\%$ | $p=0$ | 100.00 | 100.31 | 100.23 | 96.68 | |

$r=5\%$ | $p=0$ | 99.67 | 99.97 | 99.71 | 96.36 | |

$r=1\%$ | $p=0.05$ | 91.23 | 94.23 | 86.88 | 87.73 | |

$r=5\%$ | $p=0.05$ | 90.92 | 93.99 | 86.47 | 87.51 |

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**MDPI and ACS Style**

Härdle, W.K.; Osipenko, M.
A Dynamic Programming Approach for Pricing Weather Derivatives under Issuer Default Risk. *Int. J. Financial Stud.* **2017**, *5*, 23.
https://doi.org/10.3390/ijfs5040023

**AMA Style**

Härdle WK, Osipenko M.
A Dynamic Programming Approach for Pricing Weather Derivatives under Issuer Default Risk. *International Journal of Financial Studies*. 2017; 5(4):23.
https://doi.org/10.3390/ijfs5040023

**Chicago/Turabian Style**

Härdle, Wolfgang Karl, and Maria Osipenko.
2017. "A Dynamic Programming Approach for Pricing Weather Derivatives under Issuer Default Risk" *International Journal of Financial Studies* 5, no. 4: 23.
https://doi.org/10.3390/ijfs5040023