# Risk Management of Interest Rate Derivative Portfolios: A Stochastic Control Approach

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## Abstract

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## 1. Introduction

**underlying interest rate model**that is used to price the various deals. On the other hand, we must specify the

**financial institution’s objectives**(usually expressed as a real function, known as the utility, over the whole FIM portfolio) and the

**risk limits**imposed by risk management. These risk limits are in turn functions of the general position of the financial institution in the market place, of the its attitude towards risk and of various regulatory constraints. The outputs of this system are the various positions (controls) in each derivative so that the it meets its objectives. Risk is quantified using the sensitivity approach, i.e., through the various portfolio parameter sensitivities (delta, gamma, vega, theta, etc.).

## 2. Preliminaries

**value process**of the portfolio for an initial investment v is defined to be the process given by the strong solution of the linear stochastic differential equation

**admissible**for the portfolio F and is denoted by $\mathcal{A}(T,v)$. Throughout this paper we use the portfolio process concept or the equivalent trading positions concept according to our needs.

## 3. The Risk Management Control Problem

#### 3.1. The Control Sets

#### 3.2. Problem Formulation

#### 3.3. The Numerical Approximation Scheme

- We discretize the state-space Q taking the boundaries into consideration.
- The state and the control at time l are denoted by ${x}^{l}$, ${\pi}^{l}$ respectively. These are discrete-time stochastic processes such that ${x}^{l}\in O$ and ${\pi}^{l}\in U(l,{x}^{l})$. The ${x}^{l}$ is a discrete-time Markov chain (something that is natural given that the solution of the stochastic differential equation for the state variable x is a Markov process). The state dynamics are governed by the one-step transition probabilities ${p}_{l}^{{\pi}^{l}}(x,y)$, which describe the probability of going from the state x at time l to the state y at time $l+1$ given that the information structure is ${\mathcal{F}}_{l}$ and the Markov control is ${\pi}^{l}.$
- We solve recursively (backwards) the discrete dynamic programming equation.

- Boundary points are absorbing: ${p}^{\pi}(x,x)=0,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}x\in \partial {\Sigma}^{h}$
- If the points are interior points, i.e., $x\in {\Sigma}^{h}\backslash \partial {\Sigma}^{h}$, then the one step transition probabilities are defined as follows:

## 4. Application in the Single Factor Interest Rate Market

- Step 1
- We first calculate the value of the derivatives and the portfolio sensitivities at each point $(t,r(t\left)\right)$ of the side $OABF$.
- Step 2
- Since we know the value function over the sides $ABCD$, $OABF$, $DCEG$, $OADG$, we opt to calculate the value function starting from top to bottom (backwards in time). Taking a typical “slice” in time between $t-h$ and t (Figure 2), we see that the arrows starting from $(t-h,x,r)$ are the transition probabilities to time t. These are non-linear functions of the portfolio positions at time $t-h$, and their values are given according to equation (9). The value function at time $t-h$ and at state $(x,r)$ is given by the dynamic programming formula:$$V(t-h,x,r)=\underset{\theta (t-h)\in U(t-h,x,r)}{min}\sum _{i=1}^{9}{p}_{i}{V}_{i}$$
- Step 3
- For initial wealth v and for a particular level of the interest rate, we find the optimal portfolio positions that we adopt in order to meet our objectives. The whole evolution of the state dynamics can be found up to time T simply by following the tree created-by the state evolution probabilities. In addition, the procedure gives us, for every initial wealth and for every level of the interest rate, the evolution of the value function according to the state transition probabilities.

#### 4.1. Numerical Implementation and Results

bond | swap | ||
---|---|---|---|

Starting Period | 0 | Starting Period | 0 |

Maturity Period | 19 | Maturity Period | 19 |

Coupon Period | 2 | Coupon Period | 2 |

Coupon Rate | 0.10 | Coupon Rate | 0.10 |

Principal | 100.0 | Notional Principal | 100. |

European bond call | European bond put | ||

Starting Period | 0 | Starting Period | 0 |

Maturity Period | 10 | Maturity Period | 10 |

Exercise Price | 80.5 | Exercise Price | 80.5 |

Bond Staring Period | 0 | Bond Staring Period | 0 |

Bond Maturity Period | 19 | Bond Maturity Period | 19 |

Bond Coupon Period | 2 | Bond Coupon Period | 2 |

Bond Coupon Rate | 0.1 | Bond Coupon Rate | 0.1 |

Bond Principal | 100 | Bond Principal | 100 |

American bond call | American bond put | ||

Starting Period | 0 | Starting Period | 0 |

Maturity Period | 10 | Maturity Period | 10 |

Exercise Price | 80.5 | Exercise Price | 80.5 |

Bond Staring Period | 0 | Bond Staring Period | 0 |

Bond Maturity Period | 19 | Bond Maturity Period | 19 |

Bond Coupon Period | 2 | Bond Coupon Period | 2 |

Bond Coupon Rate | 0.1 | Bond Coupon Rate | 0.1 |

Bond Principal | 100 | Bond Principal | 100 |

European receiver’s Swaoption | European payer’s put | ||

Starting Period | 0 | Starting Period | 0 |

Maturity Period | 10 | Maturity Period | 10 |

Exercise Price | 6.5 | Exercise Price | 80.5 |

Swap Staring Period | 0 | Bond Maturity Period | 0 |

Swap Maturity Period | 19 | Bond Staring Period | 19 |

Swap Reset Period | 2 | Bond Coupon Period | 2 |

Swap Fixed Rate | 0.1 | Bond Coupon Rate | 0.1 |

Swap Notional Principal | 100 | Bond Principal | 100 |

American receiver’s Swaoption | American payer’s put | ||

Starting Period | 0 | Starting Period | 0 |

Maturity Period | 10 | Maturity Period | 10 |

Exercise Price | 6.5 | Exercise Price | 80.5 |

Swap Staring Period | 0 | Bond Staring Period | 0 |

Swap Maturity Period | 19 | Bond Maturity Period | 19 |

Swap Reset Period | 2 | Bond Coupon Period | 2 |

Swap Fixed Rate | 0.1 | Bond Coupon Rate | 0.1 |

Swap Notional Principal | 100 | Bond Principal | 100 |

cap | floor | ||

Starting Period | 0 | Starting Period | 0 |

Maturity Period | 10 | Maturity Period | 10 |

Reset Period | 2 | Reset Period | 2 |

Cap rate | 0.1 | Floor rate | 0.1 |

Cap Notional Principal | 100 | Floor Notional Principal | 100 |

Declining Initial Term Structure and Initial Volatility Structure | |
---|---|

Time Horizon $\left(N\right)$ | 20 |

Time Step $(\Delta t)$ | 1 |

β | 1 |

Initial Volatility $\left(V\right(0\left)\right)$ | 0.14 |

Initial Term Structure at time $i\Delta t$ | $0.1(1+(i\Delta t)/70)$ |

Initial Volatility Structure at at time $i\Delta t$ | $0.14(1+(i\Delta t)/110)$ |

Risk Limits | |
---|---|

Delta limit $\left(\delta \right)$ | 0.009211 |

Gamma limit $\left(\gamma \right)$ | 0.021584 |

Theta limit $\left(\theta \right)$ | 1.696839 |

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Kiriakopoulos, K.; Koulis, A.
Risk Management of Interest Rate Derivative Portfolios: A Stochastic Control Approach. *J. Risk Financial Manag.* **2014**, *7*, 130-149.
https://doi.org/10.3390/jrfm7040130

**AMA Style**

Kiriakopoulos K, Koulis A.
Risk Management of Interest Rate Derivative Portfolios: A Stochastic Control Approach. *Journal of Risk and Financial Management*. 2014; 7(4):130-149.
https://doi.org/10.3390/jrfm7040130

**Chicago/Turabian Style**

Kiriakopoulos, Konstantinos, and Alexandros Koulis.
2014. "Risk Management of Interest Rate Derivative Portfolios: A Stochastic Control Approach" *Journal of Risk and Financial Management* 7, no. 4: 130-149.
https://doi.org/10.3390/jrfm7040130