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.
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.
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
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.
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 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.
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.
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.
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.
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
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.
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 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.
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.
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.
1) The document discusses pricing models for derivatives such as options and interest rate swaps. It introduces concepts such as local volatility, which models implied volatility as a function of strike price and time to maturity.
2) Black-Scholes pricing is based on the assumption of a perfect hedging strategy, but the document notes this is formally incorrect as the hedging portfolio defined does not satisfy the required equations.
3) Local volatility presents the option price as a function of strike and time to maturity, with the diffusion coefficient estimated from option price data, whereas Black-Scholes models the price as a function of the underlying and time, with volatility as an input.
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
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 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.
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.
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 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.
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 is equal to the risk-neutral expected value of the average variance over the swap period, where the expectation is calculated using Black-Scholes option pricing.
3. The value of a variance swap is defined as the notional amount multiplied by the difference between realized variance, estimated from historical stock prices, and the strike variance estimated from option prices. The variance swap thus specifies the value of differences between two estimates of the true variance.
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.
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
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.
Pricing Exotics using Change of NumeraireSwati Mital
The intention of this essay is to show how change of numeraire technique is used in pricing derivatives with complex payoffs. In the first instance, we apply the technique to pricing European Call Options and then use the same method to price an exotic Power Option.
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.
This document describes an uncertain volatility model for pricing equity option trading strategies when the volatilities are uncertain. It uses the Black-Scholes Barenblatt equation developed by Avellaneda et al. to derive price bounds. The model is implemented in C++ using recombining trinomial trees to discretize the asset prices over time and space. The code computes the upper and lower price bounds by solving the Black-Scholes Barenblatt PDE using numerical techniques, with the volatility set based on the sign of the option gamma.
Stochastic Local Volatility Models: Theory and ImplementationVolatility
1) Hedging and volatility
2) Review of volatility models
3) Local volatility models with jumps and stochastic volatility
4) Calibration using Kolmogorov equations
5) PDE based methods in one dimension
5) PDE based methods in two dimensions
7) Illustrations
Presentation to the Centre for Veterinary Education (CVE) tutors forum in Melbourne, April 2014. Focus on exploring the functionality of LMS systems to support good online pedagogy.
Michelle Bloomberg earned her Master of Science in Cardiovascular Disease Epidemiology from Harvard School of Public Health in 2007. She is currently a PhD candidate at Johns Hopkins Bloomberg School of Public Health, expected to graduate in 2010. Her research and academic interests include cardiovascular disease, health disparities, and community health. She has over 3 years of experience conducting epidemiological research and has authored or co-authored several peer-reviewed publications and conference presentations. Ms. Bloomberg is also actively involved in various professional organizations, community service projects, and international health initiatives focused on topics such as malaria prevention, neighborhood effects on health, and chronic disease.
This document discusses cerebrovascular disorders such as stroke. It begins by defining cerebrovascular disorders as any functional abnormality of the central nervous system caused by disrupted blood supply to the brain. Stroke is the primary cerebrovascular disorder. The document then covers the anatomy of the nervous system, definitions of stroke, risk factors, types of stroke (ischemic and hemorrhagic), clinical manifestations, diagnostic findings, and management approaches including medical, surgical, and nursing considerations.
This document provides details on examining patients with cardiovascular symptoms. It describes how to take a history, including presenting symptoms, previous illnesses, habits, and family history. The physical exam involves inspection, pulse examination, blood pressure measurement, jugular vein examination, chest examination, and heart auscultation. Specific cardiovascular conditions can cause chest pain, dyspnea, fatigue, palpitations, and syncope. Findings on exam include pulses, jugular vein pressure, heart sounds, murmurs, and peripheral signs. The goal is to evaluate symptoms, signs, and history to understand a patient's cardiovascular condition.
This document provides information on cerebrovascular accidents (strokes). It defines a stroke as occurring when blood supply to the brain is interrupted, usually due to a blood clot or burst blood vessel. Strokes can be ischemic, caused by a clot cutting off blood flow, or hemorrhagic, caused by a ruptured blood vessel. Warning signs include sudden weakness, numbness, trouble speaking, and loss of vision. Acting FAST (Facial drooping, Arm weakness, Speech difficulties, Time to call for help) can help identify a stroke and get immediate medical attention, as rapid treatment improves outcomes. Risk factors include age, race, family history, high blood pressure, diabetes, smoking and more.
1) The document discusses pricing models for derivatives such as options and interest rate swaps. It introduces concepts such as local volatility, which models implied volatility as a function of strike price and time to maturity.
2) Black-Scholes pricing is based on the assumption of a perfect hedging strategy, but the document notes this is formally incorrect as the hedging portfolio defined does not satisfy the required equations.
3) Local volatility presents the option price as a function of strike and time to maturity, with the diffusion coefficient estimated from option price data, whereas Black-Scholes models the price as a function of the underlying and time, with volatility as an input.
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
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 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.
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.
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 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.
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 is equal to the risk-neutral expected value of the average variance over the swap period, where the expectation is calculated using Black-Scholes option pricing.
3. The value of a variance swap is defined as the notional amount multiplied by the difference between realized variance, estimated from historical stock prices, and the strike variance estimated from option prices. The variance swap thus specifies the value of differences between two estimates of the true variance.
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.
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
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.
Pricing Exotics using Change of NumeraireSwati Mital
The intention of this essay is to show how change of numeraire technique is used in pricing derivatives with complex payoffs. In the first instance, we apply the technique to pricing European Call Options and then use the same method to price an exotic Power Option.
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.
This document describes an uncertain volatility model for pricing equity option trading strategies when the volatilities are uncertain. It uses the Black-Scholes Barenblatt equation developed by Avellaneda et al. to derive price bounds. The model is implemented in C++ using recombining trinomial trees to discretize the asset prices over time and space. The code computes the upper and lower price bounds by solving the Black-Scholes Barenblatt PDE using numerical techniques, with the volatility set based on the sign of the option gamma.
Stochastic Local Volatility Models: Theory and ImplementationVolatility
1) Hedging and volatility
2) Review of volatility models
3) Local volatility models with jumps and stochastic volatility
4) Calibration using Kolmogorov equations
5) PDE based methods in one dimension
5) PDE based methods in two dimensions
7) Illustrations
Presentation to the Centre for Veterinary Education (CVE) tutors forum in Melbourne, April 2014. Focus on exploring the functionality of LMS systems to support good online pedagogy.
Michelle Bloomberg earned her Master of Science in Cardiovascular Disease Epidemiology from Harvard School of Public Health in 2007. She is currently a PhD candidate at Johns Hopkins Bloomberg School of Public Health, expected to graduate in 2010. Her research and academic interests include cardiovascular disease, health disparities, and community health. She has over 3 years of experience conducting epidemiological research and has authored or co-authored several peer-reviewed publications and conference presentations. Ms. Bloomberg is also actively involved in various professional organizations, community service projects, and international health initiatives focused on topics such as malaria prevention, neighborhood effects on health, and chronic disease.
This document discusses cerebrovascular disorders such as stroke. It begins by defining cerebrovascular disorders as any functional abnormality of the central nervous system caused by disrupted blood supply to the brain. Stroke is the primary cerebrovascular disorder. The document then covers the anatomy of the nervous system, definitions of stroke, risk factors, types of stroke (ischemic and hemorrhagic), clinical manifestations, diagnostic findings, and management approaches including medical, surgical, and nursing considerations.
This document provides details on examining patients with cardiovascular symptoms. It describes how to take a history, including presenting symptoms, previous illnesses, habits, and family history. The physical exam involves inspection, pulse examination, blood pressure measurement, jugular vein examination, chest examination, and heart auscultation. Specific cardiovascular conditions can cause chest pain, dyspnea, fatigue, palpitations, and syncope. Findings on exam include pulses, jugular vein pressure, heart sounds, murmurs, and peripheral signs. The goal is to evaluate symptoms, signs, and history to understand a patient's cardiovascular condition.
This document provides information on cerebrovascular accidents (strokes). It defines a stroke as occurring when blood supply to the brain is interrupted, usually due to a blood clot or burst blood vessel. Strokes can be ischemic, caused by a clot cutting off blood flow, or hemorrhagic, caused by a ruptured blood vessel. Warning signs include sudden weakness, numbness, trouble speaking, and loss of vision. Acting FAST (Facial drooping, Arm weakness, Speech difficulties, Time to call for help) can help identify a stroke and get immediate medical attention, as rapid treatment improves outcomes. Risk factors include age, race, family history, high blood pressure, diabetes, smoking and more.
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 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.
This document discusses the Black-Scholes pricing concept for options. It summarizes two popular derivations of the Black-Scholes equation, the original derivation and an alternative presented in other literature. It also discusses ambiguities that have been noted in the derivation of the Black-Scholes equation and proposes corrections to the derivation using modern stochastic calculus. Specifically, it introduces a hedged portfolio function defined over two variables to accurately represent the value and dynamics of the hedged portfolio. The document concludes that the Black-Scholes pricing concept only guarantees a risk-free return at a single point in time and does not necessarily reflect market prices.
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 point in time, it does not necessarily reflect market prices and there is market risk for the option seller at future times.
equity, implied, and local volatilitiesIlya Gikhman
This document discusses connections between stock volatility, implied volatility, and local volatility in option pricing models. It provides an overview of the Black-Scholes pricing model, which assumes stock volatility is known. However, implied volatility estimated from market option prices does not match the true stock volatility. The local volatility model develops implied volatility as a function of underlying variables to better match market prices, without relying on an assumed stock process.
This document summarizes a research paper that examines the optimal investment, consumption, and life insurance selection problem for a wage earner. The problem is modeled using a financial market with one risk-free asset and one risky jump-diffusion asset, along with an insurance market composed of multiple life insurance companies. The goal is to maximize the wage earner's expected utility from consumption during life, wealth at retirement or death, by choosing an optimal investment, consumption, and insurance strategy. The authors use dynamic programming to characterize the optimal solution and prove existence and uniqueness of a solution to the associated nonlinear Hamilton-Jacobi-Bellman equation.
In this paper, we consider an AAI with two types of insurance business with p-thinning dependent
claims risk, diversify claims risk by purchasing proportional reinsurance, and invest in a stock with Heston
model price process, a risk-free bond, and a credit bond in the financial market with the objective of maximizing
the expectation of the terminal wealth index effect, and construct the wealth process of AAI as well as the the
model of robust optimal reinsurance-investment problem is obtained, using dynamic programming, the HJB
equation to obtain the pre-default and post-default reinsurance-investment strategies and the display expression
of the value function, respectively, and the sensitivity of the model parameters is analyzed through numerical
experiments to obtain a realistic economic interpretation. The model as well as the results in this paper are a
generalization and extension of the results of existing studies.
This document summarizes a mathematical economics model of stock prices using differential equations. It begins by modeling stock prices based on dividends and interest rates, assuming stock prices converge to the fundamental price over time. It then models stock prices under rational expectations, finding stock prices instantly jump to the fundamental price. Finally, it briefly introduces a dynamic IS-LM model to analyze output and interest rates over time.
The Black-Scholes-Merton model prices options using the assumption that stock price changes are lognormally distributed. It derives the Black-Scholes differential equation by constructing a riskless portfolio of stock and options and requiring its value to earn the risk-free rate of return. The model uses risk-neutral valuation, which assumes the expected return on the stock is the risk-free rate, to calculate the expected payoff of the option. Discounting this expected payoff at the risk-free rate gives the option price. The model provides closed-form formulas for European call and put option prices in terms of the stock price, strike price, risk-free rate, time to maturity, and volatility.
An Approximate Distribution of Delta-Hedging Errors in a Jump-Diffusion Model...Volatility
1) Analyse the distribution of the profit&loss (P&L) of delta-hedging strategy for vanilla options in Black-Scholes-Merton (BSM) model and an extension of the Merton jump-diffusion (JDM) model assuming discrete trading and transaction costs
2) Examine the connection between the realized variance and the realized P&L
3) Find approximate solutions for the P&L volatility and the expected total transaction costs
4) Apply the mean-variance analysis to find the trade-off between the costs and P&L variance given hedger's risk tolerance
5) Consider hedging strategies to minimize the jump risk
Pricing of fx target redemption note by simulationcaplogic-ltd
This document describes using a Monte Carlo simulation to price a foreign exchange (FX) target redemption note. The note pays annual coupons based on the exchange rate between two currencies, with the first coupon fixed and subsequent coupons varying. It terminates if accumulated coupons reach a cap. The simulation models the FX rate and domestic/foreign interest rates as correlated stochastic processes. It runs trials simulating the rates over time, calculates coupons, and discounts cash flows to value the note. The Hull-White model is used to simulate the interest rate processes.
This document summarizes research on strong duality analysis for discrete-time constrained portfolio optimization problems. It begins by introducing the mathematical formulation of a discrete-time portfolio selection model with constraints expressed as convex inequalities. It then discusses a risk neutral computational approach based on embedding the primal constrained problem into a family of unconstrained problems in auxiliary markets. Weak duality is shown to hold, relating the optimal values of the primal and auxiliary problems. The document defines a dual problem, known as Pliska's κ dual, that seeks to minimize the optimal values of the auxiliary problems. Conditions for strong duality are presented, under which the optimal solution to the dual problem also solves the primal constrained problem.
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বাংলাদেশ অর্থনৈতিক সমীক্ষা (Economic Review) ২০২৪ UJS App.pdf
Bond Pricing and CVA
1. 1
BOND PRICING AND CVA.
Ilya I. Gikhman
6077 Ivy Woods Court,
Mason, OH 45040, USA
ph. 513-573-9348
email: ilyagikhman@mail.ru
JEL : G12, G13
Key words. Bond, mark-to-market, counterparty risk, CVA.
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.
Default at Maturity. Let us briefly recall the reduced form of risky bond valuation. Denote R ( t , T )
and B ( t , T ) prices of the risky and risk free bonds correspondingly at date t with expiration dates at T.
Both bonds promise $1 at the date T. The difference
B ( t , T ) – R ( t , T ) 0
is known as credit spread. It is a characteristic of a chance of default of the corporate bond. We illustrate
default pricing by using an example [ me] in which default of the corporate bond may occurred only at
expiration date. Our interpretation differs from the original interpretation [1] where only a single value
was admitted at the default moment. Suppose that risky bond might default only at expiration date T and
R ( T , T , ) = j , where D j = { : R ( t , T , ) = j } , P { D j } = p j
where 0 ≤ n < n – 1 < … < 0 = 1,
n
1j
p j = 1. Default distribution p j and recovery rate δ j are
assumed to be known. Default values imply the formula
2. 2
R ( t , T ; ω ) = B ( t , T )
n
1j
j χ ( D j ) (1)
Formula (1) shows that market price of the risky bond at t is a random variable. The spot price of the bond
R spot ( t , T ) is the price which is used for buying or selling bonds at the market at t. The spot price is
established as a settlement between buyers and sellers by weighting their estimates of possible profits and
losses based on (1). Applying spot price buyers and sellers are subject of the market risk. Buyer market
risk is measured by the probability
P { R ( t , T ; ω ) < R spot ( t , T ) }
This probability represents the value of a chance that spot price is higher than it would be realized by
market scenarios. Average and dispersion of the losses
m b risk = E R ( t , T ; ω ) χ { R ( t , T ; ω ) < R spot ( t , T ) }
D b risk = E R 2
( t , T ; ω ) χ { R ( t , T ; ω ) < R spot ( t , T ) } – m 2
riskb
are primary characteristics of the buyer credit risk at t. Similarly market risk of the bond seller is equal to
P { R ( t , T ; ω ) > R spot ( t , T ) }
and variables
m s risk = E R ( t , T ; ω ) χ { R ( t , T ; ω ) > R spot ( t , T ) }
D s risk = E R 2
( t , T ; ω ) χ { R ( t , T ; ω ) > R spot ( t , T ) } – m 2
risks
define market risk of the bond seller. Note later variables specify market risk as far as buyer of the bond
does not responsible for the bond default. The later formulas specify the chance that seller of the bond
receives less than it is implied by the market. In theory, given market and spot prices we enable to
estimate market risks, expected returns, and possible market deviations. Suppose that spot price is defined
by the formula
R spot ( t , T ) = E R ( t , T ; ω )
Then
R spot ( t , T ) = B ( t , T )
n
1j
j p j , E R 2
( t , T ; ω ) = B 2
( t , T )
n
1j
2
j p 2
j
D R ( t , T ; ω ) = E R 2
( t , T ; ω ) – E [ R spot ( t , T ) ] 2
Randomness of the date-t market price of the corporate bond which is stipulated by a chance of default
represents credit risk of the contract.
Remark. Let us recall the benchmark risk neutral valuation following by [1]. Let A ( t ) denote value of
the money market account (MMA) which is constructed as following.
3. 3
By definition we put A ( 0 ) = $1 and next by induction define A ( t + 1 ) = A ( t ) R , t = 0, 1, …
where R denotes risk free interest rate over a single period. R – 1
is used as risk-free discount factor in
binomial scheme option valuation. In a single period economy t = 0, T = 1 we assume that underlying
security admits two values S d < S u at maturity t + 1 and
S d < R S ( t ) < S u (2)
Condition (2) guarantees no arbitrage opportunity. For given R there exists unique weights ( 1 – ) ,
which depends on t such that
E π
S ( t + 1 )
def
( 1 – ) S d + S u = R S ( t ) =
)t(A
)1t(A
S ( t )
Then
E
{
)1t(A
)1t(S
| F t } =
)t(A
)t(S
(3)
Here F t is the - algebra generated by values of the stock up to the moment t and weights ( 1 – ) ,
are called unique equivalent martingale probabilities [1]. This equality shows that random process
S ( t ) A – 1
( t ) is a martingale with respect to risk neutral measure { ( 1 - ) , }.
Equality (3) can be also represented in the form
E
)t(S
)t(S)1t(S
=
)t(A
)t(A)1t(A
It is proved that equality (3) follows from (2) for the case when S ( T ) takes two values. If S ( T ) takes
arbitrary finite number of states the existence of the unique equivalent martingale probabilities did not be
proved. If uniqueness of the risk neutral probabilities does not exist then the two state stock problem can
be considered as degenerating case.
In the latter construction investor buying a stochastic stock is effected by the market risk. This risk is
represented by the lower price S d at maturity in no arbitrage price setting. Therefore no arbitrage price
does not eliminate the market risk. The construction of the unique equivalent martingale probabilities can
be briefly formulated as following.
Statement. Given stock price with two states at T which is defined on original probability space there
exists a new measure π such that the stock expected rate of return is equal to risk free interest rate.
Actually risk neutral distribution construction does not represent a new pricing model of pricing. It is used
for Black-Scholes option pricing to reveal a connection between risk neutral Black Scholes pricing
underlying and real security used for the option pricing.
The binomial scheme can be applied for a corporate bond. Assume that risky bond at maturity is
defined by equality (1). Let n = 0, 1 and bond takes only two values1 and δ [ 0 , 1 ). Let us for
simplicity default may occur only at maturity. The spot price of the risky bond is assumed to be defined
by unique equivalent martingale probabilities (3). Then
E
{
)T(A
)ω;T,T(R
| F t } =
)t(A
)ω,T,t(R
(3)
4. 4
Here randomness of the R ( t , T , ω ) is stipulated by the randomness of the face value at maturity
T = t + 1. Thus date-t price of the risky bond is always a random variable. If buyer purchases bond for a
spot price R spot ( t , T ) then
δ ≤ A R spot ( t , T ) ≤ 1
If we calculate risk neutral probabilities we arrive at risk neutral distribution and 1 – .
Assume that the random pricing process R ( t , T ; ω ) is constructed for any t , t [ 0 , T ]. Let
us estimate recovery rate (RR) and correspondent default probability (DP). By definition it follows that
)T,t(B
)ω,T,t(R
1 = ( 1 – ) χ ( ω , D ) (4)
Hence, the relative value of the corporate bond with respect to its risk free counterpart at t can be
presented as
1 , for ω ≠ D
δ = δ ( ω ) = {
)T,t(B
)ω,T,t(R
, for ω D
Formulas
P { δ ( ω ) < u } = P {
)T,t(B
)ω,T,t(R
< u } , u [ 0 , 1 )
(5)
P ( D ) = 1 – P {
)T,t(B
)ω,T,t(R
= 1 }
represent date-t stochastic recovery rate δ ( ω ). For zero default scenario price of the corporate bond
coincides with the risk free bond price for any t ≤ T. The distribution of the random variable
δ ( ω ) = δ ( t , T ; ω ) defined by the formula (5) is the distribution of the recovery rate which depends
on parameters t and T. If the function B ( t , T ) is a function of the time to maturity T – t then risky bond
price would be represented by a random function of the variable T – t.
Remark. It is popular in applications to assume a reduction of the RR in which δ is a given number equal
to for example 30%, 40% or other known fraction. Such assumption is extended on a set of corporate
bonds. From formula (4) one can easy find that given δ the probability of default PD is equal
P ( D ) = ( 1 - ) – 1
E [
)T,t(B
)ω,T,t(R
1 ]
The known δ helps to overcome the problem of solving one equation with two unknowns RR and DP. By
taking expectation from both sides of the equation (1) we arrive at the primary equation of the reduced
form of default theory. The assumption of known recovery rate looks over simplified and correspondent
value of the default probability does not represent a sufficiently reliable estimate.
A popular statistical pointwise estimate of a random variable is the expected value. The pointwise
estimate should be replaced by a confidence interval when one takes into account deviation or volatility of
the random variable does not very small. Default at maturity T is by definition associated with the event
5. 5
D T = { ω : R ( T , T , ) < 1 }
Introduce a random function
θ ( t , T ; ω ) = 1 –
)T,t(B
)ω,T,t(R
(6)
This function represents estimate of the loss given default, LGD. Using function θ ( t , T ; ω ) we note that
P { δ < x } = P {
)T,t(B
)ω,T,t(R
< x } (6′)
Cumulative distribution function (6′) performs complete information about RR and PD. The use of a
nonrandom constant δ as a point estimate of the RR implies market risk. From buyer perspective it is
measured by probability that the realized at default recovery rate would be bellow than the chosen
constant δ. This risk represents additional losses for the buyer. At the same time buyer’s risk implies
profit for the bond seller. Assuming that recovery rate δ is an unknown nonrandom constant < δ > it
follows from equality (4) that
E θ ( t , T ; ω ) = ( 1 – < δ > ) P ( D )
E θ ² ( t , T ; ω ) = ( 1 – < δ > ) ² P ( D )
This is a system of two equations for two unknowns < δ >, P ( D ). Solving system for RR and PD we
arrive at the values
)ω;T,t(θE
])T,t(θE[
)D(P
)7(
)ω;T,t(θE
)ω;T,t(θE
1δ
2
2
2
Here RR and PD depend on parameters t , T.
Randomization. Our goal is a construction of the random price of the corporate bond. This
randomization is stipulated by the credit risk of the bond. Recall that only historical data R spot ( t , T ) at
each moment t is available. One can use historical data to construct stochastic prices function
R ( t , T , ). It is common practice to use close prices as a date-t asset price. Such reduction can be
accurate when the difference max - min of the date-t asset prices is sufficiently small. Formulas (4) – (7)
present a solution of the default problem when distribution R ( T , T , ) is known and default time
= T. In practice, distributions of the default time and prices of the bond at default are unknown.
Suppose for simplicity that default may occur only at maturity date T and distribution R ( T , T , ) is
unknown. Formula (6) uses stochastic price R ( t , T ; ω ) to derive RR and PD. Thus randomization
problem is a construction random variable R ( t , T ; ω ) by using historical data. It is clear that a single
6. 6
number the close price R close ( t , T ) at the date t does not sufficient to present distribution of the bond
price at t and construction recovery rate at maturity.
Usually one chooses a particular period and historical close prices over this period. Following
mathematical statistics historical prices are interpreted as independent observations in equal market
conditions. Such observations in statistics are known as a random sample taking from the total population.
A random sample is interpreted then as a set of independent equally distributed random variables. If
observations over a random variable cannot be drawn in equal conditions then sample is not a random and
it cannot adequately represent general population and statistical theory fails to make a good statistical
conclusion. Taking into account latter remark we note that historical data which is represented by the set
of close prices over extended period of time may not be interpreted as a set of equally distributed random
variables as far as market conditions change over the time. Hence, historical close price data may not
represent independent equally distributed observations and therefore the estimated parameters of a pricing
model can be bad. On the other hand if max - min spread of the historical prices over a single date does
not small enough stochastic effect of the price over a single date can not be ignored.
In order to present stochastic price R ( t , T , ) at t we first should specify the meaning of the date-t price
notion. Dealing with close prices historical data our statistical forecast of the date-T future price would
relate to close price at T. If deviation of the prices during a date can not be ignored then a single number
reduction (close price) can be oversimplified.
Let us interpret the price of the bond at t as a random variable having for example the uniform distribution
on interval [ d – , d + ] where
d – = R min ( t ) =
}t{u
min
R ( u , T ) , d + = R max ( t ) =
}t{u
max
R ( u , T )
Here { t } denotes the trading period associated with the date t. Date t can be a trading day, or a week or
other convenient period. Assumption regarding uniform distribution is taking for simplicity. One can also
apply other type of distribution though it does not guarantee a higher accuracy. Consider two possible
scenarios
( α ) 0 ≤ R min ( t ) ≤ R max ( t ) ≤ B ( t , T ) ≤ 1
( β ) R min ( t ) ≤ B ( t , T ) ≤ R max ( t ) ≤ 1
Case ( α )
Assume that occurs at maturity, i.e. P { = T } = 1 and that the random variable δ = δ ( ω ) is
uniformly distributed on the interval
[ B – 1
( t , T ) d –_( t ) , B – 1
( t , T ) d + ( t ) ]
Then the first two moments of the random variable which is defined in (6) are equal to
E θ ( t , T ; ω ) =
dd
1
d
d
[ 1 –
)T,t(B
u
] d u =
=
)dd(2
1
{ [ 1 –
)T,t(B
d
] 2
– [ 1 –
)T,t(B
d
] 2
}
E θ ² ( t , T ; ω ) =
dd
1
d
d
[ 1 –
)T,t(B
u
] 2
d u =
7. 7
=
)dd(3
1
{ [ 1 –
)T,t(B
d
] 3
– [ 1 –
)T,t(B
d
] 3
}
Then from (7) it follows that point estimate < δ > of the recovery rate and correspondent probability of
default, PD at t can be represented by the formulas
< δ > = 1 –
3
2
22
33
]
)T,t(B
d
1[]
)T,t(B
d
1[
]
)T,t(B
d
1[]
)T,t(B
d
1[
PD =
)dd(4
3
33
222
]
)T,t(B
d
1[]
)T,t(B
d
1[
}]
)T,t(B
d
1[]
)T,t(B
d
1[{
Case ( β )
In this case we supposed that d –_( t ) ≤ B ( t , T ) ≤ d + ( t ) ≤ 1 and P { = T } = 1. The date-t
market implied estimate of the recovery rate is a random variable
1 , with probability P { R ( t , T , ) [ B ( t , T ) , d + ( t ) ] }
δ ( ω ) = {
u , with density P { δ ( ω ) [ u , u + ∆ u ] } =
dd
uΔ
where u < 1. Then the values of the first and second moments of the θ ( t , T ; ω ) can be represented by
formulas
E θ ( t , T ; ω ) =
dd
1
)T,t(B
d
[ 1 –
)T,t(B
u
] d u =
)dd(2
]
)T,t(B
d
1[ 2-
E θ ² ( t , T ; ω ) =
dd
1
)T,t(B
d
[ 1 –
)T,t(B
u
] 2
d u =
)dd(3
]
)T,t(B
d
1[ 3-
Therefore
< δ > = 1 –
3
2
2
3
]
)T,t(B
d
1[
]
)T,t(B
d
1[
, PD =
)dd(4
3
3
4
]
)T,t(B
d
1[
]
)T,t(B
d
1[
8. 8
Default in Continuous Time. In continuous time corporate bond admits default at any future moment
prior to expiration date. Introduce a discrete time approximation of the continuous default moment
λ ( ω ) =
n
1k
t k χ { ( ω ) ( t k – 1 , t k ] } + T χ { > T }
where 0 = t 0 < … < t n = T is a λ-partition of the interval [ 0 , T ] , where λ = t k – t k – 1 does not
depend on k. Continuous real time cash flow CF A ( ω ) associated with zero coupon corporate bond from
bond buyer perspective can be then approximated then by a discrete time cash flow
CF A ( λ ) = – R ( t 0 , T ) χ { t = t 0 } +
n
1k
R ( , T , ) χ { ( t k – 1 , t k ] } χ { t = t k } +
+ χ { > T } χ { t = T } = – R ( t 0 , T ) χ { λ > 0 } χ { t = t 0 } + (8)
+
n
1k
R ( t k , T , ) χ { λ = t k } χ { t = t k } + χ { λ > T } χ { t = T }
Cash flow CF A ( λ ) corresponds to transactions:
*) A pays R ( t 0 , T ) to B at t 0 ,
**) A receives R ( , T ) from B at t k if ( ω ) ( t k – 1 , t k ] , k = 1, 2, … n ,
***) A receives $1 at T for the no default scenario { > T }.
For each market scenario ω only one term on the right hand side (8) does not equal to zero. On the right
hand side of the formula (8) the value of the bond at the moment λ = t k is defined as
R ( t k , T ) = δ k ( ω ) B ( t k , T ) , k = 1, 2, … n
Assume that corporate bonds with different maturities default simultaneously with equal recovery rate
with respect to risk free bonds with the same maturities, i.e.
δ k ( ω ) = δ k ( t 0 , ) , { λ = t k }
and δ k ( ω ) does not depend on T. Rates δ k ( ω ) , k = 1, 2, … n do not known at initiation. Thus
R ( t 0 , T , ω ) =
n
1k
B ( t 0 , t k ) δ k B ( t k , T ) χ { λ = t k } + B ( t 0 , T ) χ { λ > T } (9)
We make additional assumptions which will help us to reduce discrete in time valuation problem to the
set of default at maturity problems. Suppose that
*) if a corporate bond with maturity t k , t k < T does not exist then we will use market implied bond
**) random variables δ k ( ω ) = δ ( ω ) does not depend on k.
One can interpret later assumption as an assignment of a particular fixed credit rating of the bond. Bearing
in mind market implied forward discount rate we define the market implied bond value at t k
B ( t k , T ; t 0 ) by the equality
B ( t 0 , T ) = B ( t k , T ; t 0 ) B ( t 0 , t k )
Then (9) can be rewritten as
9. 9
R ( t 0 , T , ω ) = δ ( ω )
n
1k
B ( t 0 , t k ) B ( t k , T ; t 0 ) χ { λ = t k } + B ( t 0 , T ) χ { λ > T } =
(10)
= δ ( ω ) B ( t 0 , T ) χ { λ ≤ T } + B ( t 0 , T ) χ { λ > T }
The market implied forward discount rate B ( t k , T ; t 0 ) is a market estimate of the unknown at t 0
date- t k forward starting bond value B ( t k , T ). Unknown forward discount rate is assumed to be random
variable and therefore the replacement of the real random forward rate by its estimate implies market risk
for both counterparties. Dividing both sides of the latter equation by B ( t 0 , T ) we arrive at the equation
)T,t(B
)ω,T,t(R
1
0
0
= ( 1 – ) χ { λ ≤ T } (11)
Equation (11) is similar to (4). Consider case when recovery rate is random δ ( ω ). Similar to (5) we
conclude that
1 , for ω { λ > T }
δ = δ ( ω ) = {
)T,t(B
)ω,T,t(R
0
0
, for ω { λ ≤ T }
P { δ ( ω ) < u } = P {
)T,t(B
)ω,T,t(R
0
0
< u } , u [ 0 , 1 )
P ( λ ≤ T ) = 1 – P {
)T,t(B
)ω,T,t(R
0
0
= 1 }
Then taking into account formula (7) we arrive at formulas
< δ > = 1 –
]
)T,t(B
)ω,T,t(R
-1[E
]
)T,t(B
)ω,T,t(R
-1[E
0
0
2
0
0
, P ( λ ≤ T ) =
2
0
0
2
0
0
]
)T,t(B
)ω,T,t(R
-1[E
}]
)T,t(B
)ω,T,t(R
-1[E{
(12)
Formulas (12) represent deterministic market estimates of the recovery rate and correspondent default
probability over [ t 0 , T ]. These formulas do not depend on a particular distribution P ( = t k ).
Thus the initial continuous time default distribution we approximated by a discrete time default
distribution which produces nonrandom recovery rate.
The assumption δ k ( ω ) = δ ( ω ) can be too strong for a long term bonds. Let us consider more accurate
construction. Using market implied estimate of the forward discount rate it follows from (9) that
R ( t 0 , T , ω ) = B ( t 0 , t k ) δ k B ( t k , T ) B ( t 0 , t k ) δ k B ( t k , T , t 0 )
for ω { λ = t k }. Then
10. 10
B – 1
( t k , T , t 0 ) R ( t 0 , T , ω ) χ { λ = t k } = B ( t 0 , t k ) δ k χ { λ = t k } =
= R ( t 0 , t k , ω ) χ { λ = t k }
k = 1, 2, … n. Right hand side of the later equality represents market implied estimate of the risky bond
price with expiration date which might not exist on the market. This estimate is constructed similar to
market implied forward rate which is an important estimate of the trading fixed income forward type
contracts. Market implied corporate bond R ( t 0 , t k , ω ) with expiration date t k admits defaults at any of
the dates t j = 1, 2, … k. Then we note that
R ( t 0 , t k , ω ) χ { λ = t k } = B ( t 0 , t k – 1 ) R ( t k – 1 , t k , t 0 , ω ) χ { λ = t k }
where forward market implied corporate bond price is defined by equality
R ( t k – 1 , t k , t 0 , ω ) = B – 1
( t 0 , t k – 1 ) R ( t 0 , t k , ω )
Latter equality takes place for the scenarios ω { > t k – 1 }. Therefore
B ( t k – 1 , t k , t 0 ) , for { λ > t k }
R ( t k – 1 , t k , t 0 , ω ) = {
B – 1
( t 0 , t k – 1 ) R ( t 0 , t k , ω ) , for { λ = t k }
The latter equality can be rewritten in the form
θ k = B ( t k – 1 , t k , t 0 ) –
)t,t(B
)ω,t,t(R
1-k0
k0
= ( 1 – k ) χ ( λ = t k ) (13)
Hence, conditional probability of default and conditional distribution of the recovery rate are defined by
the formulas
P ( λ = t k | > t k – 1 ) = P {
)t,t(B
)ω,t,t(R
1-k0
k0
< B ( t k – 1 , t k , t 0 ) | > t k – 1 }
(14)
P { δ k ( ω ) < u | > t k – 1 } = P {
)t,t(B
)ω,t,t(R
1-k0
k0
< u | > t k – 1 } , u [ 0 , B ( t k – 1 , t k , t 0 ))
Let us reduce the default problem. Let recovery rate δ k be a nonrandom constant. Taking conditional
expectation E k – 1 { ∙ } = E { ∙ | > t k – 1 } from both sides (13) we arrive at the system
E k – 1 θ k = ( 1 – < δ k > ) P { = t k | > t k – 1 }
E k – 1 θ 2
k = ( 1 – < δ k > ) ² P { = t k | > t k – 1 }
Solving the system for RR and PD we arrive at the values
11. 11
(RR) k = < δ k > = 1 –
)ω;t,t(θE
)ω;t,t(θE
k0k1-k
k0
2
k1-k
(15)
(PD) k = P { = t k | > t k – 1 } =
)ω;t,t(θE
])ω;t,t(θE[
k0
2
k1-k
2
k0k1-k
As far as P ( > t 0 ) = 1 then
(PD) 1 = P { = t 1 | > t 0 } = P ( = t 1 )
Next for k = 2 we note that
(PD) 2 = P { = t 2 | > t 1 } =
)tτ(P
)tτtτ(P
1
12
=
)tτ(P
)tτ(P
1
2
where P ( > t 1 ) = 1 – P ( = t 1 ). Hence
P ( = t 2 ) = P { = t 2 | > t 1 } [ 1 – P ( = t 1 ) ]
Applying induction method let us assume that P ( = t k – 1 ), k = 2, 3, … is known. Then similar to the
case k = 2 we note that
P { = t k | > t k – 1 } =
)tτ(P1
)tτ(P
1-k
k
and therefore
P ( = t k ) = P { = t k | > t k – 1 } [ 1 – P ( = t k – 1 ) ] (16)
Formulas (16) and (15) define probability of default and deterministic recovery rate at date t k ,
k = 1, 2, … n which represent complete information regarding date-t corporate bond price.
Implementation of this pricing model interprets date-t as a day of trade rather than the price at the say
t = 10 am, 3.30 pm, or close price at day t. Therefore the date-t price can be thought as a random variable
R ( t , T , ω ). If an investor pays a particular deterministic price R ( t , T ) for the bond during trading
time at t then there exists market risk the price R ( t , T ) is higher than it is implied by the market. Such
market risk is valued by P { R ( t , T ) > R ( t , T , ω ) } and average of the overpayment is equal to
E [ R ( t , T ) – R ( t , T , ω ) ] χ { R ( t , T ) > R ( t , T , ω ) }
Mark-to-Market (MtM) bond pricing.
Bonds are single party risky instruments. Buyer, A pays bond price at initiation of the deal and expects to
receive the face value at expiration date. If the bond price remarkably falls at some point of time the seller
of the bond may declare default in obligation to deliver face value at the expiration date. In this case
issuer of the bond pays a settlement price of the bond. Note that the bond itself might not default at this
moment. By using MtM account bond buyer is guaranteed to receive higher price at the default settlement
than in the standard plain trading.
12. 12
Assume that counterparties of the bond trading agree to use MtM account to hedge the risk of default.
They also need to establish a benchmark risk rate to keep MtM account until expiration date. For US
corporate bonds such risk free benchmark can be represented by T-bond rate.
Let t 0 and t j + 1 = t j + 1, j = 0, 1, … n, t n = T denote initiation date and MtM adjustment moments
correspondingly. Let us present a sketch of the MtM account transactions. At initiation date t 0 bond
buyer pays R ( t 0 , T ), R ( t 0 , T ) < B ( t 0 , T ) and receives corporate bond that promises $1 at T.
At the date t 0 the value of the MtM account is set to be equal
MtM ( t 0 ) = R ( t 0 , T ) + ad ( t 0 ) = B ( t 0 , T )
where R ( t 0 , T ) is paid by A and the value of adjustment at t 0 is
ad ( t 0 ) = B ( t 0 , T ) – R ( t 0 , T )
is paid by B. The MtM account is under bond seller supervision. Hence initial payment made by bond
buyer does not go directly to bond seller. It may or may not go to bond seller, B later at the settlement
moment. Counterparty B as owner of MtM account pays interest i MtM ( t 0 , t 1 ) for keeping money during
[ t 0 , t 1 ) period. Hence just before the new prices comes to the market at the date t 1 the value of the MtM
account is equal to
MtM ( t 1 – 0 ) = B ( t 0 , T ) ( 1 + i MtM ( t 0 , t 1 ) )
where = t j + 1 – t j does not depend on j and interest rate i MtM ( t 0 , t 1 ) may or may not coincide with
the risk free bond rate. At thee date t 1 the value of the MtM account is equal to
MtM ( t 1 ) = B ( t 1 , T ) = MtM ( t 1 – 0 ) + ad ( t 1 )
where
ad ( t 1 ) = B ( t 1 , T ) – MtM ( t 1 – 0 )
is date-t 1 adjustment of the MtM account. If ad ( t 1 ) > 0 then B deposits ad ( t 1 ) and if ad ( t 1 ) < 0 then
B withdraws this sum from MtM account. In general if MtM ( t j ) is known then
MtM ( t j + 1 – 0 ) = B ( t j , T ) ( 1 + i MtM ( t j , t j + 1 ) ) ,
MtM ( t j + 1 ) = B ( t j + 1 , T ) = MtM ( t j + 1 – 0 ) + ad ( t j + 1 ) ,
ad ( t j + 1 ) = B ( t j + 1 , T ) – MtM ( t j + 1 – 0 )
j = 0, 1, … n. The value of ad ( t j + 1 ) is added by B to MtM account when it is positive and it is
withdrawn from MtM account if it is negative. Define bond seller default time . For writing simplicity
low index bellow will be omitted. The cash flows to MtM account can be represented by the formula
CF MtM = MtM ( t 0 ) χ ( t = t 0 ) +
n
1k
χ { = t k } {
1-k
1j
ad ( t j ) χ ( t = t j ) +
+ δ k B ( t k , T ) χ ( t = t k ) } + χ { > T } [ – 1 + MtM ( T – 0 ) ] χ ( t = T )
13. 13
From bond buyer and seller perspective the values of MtM transactions are different. They can be
represented by the following formulas correspondingly
CF A = – R ( t 0 , T ) χ ( t = t 0 ) +
n
1k
χ { = t k } [ MtM ( t k – 0 ) χ ( t = t k ) + δ k B ( t k , T ) ] +
+ χ { > T } 1 χ ( t = T ) = – R ( t 0 , T ) χ ( t = t 0 ) +
n
1k
χ ( = t k ) [ B ( t k – 1 , T )
[ 1 + i MtM ( t k – 1 , t k ) ] + δ k B ( t k , T ) ] χ ( t = t k ) + χ ( > T ) χ ( t = T ) ,
CF B = – ad ( t 0 ) χ ( t = t 0 ) +
+
n
1k
χ ( = t k ) {
1-k
1j
ad ( t j ) χ ( t = t j ) – MtM ( t k – 0 ) χ ( t = t k ) } +
+ χ ( > T ) [ – 1 + B ( t n – 1 , T ) ( 1 + i MtM ( t n – 1 , t n ) ) ] χ ( t = T) =
= χ ( > T ) { B ( t 0 , T ) χ ( t = t 0 ) + B ( t n – 1 , T ) ( 1 + i MtM ( t n – 1 , t n ) ] χ ( t = T ) }
Note that MtM account applies interest rate i MtM which may be lower than interest rate implied by the
risk free bond prior to counterparty default moment or bond expiration date which one comes first.
Forward interest rate i MtM and forward risk free discount rate B ( t j , T ) are unknown at initiation date.
We assume that these forward rates are random processes. It is market rule to use market implied forward
estimates for pricing of these contracts. The use of market estimates versus stochastic rates implies market
risk. Buyer and seller expected value of the spot prices can be represented by the EPV of the
correspondent cash flows. Applying formulas (12) we receive formulas
EPV CFA ( ω ) = – R ( t 0 , T ) +
n
1k
P ( = t k ) { B ( t k – 1 , T , t 0 ) [ 1 + i MtM ( t k – 1 , t k , t 0 ) ] +
+ < δ k > B ( t k , T , t 0 ) } B ( t k , T , t 0 ) + P ( > T ) B ( t 0 , T )
EPV CFB = P ( > T ) { B ( t 0 , T ) + B ( t n – 1 , T , t 0 ) [ 1 + i MtM ( t n – 1 , t n , t 0 ) ] B ( t 0 , T ) }
where default distribution P ( = t k ) and recovery rates < δ k > are defined in (15) and (16). E PV CF A
and E PV CF B are associated with bid and ask bond prices in MtM trading format. The gap between
bid and ask prices is a measure of illiquidity of the corporate bonds on MtM bond trading.