1) The document discusses models for pricing corporate bonds, specifically comparing a reduced form default model to the author's proposed model.
2) In the author's model, the date-t bond price is a random variable between the minimum and maximum price on date t, rather than a single number. This models the bond price as the present value of the recovery rate assuming default occurs at maturity.
3) With the bond price as a random variable, the recovery rate can be assumed to be a non-random constant, reducing the default problem to finding the unknown recovery rate and default probability. Equations for the first and second moments of the bond price can then be derived.
This document discusses no arbitrage pricing theory and market risk. It begins by defining no arbitrage pricing as having a zero initial and expiration value. However, it notes that this definition does not guarantee a zero expiration value when holding coupon payments. It then introduces the concepts of present value and forward value, and defines no arbitrage prices that set the present and forward values equal to zero. However, it notes that this introduces market risk, as forward rates are random variables. It concludes by providing examples of interest rate swap valuation and defining market risk probabilities.
There are several significant drawbacks in derivative price modeling which relate to global regulations of the derivatives market. Here we present a unified approach which in stochastic market interprets option price as a random variable. Therefore spot price does not complete characteristic of the price in stochastic environment. Complete derivatives price includes the spot price as well as thevalue of market risk implied by the use of the spot price. This interpretation is similar to the notion of therandom variable in Probability Theory in which an estimate of the random variable completely defined by its cumulative distribution function
In this short notice, we present structure of the perfect hedging. Closed form formulas clarify the fact that Black-Scholes (BS) portfolio which provides perfect hedge only at initial moment. Holding portfolio over a certain period implies additional cash flow, which could not be imbedded in BS pricing scheme, and therefore BS option price cannot be derived without additional cash flow which affects BS option price.
Abstract. Regulations of the market require disclosure of information about the nature and extent of risks arising from the trades of the market instruments. There are several significant drawbacks in fixed income pricing modeling. In this paper we interpret a corporate bond price as a random variable. In this case the spot price does not a complete characteristic of the price. The price should be specified by the spot price as well as its value of market risk. This interpretation is similar to a random variable in Probability Theory where an estimate of the random variable completely defined by its cumulative distribution function. The buyer market risk is the value of the chance that the spot price is higher than it is implied by the market scenarios. First we quantify credit risk of the corporate bonds and then consider marked-to-market pricing adjustment to bond price. In the case when issuer of the corporate bond is the counterparty of the bond buyer counterparty and credit risks are coincide.
1. The document discusses the concept of forward rates and their use in pricing forward rate agreements (FRAs). It proposes models for randomizing both the future LIBOR rate and the implied forward rate used in FRA pricing.
2. FRAs are over-the-counter derivatives where the payoff depends on the difference between the realized LIBOR rate and the fixed FRA rate. However, LIBOR is unknown at pricing date so the implied forward rate is used as an approximation, introducing market risk.
3. The document presents stochastic differential equations to model the future LIBOR rate and the implied forward rate as random processes. This allows calculation of market risk metrics like expected losses for FRA buyers and sellers.
A new derivation of the Black Scholes Equation (BSE) based on integral form stochastic calculus is presented. Construction of the BSE solution is based on infinitesimal perfect hedging. The perfect hedging on a finite time interval is a separate problem that does not change option pricing. The cost of hedging does not present an adjustment of the BS pricing. We discuss a more profound alternative approach to option pricing. It defines option price as a settlement between counterparties and in contrast to BS approach presents the market risk of the option premium.
This document discusses the construction of riskless derivatives portfolios as proposed by Black and Scholes. It summarizes Black and Scholes' approach and then argues that their portfolio is not truly riskless, as it takes on risk at each discrete time interval. Specifically, the portfolio requires reconstruction at each time point to eliminate risk, and in the limit of infinitesimally small time intervals, the portfolio retains risk at all times. The document makes a similar argument against the claim that portfolios of multiple derivatives can be constructed to be riskless.
Market risk and liquidity of the risky bondsIlya Gikhman
This document discusses modeling the effect of liquidity on risky bond pricing using a reduced form approach. It begins by presenting a simplified model where default can only occur at maturity. It then extends this to a discrete time approximation for default occurrence. The key concepts discussed are:
- Defining bid and ask prices for risk-free and corporate bonds to model liquidity spread
- Using a single price framework and extending it to account for liquidity spread
- Modeling the corporate bond price as a random variable based on default/no default scenarios
- Defining market and spot prices of bonds and the associated market risks for buyers and sellers
- Estimating the recovery rate and default probability given observations of spot prices over time
This document discusses no arbitrage pricing theory and market risk. It begins by defining no arbitrage pricing as having a zero initial and expiration value. However, it notes that this definition does not guarantee a zero expiration value when holding coupon payments. It then introduces the concepts of present value and forward value, and defines no arbitrage prices that set the present and forward values equal to zero. However, it notes that this introduces market risk, as forward rates are random variables. It concludes by providing examples of interest rate swap valuation and defining market risk probabilities.
There are several significant drawbacks in derivative price modeling which relate to global regulations of the derivatives market. Here we present a unified approach which in stochastic market interprets option price as a random variable. Therefore spot price does not complete characteristic of the price in stochastic environment. Complete derivatives price includes the spot price as well as thevalue of market risk implied by the use of the spot price. This interpretation is similar to the notion of therandom variable in Probability Theory in which an estimate of the random variable completely defined by its cumulative distribution function
In this short notice, we present structure of the perfect hedging. Closed form formulas clarify the fact that Black-Scholes (BS) portfolio which provides perfect hedge only at initial moment. Holding portfolio over a certain period implies additional cash flow, which could not be imbedded in BS pricing scheme, and therefore BS option price cannot be derived without additional cash flow which affects BS option price.
Abstract. Regulations of the market require disclosure of information about the nature and extent of risks arising from the trades of the market instruments. There are several significant drawbacks in fixed income pricing modeling. In this paper we interpret a corporate bond price as a random variable. In this case the spot price does not a complete characteristic of the price. The price should be specified by the spot price as well as its value of market risk. This interpretation is similar to a random variable in Probability Theory where an estimate of the random variable completely defined by its cumulative distribution function. The buyer market risk is the value of the chance that the spot price is higher than it is implied by the market scenarios. First we quantify credit risk of the corporate bonds and then consider marked-to-market pricing adjustment to bond price. In the case when issuer of the corporate bond is the counterparty of the bond buyer counterparty and credit risks are coincide.
1. The document discusses the concept of forward rates and their use in pricing forward rate agreements (FRAs). It proposes models for randomizing both the future LIBOR rate and the implied forward rate used in FRA pricing.
2. FRAs are over-the-counter derivatives where the payoff depends on the difference between the realized LIBOR rate and the fixed FRA rate. However, LIBOR is unknown at pricing date so the implied forward rate is used as an approximation, introducing market risk.
3. The document presents stochastic differential equations to model the future LIBOR rate and the implied forward rate as random processes. This allows calculation of market risk metrics like expected losses for FRA buyers and sellers.
A new derivation of the Black Scholes Equation (BSE) based on integral form stochastic calculus is presented. Construction of the BSE solution is based on infinitesimal perfect hedging. The perfect hedging on a finite time interval is a separate problem that does not change option pricing. The cost of hedging does not present an adjustment of the BS pricing. We discuss a more profound alternative approach to option pricing. It defines option price as a settlement between counterparties and in contrast to BS approach presents the market risk of the option premium.
This document discusses the construction of riskless derivatives portfolios as proposed by Black and Scholes. It summarizes Black and Scholes' approach and then argues that their portfolio is not truly riskless, as it takes on risk at each discrete time interval. Specifically, the portfolio requires reconstruction at each time point to eliminate risk, and in the limit of infinitesimally small time intervals, the portfolio retains risk at all times. The document makes a similar argument against the claim that portfolios of multiple derivatives can be constructed to be riskless.
Market risk and liquidity of the risky bondsIlya Gikhman
This document discusses modeling the effect of liquidity on risky bond pricing using a reduced form approach. It begins by presenting a simplified model where default can only occur at maturity. It then extends this to a discrete time approximation for default occurrence. The key concepts discussed are:
- Defining bid and ask prices for risk-free and corporate bonds to model liquidity spread
- Using a single price framework and extending it to account for liquidity spread
- Modeling the corporate bond price as a random variable based on default/no default scenarios
- Defining market and spot prices of bonds and the associated market risks for buyers and sellers
- Estimating the recovery rate and default probability given observations of spot prices over time
The document provides an overview of the Black-Scholes option pricing model (BSOPM). It describes the key assumptions of the BSOPM, including that the underlying stock pays no dividends, markets are efficient, and prices are lognormally distributed. It also outlines how the BSOPM can be used to calculate theoretical option prices from historical data on the stock price, strike price, time to expiration, interest rate, and volatility. The document discusses implied volatility and how it differs from historical volatility, as well as limitations of the BSOPM.
1. The Black-Scholes option pricing model assumes a perfect hedge using a dynamic trading strategy, but this requires frequent rebalancing of the hedge portfolio which incurs transaction costs that are not accounted for in the model.
2. When the hedge portfolio is rebalanced over discrete time intervals, an adjustment cost arises at each interval that affects the expected present value of the cash flows and thus the derived option price.
3. For the Black-Scholes model to accurately price options, it must account for the expected costs of dynamically rebalancing the hedge portfolio over the life of the option.
The document discusses key concepts related to option pricing models. It provides an overview of the binomial option pricing model (BOPM) and Black-Scholes option pricing model (BSOPM). The BOPM values options using a discrete time approach where the underlying asset price can move up or down over time. The BSOPM uses a continuous time approach to value options based on the stochastic behavior of the underlying asset price over time. Both models are based on the principle of risk neutral valuation and creating a riskless hedge to determine the appropriate discount rate.
Solution to Black-Scholes P.D.E. via Finite Difference Methods (MatLab)Fynn McKay
Simple implementable of Numerical Analysis to solve the famous Black-Scholes P.D.E. via Finite Difference Methods for the fair price of a European option.
This document discusses the Black-Scholes pricing concept for options. It summarizes two common derivations of the Black-Scholes equation, the original by Black and Scholes which uses a hedged position consisting of a long stock position and short option position. An alternative derivation presented in some textbooks uses a hedged position consisting of a long option position and short stock position. The document also notes that while the Black-Scholes price guarantees a risk-free return at a single moment in time, it does not necessarily reflect market prices and there is no guarantee the market will use the Black-Scholes price.
In this paper we show how the ambiguities in derivation of the BSE can be eliminated.
We pay attention to option as a hedging instrument and present definition of the option price based on market risk weighting. In such approach we define random market price for each market scenario. The spot price then is interpreted as a one that reflect balance between profit-loss expectations of the market participants
In this paper, we show how the ambiguities in derivation of the BSE can be eliminated.
We pay attention to option as a hedging instrument and present definition of the option price based on market risk weighting. In such approach, we define random market price for each market scenario. The spot price then is interpreted as a one that reflects balance between profit-loss expectations of the market participants.
In this paper, we show how the ambiguities in derivation of the BSE can be eliminated.
We pay attention to option as a hedging instrument and present definition of the option price based on market risk weighting. In such approach, we define random market price for each market scenario. The spot price then is interpreted as a one that reflects balance between profit-loss expectations of the market participants.
. In some papers it have been remarked that derivation of the Black Scholes Equation (BSE) contains mathematical ambiguities. In particular in [2,3 ] there are two problems which can be raise by accepting Black Scholes (BS) pricing concept. One is technical derivation of the BSE and other the pricing definition of the option.
In this paper, we show how the ambiguities in derivation of the BSE can be eliminated.
We pay attention to option as a hedging instrument and present definition of the option price based on market risk weighting. In such approach, we define random market price for each market scenario. The spot price then is interpreted as a one that reflects balance between profit-loss expectations of the market participants.
Black-Scholes Model
Introduction
Key terms
Black Scholes Formula
Black Scholes Calculators
Wiener Process
Stock Pricing Model
Ito’s Lemma
Derivation of Black-Sholes Equation
Solution of Black-Scholes Equation
Maple solution of Black Scholes Equation
Figures
Option Pricing with Transaction costs and Stochastic Volatility
Introduction
Key terms
Stochastic Volatility Model
Quanto Option Pricing Model
Key Terms
Pricing Quantos in Excel
Black-Scholes Equation of Quanto options
Solution of Quanto options Black-Scholes Equation
The document summarizes key concepts related to the Black-Scholes partial differential equation. It introduces Black-Scholes, which revolutionized finance by finding the fair price of derivatives. The formula was derived from the heat equation and allowed investors to earn maximum profits without risk. It discusses the variables in the Black-Scholes equation like stock price, exercise price, volatility and risk-free rate. An example valuation of a call and put option is shown. The document also covers fundamental concepts like interest rates, probability, expected value, and continuous random variables.
This document discusses pricing models for American option contracts. It begins by outlining the standard model, which values American options based on the moment that guarantees maximum option value. However, the author proposes an alternative view, where the optimal exercise time is when the underlying asset reaches its maximum value on [0,T]. Exercising at this maximum value ensures a payoff equal to the selling price, avoiding arbitrage. The document formalizes this idea using concepts like risk-neutral probabilities and derivations of put-call parity relationships to define fair option prices.
Last my paper equity, implied, and local volatilitiesIlya Gikhman
In this paper we present a critical point on connections between stock volatility, implied
volatility, and local volatility. The essence of the Black Scholes pricing model is based on assumption
that option piece is formed by no arbitrage portfolio. Such assumption effects the change of the real
underlying stock by its risk neutral counterpart. Market practice shows even more. The volatility of the
underlying should be also changed. Such practice calls for implied volatility. Underlying with implied
volatility is specific for each option. The local volatility development presents the value of implied
volatility.
Notes I made in June 2013 on the derivation and use of the Black-Scholes equation. If you can forgive the terseness, you can look forward to some nifty stochastic partial differential equation twirling!
Any and all corrections are welcome!
The document discusses option pricing models. It covers the Binomial model and the Black-Scholes model. The Black-Scholes model assumes the stock price follows a geometric Brownian motion and uses a partial differential equation to derive a closed-form solution for pricing European stock options. It requires parameters such as the stock price, exercise price, risk-free interest rate, time to maturity, and volatility. The model provides a theoretical fair value for options.
This document provides an introduction to Monte Carlo simulations in finance. It discusses how Monte Carlo methods can be used to value financial derivatives by simulating asset price paths over time based on stochastic processes, and taking the average of the resulting payoffs. It also describes how Monte Carlo integration can be applied to problems involving the numerical evaluation of multi-dimensional integrals. The document outlines the basic concepts and provides examples of applying Monte Carlo techniques to price European options and estimate the value of pi.
A short remark on Feller’s square root condition.Ilya Gikhman
This document presents a proof of Feller's square root condition for the Cox-Ingersoll-Ross model of short interest rates.
The CIR model describes the dynamics of the short rate r(t) as a scalar SDE with parameters k, θ, and σ.
The theorem states that if the Feller condition 2kθ > σ^2 is satisfied, then there exists a unique positive solution r(t) on each finite time interval t ∈ [0, ∞).
The proof uses Ito's formula and Gronwall's inequality to show that as ε approaches 0, the probability that the solution falls below ε approaches 0 as well.
Critical thoughts about modern option pricingIlya Gikhman
1. The document discusses issues with the Black-Scholes pricing model and proposes an alternative approach. Specifically, it argues that the B-S equation does not necessarily hold at each point in time and adjustments are needed when moving between time intervals.
2. The alternative approach defines option price stochastically based on possible future stock prices at expiration. Price is set to 0 if stock price is below strike, and a fraction of the in-the-money amount otherwise. This defines a distribution for option price based on the underlying stock distribution.
3. In the alternative approach, option price has an associated market risk defined by the probabilities of the traded price being above or below the stochastically defined price.
The document provides an overview of the Black-Scholes option pricing model (BSOPM). It describes the key assumptions of the BSOPM, including that the underlying stock pays no dividends, markets are efficient, and prices are lognormally distributed. It also outlines how the BSOPM can be used to calculate theoretical option prices from historical data on the stock price, strike price, time to expiration, interest rate, and volatility. The document discusses implied volatility and how it differs from historical volatility, as well as limitations of the BSOPM.
1. The Black-Scholes option pricing model assumes a perfect hedge using a dynamic trading strategy, but this requires frequent rebalancing of the hedge portfolio which incurs transaction costs that are not accounted for in the model.
2. When the hedge portfolio is rebalanced over discrete time intervals, an adjustment cost arises at each interval that affects the expected present value of the cash flows and thus the derived option price.
3. For the Black-Scholes model to accurately price options, it must account for the expected costs of dynamically rebalancing the hedge portfolio over the life of the option.
The document discusses key concepts related to option pricing models. It provides an overview of the binomial option pricing model (BOPM) and Black-Scholes option pricing model (BSOPM). The BOPM values options using a discrete time approach where the underlying asset price can move up or down over time. The BSOPM uses a continuous time approach to value options based on the stochastic behavior of the underlying asset price over time. Both models are based on the principle of risk neutral valuation and creating a riskless hedge to determine the appropriate discount rate.
Solution to Black-Scholes P.D.E. via Finite Difference Methods (MatLab)Fynn McKay
Simple implementable of Numerical Analysis to solve the famous Black-Scholes P.D.E. via Finite Difference Methods for the fair price of a European option.
This document discusses the Black-Scholes pricing concept for options. It summarizes two common derivations of the Black-Scholes equation, the original by Black and Scholes which uses a hedged position consisting of a long stock position and short option position. An alternative derivation presented in some textbooks uses a hedged position consisting of a long option position and short stock position. The document also notes that while the Black-Scholes price guarantees a risk-free return at a single moment in time, it does not necessarily reflect market prices and there is no guarantee the market will use the Black-Scholes price.
In this paper we show how the ambiguities in derivation of the BSE can be eliminated.
We pay attention to option as a hedging instrument and present definition of the option price based on market risk weighting. In such approach we define random market price for each market scenario. The spot price then is interpreted as a one that reflect balance between profit-loss expectations of the market participants
In this paper, we show how the ambiguities in derivation of the BSE can be eliminated.
We pay attention to option as a hedging instrument and present definition of the option price based on market risk weighting. In such approach, we define random market price for each market scenario. The spot price then is interpreted as a one that reflects balance between profit-loss expectations of the market participants.
In this paper, we show how the ambiguities in derivation of the BSE can be eliminated.
We pay attention to option as a hedging instrument and present definition of the option price based on market risk weighting. In such approach, we define random market price for each market scenario. The spot price then is interpreted as a one that reflects balance between profit-loss expectations of the market participants.
. In some papers it have been remarked that derivation of the Black Scholes Equation (BSE) contains mathematical ambiguities. In particular in [2,3 ] there are two problems which can be raise by accepting Black Scholes (BS) pricing concept. One is technical derivation of the BSE and other the pricing definition of the option.
In this paper, we show how the ambiguities in derivation of the BSE can be eliminated.
We pay attention to option as a hedging instrument and present definition of the option price based on market risk weighting. In such approach, we define random market price for each market scenario. The spot price then is interpreted as a one that reflects balance between profit-loss expectations of the market participants.
Black-Scholes Model
Introduction
Key terms
Black Scholes Formula
Black Scholes Calculators
Wiener Process
Stock Pricing Model
Ito’s Lemma
Derivation of Black-Sholes Equation
Solution of Black-Scholes Equation
Maple solution of Black Scholes Equation
Figures
Option Pricing with Transaction costs and Stochastic Volatility
Introduction
Key terms
Stochastic Volatility Model
Quanto Option Pricing Model
Key Terms
Pricing Quantos in Excel
Black-Scholes Equation of Quanto options
Solution of Quanto options Black-Scholes Equation
The document summarizes key concepts related to the Black-Scholes partial differential equation. It introduces Black-Scholes, which revolutionized finance by finding the fair price of derivatives. The formula was derived from the heat equation and allowed investors to earn maximum profits without risk. It discusses the variables in the Black-Scholes equation like stock price, exercise price, volatility and risk-free rate. An example valuation of a call and put option is shown. The document also covers fundamental concepts like interest rates, probability, expected value, and continuous random variables.
This document discusses pricing models for American option contracts. It begins by outlining the standard model, which values American options based on the moment that guarantees maximum option value. However, the author proposes an alternative view, where the optimal exercise time is when the underlying asset reaches its maximum value on [0,T]. Exercising at this maximum value ensures a payoff equal to the selling price, avoiding arbitrage. The document formalizes this idea using concepts like risk-neutral probabilities and derivations of put-call parity relationships to define fair option prices.
Last my paper equity, implied, and local volatilitiesIlya Gikhman
In this paper we present a critical point on connections between stock volatility, implied
volatility, and local volatility. The essence of the Black Scholes pricing model is based on assumption
that option piece is formed by no arbitrage portfolio. Such assumption effects the change of the real
underlying stock by its risk neutral counterpart. Market practice shows even more. The volatility of the
underlying should be also changed. Such practice calls for implied volatility. Underlying with implied
volatility is specific for each option. The local volatility development presents the value of implied
volatility.
Notes I made in June 2013 on the derivation and use of the Black-Scholes equation. If you can forgive the terseness, you can look forward to some nifty stochastic partial differential equation twirling!
Any and all corrections are welcome!
The document discusses option pricing models. It covers the Binomial model and the Black-Scholes model. The Black-Scholes model assumes the stock price follows a geometric Brownian motion and uses a partial differential equation to derive a closed-form solution for pricing European stock options. It requires parameters such as the stock price, exercise price, risk-free interest rate, time to maturity, and volatility. The model provides a theoretical fair value for options.
This document provides an introduction to Monte Carlo simulations in finance. It discusses how Monte Carlo methods can be used to value financial derivatives by simulating asset price paths over time based on stochastic processes, and taking the average of the resulting payoffs. It also describes how Monte Carlo integration can be applied to problems involving the numerical evaluation of multi-dimensional integrals. The document outlines the basic concepts and provides examples of applying Monte Carlo techniques to price European options and estimate the value of pi.
A short remark on Feller’s square root condition.Ilya Gikhman
This document presents a proof of Feller's square root condition for the Cox-Ingersoll-Ross model of short interest rates.
The CIR model describes the dynamics of the short rate r(t) as a scalar SDE with parameters k, θ, and σ.
The theorem states that if the Feller condition 2kθ > σ^2 is satisfied, then there exists a unique positive solution r(t) on each finite time interval t ∈ [0, ∞).
The proof uses Ito's formula and Gronwall's inequality to show that as ε approaches 0, the probability that the solution falls below ε approaches 0 as well.
Critical thoughts about modern option pricingIlya Gikhman
1. The document discusses issues with the Black-Scholes pricing model and proposes an alternative approach. Specifically, it argues that the B-S equation does not necessarily hold at each point in time and adjustments are needed when moving between time intervals.
2. The alternative approach defines option price stochastically based on possible future stock prices at expiration. Price is set to 0 if stock price is below strike, and a fraction of the in-the-money amount otherwise. This defines a distribution for option price based on the underlying stock distribution.
3. In the alternative approach, option price has an associated market risk defined by the probabilities of the traded price being above or below the stochastically defined price.
1. The document discusses the pricing of variance swaps using risk neutral valuation. It defines variance swaps as transactions where the payout is based on the difference between realized variance and a prespecified strike variance.
2. It derives a formula for the strike variance that equates it to the risk-neutral expected value of the integrated variance process over the swap period, where the expectation is calculated using Black-Scholes option prices.
3. The document explains that variance swaps allow parties to hedge differences between estimates of ex-ante variance derived from option prices and ex-post variance calculated from realized stock returns over the swap period.
1. The document presents a new approach to proving comparison theorems for stochastic differential equations (SDEs) using differentiation of solutions with respect to initial data.
2. It proves that if the drift term of one SDE is always greater than or equal to the other, and their initial values satisfy the same relation, then the solutions will also satisfy this relation for all time.
3. Two methods are provided: the first uses explicit solutions, the second avoids this by showing the difference process cannot reach zero in finite time based on its behavior.
This document discusses modeling fixed interest rates and summarizes the key concepts. It proposes a new approach to constructing variable deterministic and stochastic interest rates based on randomizing the forward rate concept. It defines basic terms like zero-coupon bond prices, interest rates, and cash flows. It then summarizes the pricing of forward rate agreements and interest rate swaps, highlighting the difference between the benchmark approach and the proposed stochastic approach, which accounts for market risk.
1) The document outlines drawbacks in the Black-Scholes option pricing theory, including mathematical errors in its derivations. Specifically, the assumption that a hedging portfolio eliminates risk is incorrect as a third term was omitted from the change in the portfolio value.
2) It also discusses issues with the local volatility adjustment concept, noting that transforming the constant diffusion coefficient to a local volatility surface does not actually explain the smile effect observed in options data.
3) While local volatility aims to match implied volatilities observed in the market, the theory suggests the local volatility surface should actually be equal to the original constant diffusion coefficient.
This document discusses pricing models for American options. It specifies that American options can be exercised at any time prior to maturity, unlike European options which can only be exercised at maturity. The value of an American option is defined as the expected value of the European option price using the random exercise time. American options can be decomposed into their European counterpart plus an early exercise premium. Determining the optimal early exercise time is formulated as finding the stopping time that maximizes the expected discounted payoff over the lifetime of the contract. References for further reading on pricing American options are also provided.
We use stochastic methods to present mathematically correct representation of the wave function. Informal construction was developed by R. Feynman. This approach were introduced first by H. Doss Sur une Resolution Stochastique de l'Equation de Schrödinger à Coefficients Analytiques. Communications in Mathematical Physics
October 1980, Volume 73, Issue 3, pp 247–264.
Primary intention is to discuss formal stochastic representation of the Schrodinger equation solution with its applications to the theory of demolition quantum measurements.
I will appreciate your comments.
This document discusses issues with the derivation of the Black-Scholes equation and option pricing model. It highlights two popular derivations of the Black-Scholes equation, noting ambiguities in the original derivation. It proposes defining the hedged portfolio over a variable time interval to address these ambiguities. The document also notes drawbacks of the Black-Scholes price, including that it only guarantees a risk-free return over an infinitesimal time period and does not reflect market prices which may incorporate other strategies.
This document discusses discount factors and mark-to-market valuation of cross currency swaps. It begins by explaining how discount factors are derived from risk-free bonds and how rates like Libor and OIS are used as proxies. However, it notes that swap rates cannot be directly used as discount factors since they do not guarantee a fixed payment amount at maturity. The document then discusses how to model the cash flows of interest rate swaps and cross currency swaps, and how to calculate stochastic and implied swap rates to value them using mark-to-market approaches.
This document summarizes key concepts related to derivatives and risk management. It discusses forwards, futures, swaps, and options contracts. It explains how forwards, futures, and swaps work to transfer risk, while options provide choice. The cost-of-carry model for pricing forwards is described. Forward rate agreements are introduced as interest rate derivatives. Forward exchange rates are projected using interest rate parity.
Macrodynamics of Debt-Financed Investment-Led Growth with Interest Rate Rulespkconference
This document provides an overview of a macroeconomic model that examines debt-financed investment-led growth. Some key points:
- The model explores whether financial factors can provide stability to an otherwise unstable demand-constrained economy, and whether they can generate growth cycles.
- Investment is determined by a post-Keynesian accelerator function where the sensitivity of investment to capacity utilization depends on financial factors like the risk of default and interest rates.
- Debt dynamics are modeled, where borrowing, repayment, and the debt stock over time are determined. Financial fragility and creditworthiness indicators are also developed.
- Monetary policy follows a Taylor-type rule where the central bank adjusts interest rates in response
This document discusses three main approaches to modeling credit risk: structural, reduced form, and incomplete information. It provides details on the structural approach using the Merton and first passage models and the reduced form approach using a Poisson process for default. It also discusses extending these models to value bank loans, specifically comparing the structural KMV model and reduced form CreditRisk+ model. The critiques note limitations like non-observability of variables, lack of dynamics, and potential underestimation of risk.
Chapter 05_How Do Risk and Term Structure Affect Interest Rate?Rusman Mukhlis
The document discusses two factors that affect interest rates: risk structure and term structure. For risk structure, it explains how default risk, liquidity, and taxes can cause different interest rates for bonds with different levels of risk. For term structure, it presents the expectations theory, which states that interest rates of different maturities should equal the average expected future short-term rates. It also discusses empirical findings about the typical upward slope of the yield curve.
Chapter 03_What Do Interest Rates Mean and What Is Their Role in Valuation?Rusman Mukhlis
This chapter discusses interest rates and their role in valuation. It defines key terms like yield to maturity, which is the most accurate measure of interest rates. It examines how to measure and understand different interest rates, the distinction between real and nominal rates, and the relationship between interest rates and returns. It also covers how the concept of present value is used to evaluate debt instruments and how duration is used to measure interest rate risk.
This document discusses different types of bonds and bond characteristics such as par value, coupon rate, maturity date, and callability. It explains how to calculate bond yields such as current yield, yield to maturity, and yield to call. The key valuation models discussed are the present value model, which values a bond based on its expected cash flows discounted at the yield to maturity, and the yield model, which solves for the yield given the bond's market price. The document also covers factors that impact bond prices such as interest rates, maturity, and coupon rate. Bond duration, a measure of price sensitivity to interest rates, is introduced along with related concepts like modified duration.
This document discusses different types of bonds and bond characteristics such as par value, coupon rate, maturity date, and callability. It explains how to calculate bond yields such as current yield, yield to maturity, and yield to call. Factors that influence bond prices such as interest rates, maturity, and coupon are also covered. The concept of duration, which measures a bond's price sensitivity to interest rate changes, is introduced.
This document discusses different types of bonds and bond characteristics such as par value, coupon rate, maturity date, and callability. It explains how to calculate bond yields such as current yield, yield to maturity, and yield to call. Factors that influence bond prices such as interest rates, maturity, and coupon are also covered. The concept of duration, which measures a bond's price sensitivity to interest rate changes, is introduced.
The document discusses bond valuation and interest rates. It defines key bond concepts like yield to maturity and explains how spot and forward rates are used to value pure discount bonds. The document also explores yield curves and theories for why they take different shapes. Additional topics covered include credit risk, bond ratings, junk bonds, and embedded options in bonds like calls, conversions, and their impact on convertible bond valuation.
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Options Pricing The Black-Scholes ModelMore or .docxhallettfaustina
*
Options Pricing: The Black-Scholes ModelMore or less, the Black-Scholes (B-S) Model is really just a fancy extension of the Binomial Model.
(Fancy enough, however, to win a Nobel Prize…).
*
How B-S extends the Binomial Model1. Instead of assuming two possible states for future exchange rates, and thus returns (i.e., “up” and “down”), B-S assumes a continuous distribution of returns, R, so that returns can take on a whole range of values.
Binomial B-S
*
How B-S extends the Binomial ModelIn fact, exchange rate returns are approximately normally distributed, so this is a “reasonable” assumption:
*
How B-S extends the Binomial Model2. Instead of just one time period, B-S assumes multiple time periods and that the time between periods is instantaneous (i.e., continuous).
(See lecture)
Also, the time between periods t=0, t=1, t=2, etc. shrinks to zero, so that spot rate is changing at every instant.
*
How B-S extends the Binomial ModelThis is more realistic, since actual currency trades take place on a second-to-second, nearly continuous basis.
*
How B-S extends the Binomial ModelIt turns out that these two extensions are enough to make the math very hard. Thus, deriving the B-S model is no easy task.
The most important thing to recognize is that despite the above complications, the basic underlying approach of the B-S model remains the same…
*
How B-S extends the Binomial Model3. Create a replicating portfolio and price the option using a no-arbitrage argument.Calculate NS and NB: Now, since these are constantly changing over time, this process is called “dynamic hedging”.Replicating portfolio:It turns out that it is possible to use a combination of foreign currency and USD, and now in addition, options themselves, to form a riskless portfolio (i.e., return is known for sure).No-arbitrage: Riskless portfolios must have the same price as risk-free securities, otherwise arbitrage is possible. Use this fact to figure out c.
*
The Black-Scholes Options Pricing FormulaPutting the above all together, we get the Black-Scholes formula for pricing a European call option on foreign currency:
where
and S, X, T as before
r = domestic risk-free rate, r* = foreign risk-free rate
s = volatility of the foreign currency (sd of returns).
*
The Black-Scholes Options Pricing Formula
Also, N(x) = Prob that a random variable will be less than x under the standard normal distribution (i.e., cumulative distribution function).Calculate in EXCEL using “=NORMSDIST(x)”.
represents discounting when interest rates are continuously compounded, so basically it corresponds to:
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My study abroad in Bali, Indonesia, inspired this research topic as I noticed how globalization is changing the culture of its people. I learned their language and way of life which helped me understand the beauty and importance of cultural preservation. I believe we could all benefit from learning new perspectives as they could help us ideate solutions to contemporary issues and empathize with others.
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Detailed power point presentation on compound interest and how it is calculated
Basic of pricing 2
1. 1
BASIC OF PRICING 2.
Ilya Gikhman
6077 Ivy Woods Court
Mason OH 45040 USA
Ph. 513-573-9348
Email: ilyagikhman@mail.ru
Abstract. In this paper we develop a model of corporate bonds pricing. We begin with default definition
which is similar to one that is used in the standard reduced form of default model. The primary distinction
between our model and reduced form of default model is interpretation of the date-t price of the bond. In
reduced form model date-t corporate bond price is a single number which in practical applications is the
close price of the bond at date t. It could be a reasonable reduction when deviation between maximum and
minimum of the bond prices at date t is sufficiently small. Otherwise the reduction’s error could be
remarkable. In our interpretation date-t bond price is a random variable taking values between minimum -
maximum prices at the date t. In such setting random value of the bond is considered as the present value
of the recovery rate assuming that default occurs at maturity of the bond. Random format of the recovery
rate does not convenient to compare different risky bonds. It makes sense to assume that recovery rate is a
nonrandom constant. This reduction reduced default problem to finding unknown recovery rate and
correspondent default probability. This is the case of two unknowns and one can derive equations for the
first and second moments of the bond price to present a solution of the problem. This approach can be
extend to resolve default problem in more general cases.
JEL : G13.
Keywords: no arbitrage, mark-to-market, cash flow, market risk, credit risk, reduced form pricing, credit
risk, interest rate swap.
1. Risk Free Bond Pricing.
Introduction. The price notion is the basis of the finance theory and practice. In standard trading of the
risk free bond investors pay the bond price B ( t , T ) at date t ≥ 0 and receive bond which promises its
face value of $1 at bond’s expiration date T. In [1] we presented a formal definition of the no arbitrage
pricing. The no arbitrage pricing of a financial instrument is defined as pricing which starts with zero
value of the initial investment at initiation date. The zero value at initiation is formed by buying the
instrument and borrowing this amount at risk free interest rate from the bank. Conditioning on zero initial
value of the investor’s position investor should arrive at the 0 value at expiration date T. Such
2. 2
interpretation of the no arbitrage pricing is an ideal scheme. It should be adjusted for a stochastic market.
Interest rates at the future dates are unknown which are interpreted as random variables. It can be higher
or lower than market implied forward rates estimated at initiation date. By using market implied forward
rates one can produce market implied forward estimate of the future coupon bond price, which does not
contain arbitrage opportunity with respect to the spot price. No arbitrage price of the risk free coupon
bond does not eliminate market risk. It stems from the fact that coupon payments received by bond buyer
at future dates should be invested at unknown at initiation real forward risk free rates while on no
arbitrage price set at initiation uses market implied forward rates. The difference between real market
forward rates and its market implied forward estimates defines market risk of the risk free bond during
lifetime of the bond prior to its expiration date.
We use cash flow as a formal definition of the financial instrument. Risk free coupon bond from the
buyer perspective can be defined by the cash flow
CF = – B c ( t 0 , T ) χ { t = t 0 } +
n
1j
c χ { t = t j } + 1 χ { t = T } (1)
Here c > 0 is a coupon payment taking place at the dates t j , j = 1, 2, … n and t n = T Function
χ { t = T } denotes indicator of the event { t = T }. One usually interprets the CF’s portion
n
1j
c χ { t = t j } + 1 χ { t = T } (2)
as a portfolio of the risk free bonds with face values c at t j , j < n and 1 at T correspondingly. Portfolio
interpretation of the coupon bond price B c ( t 0 , T ) makes it possible reduce no arbitrage of the coupon
bond to a sum of no arbitrage prices of the zero coupon bonds in the portfolio. The equivalence of the
coupon bond and portfolio of the zero coupon bonds with different maturities takes place in the perfect
market. Indeed, in the perfect market buyer of the bond can go short at initiation with the portfolio of the
zero coupon bonds to compose the value B c ( t 0 , T ). More realistic setting of the bond pricing problem
implies borrowing funds equal to the bond price from the bank at risk free interest at date t 0 and then
return borrowing amount plus interest at the bond maturity T. In other words, we do not assume that
money borrowed from bank can be returned to the bank by parts. Cash flow (1.2) admits two types
representation of the date-t 0 price of the bond. One representation uses the real world market scenarios to
define B ( t j , T , ω ). It will be used to present stochastic market price at the date-t 0 . Other
representation uses market implied forward rates to present the current spot price of the bond. Here
B c ( t j , T ) , j > 0 denote future values of the bond at t j that are unknown at t 0 . One usually apply a
stochastic equation as a theoretical model representing future values of the bond B c ( t j , T , ω ). Using a
model of the B c ( t j , T , ω ) we can construct date-T future value FV of the CF
n
1j
c B – 1
( t j , T , ω ) + 1
Market risk of the coupon bond investment over [ t 0 , T ] period. Bond buyer loses money if
3. 3
B c ( t 0 , T ) > B ( t 0 , T ) [
n
1j
c B – 1
( t j , T , ω ) + 1 ]
and he gets money otherwise. Nevertheless applying implied market forward estimate to valuation of the
PV of the cash flow (2) shows that B c ( t 0 , T ) is indeed no arbitrage price. Indeed one can easy verify
the equality
B c ( t 0 , T ) = B ( t 0 , T ) [
n
1j
c B – 1
( t j , T , t 0 ) + 1 ]
where date-t 0 market implied forward discount rate B ( t j , T , t 0 ) is defined as
B ( t 0 , T ) = B ( t 0 , t j ) B ( t j , T , t 0 )
j = 1, 2, … n. This equality confirms the fact that the value B c ( t 0 , T ) borrowed from the bank to buy
coupon bond at t 0 at risk free rate will be covered by the date-T value of the cash flow generated by the
bond.
In stochastic setting no arbitrage pricing does not eliminate market risk. In finance theory market risk
does not formally defined. Informally it can be associated with profit-loss diagram which represents profit
and loss of an investment based on a market scenario. Market scenario is associated with the future price
at a particular date or at a series of dates. The modern finance theory deals with modeling of the spot
price. Spot price of an instrument is interpreted as present value, PV of the future cash flow associated
with CF. For example, for the coupon bond defined by cash flow (0.1) we can write the formula
B c ( t 0 , T ) = PV CF
Market risk of the bond buyer or seller is associated with lower return than it is expected at the initial
moment. If risk free bond is used for financing of a business than going short and buying back bond
during lifetime of the bond is a market risky deal. On the other hand buying risk free bond and holding it
until expiration does not risky purchase.
Corporate bond pricing. Let { Ω , F , P }be a complete probability space. Elements of the set Ω
represent market scenarios which represent stochastic prices of the debt instruments, F is the sigma
algebra generated by the observed market scenarios, and P is a complete probabilistic measure. Consider
credit risk effect on bonds pricing. Credit risk is associated with bond default. Other risk that also effects
on pricing is counterparty risk. Counterparty risk is a risk when one of the participants could not fulfill its
obligations. In such a case, underlying financial instrument does not default. In case of a corporate bond
the credit risk is associated with the issuer of the bond that could not pay face value of the bond to bond
buyer at maturity. In this case counterparty risk coincides with the credit risk. For more complex financial
instruments credit and counterparty risks are different.
A corporate bond is a risky instrument in which only buyer of the bond is exposed to risk. This is credit
risk which implies that bond’s seller could not accomplish its obligations to pay notional value of the
bond at its expiration. In standard ’cash-and-carry’ trading buyer if the bond pays the spot price R ( t , T )
4. 4
and gets the bond which promises say $1 at T. If default occurs prior to maturity buyer of the bond
receives bond’s recovery rate, RR = δ < $1.
Reduced form of default is probably the most popular model of the corporate bonds pricing theory. There
are two primary parameters such as recovery rate and probability of default that effect on corporate bond
price. Benchmark reduced form of default theory does not present simultaneously calculations of the
probability of default and recovery rate values. In practice, agencies heuristically assign 40% or other
popular percentage to the bond’s face value of the recovery rate. This assumption simplifies the reduced
form default model and makes it possible calculation of the probability of default that corresponds to a
chosen recovery rate. In contrast to the benchmark approach, we do not use known recovery rate
assumption. We considered more general approach to default problem in [2,3]. For example based on
pricing data at date t , t < T and assuming that default takes place only at maturity we present closed
form formulas for recovery rate and probability of default. Here we present more detailed study regarding
default.
Let briefly recall basic corporate coupon bond valuation formulas. Let corporate bond admits default
only at maturity. Then at maturity T the value of the bond can be defined by the formula
R ( T , T , ω ) = χ { > T } + δ χ { = T }
where the constant δ [ 0 , 1 ) is assumed to be known depending on t. Functions χ { > T } and
χ { = T } denote indicators of the no default and default at T scenarios correspondingly. Probabilities
of the default and no default events are also depend on t. Value δ and ( 1 – δ ) are referred to as recovery
rate and loss given default . Stochastic value of the corporate bond at maturity T implies stochastic price
of the bond at date t
R ( t , T , ω ) = B ( t , T ) [ χ { > T } + δ χ { = T } ] (3)
= B ( t , T ) [ 1 – ( 1 – δ ) χ { = T } ]
Formula (3) defines date-t ‘fair’ price for each market scenario ω. Denote R spot ( t , T ) the spot price of
the bond at date t. It is a statistics of the observed data during a period associated with date t. One can
interpret spot price of the day t as open, close, middle, or expected value of the random market price (1.3).
For given spot price probabilities
P { R ( t , T , ω ) > R spot ( t , T ) } , P { R ( t , T , ω ) < R spot ( t , T ) }
define seller’s and buyer’s market values of risk correspondingly. First probability represents the chance
that bond is underpriced at t while the second probability represents the chance that bond at t is overpriced
by the market. In practice the value of recovery rate is unknown and should be estimated by historical
data. Note that our interpretation of the notion price significantly broader than it is used in modern finance
theory. For example in the standard reduced form model of default the only spot price is defined, which is
expectation of the random price (3).
Formula (3) presents date-t bond price given value of the bond at maturity. There are two values of the
bond ( δ , 1 ) at T with known probability distribution. To each value of the bond corresponds a unique
value of the bond ( δ B ( t , T ) , B ( t , T ) ) at t and probability distribution does not changed. This
5. 5
construction of the stochastic bond price does not be changed if we assume that value of the bond at T is a
discrete or continuous distributed random variable taking values on interval [ 0 , 1 ]. In practice we deal
with inverse problem. We observe distribution of the bond prices during a date t and our goal is to make a
conclusion regarding default distribution. Note that given price R ( t , T ; ω ) distribution at t we can
define market implied forward value δ ( ω ) = B – 1
( t , T ) R ( t , T ; ω ) to value of the bond at maturity
which represents market implied stochastic recovery rate at T assuming that default occurs at maturity. It
is not convenient to justify which bond is more reliable dealing with continuous distribution of recovery
rates.
In order to present valuation in more explicit form it is necessary to make a discrete reduction of the
recovery rates. We begin with theoretical model where values of the bond are assumed to be known at
maturity. Given that default can be only at maturity T we introduce a total set of scenarios Ω =
m
1j
ω j ,
where ω j = { ω : δ j – 1 ≤ R ( T , T ; ω ) < δ j } . Here 0 = δ 0 ≤ ≤ δ j < δ j + 1 , j = 1, 2, … m – 1
and δ m = 1. Thus a continuously distributed recovery rate δ ( ω ) can be approximated by a discrete
random variable δ λ ( ω ) =
m
0j
δ j χ ( ω j ) where
P ( ω j ) = p j = P { R ( T , T ; ω ) [ δ j , δ j + 1 ) }
Let χ D ( ) denote indicator of the default. We can present price of the corporate bond as
R ( t , T ; ω ) = B ( t , T ) [ 1 – ( 1 – ( ω ) ) χ D ( ) ]
This equation defines the price for each market scenario ω. We can approximate latter equality
1 – λ ( t , T ; ) = ( 1 – λ ) χ ( ) =
m
1j
( 1 – δ j ) χ ( ω j )
where
λ ( t , T ; ) =
)T,t(B
)ω;T,t(R
λ
λ
Therefore
E k
λ ( t , T ; ) =
m
1j
( 1 – δ j ) k
p j
k = 1, 2, … m. This is a linear system of the m order with respect to unknown p j . It could be solved by
the standard methods of the Linear Algebra. The probability of the no default is equal to
p 0 = 1 – p 1 – p 2 – … – p m – 1
6. 6
The solution of the system represents the approximation of the default distribution corresponding to the
stochastic recovery rate.
Corporate bond pricing.
Let us assume that recovery rate δ = δ ( ω ) is a continuously distributed random variables on [ 0 , 1 ).
Introduce a discrete approximation of the recovery rate given that default occurs only at maturity date T.
Denote
p j = P { R ( T , T , ω ) [ δ j – 1 , δ j ) }
(4)
ω j = { ω : R ( T , T , ω ) [ δ j – 1 , δ j ) } , Ω =
m
1j
ω j
where 0 ≤ δ j < δ j + 1 , j = 1, 2, … m – 1, δ 0 = 0 and δ m = 1 . In theory we suppose that distribution
of the random variable R ( T , T ; ω ) is known and therefore probabilities p j are known too. From (4) it
follows that market price of the bond that admits default at maturity with known recovery rate δ j is
defined as
R mkt ( t 0 , T ; Ω j ) = δ j B ( t 0 , T )
Hence a discrete approximation of the bond can be written as
R mkt ( t 0 , T , ω ) =
m
1j
δ j B ( t 0 , T ) χ Ω j ( ω )
We have introduced theoretical valuation formulas of corporate bond. In practice we have only historical
data available. Recovery rate is unknown and should be estimated based on observed data. Hence the
problem is how using observed data present stochastic market price and recovery rate estimate.
Let us first introduce randomization of the bond’s price. Recall that market risk of the no arbitrage price
of the default free coupon bond is stipulated by the unknown at t 0 future values of the bond
B ( t j , T , ω ) [1] which are estimated by market implied forward rates. On the other hand later model of
the corporate zero coupon bond given possibility of default at maturity moment the date-t 0 uncertainty of
the PV reduction of the corporate bond comes with unknown recovery rate. In [1] we presented a
theoretical solution of the problem.
Remark. Benchmark reduced form model of default begins with the similar definition of default. Next
they introduce date-t spot price ignoring stochastic price at maturity of the bond. Recall that the price of
the risky bond is a random variable which takes two values 1 and δ with known probabilities. Difference
of these two approaches relates to the ways how one interprets date-t price of the bond. In theory asset
price is interpreted as a continuous time random process. In practice we usually use close price of asset as
the price at date-t. In such reduction of the price we can consider future price at T as close price at T. It
fallows from the fact that all data we use represent only close historical prices. In other interpretation of
the asset price close price of the date t can be interpreted as a good approximation of the whole trading
7. 7
period which we associate with date-t. Good approximation can be justified by a low volatility of the
date-t asset prices. If volatility cannot by considered as sufficiently small then use deterministic statistics
like close, open admits additional risk leading to possible losses. Standard reduced form model deals with
one equation and two unknowns. It could not present a unique solution of the default problem in simplest
setting when default might occur only at maturity. That is why the primary financial institutions and
rating agencies need to make additional assumption that recovery rate is a known such as for example
30%, 40% or other pre-specified value of recovery rate. Using a heuristic recovery rate helps to produce
heuristic probability of default. It is clear that such simplification of the problem leads to distortion of the
estimates of the recovery rate as well as probability of default which are the primary quantitative rating
parameters.
Our approach to default is based on stochastic market price was introduced in [1]. Stochastic price at t is
defined for each market scenario which associated with a particular value of the bond during date-t
period. With the help of stochastic price one can present independent equations for higher moments of the
market price. The system of two equations for the first and the second moments is sufficient to calculate
nonrandom recovery rate along with correspondent default probability. Following [1] let us briefly recall
this construction. From equality (3) it follows that
1 –
)T,t(B
)ω,T,t(R
= ( 1 – δ ) χ { = T } (5)
Then recovery rate can be written in the form
)T,t(B
)ω,T,t(R
, for ω { ( ω ) = T }
δ = δ ( t , T , ω ) = {
1 , for ω { ( ω ) > T }
The distribution of the random variable δ ( ω ) is the distribution of the stochastic recovery rate. One
choice of the statistical estimate Δ of the random recovery rate δ ( ω ) is its expectation Δ = E δ ( ω ).
Stochastic interpretation of the recovery rate implies that reduction Δ of the rate δ ( ω ) implies market
risk. This risk for the bond buyer is the higher value of the Δ than recovery rate which is specified by a
scenario at the default moment. Buyer and seller risks are measured by probabilities of undervalue or
overvalue of the recovery rate
P { δ ( ω ) < Δ } , P { δ ( ω ) > Δ }
correspondingly. Consider for example the estimate δ ( ω ) χ ( ω D ), where D = { ω : δ ( ω ) < 1 }.
This estimate specifies credit risk when bond seller could not pay initially promised amount of $1 at
expiration date. At the same time this risk is appealing bond sellers. Define normalized spread function
θ ( t , T ; ω ) by the formula
)T,t(B
)ω,T,t(R
1)ω,T,t(θ
8. 8
Note that this random function associated with normalized estimate of the LGD. The random function
θ ( t , T ; ω ) is an observable function. Stochastic recovery rate is complete credit information of the
bond. Nevertheless it is difficult to compare two distribution functions. To present credit characteristics in
more comparable form we assume that observations on bond’s prices at t correspond to unknown
deterministic recovery rate in case of default. We use function θ ( t , T ; ω ) to calculate a nonrandom
market implied recovery rate < Δ > and its correspondent probability of default. From (5) it follows that
E θ ( t , T , ω ) = ( 1 – < Δ > ) P ( D )
E θ ² ( t , T , ω ) = ( 1 – < Δ > ) ² P ( D )
Solving the system for < Δ > and P ( D ) we arrive at the solution
< Δ > = 1 –
)ω,T,t(θE
)ω,T,t(θE 2
, P ( t , T , D ) =
)ω,T,t(θE
])ω,T,t(θE[
2
2
(6)
Value < Δ > is market implied estimate and it does not equal to expected value of the stochastic recovery
rate
δ ( t , T , ω ) =
)T,t(B
)ω,T,t(R
Value < Δ > depends on distribution of the random process R ( t , T ; ω ) which is an assumption of the
model. Note that recovery rate < Δ > and correspondent probability of default represents estimates of the
real credit risk. In general
E θ p
( t , T , ω ) ≠ ( 1 – < Δ > ) p
P ( D )
p = 3, 4, … . Hence the approach is good for price distribution which completely defined by its first and
second moments. More crude but more explicit estimate can be presented as following. Assume for
example that we fix a particular recovery rate < Δ > . Applying first moment equation we arrive at the
formula
P ( D ) = ( 1 – < Δ > ) – 1
E θ ( t , T , ω )
Thus one can compare probabilities of default for different bonds given the same value of the recovery
rate. On the other hand one can fix probability of default and consider values of recovery rate of different
bonds. For example one can fix 45% recovery rate and determine bonds which probabilities of default
less than 0.2. Then fixing probability of default 0.3 one can find the bond which recovery rate is maximal
or a set of bonds which recovery rates exceed 80%.
Let us consider an implementation of randomization of the bond price. Let { t 0 } denote trading time
interval of the date t 0 . Value of { t 0 } can be either a day, week , or other appropriate period. Define
minimum and maximum values of the bond prices over the day period { t 0 }. Denote
9. 9
D min
( t 0 , T ) =
}t{u 0
min
D spot ( u , T ) , D max
( t 0 , T ) =
}t{u 0
max
D spot ( u , T )
Symbol D spot is interpreted as spot price of the corporate bond. We interpret date-t 0 bond price as a
random variable taking values on the interval [ D min
( t 0 , T ) , D max
( t 0 , T ) ] . There are different ways
to assign distribution to random bond price D mkt ( t 0 , T , ω ). A simple distribution that can be used for
D mkt ( t 0 , T , ω ) is uniform distribution. This distribution actually does not have any advantages or
drawbacks with respect to other types of probability distributions that can be applied for randomization.
Discrete approximation of the uniform distribution can be introduced as following. Fix a number m and
denote δ j = j / m , j = 0, 1, … m Defined numbers k and q are defined by inequalities
δ k ≤ R min ( t 0 , T ) < δ k + 1 and δ k + q – 1 ≤ R max ( t 0 , T ) < δ k + q
Then putting Ω j = { u { t 0 } : δ j ≤ R spot ( u , T ) < δ j + 1 } we put P { Ω j } = q – 1
,
j = k + 1, k + 2, … k + q – 1. Note that numbers k and q depends on t 0 . This is a heuristic distribution
that approximates real prices of the bond observed over the date t 0 .
Other more practical way of randomization of the date-t 0 pricing data is an assumption that distribution
P ( Ω j ) is proportional to the time when spot price
P ( Ω j ) ~ measure { u { t 0 } : R spot ( u , T ) [ δ j , δ j + 1 ] }
We also can use other distributions that approximated prices of the bond. These randomizations are
assigned to a particular date. The benchmark pricing uses one close price to represent a date t 0 bond
price. Such approximation can be good if the value D max ( t 0 , T ) – D min ( t 0 , T ) is sufficiently small.
Otherwise a lot of real pricing information will be lost. Pricing in stochastic environment implies market
risk regardless the chance of default. For example buyer’s market risk is measured by the probability that
buyer pays a higher price for the bond than it is implied by the market. If spot price at t 0 is identified as
the close price of the date t 0 then market risk of the buyer is the probability
P { R close ( t 0 , T ) > R ( t 0 , T , ω ) } =
)T,t(R-)T,t(R
)T,t(R-)T,t(R
0min0max
0close0max
Let us define recovery rate and default distribution implied by the stochastic price of the corporate bond
R ( t 0 , T ; ω ). Assuming for example uniform distribution of the recovery rate at maturity we arrive at
the formula
R ( t 0 , T , ω ) = δ j B ( t 0 , T ) , ω Ω j
P ( Ω j ) = q – 1
, j = k + 1, … k + q – 1
Hence given the date-t 0 pricing data of the risky bond δ j = δ j ( t 0 ) , k = k ( t 0 ) , q = q ( t 0 ) we can
calculate credit risk based on prior assumed uniform distribution.
Remark. Randomization of the date t spot price of the bond is a primary assumption of our approach to
credit risk valuation. In modern finance it is common rule to use historical data time series as independent
10. 10
observation. It is not quite accurate interpretation of the statistical sample. Indeed consider for example
rate of return on an asset S
i ( t , T ) =
)t(S
)t(S)T(S
and let t j < t j + 1 , j = 0 , 1, … n – 1 be a partition of the interval [ t , T ]. Then historical data
represented by time series i ( t j , t j + 1 ) is usually interpreted as independent observation over real rate of
return. Given historical data one should first to complete test of independence. Second unknown historical
data such as rate of return is assumed to present independent observation of the random variable. Hence
unknown parameters mean and variance are assumed to be constant. If not one should expect a large
deviation between model and real data. Also if parameters of the model could not be assume to be
constant one can observe other effects such as ‘fat’ tails. It can occur because variance depends on time
and fuzzifies data.
Consider corporate zero coupon bonds that admit default at any moment during its lifetime. Recall
approach that leads us to exponential distribution of the default moment. Introduce no default probability
distribution function P ( t ) = P { > t } and let t 0 < t 1 <…< t n = T be a partition of the interval
[ t 0 , T ]. Function P ( t ) is monotonic decreasing function in variable t. Our problem is: given no default
up to the moment t j to calculate probability that there is no default up to the future moment t k , k > j. A
solution of the problem can be represented by the conditional probability
P ( t k , t j ) = P { > t k | > t j }
Bearing in mind that default time depends on initiation date t 0 all functions defined bellow are also
depend on t 0 . Bearing in mind that
P { > t k > t j } = P { > t k } = P ( t k )
we note that
P ( t k , t j ) =
}tτ{P
}tτ{P
j
k
=
)t(P
)t(P
j
k
The probability of default on ( t j , t k ] is then equal to
Q ( t k , t j ) = 1 – P ( t k , t j ) = 1 –
)t(P
)t(P
j
k
=
)t(P
)t(P-)t(P
j
kj
Putting t j = t and t k = t + Δ t we arrive at the equality
Q ( t + Δ t , t ) = –
)t(P
)t('P
Δ t + o ( Δ t )
11. 11
Denote λ ( t ) =
)t(P
)t('P
. Then
P ( t ) = exp –
t
t 0
λ ( s ) d s
There are other types of distributions that can be applied for default time modeling. These types are
Weibull distribution, Lognormal, Power, Gamma distributions. These distributions have multiple
parameters that can be used for better than exponential approximations of the default time. Function
λ ( s ) in exponential distribution is known as a hazard rate.
A discrete time approximation of the continuously distributed default time can be represented in the form
λ ( ω ) =
n
1k
t k χ { ( ω ) ( t k – 1 , t k ] } + T χ { ( ω ) > T }
For simplicity we assumes that zero coupon bond admits default at the dates t j . There are a few pricing
settlements of the corporate bond at the time of default. Fractional recovery of the market value
δ j B ( t j , T ) that is paid at t j . Other types of default settlements are the contingent claim of value
defined as δ j B – 1
( t j , T ) paid at T and the fractional recovery of the Treasury value δ j at T. Given a
particular choice of the default settlement a discrete time approximation of the cash flows of the corporate
bond can be represented in one of the forms
CF A 1 ( ω ) =
n
1j
χ ( λ = t j ) δ j B ( t j , T ) χ ( t = t j ) + χ ( λ > T ) χ ( t = T )
CF A 2 ( ω ) =
n
1j
χ ( λ = t j ) δ j B – 1
( t j , T ) χ ( t = T ) + χ ( λ > T ) χ ( t = T )
CF A 3 ( ω ) =
n
1j
χ ( λ = t j ) δ j χ ( t = t j ) + χ ( t > T ) χ ( t = T )
Here χ ( t = t j ) is indicator function in t which specifies value of transaction between buyer and seller
which takes place at t j . Cash flow CF A 1 defines recovery rates δ j of the corporate bond with respect to
the value of the risk free bond with equal maturity. Also note that as far as the bid and ask prices are
assumed in theory to be equal for a fixed maturity date T the bond price at a future date t j , t j < T does
not depend on time when bond was issued. In other words on the run risky or risk free bond issued at t j
and the similar bond issued prior to the date t j have the same price at the date t j . Note that theoretically
cash flows CF A k , k = 2, 3 can be rewritten in the CF A 1 form. Hence we assume that bond is
represented by the formula CF A 1 . The stochastic market price of the bond at t 0 can be represented as
the PV of the cash flow CF A 1 . Thus
12. 12
PV { CF ( ω ) } = R ( t 0 , T , ω ) =
n
1j
χ { = t j } R ( t 0 , T , ω ) + χ { > T } B ( t 0 , T ) =
=
n
1j
χ { = t j } B ( t 0 , t j ) δ j B ( t j , T ) + χ { > T } B ( t 0 , T ) = (7)
=
n
1j
χ { = t j } R ( t 0 , t j , ω ) B ( t j , T ) + χ { > T } B ( t 0 , T )
Next for writing simplicity we omit index λ and low index A 1 which specifies cash flow. For the fixed t 0
ratio δ j on the right hand side (7) depends on t j and does not depend on T . The recovery rate of the bond
at the date of default δ j B ( t j , T ) depends on T .
Risk factor that affects bond price (7) associated with default time distribution is referred to as credit risk.
Values B ( t j , T ) which are unknown at initiation date t 0 specify market risk of the recovery rate of the
corporate bond. Indeed replacing values B ( t j , T ) in formula (7) by market implied forward discounting
rate B ( t j , T , t 0 ) we use date-t 0 estimate of the real rate which arrives at the date of default. Future
value of the CF ( ω ) is equal to
FV { CF ( ω ) } =
n
1j
χ ( λ = t j ) δ j B ( t j , T ) B – 1
( t j , T ) + χ ( λ > T ) =
=
n
1j
χ ( = t j ) δ j ( ω ) + χ ( > T )
No arbitrage pricing on [ t 0 , T ] we defined in [1] by equality
B – 1
( t 0 , T ) PV { CF ( ω ) } = FV { CF ( ω ) }
where cash flow CF was associated with risk free bond. Spot price of the risk free bond is assumed here
to be a constant while risky bond spot price is a random variable. Therefore
B – 1
( t 0 , T ) PV { CF ( ω ) } = B – 1
( t 0 , T ) R ( t 0 , T , ω ) =
= B – 1
( t 0 , T ) [
n
1j
χ ( = t j ) B ( t 0 , t j ) δ j B ( t j , T ) + χ ( > T ) B ( t 0 , T ) ] =
=
n
1j
χ ( = t j ) B – 1
( t j , T , t 0 ) δ j B ( t j , T ) + χ ( > T ) =
=
n
1j
χ ( = t j ) δ j ( ω ) + χ ( > T ) = FV { CF ( ω ) }
13. 13
Therefore PV of the stochastic cash flow represents no arbitrage (stochastic) market price of the bond.
There is no market risk on [ t 0 , T ] as no arbitrage pricing excludes market risk. Nevertheless early
selling of the bond might represent market risk to counterparty. Indeed let us define market risk of the
corporate bond. We defined spot price as a random variable. By definition scenario ω is equal to a
particular price of the bond during trade time of the date t 0 . Note that prices of the bond at different
moments can be equal to each other. Investor buying bond at date t 0 does not know either exact moment
of default or recovery rate at this moment. Such uncertainty implies that counterparty could loss or makes
a profit based on realized market scenario. On the other hand price defined by (7) does not eliminate
credit risk which can be specified by a choice of default time distribution.
Remark. Recall that standard approaches to corporate bond pricing deal with expected value of the
market implied PV of the CF. This approach replace future rate by its market implied estimate. Such
reduction incorporates market risk. This risk is specified by the difference between real future rate and its
market implied estimate. For a small standard deviation it might be reasonable approximation of the
stochastic price R ( t 0 , T , ω ) to a single number R spot ( t 0 , T ). It can be open, close price, or for
example
R spot ( t 0 , T ) = E R ( t 0 , T , ω )
For each future date t j market risk is implied by the fact that B ( t j , T , t 0 ) ≠ B ( t j , T ). Applying a
stochastic model of the future rates [4] one can estimate a chance P { B ( t j , T , t 0 ) > B ( t j , T ) } or
P { B ( t j , T , t 0 ) < B ( t j , T ) } at the date t 0 . Such estimates can justify reduction of the stochastic
price to single number as a representation of the date-t 0 bond price.
Now let us consider a generalization of the method used for calculation recovery rate and probability of
default (6) to cover general discrete time pricing model. In formula (7) dates t j , j = 1,2 … n represent
possible dates of default. Suppose (in theory) that bonds with expiration dates at t j , j = 1, 2, … n are
available on the market. Applying (3) we arrive at the formula
( 1 – δ 1 ) χ { = t 1 } = 1 –
)t,t(B
)ω,t,t(R
10
10
From this equality it follows that
( 1 – δ 1 ) P { = t 1 } = E [ 1 –
)t,t(B
)ω,t,t(R
10
10
]
( 1 – δ 1 ) 2
P { = t 1 } = E [ 1 –
)t,t(B
)ω,t,t(R
10
10
] 2
Solving the system for recovery rate and default probability we arrive at the values
14. 14
δ 1 = 1 –
]
)t,t(B
)ω,t,t(R
1[E
]
)t,t(B
)ω,t,t(R
1[E
10
10
2
10
10
,
P { = t 1 } = ( 1 – δ 1 ) – 1
E [ 1 –
)t,t(B
)ω,t,t(R
10
10
] = (8)
= 2
1010
2
1010
])ω,t,t(R)t,t(B[E
}])ω,t,t(R)t,t(B[E{
,
P { > t 1 } = 1 – P { = t 1 }
Bearing in mind Jensen’s inequality it is easy to verify inequality
P { = t 1 } = 2
1010
2
1010
])ω,t,t(R)t,t(B[E
}])ω,t,t(R)t,t(B[E{
≤ 1
Formulas (8) represent first term on the right hand side (7) which corresponds to a set of scenarios
D 1 = { ω : = t 1 }. Next we note that value of probabilities P ( D j ) = P { ω : = t j } and
correspondent recovery rates could be adjusted by taking into account that { > t j – 1 }, j = 2, 3, … n .
Such adjustment implies on credit risk exposure and it can be realized by applying conditional probability
P { λ = t j | > t j – 1 }. Taking into account inclusion
{ > t j } { > t j – 1 }
we can conclude that
P { λ = t j + 1 | > t j , … > t 1 } = P { λ = t j + 1 | > t j }
Given that { ( ω ) > t j – 1 }, j = 2, 3, … n we use estimate of the price of the bond at the moment t j – 1
provided by market implied forward value. Definition of the market implied forward risky discount rate
which in contrast to the risk free discount rate should take into account default time distribution. It
follows from (7) that
R ( t 0 , T , ω ) χ { = t j } = R ( t 0 , t j , ω ) B ( t j , T ) χ { = t j } (9)
j = 1, 2, … n. Here B ( t j , T ) = B ( t j , T , ω ) is future rate known only at t j and it is unknown at t 0.
Let us apply market implied forward rate as an estimate of the real forward rate. Then we arrive at the
reduction which decompose bond price
R ( t 0 , T , ω ) χ { = t j } = [ δ j B ( t 0 , t j ) ] B ( t j , T , t 0 ) χ { = t j } =
= R ( t 0 , t j , ω ) B ( t j , T , t 0 ) χ { = t j }
Solving latter equation for R ( t 0 , t j , ω ) brings us the formula
15. 15
R ( t 0 , t j , ω ) χ { = t j } = B – 1
( t j , T , t 0 ) R ( t 0 , T , ω ) χ { = t j } (9)
Note that transition from (9) to (9) implies market risk and the fact that bonds R ( t 0 , t j , ω ) , j = 1, 2,
… n – 1 might not exist on the market. On the other hand taking into account equality
R ( t 0 , t j – 1 , ω ) χ { = t j } = B ( t 0 , t j – 1 ) χ { = t j }
we define market implied forward risky discount rate R ( t j – 1 , t j ; t 0 , ω ) , ω { = t j } by equality
R ( t 0 , t j , ω ) χ { = t j } = R ( t 0 , t j – 1 , ω ) R ( t j – 1 , t j , t 0 , ω ) χ { = t j } =
= B ( t 0 , t j – 1 ) R ( t j – 1 , t j , t 0 , ω ) χ { = t j }
Bearing in mind that
R ( t 0 , t j – 1 , ω ) χ { > t j – 1 } = B ( t 0 , t j – 1 ) χ { > t j – 1 }
we conclude that
R ( t j – 1 , t j , t 0 , ω ) χ { = t j } = B – 1
( t 0 , t j – 1 ) R ( t 0 , t j , ω ) χ { = t j } = (10)
= B – 1
( t 0 , t j – 1 ) R ( t 0 , T , ω ) B – 1
( t j , T , t 0 ) χ { = t j }
Market implied forward discount rate R ( t j – 1 , t j , t 0 , ω ) is represented by forward starting bond that
admits default only at its maturity t j . We can now apply estimates represented in formula (8).
Applying formulas (8) to the function R ( t j – 1 , t j , t 0 , ω ) defined by (10) we arrive at the formulas
δ j ( t 0 ) = 1 –
]
)t,t,t(B
)t,T,t(B)ω,T,t(R)t,t(B
1[E
]
)t,t,t(B
)t,T,t(B)ω,T,t(R)t,t(B
1[E
0j1-j
0j
1
01-j0
1
R
2
0j1-j
0j
1
01-j0
1
R
(11)
with correspondent conditional probabilities of default
P { = t j | > t j – 1 } = ( 1 – δ j ( t 0 ) ) – 1
[ 1 – E
)t,t,t(B
)t,T,t(B)ω,T,t(R)t,t(B
0j1-j
0j
1
01-j0
1
]
No default conditional probability is then equal to
P { > t j | > t j – 1 } = 1 – P { = t j | > t j – 1 }
Unconditional probability of default at the date t j and no default over lifetime of the bond could be
defined by the formula
P { = t j } = P ( = t j | > t j – 1 )
1j
0k
[ 1 – P ( = t k | > t k – 1 ) ]
16. 16
(11)
P { > T } = 1 –
n
1j
P ( = t j | > t j – 1 )
1j
0k
[ 1 – P ( = t k | > t k – 1 ) ]
Formulas (1.11), (1.11) represent recovery rate and probability of default values of the zero coupon
corporate bond in discrete time setting.
Remark. Let us add some remarks on randomization. We generalized spot price notion of the bond.
Usually date-t price of the bond is associated with a particular price such as close price of the bond at the
date t. Our approach interprets bond price at date t as a random variable R ( t , T , ω ) taking values in the
interval
[
}t{u
min
R ( u , T ) ,
}t{u
max
R ( u , T ) ]
Therefore market scenario associated with a particular value of the bond, i.e. ω = R ( u , T ) for a some
moment u { t }. The use of a fixed price such as close price at date t should be interpreted as an
approximation of the random variable implies market risk. If we consider random price at a future
moment t + Δ t then the probability space
[
}tt{u
min
R ( u , T ) ,
}tt{u
max
R ( u , T ) ]
does not coincide with the one that is defined at t. Therefore dealing with dynamic market we should
introduce a unique probability space. Next we associate probability space Ω R with the set of measurable
functions ω ( t ) such that
}t{u
min
R ( u , T ) ≤ ω ( t ) ≤
}t{u
max
R ( u , T ) }
t [ t 0 , T ]. Prior to expiration date values of the corporate bond does not represent default event. Hence
time of default of the bond can be thought as an internal company factor and it can be interpreted as a
random variable. As far as values of the bond do not completely define time of default distribution the
probability space Ω ( R ) should not interpret default time ( ω ) ( t 0 , + ) , ω Ω . The term ‘
should not interpret default time’ suggests that in general ( ω ) does not measurable function on Ω ( R ).
We introduce measurable space { Ω , B } where Ω = Ω ( R ) Ω , B = B ( R ) B . Here
symbol B denote σ-algebra of Borel sets on correspondent space. Probability measure P on measurable
space { Ω , B } is defined by equality
P ( A Δ ) = P { R ( t , T , ω ) A , ( ω ) Δ }
A B ( R ), Δ B . For a random variable ξ ( ω ) on { Ω , B , P } denote
expectation with respect to market scenarios generated by the values of the bond while default time
remains stochastic. Such construction implies that values of the bond and default time are mutually
17. 17
independent random variables and symbol E R denotes conditional expectation with respect to -algebra
generated by the default time
E R ξ ( ω ) = E { ξ ( ω ) | B }
Note that it is quite a strong assumption and it is very popular in credit risk valuations.
Corporate coupon bearing bond pricing. Zero coupon corporate bonds pricing scheme can be applied
to coupon bearing bond. Suppose for simplicity that default dates coincide with coupon payment dates
T 1 < T 2 < … < T m = T. Then cash flow of the coupon bearing bond from bond to bond buyer
perspective can be written in the form
CF ( ω ) =
m
1j
χ ( = T j ) {
1-j
1k
c χ ( t = T k ) + δ j [ c
m
ji
B ( T j , T i ) +
+ B ( T j , T m ) ] χ ( t = T j ) + χ ( > T ) [
m
1j
c χ ( t = T k ) + χ ( t = T m ) ]
In this formula recovery rate is assigned to outstanding debt value defined at default moment. The PV
reduction at t of the cash flow CF ( ω ) defines market price of the corporate bond depending on market
scenario which incorporate unknown values of the risk free bonds B ( T j , T i ) at future moments T j , T i
and time of default. Assume that corporation simultaneously issued zero coupon and coupon bonds. In
this case investors can expect that loss-ratio due to default of the zero and nonzero coupon bonds are
equal. Otherwise there exists hypothetical arbitrage opportunity. The notion ‘equal loss-ratio’ in latter
statement should be refined. We interpret bond default as default of the corporation and therefore default
of the corporation affects on all corporate bonds issued by this corporation. One should assume that
probability distribution of default time for corporate zero and nonzero coupon bonds are the same. Recall
that recovery rate of the zero coupon bond conditional on default at T j can be represented in the form
c δ j ( t , ω ) B ( T j , T , ω ), which is unknown at t and can be estimated by deterministic value
δ j ( t ) B ( T j , T , t ) . Hence recovery rate of the zero coupon corporate bond is interpreted as a portion
δ j ( t ) of the date-T j PV of the face value of the bond. Following this idea we define recovery value of
the coupon bond at T j as a portion δ
c
j ( t ) of the outstanding balance of the coupon bond on [ T j , T ].
Hence, recovery rate of the coupon bond can be defined as
δc
j [ 1 + c
m
1ji
B – 1
( T i , T ) + c B – 1
( T j , T ) ]
The value δc
j we call recovery ratio. For the zero coupon bond, c = 0 recovery rate notion coincides with
recovery ratio. Total losses due to default are equal to
( 1 – δ
c
j ) [ 1 + c
m
1ji
B – 1
( T i , T ) + c B – 1
( T j , T ) ]
18. 18
Theoretically the value of the losses at T of the zero and nonzero coupon bonds of the corporation that
defaults at T j should be equal for each scenario. In this case we eliminate arbitrage opportunity between
zero and non zero coupon bonds for each scenario. Hence
1
δ1
)T,T(Bc1
])T,T(Bc)T,T(Bc1[)δ1(
j
i
1
m
1ji
j
1
i
1
m
1ji
c
j
Solving latter equation for the recovery rate of the coupon bond rate we arrive at the formula
δc
j ( t , ω ) = 1 –
)T,T(B)T,T(Bc1
])T,T(Bc1[))ω,t(δ1(
j
1
i
1
m
1ji
i
1
m
1ji
j
Market implied estimate of the above stochastic recovery rate can be represented as
δc
j = 1 –
)t,T,T(B)T,T(Bc1
])t,T,T(Bc1[)δ1(
j
1
i
1
m
1ji
i
1
m
1ji
j
Recall that using market implied forward estimate δc
j implies market risk as far as δc
j ( t , ω ) ≠ δ c
j .
19. 19
References.
1. I. Gikhman, BASIC OF PRICING 1, http://www.slideshare.net/list2do/basic-pricing ,
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2428024.
2. I. Gikhman, Corporate Debt Pricing, http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1163195
3. Multiple Risky Securities Valuation I,
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1944171
4. Fixed Rates Modeling. 2013 p. 25,
http://www.slideshare.net/list2do/fixed-rates-modeling
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2287165