Pricing Index Options with Stochastic Volatility

The objective of this paper is to investigate the pricing accuracy under stochastic volatility where the volatility follows a square root process. The theoretical prices are compared with market price data (the German DAX index options market) by using two different techniques of parameter estimation, the method of moments and implicit estimation by inversion. Standard Black & Scholes pricing is used as a benchmark. The results indicate that the stochastic volatility model with parameters estimated by inversion using the available prices on the preceding day, is the most accurate pricing method of the three in this study and can be considered satisfactory. However, as the same model with parameters estimated using a rolling window (the method of moments) proved to be inferior to the benchmark, the importance of stable and correct estimation of the parameters is evident.

An efficient pricing based protocol for broadcasting in wireless ad hoc networks

In many applications of wireless ad hoc networks, wireless nodes are owned by rational and intelligent users. In this paper, we call nodes selfish if they are owned by independent users and their only objective is to maximize their individual goals. In such situations, it may not be possible to use the existing protocols for wireless ad hoc networks as these protocols assume that nodes follow the prescribed protocol without deviation. Stimulating cooperation among these nodes is an interesting and challenging problem. Providing incentives and pricing the transactions are well known approaches to stimulate cooperation. In this paper, we present a game theoretic framework for truthful broadcast protocol and strategy proof pricing mechanism called Immediate Predecessor Node Pricing Mechanism (IPNPM). The phrase strategy proof here means that truth revelation of cost is a weakly dominant-strategy (in game theoretic terms) for each node. In order to steer our mechanism-design approach towards practical implementation, we compute the payments to nodes using a distributed algorithm. We also propose a new protocol for broadcast in wireless ad hoc network with selfish nodes based on IPNPM. The features of the proposed broadcast protocol are reliability and a significantly reduced number of packet forwards compared to the number of network nodes, which in turn leads to less system-wide power consumption to broadcast a single packet. Our simulation results show the efficacy of the proposed broadcast protocol.

Risk Minimizing Option Pricing for a Class of Exotic Options in a Markov-Modulated Market

We address risk minimizing option pricing in a regime switching market where the floating interest rate depends on a finite state Markov process. The growth rate and the volatility of the stock also depend on the Markov process. Using the minimal martingale measure, we show that the locally risk minimizing prices for certain exotic options satisfy a system of Black-Scholes partial differential equations with appropriate boundary conditions. We find the corresponding hedging strategies and the residual risk. We develop suitable numerical methods to compute option prices.

An optimal weighted-average congestion based pricing scheme for enhanced QoS, , LNCS 4882, 2007

Pricing is an effective tool to control congestion and achieve quality of service (QoS) provisioning for multiple differentiated levels of service. In this paper, we consider the problem of pricing for congestion control in the case of a network of nodes under a single service class and multiple queues, and present a multi-layered pricing scheme. We propose an algorithm for finding the optimal state dependent price levels for individual queues, at each node. The pricing policy used depends on a weighted average queue length at each node. This helps in reducing frequent price variations and is in the spirit of the random early detection (RED) mechanism used in TCP/IP networks. We observe in our numerical results a considerable improvement in performance using our scheme over that of a recently proposed related scheme in terms of both throughput and delay performance. In particular, our approach exhibits a throughput improvement in the range of 34 to 69 percent in all cases studied (over all routes) over the above scheme.

Pricing a Shared Access Link for Fair and Efficient Operation with Variable User Data Rates

The literature on pricing implicitly assumes an "infinite data" model, in which sources can sustain any data rate indefinitely. We assume a more realistic "finite data" model, in which sources occasionally run out of data; this leads to variable user data rates. Further, we assume that users have contracts with the service provider, specifying the rates at which they can inject traffic into the network. Our objective is to study how prices can be set such that a single link can be shared efficiently and fairly among users in a dynamically changing scenario where a subset of users occasionally has little data to send. User preferences are modelled by concave increasing utility functions. Further, we introduce two additional elements: a convex increasing disutility function and a convex increasing multiplicative congestion-penally function. The disutility function takes the shortfall (contracted rate minus present rate) as its argument, and essentially encourages users to send traffic at their contracted rates, while the congestion-penalty function discourages heavy users from sending excess data when the link is congested. We obtain simple necessary and sufficient conditions on prices for fair and efficient link sharing; moreover, we show that a single price for all users achieves this. We illustrate the ideas using a simple experiment.

Resource Pricing for Fair and Efficient Link Sharing with a Finite Data Model

The literature on pricing implicitly assumes an "infinite data" model, in which sources can sustain any data rate indefinitely. We assume a more realistic "finite data" model, in which sources occasionally run out of data. Further, we assume that users have contracts with the service provider, specifying the rates at which they can inject traffic into the network. Our objective is to study how prices can be set such that a single link can be shared efficiently and fairly among users in a dynamically changing scenario where a subset of users occasionally has little data to send. We obtain simple necessary and sufficient conditions on prices such that efficient and fair link sharing is possible. We illustrate the ideas using a simple example

Network pricing for QoS: A 'Regulation' approach

V. S. Borkar’s work was supported in part by grant number III.5(157)/99-ET from the Department of Science and Technology, Government of India. D. Manjunath’s work was supported in part by grant number 1(1)/2004-E-Infra from the Ministry of Information Technology, Government of India.

Reinforcement learning applications in dynamic pricing of retail markets

In this paper, we investigate the use of reinforcement learning (RL) techniques to the problem of determining dynamic prices in an electronic retail market. As representative models, we consider a single seller market and a two seller market, and formulate the dynamic pricing problem in a setting that easily generalizes to markets with more than two sellers. We first formulate the single seller dynamic pricing problem in the RL framework and solve the problem using the Q-learning algorithm through simulation. Next we model the two seller dynamic pricing problem as a Markovian game and formulate the problem in the RL framework. We solve this problem using actor-critic algorithms through simulation. We believe our approach to solving these problems is a promising way of setting dynamic prices in multi-agent environments. We illustrate the methodology with two illustrative examples of typical retail markets.

Optimal multi-layered congestion based pricing schemes for enhanced QoS

Pricing is an effective tool to control congestion and achieve quality of service (QoS) provisioning for multiple differentiated levels of service. In this paper, we consider the problem of pricing for congestion control in the case of a network of nodes with multiple queues and multiple grades of service. We present a closed-loop multi-layered pricing scheme and propose an algorithm for finding the optimal state dependent price levels for individual queues, at each node. This is different from most adaptive pricing schemes in the literature that do not obtain a closed-loop state dependent pricing policy. The method that we propose finds optimal price levels that are functions of the queue lengths at individual queues. Further, we also propose a variant of the above scheme that assigns prices to incoming packets at each node according to a weighted average queue length at that node. This is done to reduce frequent price variations and is in the spirit of the random early detection (RED) mechanism used in TCP/IP networks. We observe in our numerical results a considerable improvement in performance using both of our schemes over that of a recently proposed related scheme in terms of both throughput and delay performance. In particular, our first scheme exhibits a throughput improvement in the range of 67-82% among all routes over the above scheme. (C) 2011 Elsevier B.V. All rights reserved.

Pricing Defaultable Bonds in a Markov Modulated Market

We address the problem of pricing defaultable bonds in a Markov modulated market. Using Merton's structural approach we show that various types of defaultable bonds are combination of European type contingent claims. Thus pricing a defaultable bond is tantamount to pricing a contingent claim in a Markov modulated market. Since the market is incomplete, we use the method of quadratic hedging and minimal martingale measure to derive locally risk minimizing derivative prices, hedging strategies and the corresponding residual risks. The price of defaultable bonds are obtained as solutions to a system of PDEs with weak coupling subject to appropriate terminal and boundary conditions. We solve the system of PDEs numerically and carry out a numerical investigation for the defaultable bond prices. We compare their credit spreads with some of the existing models. We observe higher spreads in the Markov modulated market. We show how business cycles can be easily incorporated in the proposed framework. We demonstrate the impact on spreads of the inclusion of rare states that attempt to capture a tight liquidity situation. These states are characterized by low risk-free interest rate, high payout rate and high volatility.

Pricing credit derivatives

The financial crisis set off by the default of Lehman Brothers in 2008 leading to disastrous consequences for the global economy has focused attention on regulation and pricing issues related to credit derivatives. Credit risk refers to the potential losses that can arise due to the changes in the credit quality of financial instruments. These changes could be due to changes in the ratings, market price (spread) or default on contractual obligations. Credit derivatives are financial instruments designed to mitigate the adverse impact that may arise due to credit risks. However, they also allow the investors to take up purely speculative positions. In this article we provide a succinct introduction to the notions of credit risk, the credit derivatives market and describe some of the important credit derivative products. There are two approaches to pricing credit derivatives, namely the structural and the reduced form or intensity-based models. A crucial aspect of the modelling that we touch upon briefly in this article is the problem of calibration of these models. We hope to convey through this article the challenges that are inherent in credit risk modelling, the elegant mathematics and concepts that underlie some of the models and the importance of understanding the limitations of the models.