scholarly journals Pricing Formula for Exotic Options with Assets Exposed to Counterparty Risk

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Li Yan
Keyword(s):  
Asset Price ◽  
Exotic Options ◽  
Barrier Options ◽  
Counterparty Risk ◽  
Default Time ◽  
Novel Technique ◽  
Analytical Formulas ◽  
Pricing Formula

This paper gives analytical formulas for lookback and barrier options on underlying assets that are exposed to a counterparty risk. The counterparty risk induces a drop in the asset price, but the asset can still be traded after this default time. A novel technique is developed to valuate the lookback and barrier options by first conditioning on the predefault and the postdefault time and then obtain the unconditional analytic formulas for their prices.

scholarly journals Explicit Pricing Formulas for European Option with Asset Exposed to Double Defaults Risk

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Taoshun He
Keyword(s):  
Default Risk ◽  
Asset Price ◽  
European Option ◽  
Put Options ◽  
Default Time ◽  
Novel Technique ◽  
Endogenous Default ◽  
Option Model ◽  
Analytical Formulas

We derive analytical formulas for European call and put options on underlying assets that are exposed to double defaults risks which include exogenous counterparty default risk and endogenous default risk. The endogenous default risk leads the asset price to drop to zero and the exogenous counterparty default risk induces a drop in the asset price, but the asset can still be traded after this default time. A novel technique is developed to evaluate the European call and put options by first conditioning on the predefault and the postdefault time and then obtaining the unconditional analytic formulas for their price. We also compare the pricing results of our model with default-free option model and counterparty default risk option model.

scholarly journals Option Pricing for Path-Dependent Options with Assets Exposed to Multiple Defaults Risk

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Taoshun He
Keyword(s):  
Option Pricing ◽  
Default Risk ◽  
Asset Price ◽  
Default Time ◽  
Original Technique ◽  
Endogenous Default ◽  
Lookback Options ◽  
Option Model ◽  
Analytical Formulas ◽  
Path Dependent

In the present paper, we derive analytical formulas for barrier and lookback options with underlying assets exposed to multiple defaults risks which include exogenous counterparty default risk and endogenous default risk. The endogenous default risk leads the asset price drop to zero and the exogenous counterparty default risk induces a drop in the asset price, but the asset can still be traded after this default time. An original technique is developed to valuate the barrier and lookback options by first conditioning on the predefault and the afterdefault time and then obtaining the unconditional analytic formulas for their price. We also compare the pricing results of our model with the default-free option model and exogenous counterparty default risk option model.

scholarly journals Reverse Bonus Certificate Design and Valuation Using Pricing by Duplication Methods

2015 ◽  
Vol 62 (3) ◽  
pp. 277-289
Author(s):  
Martina Bobriková ◽  
Monika Harčariková
Keyword(s):  
Option Pricing ◽  
Exotic Options ◽  
Pricing Models ◽  
Underlying Asset ◽  
Replication Method ◽  
Maturity Date ◽  
Derivative Market ◽  
Replicating Portfolio ◽  
Pricing Formula

Abstract In this paper we perform an analysis of a capped reverse bonus certificate, the value of which is derived from the value of an underlying asset. A pricing formula for the portfolio replication method is applied to price the capped reverse bonus certificate. A replicating portfolio has profit that is identical to profit from a combination of positions in spot and derivative market, i.e. vanilla and exotic options. Based upon the theoretical option pricing models, the replicating portfolio for capped reverse bonus certificate on the Euro Stoxx 50 index is engineered. We design the capped reverse bonus certificate with various parameters and calculate the issue prices in the primary market. The profitability for the potential investor at the maturity date is provided. The relation between the profit change of the investor and parameters’ change is detected. The best capped reverse bonus certificate for every estimated development of the index is identified.

CLOSED FORM FORMULAS FOR EXOTIC OPTIONS AND THEIR LIFETIME DISTRIBUTION

1999 ◽  
Vol 02 (01) ◽  
pp. 17-42 ◽  
Author(s):  
RAPHAËL DOUADY
Keyword(s):  
Interest Rates ◽  
Closed Form ◽  
Term Structure ◽  
Exit Time ◽  
Exotic Options ◽  
Barrier Options ◽  
Double Barrier ◽  
Knock Out ◽  
The Mean ◽  
Mirror Principle

We first recall the well-known expression of the price of barrier options, and compute double barrier options by the mean of the iterated mirror principle. The formula for double barriers provides an intraday volatility estimator from the information of high-low-close prices. Then we give explicit formulas for the probability distribution function and the expectation of the exit time of single and double barrier options. These formulas allow to price time independent and time dependent rebates. They are also helpful to hedge barrier and double barrier options, when taking into account variations of the term structure of interest rates and of volatility. We also compute the price of rebates of double knock-out options that depend on which barrier is hit first, and of the BOOST, an option which pays the time spent in a corridor. All these formulas are either in closed form or double infinite series which converge like e-α n2.

scholarly journals Pricing Exotic Options Using Some Lattice Procedures

2021 ◽  
Vol 41 (1) ◽  
pp. 26-40
Author(s):  
Sadia Anjum Jumana ◽  
ABM Shahadat Hossain
Keyword(s):  
Stock Price ◽  
Lattice Models ◽  
Exotic Options ◽  
Barrier Options ◽  
Tree Model ◽  
Binomial Tree ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Binomial Tree Model ◽  
Trinomial Tree Model

In this work, we discuss some very simple and extremely efficient lattice models, namely, Binomial tree model (BTM) and Trinomial tree model (TTM) for valuing some types of exotic barrier options in details. For both these models, we consider the concept of random walks in the simulation of the path which is followed by the underlying stock price. Our main objective is to estimate the value of barrier options by using BTM and TTM for different time steps and compare these with the exact values obtained by the benchmark Black-Scholes model (BSM). Moreover, we analyze the convergence of these lattice models for these exotic options. All the results have been shown numerically as well as graphically. GANITJ. Bangladesh Math. Soc.41.1 (2021) 26-40

THE VALUATION OF THE BASKET CDS IN A PRIMARY-SUBSIDIARY MODEL

2011 ◽  
Vol 28 (02) ◽  
pp. 213-238 ◽  
Author(s):  
JIANWEI LIN ◽  
GECHUN LIANG ◽  
SEN WU ◽  
HARRY ZHENG
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Method ◽  
Numerical Results ◽  
Joint Density ◽  
Hazard Rates ◽  
Counterparty Risk ◽  
Default Time ◽  
The Monte Carlo Method ◽  
Basket Cds ◽  
Homogenous Case

This paper considers the valuation problem of basket CDSs. Based on the construction of total hazard rates, the paper develops the work of Zheng and Jiang Zheng and Jiang (2009) from the homogenous case to the primary-subsidiary heterogenous case in the interacting intensity framework, and obtains the corresponding joint density of the default time. Moreover, the paper derives the valuation formulae for the basket CDSs with and without counterparty risk. Numerical results robustly show that, under certain conditions, using the analytical pricing formulae derived in this paper is more efficient than the Monte Carlo method for the basket CDS valuation.

scholarly journals Pricing Various Types of Power Options under Stochastic Volatility

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1911
Author(s):  
Youngrok Lee ◽  
Yehun Kim ◽  
Jaesung Lee
Keyword(s):  
Financial Markets ◽  
Stochastic Volatility ◽  
Asset Price ◽  
Stochastic Volatility Model ◽  
Exotic Options ◽  
Option Prices ◽  
Black Scholes Model ◽  
Volatility Model ◽  
Black Scholes ◽  
Pricing Power

The exotic options with curved nonlinear payoffs have been traded in financial markets, which offer great flexibility to participants in the market. Among them, power options with the payoff depending on a certain power of the underlying asset price are widely used in markets in order to provide high leverage strategy. In pricing power options, the classical Black–Scholes model which assumes a constant volatility is simple and easy to handle, but it has a limit in reflecting movements of real financial markets. As the alternatives of constant volatility, we focus on the stochastic volatility, finding more exact prices for power options. In this paper, we use the stochastic volatility model introduced by Schöbel and Zhu to drive the closed-form expressions for the prices of various power options including soft strike options. We also show the sensitivity of power option prices under changes in the values of each parameter by calculating the resulting values obtained from the formulas.

scholarly journals PORTFOLIO OPTIMIZATION IN A DEFAULT MODEL UNDER FULL/PARTIAL INFORMATION

2015 ◽  
Vol 29 (4) ◽  
pp. 565-587 ◽  
Author(s):  
Thomas Lim ◽  
Marie-Claire Quenez
Keyword(s):  
Utility Function ◽  
Partial Information ◽  
Backward Stochastic Differential Equation ◽  
Asset Price ◽  
Incomplete Market ◽  
Exponential Utility ◽  
Indifference Pricing ◽  
Insider Information ◽  
Default Time ◽  
Default Intensity

In this paper, we consider a financial market with an asset exposed to a risk inducing a jump in the asset price, and which can still be traded after the default time. We use a default-intensity modeling approach, and address in this incomplete market context the problem of maximization of expected utility from terminal wealth for logarithmic, power and exponential utility functions. We study this problem as a stochastic control problem both under full and partial information. Our contribution consists of showing that the optimal strategy can be obtained by a direct approach for the logarithmic utility function, and the value function for the power (resp. exponential) utility function can be determined as the minimal (resp. maximal) solution of a backward stochastic differential equation. For the partial information case, we show how the problem can be divided into two problems: a filtering problem and an optimization problem. We also study the indifference pricing approach to evaluate the price of a contingent claim in an incomplete market and the information price for an agent with insider information.

A closed-form pricing formula for catastrophe equity options

2021 ◽  
pp. 1-13
Author(s):  
Puneet Pasricha ◽  
Anubha Goel ◽  
Song-Ping Zhu
Keyword(s):  
Closed Form ◽  
Asset Price ◽  
Distribution Functions ◽  
Equity Options ◽  
Put Options ◽  
Stochastic Interest ◽  
Joint Characteristic ◽  
Infinite Sums ◽  
Closed Form Formula ◽  
Pricing Formula

In this article, we derive a closed-form pricing formula for catastrophe equity put options under a stochastic interest rate framework. A distinguishing feature of the proposed solution is its simplified form in contrast to several recently published formulae that require evaluating several layers of infinite sums of $n$ -fold convoluted distribution functions. As an application of the proposed formula, we consider two different frameworks and obtain the closed-form formula for the joint characteristic function of the asset price and the losses, which is the only required ingredient in our pricing formula. The prices obtained by the newly derived formula are compared with those obtained using Monte-Carlo simulations to show the accuracy of our formula.

MAXIMUM DRAWDOWN INSURANCE

2011 ◽  
Vol 14 (08) ◽  
pp. 1195-1230 ◽  
Author(s):  
PETER CARR ◽  
HONGZHONG ZHANG ◽  
OLYMPIA HADJILIADIS
Keyword(s):  
Risk Measure ◽  
Asset Price ◽  
Spot Price ◽  
Barrier Options ◽  
Double Barrier ◽  
Fixed Amount ◽  
Model Free ◽  
Running Maximum ◽  
Large Maximum ◽  
Maximum Drawdown

The drawdown of an asset is a risk measure defined in terms of the running maximum of the asset's spot price over some period [0, T]. The asset price is said to have drawn down by at least $K over this period if there exists a time at which the underlying is at least $K below its maximum-to-date. We introduce insurance against a large realization of maximum drawdown and a novel way to hedge the liability incurred by underwriting this insurance. Our proposed insurance pays a fixed amount should the maximum drawdown exceed some fixed threshold over a specified period. The need for this drawdown insurance would diminish should markets rise before they fall. Consequently, we propose a second kind of cheaper maximum drawdown insurance that pays a fixed amount contingent on the drawdown preceding a drawup. We propose double barrier options as hedges for both kinds of insurance against large maximum drawdowns. In fact for the second kind of insurance we show that the hedge is model-free. Since double barrier options do not trade liquidly in all markets, we examine the assumptions under which alternative hedges using either single barrier options or standard vanilla options can be used.

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