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 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.
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.
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.
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.
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.
. 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.
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.
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.
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.
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.
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.
. 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.
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 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 reflect balance between profit-loss expectations of the market participants
1) The Savage-Dickey ratio provides a specific representation of the Bayes factor by using only the posterior distribution under the alternative hypothesis at the null value.
2) Verdinelli and Wasserman (1995) proposed a generalization of the Savage-Dickey ratio that avoids a "void" constraint in the prior.
3) The paper demonstrates that the Savage-Dickey ratio is a generic representation of the Bayes factor that relies on specific measure-theoretic versions of the densities, rather than imposing a mathematically void constraint on the prior. It clarifies the measure-theoretic foundations of both the Savage-Dickey ratio and its generalization by
The document discusses pricing the Margrabe option using Monte Carlo simulation and an explicit closed-form solution. It begins by defining the Margrabe option and explaining its use. It then presents Margrabe's closed-form solution, which prices the option as a European call using a change of numeraire approach. Next, it analyzes the option's sensitivity to various parameters. Finally, it outlines different option pricing methods and focuses on Monte Carlo simulation and the change of numeraire approach.
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 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.
This slideshow teaches you some basic ideas about signals and types of signals. Some basic types are discussed briefly along with images for your better understanding.
Lecture slides for Auction Theory (for graduate students) at Osaka University in 2016, 2nd semester. Complementary materials and related information can be obtained from the course website below:
https://sites.google.com/site/yosukeyasuda2/home/lecture/auction16
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 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.
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.
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.
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
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.
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.
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.
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 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 reflect balance between profit-loss expectations of the market participants
1) The Savage-Dickey ratio provides a specific representation of the Bayes factor by using only the posterior distribution under the alternative hypothesis at the null value.
2) Verdinelli and Wasserman (1995) proposed a generalization of the Savage-Dickey ratio that avoids a "void" constraint in the prior.
3) The paper demonstrates that the Savage-Dickey ratio is a generic representation of the Bayes factor that relies on specific measure-theoretic versions of the densities, rather than imposing a mathematically void constraint on the prior. It clarifies the measure-theoretic foundations of both the Savage-Dickey ratio and its generalization by
The document discusses pricing the Margrabe option using Monte Carlo simulation and an explicit closed-form solution. It begins by defining the Margrabe option and explaining its use. It then presents Margrabe's closed-form solution, which prices the option as a European call using a change of numeraire approach. Next, it analyzes the option's sensitivity to various parameters. Finally, it outlines different option pricing methods and focuses on Monte Carlo simulation and the change of numeraire approach.
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 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.
This slideshow teaches you some basic ideas about signals and types of signals. Some basic types are discussed briefly along with images for your better understanding.
Lecture slides for Auction Theory (for graduate students) at Osaka University in 2016, 2nd semester. Complementary materials and related information can be obtained from the course website below:
https://sites.google.com/site/yosukeyasuda2/home/lecture/auction16
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 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.
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.
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.
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
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.
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.
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.
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.
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 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.
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.
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.
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.
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 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.
The document discusses a stochastic volatility model that incorporates jumps in volatility and the possibility of default. It describes the dynamics of the model and how it can be used to price volatility and credit derivatives. Analytical and numerical methods are presented for solving the pricing problem. As an example application, the model is fit to data on General Motors to analyze the implications.
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.
This document describes an analytic pricing methodology for Asian cap/floor options on Constant Maturity Swap (CMS) rates with fixed strike rates. The methodology prices these derivatives using a lognormal model for forward swap rates, with adjustments to account for valuing multiple CMS rates under a common forward measure. The methodology can price both vanilla Asian cap/floor options (with fixed averaging periods) as well as plain vanilla cap/floor options. The price of an Asian cap/floor is calculated as the sum of the prices of its constituent caplets/floorlets, where each caplet/floorlet price is determined using an adjusted Black formula that approximates the distribution of the average CMS rate.
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
The document discusses pricing interest rate derivatives using the one factor Hull-White short rate model. It begins with an introduction to short rate models and the Hull-White model specifically. It describes how the Hull-White model can be calibrated to market prices by relating its parameter θ to the market term structure. The document then discusses implementing the Hull-White model using trinomial trees and pricing constant maturity swaps.
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1. 1
FIXED RATES MODELING.
Ilya Gikhman
6077 Ivy Woods Court
Mason OH 45040 USA
Ph. 513-573-9348
Email: ilyagikhman@yahoo.com
JEL : G12, G13
Key words. Forward rate agreement, Market risk, Stochastic Libor model, Interest Rate Swap, Liquidity
of Corporate Bonds, Stochastic default intensity.
Abstract. This paper focuses on the concept of a discount rate. In [1] one expressed some concerns
regarding the popular models of the randomization of the discount rates. This paper proposes a new
approach to construction of variable deterministic and stochastic interest rates. This approach is based on
the randomization of the forward rate consept. In [2] we introduced a synthetically construction price of
the contracts associated with LIBOR. From our point of view the standard modeling of the LIBOR are
rather empirical than formal.
I. Basic notations and definitions. Denote B ( t , T ) , 0 ≤ t ≤ T the price of zero
coupon default free bond price at date t with expiration at date T and B ( T , T ) = 1. The simple interest
i s rate and discount rate i d are defined as
B ( t , T ) = [ 1 + i s ( t , T ) ( T – t ) ] – 1
= 1 - i d ( t , T ) ( T – t ) (1)
Here T – t is expressed in appropriate 365 or 360 day year format. Continuous time model of the bond
price is governed by the equation
d B ( t , T ) = r ( t , T ) B ( t , T ) d t (1′)
Here the function r ( t , T ) > 0 in is called annual interest rate. The value of a coupon bond at t which
pays coupon c at the moments t 1 < t 2 < … < t n = T is equal to
2. 2
B c ( t , T ) =
n
1j
c B ( t , t j ) + F B ( t , T )
where F is a face value of the bond. It is useful to define a path dependent financial contract with the help
of cash flow. For example the value of the coupon bond B c ( t , T ) can be interpreted as the present
value at t of the cash flow
CF =
n
1j
c χ ( t = t j ) + F χ ( t = T )
where χ ( A ) is the indicator of the event A. It is equal to 1 when A is true and 0 otherwise. Thus,
B c ( t , T ) = PV t { CF }.
II. Forward rate agreement, FRA. Though the FRA pricing is well known we pay more
attention to some details that might occasionally be missed. FRA is a two party OTC contract for a future
transaction. The value of the transaction is a notional principal multiplied by the difference between the
realized reference rate and its estimate called implied forward rate.
Let t denote initiation date and the fixed and realized and implied forward rates are assigned for a future
period [ T , T + H ], H > 0. A FRA contract can be specified as follows. Let
t 0 < t spot < t fixing < t settle < t mature be a set of the time moments. The date t 0 is called trade or deal
date. On this date the FRA contract is specified as :
the spot date t spot is usually equal t 0 + 2 is the beginning and t settle = T is the end of the m’s period,
i.e.
t settle - t spot = m (months)
Notional principal N,
FRA fixed rate, FRA ( T , T + H ; t 0 )
period specification m × H,
and the FRA time moments.
The date t fixing = T - 2 is usually two business days prior t settle = T is the date at which value of the
floating (reference) rate over the period [ T , T + H ] , t mature = T + H. At the settlement date T the
settlement (netted) sum
N [ L ( T , T + H ) – FRA ( T , T + H ; t 0 ) ] H
If the latter value is positive then the FRA seller pays it to the FRA buyer (FRA holder). If the value (2) is
negative then the FRA buyer pays the value to the FRA seller.
3. 3
The FRA pricing problem is a determination of the fixed rate FRA ( T , T + H ; t 0 ). It is common
practice to study a simplified scheme of this pricing problem. In such simplification one assumes that
t = t 0 = t spot and T = t fixing = t settle . Let us study the valuation problem in such simplified setting. A
FRA contract can be interpreted as a forward loan over [ T , T + H ]. At the date T + H a borrower of
the fund should pay the interest specified by the reference rate that usually LIBOR or other similar rate.
The rate L ( T , T + H ) is known as T and therefore we can admit that FRA payoff is scheduled on the
date T. A variation of the FRA when the payoff is scheduled for T + H. Therefore, the date-T or the
date-( T + H ) value of the contract is
V ( t , T , T + H ) = N [ L ( T , T + H ) – FRA ( T , T + H ; t ) ] H (2)
The rate FRA ( T , T + H ; t ) should be determined at t while the rate L ( T , T + H ) is unknown at
this date. It is the market practice to approximate unknown value L ( T , T + H ) by date-t implied
forward rate l ( t , T + H ; t ) over the period [ T , T + H ]. The implied forward rate l ( t , T + H ; t ) is
defined as
l ( T , T + H ; t ) = ]1
)tT()T,t(L1
)tHT()HT,t(L1
[
H
1
Here L ( t , T ) denotes date-t spot reference interest rate with expiration at T. Applying implied forward
rate l ( T , T + H ; t ) as an approximation of the unknown L ( T , T + H ) we replace the real
transaction (2) by its implied approximation
v ( t , T , T + H ) = N [ l ( T , T + H ; t ) – fra ( T , T + H ; t ) ] H (2′)
Here, fra ( T , T + H ; t ) denotes ‘no-arbitrage’ solution of the reduced FRA pricing problem
corresponding to еру payoff (2′). As far as the value of the implied form contract at the settlement for
either buyer or seller should be equal to zero then
fra ( T , T + H ; t ) = l ( T , T + H ; t )
Then the buyer price at t is the discounted payment of the contract at T. Hence,
N L ( t , T ) l ( T , T + H ; t ) H
If the settlement date is the date T + H then in the latter expression discount factor L ( t , T ) should be
replaced by the L ( t , T + H ). Note that no-arbitrage solution of the reduced FRA with the payoff (2′)
does not coincide with the real world transaction defined by the actual payoff (2). This inequality implies
the market risk of the FRA contract. The value of the risk stipulated by the value
δ l
def
L ( T , T + H ) - l ( T , T + H ; t ) ≠ 0
effective at the settlement date T. The market risk from the buyer perspective is associated with the
market scenarios for which δ l < 0 while the scenarios for which δ l > 0 specify the seller risk.
Let us now consider randomization of the FRA pricing problem. There are two functions L ( T , T + H )
and l ( T , T + H ; t ) that are available for randomization. Suppose first that
4. 4
L ( T , T + H , ) = L ( t , t + H ) +
T
t
( s ) L ( s , s + H ) d s +
(3)
+
T
t
( s ) L ( s , s + H ) d w ( s )
where coefficients , are deterministic or random functions, which satisfy the standard conditions that
ensure the existence and the uniqueness of the solution of the Ito equation (3). The solution depends on
the parameter H. Taking the limit when H tends to 0 we arrive at the instantaneous forward rate
L ( T , ) = L ( T , T + 0 , ) = P.
0H
lim L ( T , T + H , )
which is the solution of the equation
L ( T , ) = L ( t ) +
T
t
( s ) L ( s , ) d s +
T
t
( s ) L ( s , ) d w ( s )
There exists another way to present a model for the future rate L ( T , T + H , ). It is based on the
implied forward rate. At the fixed date t the function l ( T , T + H ; t ) is known. For the fixed t , T , and
H let us consider a random function l ( T , T + H ; u ) of the variable u , which represents the value of
the implied forward rate over the period [ T , T + H ] at a future date u [ t , T ]. Suppose that
l ( T , T + H ; u ) = l ( T , T + H ; t ) +
u
t
( v ) l ( T , T + H ; v ) d v +
(3′)
+
u
t
λ ( v ) l ( T , T + H ; v ) d w ( v )
In (3′) put u = T and bearing in mind that l ( T , T + H ; T ) = L ( T , T + H , ) we arrive at the
equation that defines unknown rate L ( T , T + H , ). Given randomization of the future rate
L ( T , T + H ) in the form (3) we enable to present market risk in the form P { δ l > 0 } , P { δ l < 0 }
for the FRA seller and buyer correspondingly. One can define the cumulative distribution to calculate
primary market risk characteristics such as average profit and losses as well as its standard deviations.
Remark 1. Let us first highlight the difference between the benchmark and our approaches to FRA
valuation. Let us recall the benchmark FRA pricing following [3, p.87].
5. 5
“ FRA can be valued if we:
1. Calculate the payoff on the assumption that forward rates are realized ( that is, on the assumption
that R M = R F ).
2. Discount this payoff at the risk-free rate. “
Applying our notations we get the following R F = l ( T 1 , T 2 ; t ) , R M = L ( T 1 , T 2 ). Benchmark
approach to FRA valuation makes sense in deterministic problem setting and it is ignoring the fact that
R M ≠ R F in stochastic case. In stochastic setting we interpret the date-t known rate
R F = l ( T 1 , T 2 ; t ) as an approximation of the unknown at t rate R M = L ( T 1 , T 2 ) . Let { A }
denote indicator of the event A. Then the difference
[ l ( T 1 , T 2 ; t ) - L ( T 1 , T 2 ) ] { l ( T 1 , T 2 ; t ) > L ( T 1 , T 2 ) }
is the market risk of the FRA buyer while
[ l ( T 1 , T 2 ; t ) - L ( T 1 , T 2 ) ] { l ( T 1 , T 2 ; t ) < L ( T 1 , T 2 ) }
specifies market risk of the FRA seller. Note that
1Tt
lim l ( T 1 , T 2 ; t ) = L ( T 1 , T 2 )
Nevertheless, it is easy to verify in stochastic setting by using real data that implied forward rate does not
equal to the value of the correspondent future rate, i.e. l ( T 1 , T 2 ; t ) ≠ L ( T 1 , T 2 ). The rate
L ( T 1 , T 2 ) is the real rate known at T 1 while the rate l ( T 1 , T 2 ; t ) is an approximation or a statistical
estimate of this rate. In stochastic setting the first step of the valuation is incorrect as far as
R F = l ( T 1 , T 2 ; t ) is a known constant at t while the rate R M = L ( T 1 , T 2 ) is unknown at t and
L ( T 1 , T 2 ) is assumed to be a random variable. Therefore, in general P { R M = R F } = 0. It is also
important to note that in stochastic theory the probability distribution of the random variable
L ( T 1 , T 2 ; ) is assumed to be known at t. This assumption is crucial for stochastic pricing. It leads to
new comprehension of the price. The revision of the price notion comes from the fact that a continuous
distribution prescribes probability 0 to any particular number regardless whether this number is ‘arbitrage
free’ or other meaningful spot price. Complete price of the contract consists from two components. These
are a specific spot price l ( T 1 , T 2 ; t ) along with the probability
P { L ( T 1 , T 2 ; ) < l ( T 1 , T 2 ; t ) }
The last probability represents a buyer's market risk while the probability of the inverse inequality
represents seller’s risk. Primary risk characteristic from the buyer perspective is mean of the losses
M b = E L ( T 1 , T 2 ; ) χ { L ( T 1 , T 2 ; ) < l ( T 1 , T 2 ; t ) } =
=
)t;T,T(
0
21l
y P{ L ( T 1 , T 2 ; ) d y }
6. 6
It is easy to present formula for the corresponding standard deviation of the losses
E [ L ( T 1 , T 2 ; ) χ { L ( T 1 , T 2 ; ) < l ( T 1 , T 2 ; t ) } - M b ] 2
The latter remark on pricing in stochastic setting is usually ignored as far as market risk is always omitted
from derivatives pricing concept.
Remark 2. Randomization problem was studied also in the HJM and LMM models. In [1] we discussed
the randomization problem of the instantaneous forward interest rates, which is by definition the rate
f ( t , T ) =
0H
lim l ( t ; T , T + H ; 0 ). The HJM model is the basis of the LIBOR market model presented
a randomization of the instantaneous implied forward rate in the form
f ( t , T , ) = f ( 0 , T ) +
t
0
α ( u , T ) d u +
t
0
ζ ( u , T ) d w ( u )
where α ( u , T ) , ζ ( u , T ) are known functions. Putting in the latter equation T ↓ t we arrive at the
money market rate r ( t ) = f ( t , t ), which satisfies the equation
r ( t ) = f ( 0 , t ) +
t
0
α ( u , t ) d u +
t
0
ζ ( u , t ) d w ( u )
Note that this formula does not consistent with the money market rate definition. Indeed, the bond prices
are only observable data that define r ( t ). Applying latter expression for r ( t ) to the bond definition it
follows that
B ( t , T ) = exp –
T
t
r ( u ) d u =
= exp –
T
t
{ f ( 0 , u ) +
u
0
α ( v , u ) d v +
u
0
ζ ( v , u ) d w ( v ) } du
This formula suggests that the value of the bond at any moment t depends on all observations of the bond
prices over [ 0 , T ]. This model assumption does not look accurate. Indeed, from the bond definition we
know that
B ( t , T ) = 1 –
T
t
r ( u ) B ( u , T ) d u
r ( t ) =
)t(B
)T,t(dB
Therefore, the value of the bond and the instantaneous interest rate at some date t depend on the values of
the bond on ( t , T ]. More details on instantaneous implied forward rate randomization are given in [1]. In
7. 7
particular, it was suggested that the use of the backward Ito equations for randomization problem in does
not lead to the pointed above contradictions.
III. Interest Rate Swap (IRS) valuation. In this section we investigate market risk which comes
up with a construction of the implied market price of the swap. An IRS is a cash exchange contract in
which a floating rate payments are exchanged for the fixed one at a predetermined set of dates. The
current swap price is also known as ‘spread’ is the value of the fixed rate which makes the present value
of the future fixed payments be equal to the expectation of the present value of the floating rate payments.
For calculation of the expectation of the present value of the floating rate payments the real future rates
interpreted in a theory as random variables are replaced by its expected by the market estimates. These
replacements imply market risk similar to discussed above FRA contracts.
Generic IRS is a financial contract to exchange a fixed rate q for a floating rate L based on a notional
principal. Let t = t 0 < t 1 < t 2 < … < t n = T be a known sequence of dates and N notional principal.
The buyer of a swap pays fixed rate payments N q to a swap seller and receives the amount
N L ( t j – 1 , t j ) at the reset dates t j , j = 1, 2, … , n. As far as fixed and floating transactions are
scheduled on the same reset dates the only netted values of transactions take place. The floating rate L is
one of the basic market rates like a Treasury rate, a LIBOR or other similar rates. Rates L ( t j – 1 , t j ) are
known at the dates t j – 1 , j = 1, 2, … n and therefore the only rate that is known at t is the rate
L ( t 0 , t 1 ). Real world cash flow from the swap buyer perspective A to the swap seller B can be
represented as
CF A → B ( t , T ) =
n
1j
[ L ( t j – 1 , t j ) - c ] χ ( t = t j ) (4)
Positive terms on the right hand side in (4) correspond to payments B to A while negative terms in (4)
signify payments makes by A to B. The swap valuation problem is determination a fixed rate q that
promises to counterparties A and B equality of their positions in the deal. The equality of the A and B
positions in the deal should be specified. With the deterministic setting we are dealing with known or
implied data the market risk is ignored and the problem has a unique solution. The fixed rate c is a
solution of the equality present values of the two legs of the swap . In stochastic setting we need to take
into account the market risk characteristics. There are a few types of the dollar denominated basic interest
rates. Two primary rates are Treasury and LIBOR rates. They are intensively used by the markets.
Treasury rates are specified by the T-bonds. It is a common practice to use one of these rates as discount
factor. In this case we can interpret bond price B ( t , T ) at t as a discount factor from date T to the date t.
The LIBOR rate L ( t , T ) can be used to form the discount factor D ( t , T )
D ( t , T ) = [ 1 + L ( t , T ) ( T – t ) ] – 1
which can also be interpreted as a value of a virtual bond price at t that promises $1 at T. The forward rate
implied by the discount factor D can be defined as
l ( t j – 1 , t j ; t ) = ]1
)t,t(D
)t,t(D
[
t
1
j
1-j
j
8. 8
from which it follows that
D ( t , t j – 1 ) - D ( t , t j ) = D ( t , t j ) l ( t j – 1 , t j ; t ) Δ t j
Hence, the present value of the floating leg is equal to
N
n
1j
[ D ( t , t j – 1 ) - D ( t , t j ) ] = N [ 1 - D ( t , T ) ] = N
n
1j
D ( t , t j ) l ( t j – 1 , t j ; t ) Δ t j
It is common practice to interpret equality of fixed and floating rate cash flows as equality its present
values, PVs. The PV of the fixed rate cash flow is
n
1j
c D ( t , t j ) Δ t j
In order to present PV of the floating leg we need first replace unknown at t rates L ( t j – 1 , t j ) on its
date-t estimates l ( t j – 1 , t j ; t ). Then the PV of the ‘implied’ floating rate cash flow is equal to
N
n
1j
D ( t , t j ) l ( t j – 1 , t j ; t ) Δ t j = N [ 1 - D ( t , T ) ]
Therefore, the date-t value V ( t , T ) of the swap that is by definition the PV of the difference between
floating and fixed legs is
V ( t , T ) = N [ 1 - D ( t , T ) -
n
1j
c D ( t , t j ) Δ t j ] (5)
Swap rate c is the solution of the equation V ( t , T ) = 0. Hence,
c =
jj
n
1j
t)t,t(D
)T,t(D-1
(6)
The value c = c ( t , T , n ) is called the swap spread. Equality (6) we can resolve with respect to the
discount rate D ( t , T ) = D ( t , t n ). Indeed, from (5) it follows that
1 - D ( t , t n ) -
n
1j
c D ( t , t j ) Δ t j = 0
Solving this equation for D ( t , t n ) we arrive at recursive formula
D ( t , t n ) =
n
jj
1-n
1j
tΔc1
tΔ)t,t(Dc-1
9. 9
Remark 3. Note that the formula (6) presents implied swap rate, which is known at t number. Similarly to
FRA valuation the market risk of the IRS stipulated by the stochastic rates L ( t j – 1 , t j ) can be
represented by the cash flow
n
1j
N [ L ( t j – 1 , t j , ) - l ( t j – 1 , t j ; t ) ] Δ t j { t = t j }
Expressions in brackets can be either signs positive or negative.
Assume for example, that rate L ( T , T + H , ) is governed by the equation (3). The solution of
the equation can be written in the form
L ( T , T + H , ) = L ( t , t + H ) ρ ( t , T )
where
ρ ( t , T , ) = exp {
T
t
[ ( s ) -
2
)s(σ
2
d s ] +
T
t
( s ) d w ( s ) }
Assume that Δ t j = Δ t . Then PV of the real floating leg cash flow is
N
n
1j
D ( t , t j ) L ( t j – 1 , t j ) Δ t j = N L ( t , t + Δ t )
n
1j
D ( t , t j ) ρ ( t , t j – 1 , ) Δ t
The real world value of the swap can be written as
V ( t , T , ) = N [ L ( t , t + Δ t )
n
1j
D ( t , t j ) ρ ( t , t j – 1 , ) Δ t -
n
1j
Q D ( t , t j ) Δ t ] (5′)
Therefore, the market realized swap spread depends on a market scenario and equal to
Q ( ) =
tΔ)t,t(D
tΔ)t,t(ρ)t,t(D-1
j
n
1j
1jj
n
1j
(6′)
The market risk of the swap from the buyer perspective is associated with the negative terms in (4).
Hence, swap buyer's market risk is
P { Q ( ) < c } = P {
n
1j )T,t(D
)t,t(D j
ρ ( t , t j – 1 , ) Δ t > 1 }
It represents the value of the chance that the buyer of the swap pays the higher rate than it is implied by
the market. Note that right hand side of the latter equality can be approximated by a probability, i.e.
10. 10
P {
n
1j )T,t(D
)t,t(D j
ρ ( t , t j – 1 , ) Δ t > 1 } P {
T
0 )T,t(D
)u,t(D
ρ ( t , u , ) d u > 1 }
Remark 4. Note that presented constructions have been dealt with original probability space while it is
common practice to use so-called risk-neutral space. The probability measure on risk-neutral space is
chosen such that it should replace real return of a security on risk free rate. This construction takes its
origin from Black-Scholes’ concept of the option price. It might make sense to bring a short critical
remark on Black Scholes (BS) price definition. In more details this concept is discussed in [3,4].
Let C ( t , S ( t ) ) denote BS’s call option price of a European call option with strike price K and maturity
at T. It was assumed that underlying stock price is governed by stochastic differential equation (SDE)
d S ( t ) = S ( t ) d t + ζ S ( t ) d w ( t )
with constant coefficients , ζ. Black and Schloes proposed a portfolio П with one short call option and a
number of the underlying stock shares. The value of the portfolio at t is equal to
П ( t , S ( t ) ) = - C ( t , S ( t ) ) +
S
))t(S,t(C
S ( t ) (П1)
They also supposed that infinitesimal change in value of the portfolio is given by the formula
d П ( t , S ( t ) ) = - d C ( t , S ( t ) ) +
S
))t(S,t(C
d S ( t ) (П2)
Assuming that C ( t , S ) is a smooth function and applying Ito formula to the option price we arrive at the
formula
d C ( t , S ( t ) ) = [
t
))t(S,t(C
+ 2
22
S
))t(S,t(C
2
)t(Sσ
] d t +
S
))t(S,t(C
d S ( t )
It is not difficult to note that
d П ( t , S ( t ) ) = [
t
))t(S,t(C
+ 2
22
S
))t(S,t(C
2
)t(Sσ
] d t (П3)
The right hand side of the formula (П3) does not contain risk term associated with the ‘white’ noise factor
d w ( t ). If a portfolio’s infinitesimal change in value is risk free then the only way to avoid arbitrage
opportunity is that its rate of return should be equal to risk free rate r. Therefore
d П ( t , S ( t ) ) = r П ( t , S ( t ) ) d t
Bearing in mind formula (П3) and (П1) we arrive at the Black Scholes equation
11. 11
t
))t(S,t(C
+ 2
22
S
))t(S,t(C
2
)t(Sσ
= r [ - C ( t , S ( t ) ) +
S
))t(S,t(C
S ( t ) ] (BSE)
Equation is defined in the domain ( t , S ) [ 0 , T ) ( 0 , + ∞ ) with boundary condition
C ( T , S ) = max { S - K , 0 }
Construction and correspondent assumption on portfolio’s dynamics of the portfolio in the form (П1), (П2)
is theory fundamental for derivatives pricing in no arbitrage setting.
The easiest way to convince one oneself about the Black Scholes’ error is to use explicit form of the
solution of the (BSE) and construct portfolio as it was presented in the formula (П1). Then with help of
simple calculus one can easy verify that the assumption that the infinitesimal change in value of the
portfolio follows (П2) is incorrect. The nonzero term S ( t ) d
S
))t(S,t(C
was lost on the right hand
side (П2). As an implication of the error follows the fact that portfolio in the form (П1) does not provide
‘perfect’ dynamic hedge of the option price which also serves as another version of the option price
definition. Also probabilistic interpretation of the (BSE) has lead derivatives pricing theory in replacing
real word, which is defined on original probability space by so-called risk-neutral world. The latter used
Girsanov change measure technique to replace real security return on risk free counterpart. In case when
real and risk free return are close in average over [ 0 , T ] the numeric calculations can bring close results.
Nevertheless, even in this case the replacing the real return on the risk free is formally incorrect.
The error of the Black Scholes pricing approach in stochastic setting is that they did not pay attention to
the market risk of the spot price. The market risk is an attribute of any particular model of the spot price
of a derivative instrument.
IV. Liquidity. The liquidity problem in a simple interpretation comes up as a premium problem
which one should to add to a single price model associated with the perfect liquidity price. Latter is
associated with a single price asset models, which implies equality, bid and ask prices. In theory the
liquidity premium exists when pricing uses the bid-ask format. For the last years the liquidity problem
attracted theoretical and practical attention . A measure of the liquidity premium is its bid-ask spread.
Hence, talking about assets pricing in a single pricing format one inexplicitly deals with the perfect
liquidity case. The perfect liquidity is interpreted as a trade with no delay, which follows from the
equality of the bid and ask prices at any moment time t. Indeed, in such a model a buyer who wishes to
buy an asset at t pays at this moment its price while a seller receives exactly the same amount at this
moment. There is no loss at any time for selling and buying transactions. Thus, the liquidity problem can
be interpreted as an adjustment to a single price representing by the perfect liquidity model in order to
take into account asset illiquidity. It is clear that illiquidity stems from the fact that ask price always
higher than its bid price. It is a popular practice to use the middle price of the bid-ask spread to present
premium in perfect liquidity approximation. In this case one should make an adjustment adding or
subtracting the half of the bid-ask spread in final formulas. In such interpretation the effect of illiquidity is
a half of the bid-ask spread for the buyer and seller adjustments for the price. This adjustment specifies
the liquidity premium.
12. 12
There are two primary types of liquidity. These are trading and funding liquidity. Trading
liquidity shows the ease of the asset trading while funding liquidity is a characteristic of access to
funding. Consider the trading liquidity problem. Let us define the value of the bond illiquidity at a
moment t by the spread
λ 1 , B ( t , T ) = B ask ( t , T ) - B bid ( t , T )
Indicator λ 1 , B ( t , T ) represents liquidity at t of the bond that expired at T. Obviously that similar
instruments with fixed t and different expiration dates or with different moments t with equal T would
have different liquidity. Broadly speaking the value of illiquidity represent losses on immediate buy-sell
transactions. If { t } represents a trading day period then liquidity can be interpreted as a random variable
taking values for the interval [ min λ 1 , B ( u , T ) , max λ 1 , B ( u , T ) ] , u { t }. For numerical
calculations one can use for example uniform or other simple distribution. We can also use other
definition of the liquidity. Define the profitability of the bond d B ( s , t ; T ) that represent profit or loss
ratio purchasing bond at s and selling it at a next moment t
d B ( s , t ; T ) =
)T,t(B
)T,s(B
ask
bid
and define value of liquidity as
λ B ( t , T ) =
ts
lim d B ( s , t ; T ) (7)
The perfect liquidity and complete illiquidity correspond to the value λ B ( t , T ) = 1 and λ B ( t , T ) = 0
correspondingly. Next remark does not depend on liquidity definition. We call asset A more liquid at a
moment t than asset G if λ G ( t , T ) < λ A ( t , T ). If the latter inequality takes place on [ 0 , T ] then A
is more liquid than G on the interval [ 0 , T ]. Assume that A is more liquid than G on a subinterval of the
interval [ 0 , T ] and G more liquid than A on another subinterval then the definition of liquidity should be
refined. One way to adjust the definition is to consider an average bid-ask spread. For example asset A is
more liquid than asset G on [ 0 , T ] in sense of the average if
T
0
T
1
λ G ( t , T ) d t ≤
T
0
T
1
λ A ( t , T ) d t (7′)
where λ A , λ G denote liquidity of asset A and G correspondingly. This definition of liquidity does not
comprise variety of the real market components associated with liquidity. Broadly speaking, one can
consider a liquidity adjustment including volume of trades during a particular period. Indeed, let us
imagine that assets A and G look similar in terms of (7), (7′) while the number of trades G is visibly larger
than A over [ 0 , T ] then it might makes sense to think that G is more liquid than A on this interval.
The risk free bond price at t with expiration at T is defined by its bid-ask prices
0 < B bid ( t , T ) < B ask ( t , T ) ≤ 1 , t [ 0 , T )
B bid ( T , T ) = B ask ( T , T ) = 1
13. 13
Note that in the single price format defined as
B ( t , T ) =
2
1
[ B bid ( t , T ) + B ask ( t , T ) ] (8)
is a heuristic price and it does not represent perfect liquidity either for buyer or seller of the bond. The
perfect liquidity for buyer is the bid price while the seller’s perfect liquidity is the ask price. Taking into
account this remark we note that perfect liquidity is defined for buyer and seller separately. The only real
world prices B bid ( t , T ) , B ask ( t , T ) represent liquidity or illiquidity of the bond. Therefore, liquidity
of the risk free bond which price is defined by (8) does not perfect for a counterparty. The risky bond
liquidity is more complex issue. We will discuss it later. Liquidity problem implies discovery of the
adjustment to the formula (8). On the other hand we can deal with bid-ask prices directly.
Formal randomization of liquidity problem for risk free bond can be studied by considering two random
processes for bid and ask prices. By definition we put
B k ( t , T ; ) = exp -
T
t
r k ( u , ) d u (9)
k = bid , ask. Here the risk free interest rate is defined as
r k ( u , ) =
0tΔ
lim
tΔ
)tΔt,t(B1 k
=
0tΔ
lim
tΔ
)tΔt,t(B)tΔtt,Δt(B kk
From (9) it follows that
B k ( t , T ) = 1 +
T
t
r k ( u , ) B k ( u , T ; ) d u (9′)
and
r k ( u , ) =
)T,t(B
1
k
t
)T,t(B k
From (9′) it follows that bond price and interest rate at moment t depend on values of bond on the interval
( t , T ]. It is quite common in performing randomization of the bond price (8) that the price is governed
by Linear Ito equation. In this case we can guarantee the condition B ( t , T ) > 0 while we cannot
guarantee the condition that B ( t , T ) ≤ 1. In other words we arrive at the fact that with a positive
probability for each t B ( t , T ) > 1 which does not make sense. To state that theoretical formulas
represent good approximation of the historical data makes sense if the probability P {
Tt0
sup B ( t , T ) >
1 } is sufficiently small. Following [1] consider randomization of the bond price in single price format
,i.e. ignoring liquidity aspect. Assume that r ( t , ) is govern by backward Ito equation
14. 14
r ( t , ) = r T ( ) –
T
t
λ ( u ) r ( u , ) d u –
T
t
θ ( u ) r ( u , ) d w ( u ) (10)
where stochastic integral on the right hand side is the backward Ito integral [5]. The solution of the
equation (10) is interpreted as a random F ( t , T ] = ζ { w ( s ) - w ( T ) , t < s } measurable stochastic
process for which sup E | r ( t , ) | 2
< ∞. The backward initial condition r T ( ) = r ( T + , ) is
assumed to be F T = F ( T , + ∞) measurable and thus it independent on ζ - algebra F ( t , T ]
r T =
0Δ
lim ( Δ ) – 1
r ( T , T + Δ ) (11)
It represents instantaneous rate of return (1′) over infinitesimal interval [ T , T + dot ) and it will be
known at the date T. The choice of the backward Ito equation (10) is stipulated by the connection of the
price and interest rate presented by formulas (9) , (9′). Indeed, from (9) it follows that values B ( t , T ; )
and r ( t , ) are completely determined by observations over bond prices over [ t , T ]. Modern interest
rates stochastic models assume that interest rate at t depends on r ( u , ) prior to t which implicitly
implies its dependence on B ( u , T ; ) , u ≤ t. This construction inconsistent with original definition of
the interest rate. Thus, providing randomization of the interest rate we need verify consistency random
interest rate with random bond price.
Assume that r ( t , T ) is a known function for all 0 ≤ t ≤ T < + ∞. According to market practice we
replace random r T ( ) by its market implied estimate r ( T , T + H ; t ) defined by the equation
1 + r ( t , T + H ) ( T + H - t ) = [ 1 + r ( t , T ) ( T - t ) ] [ 1 + r ( T , T + H ; t ) H ]
Solving this equation for the implied market forward rate r ( T , T + H ; t ) we arrive at
r ( T , T + H ; t ) = ]
)tT()T,t(r1
)tT()T,t(r-)tHT()HT,t(r
[
H
1
Taking into account latter formula we can present implied estimate of the future rate r T ( )
r ( T , T + 0 ; t ) =
0H
lim r ( T , T + H ; t )
Nonrandom variable r T ( t ) = r ( T , T + 0 ; t ) is known at t and the replacement of the random
variable r T ( ) on nonrandom value r T ( t ) implies market risk stipulated by the fact that in general
market scenarios reveal the fact that r T ( t ) r T ( ). Denote r ( t ; T , r T ) solution of the equation
(10). Then r ( t ; T , r T ( t )) denote solution of the equation (10) with initial value given at T
r ( T ; T , r T ( t )) = r T ( T ). Bond buyer market risk is that his price r T ( t ) = r ( T , T + 0 ; t ) at date t
is higher price than the realized price which corresponds to scenarios
{ : r ( t ; T , r T ( )) < r ( t ; T , r T ( t )) }
Suppose that random function r k ( t , ) , k = bid , ask is the solution of the backward SDE
15. 15
r k ( t , ) = r T , k ( ) –
T
t
λ k ( u , r k ( u , )) d u –
T
t
θ ( u . r k ( u , )) d w ( u ) (10′)
where λ ask < λ bid and with probability 1 r T , ask ( ) < r T , bid ( ). These conditions guarantee
correspondence between bid and ask prices of the bond. Exchange random variables r T , k ( ) on
nonrandom values r T , k ( t ) = r k ( T , T + 0 ; t ) defines the solution of the equation (10′)
r k ( t , ) = r k ( t ; T , r T , k ( t ) ) with boundary condition r T , k ( t ) = r k ( T , T + 0 ; t ), k = bid , ask.
One can assume that spot prices in bid-ask format at date t can be approximated by expectation
conditional over the observations over bond prices up to the moment t
r spot , k ( t ) = E { r k ( t , ) | F t } (12′)
where F t = F [ 0 , t ) = { B ( u , T ) , u < t }. On the other hand we can use implied market estimate
techniques to present other estimate of the spot prices. Replace values r k ( u , ) on the right hand side
(10′) by its implied non random estimate r u ( t ) = r ( u , t ). Then market implied estimate of spot
interest rate admits representation
r )im(
k
( t , ) = r T , k ( t ) –
T
t
λ k ( u , r u ( t )) d u –
T
t
θ ( u . r u ( t )) d w ( u )
k = bid , ask. In accordance to (12) one can assume that
r )im(
k,spot
( t ) = E { r )im(
k
( t , ) | F t } (12′′)
The random function r )im(
k
( t , ) is a Gaussian process with mean and volatility equal to
r T , k ( t ) –
T
t
λ k ( u , r u ( t )) d u ,
T
t
E θ 2
( u . r u ( t )) d u
correspondingly. Formulas (12′), (12′′) are the basic formulas for default free bond pricing that follow (9).
There are two types of the risks exist in our model. One is the model risk which stipulated by the
assumption presented by the formulas (10), (12′), (12′′). Other risk type is approximations of the market.
This risk is represented by the replacements by the real market unknown random future values by the
market implied forward rates know at the spot moment of time.
In our construction we specify buyer and seller liquidities separately. The buyer illiquidity is stipulated by
the fact that the ask price of a bond A is larger than its bid price. Similarly, the seller liquidity is defined
by the fact that the bid prices of the bond are lower than the ask price. Given distinctive bid and ask prices
one can define the date-t implied forward bid-ask spread at date T for the future [ T , T + H ] period.
Indeed, implied forward bid and ask rates f k ( T , T + H ; t ), k = ask, bid can be defined as
16. 16
f k ( T , T + H ; t ) = ]1
)tT()T,t(D1
)tHT()HT,t(D1
[
H
1
k
k
k = ask , bid. Note that we can use discount rates also for the interest rates that do not associate with a
bond. Dealing with LIBOR rate we put in the latter formula D k = L k and f k = l k. Hence,
fra k ( T , T + H ; t ) = l k ( T , T + H ; t )
and FRA bid-ask spread of the FRA contract by definition is a difference
λ l ( t , fra ( T , T + H ; t ) ) = l ask ( T , T + H ; t ) - l bid ( T , T + H ; t )
This formula enables to compare liquidity at t for different T , H. Recall the inverse relationship between
prices and corresponding interest rates. As it is already pointed out the implied rates represent statistical
estimates in stochastic market setting. A future period real rate depends on a market scenario. Therefore,
the use of its corresponding implied rate is subject to market risk. This risk represents the fact that the
date-t implied forward rates l bid or l ask do not equal to its real values at the future moment T. In other
words the liquidity market risk of the FRA contract implied by the bond is stipulated by the inequality
f ask ( T , T + H ; t ) - f bid ( T , T + H ; t ) ≠ B ask ( T , T + H ) - B bid ( T , T + H )
and the probability of the inequality
P { f ask ( T , T + H ; t ) - f bid ( T , T + H ; t ) > B ask ( T , T + H ) - B bid ( T , T + H ) }
is a measure of the illiquidity risk of the FRA contract. Here the FRA contract has been used to illustrate
stochastic effect on implied forward liquidity. The date-t implied liquidity spread λ f = f ask - f bid
represents a statistical estimate which can be larger or smaller than its date-T real market rate
λ = f ask ( T , T + H ) - f bid ( T , T + H )
Higher future liquidity spread corresponds to a higher liquidity risk and future buyer’s additional
expenses while lower future illiquidity suggests more liquidity in the future market. Bearing in mind
liquidity problem one can extend a single rate bid = ask model of the FRA valuation. Given risk free
curves B bid ( t , T ) , B ask ( t , T ) one can present default effect for a corporate bond pricing in bid-ask
format.
V. Liquidity of Corporate Bonds. Let B bid ( t , T ), B ask ( t , T ) and R bid ( t , T ), R ask ( t , T )
denote bid and ask prices of the risk free and risky bonds. Suppose that default can occur only at maturity
date T. Following reduced form model of default one can define default prices.
R bid ( T , T , ) = R ask ( T , T , ) = Δ < 1 , if D
R bid ( T , T , ) = R ask ( T , T , ) = Δ < 1 , if D
Here D denotes default scenarios set. The standard reduce form theory deals with the single price format
[8]. Assuming that constant Δ is known we note that
17. 17
R k ( t , T , ) = [ 1 - ( 1 - Δ ) χ ( D ) ] B k ( t , T ) (13)
k = ask , bid and t ≤ T. For other credit events more complex than bankruptcy the theory would probably
suggest that recovery rate Δ would depend on k too. For numeric calculations one usually uses either
close or open prices of the date t. Let date t is associated with a whole trading day t. In this case the close
or open prices implicitly are considered as a market approximation of the day-t prices. Such
approximation makes sense if the value
R k , max ( t , T ) - R k , min ( t , T )
is sufficiently small and k = bid, ask. Here, the latter differences denote the largest date t value of the ask
and bid spreads. Otherwise, the reduction of the daily values to the one price can be too crude estimate. In
this case, recovery rates (13) can be interpreted as random variables having an appropriate distribution.
For example, one can assume that R k ( t , T , ) has a uniform distribution on interval
I k ( t , T ) = [ R k , min ( t , T ) , R k , max ( t , T ) ]
or a Gaussian distribution with mean equal to middle point of the I k ( t , T ) and standard deviation
ζ =
3
1
[ R k , max ( t , T ) - R k , min ( t , T ) ]
If default occurs at maturity T with known recovery rates Δ k than bid and ask prices оf the bond prior to
the moment T are expressed by (8). Equality (8) does not suffice to state that the default should occur
prior to maturity T, i.e. reduced form of default model expressed by (8) is a necessary condition of the
default. In practice data R k ( t , T ) , t < T is observable. Hence, given R k ( t , T ) one needs to present
an additional assumption regarding default time η distribution and based on this assumption presents
estimates of the recovery rates Δ k , k = bid, ask. Thus, given credit spread B k ( t , T ) - R k ( t , T ),
k = bid, ask at a moment t < T we can state that corporate bond is traded at t similar to a risky bond that
might default at maturity with a particular recovery rate Δ k . The assumption that default occurs at
[ t j , t j + ε ) , ε > 0 will effect on an estimate of the value of the recovery rate. Default problem is
resolved conditioning on the assumption that default can occurred at maturity T.
Assume now that default time ( 0 , T ]. Then recovery rates implied by bid and ask bond prices in
general are different. Given corporate bond prices one need to establish an assumption regarding
distribution of the default. This distribution does not directly follows from price observations. Based on
this assumption we should present estimates of the values of recovery rates Δ k . Thus, given credit
spreads B k ( t , T ) - R k ( t , T ) , k = bid, ask at t , t < T we can state that
1. corporate bonds with unknown default time distributions are traded at t similar to a risky bonds
that might default only at maturity with a particular recovery rate Δ k ,
2. an assumption that default of a corporate bond occurs at [ t j , t j + ε ) , ε > 0 will effect on
estimate of the recovery rate.
In general theory it is a common to assume that value of the recovery rate and default time are
independent random variables. This is a technical assumption and it does not look close to real world
events.
18. 18
In the reduced form framework equality (13) represent necessary conditions of the default. One needs a
randomization of the observations in order to prescribe probability density to default at a moment t. For
example, if a portion p [ 0 , 1 ] of time during a day t the price of the corporate bond is approximately
equal to Δ B k ( t , T ) then it makes sense to state that assuming that default can be observed only at
expiration date T then with probability approximately equal to p recovery rate of the bond should be
equal Δ . Note that in this case values p and Δ are functions of t. Following this way one can present a
construction of a discrete approximation of the continuous distribution of the default at T. In bid-ask
format we observe data
B k ( t , T ) - R k ( t , T ) = B k ( t , T ) [ 1 - Δ k ]
k = bid, ask , and t < T. Then assuming that default might occur at maturity T we can state that
recovery rates implied by bid and ask prices will be equal to Δ k . Note, that
B ask ( t , T ) - R ask ( t , T ) ≠ B bid ( t , T ) - R bid ( t , T )
and therefore Δ ask ≠ Δ bid . Admitting that default might occur at t prior to expiration date T we should
assume that recovery rate is a function of t.
Remark. In practice, it is common to fixe recovery rate Δ. For example, one assumes that Δ = 40% and
default time distribution coincides with the distribution of the first jump of the Poisson process. The
intensity of the Poisson process is assumed to be consistent with B ( t , T ) - R ( t , T ).
In general theory in order to present numeric calculations it is usually supposed that moment of default
and recovery rate Δ are independent random variables. From our point of view this assumption does not
look realistic and it oversimplified stochastic setting. Recovery rate does depend on the default moment.
Recall that prior to default a holder of the bond can sell it for the bid price while a buyer of the bond
should pay ask price to purchase bond. On the default time regulatory rules, which would direct default
procedure.
We illustrate default in bid-ask format assuming that default can occur only at maturity date T. This
assumption simplifies the problem ignoring the random default time issue. We make some remarks to
reduce form of default bearing in mind liquidity setting of the problem. In single price format default
scenarios can be presented as
D = { ω : R ( T , T ; ω ) = Δ < 1 }
and put D = Ω D. Then
R ( t , T ; ω ) = B ( t , T ) [ 1 ( D ) + Δ ( D ) ] = B ( t , T ) [ 1 – ( 1 – Δ ) ( D ) ]
for t ≤ T. In bid-ask format default at T can be defined as following
D = { R bid ( T , T ; ω ) = Δ bid < 1 }
Here 0 ≤ Δ bid < 1. Then
19. 19
R k ( t , T ; ω ) = B k ( t , T ) [ 1 ( D ) + Δ k ( D ) ] = B k ( t , T ) [ 1 – ( 1 – Δ k ) ( D ) ] (14)
for k = bid, ask. Here is supposed that nonrandom function B k ( t , T ) and constants Δ k are known at t.
In this case equality (14) represents market price R k ( t , T ; ω ) of the corporate bond. The market price is
defines uniquely for each market scenario . Bearing in mind available information market participants at
form a spot price at t . Given B k ( t , T ) , Δ k , R k ( t , T ; ω ) and R
spot
k ( t , T ) we enable to estimate
market risk of a bond seller with the help of formula
P { R spot
bid
( t , T ) < R bid ( t , T ; ω ) }
This formula represent value of the chance that bond’s buyer pays less than it implies by the market. In
reality the only values B k ( t , T ) и R
spot
k
( t , T ) are observable while values Δ k are unknown.
Randomization subtends the random processes R k ( t , T ; ω ) construction based on observations
R
spot
k ( t , T ). It is common practice to identify value R
spot
k ( t , T ) with close price at date t. In the case
the forecast for the next date price is also relates to the close moment. We can think that close price
approximates the prices during the date t. This assumption can fail if the variation of the prices does not
sufficiently small. Randomization cal be realized as following. Suppose that R k ( t , T ; ω ) is a random
variable taking values from the interval [ R spot
mink
( t , T ) , R spot
maxk
( t , T ) ] where the end points of the
interval are minimum and maximum spot values during the date {t}. Date {t} can be also denote other
that day period. As model distribution of the R k ( t , T ; ω ) can be chosen uniform, triangle, or
appropriate Gaussian distributions. Given a model distribution and assuming that default possible only at
maturity one can calculate recovery rates Δ k and probability of default Р ( D ). Taking expectation in (14)
we get the equation
E R k ( t , T ; ω ) = B k ( t , T ) [ Р ( D ) + Δ k Р ( D ) ] = B k ( t , T ) [ 1 – ( 1 – Δ k ) Р ( D ) ] (14′)
In general case when recovery rate is interpreted as a random variable on [ 0 , 1 ] equation (14′) can be
rewritten as
E R k ( t , T ; ω ) = B k ( t , T ) [ 1 – Р k ( D ) +
1
0
v d Р k ( v ) ] (14′′)
where Р k ( v ) = Р { Δ k < v }. The probability Р k ( v ) on the right hand side (14′′) can be interpreted as
the portion of time during the date {t} for which
)T,t(B
)ω;T,t(R
k
k
< v
From equality (14′) it follows that
)T,t(B
)ω;T,t(RE
)T,t(B
)ω;T,t(RE
ask
ask
bid
bid
(15)
20. 20
Assuming that default can occur only at maturity we note that equalities (14′) for k = bid, ask are
dependent. Violation of the equality (15) can mean that at least one of the basic assumptions does not
correct. For example it might be incorrect to admit that default occurs at only expiration date. Let Δ k be
unknown constants. Then from (14) follows that for any n
Е [ 1 –
)T,t(B
)ω;T,t(R
k
k
] n
= ( 1 – Δ k ) n
P ( D ) (16)
Solving algebraic system (16) when n = 2 we arrive at the solution
Δ k = 1 –
)
)T,t(B
)ω;T,t(R
1(E
)
)T,t(B
)ω;T,t(R
1(E
k
k
2
k
k
, P ( D ) =
2
k
k
2
k
k
)
)T,t(B
)ω;T,t(R
1(E
])
)T,t(B
)ω;T,t(R
1(E[
(16′)
k = bid, ask. In general case we assume that the bond price immediately after default is a random variable
Δ = Δ ( t , T ; ω ) with a continuous distribution on [ 0 , 1 ]. Let us approximate it by a discrete random
variable
Δ β k ( ω ) =
1-N
1j
β j { Δ k ( ω ) [ β j , β j + 1 ) }
where = β 0 < β 1 < … < β N = 1 and Р j k = Р { Δ k ( ω ) [ β j , β j + 1 ) }. Here, Р j k are function
of the variable t. It follows from (14) that
Е [ 1 –
)T,t(B
)ω;T,t(R
k
k
] n
=
1-N
1j
( 1 – β j ) n
Р j k (16′′)
n = 1, 2, … N и k = bid, ask. For each fixed k = bid, ask there is a linear algebraic system for
unknown variables Р j k , j = 1, 2, … N. Numeric calculations unknowns can be simplified if we note that
determinant of the system is of Vandermonde’s type. Values Р j k can depend on time t. For simplicity we
can put β n = N – 1
n. Hence, the liquidity problem can be solved assuming that default occurs only at
maturity date. Let us estimate liquidity spread between the risk free and the corporate bonds. Note that
from (14) it follows that
[ R ask ( t , T ; ω ) – R bid ( t , T ; ω ) ] – [ B ask ( t , T ) – B bid ( t , T ) ] =
= [ B bid ( t , T ) ( 1 – Δ bid ) – B ask ( t , T ) ( 1 – Δ ask ) ] ( D )
and therefore
)T,t(B-)T,t(B
])T,t(B-)T,t(B[-])ω;T,t(R-)ω;T,t(R[E
bidask
bidaskbidask
=
21. 21
= – Р ( D )
)T,t(B-)T,t(B
)T,t(B-)T,t(B
bidask
bidbidaskask
This formula represents the relative change of the value corporate liquidity with respect to risk free
liquidity.
Now let us consider the case when corporate bond admits default at some fixed moment of the period
[ 0 , T ]. Let η = η ( ω ) denote the default time of the bond. The market price of the bond defined for
each market scenario can be written in the form
R ( t , T ; ω ) =
1-n
1i
R ( t , t i ; ω ) χ { η = t i } + B ( t , T ) χ { η > T } (17)
Here, R ( t , t i ; ω ) denotes the market price of the bond with expiration at t i conditioning that default can
be occurred at expiration date. For notation simplicity in (17) the index k was omitted. Assume now that
recovery rates and probability rates
Δ = Δ ( t , t i ) , P ( D ) = P ( t , t i )
are unknown. From equality (16′) it follows that market price (17) can be presented as
R ( t , T ; ω ) =
1-n
1i
Δ k ( t , t i ) В ( t , t i ; ω ) χ { η = t i } + B ( t , T ) χ { η > T } (17′)
In case when market is such that expected market loss is approximately equal to expected gain of the
bond mathematical expectation can be considered as a good estimate of the spot price at t. Then
R k spot ( t , T ) Е R k ( t , T ; ω ) =
=
1-n
1i
В k ( t , t i ; ω ) { 1 –
)
)t,t(B
)ω;t,t(R
1(E
)
)t,t(B
)ω;t,t(R
1(E
ik
ik
2
ik
ik
}
2
ik
ik
2
ik
ik
)
)t,t(B
)ω;t,t(R
1(E
])
)t,t(B
)ω;t,t(R
1(E[
+ (17′′)
+ B k ( t , T ) { 1 –
2
k
k
2
k
k
)
)T,t(B
)ω;T,t(R
1(E
])
)T,t(B
)ω;T,t(R
1(E[
}
k = bid, ask. To present liquidity effect we need to present the adjustment to the single format pricing
which is interpreted as middle price of bid and ask prices presented by (17′′).
The above formulas for recovery rate and probability of default were obtained under assumption that
default can be observed only at maturity of the bond. In order to lessen the assumption and admit that
bond can default prior to its maturity we need to make an assumption regarding distribution of the time of
22. 22
default. Different hypotheses regarding distributions of the default time lead to different statistical
characteristics of the price and liquidity of the bond. Let us briefly remind reduced form model of default.
It was assumed that distribution of the default time could be approximated be the distribution of the first
jump of a Poisson process. Let П ( t ) denote probability of the event that default does not occur up to the
moment t. Then the value П ( t ) – П ( t + h ) represents probability of event that default would first
observed during the interval [ t , t + h ]. Indeed, bearing in mind equalities
П ( t ) = P { η > t } = P { t < η ≤ t + h } + P { η > t + h } = P { t < η ≤ t + h } + П ( t + h )
it follows that unconditional probability that default occurs at [ t , t + h ] will be equal to
П ( t ) – П ( t + h ) = P { t < η ≤ t + h }
Next
Р { η ≤ t + h | η > t } = Р { η ≤ t + h ∩ η > t } [ P { η > t } ] – 1
=
)t(Π
)ht(Π)t(Π
Suppose that function П ( t ) is continuously differentiable and there exists a continuous function h ( t )
for which
td
)t(Πd
)t(Π
1
= h ( t )
Then
П ( t ) = exp –
t
0
h ( u ) d u (18)
Let us represent default intensity function in the form h ( u ) = r ( u ) + λ ( u ), where r ( u ) is the risk
free interest rate. The (18) can be rewritten in the form
П ( t ) = exp –
t
0
[ r ( u ) + λ ( u ) ] d u (18′)
This representation takes place on original probability space under original probability measure P. In (18′)
function λ ( u ) can be interpreted as a spread of the default intensity.
Remark. It is known a generalization of the representation (18). One assumes that the intensity of default
is a random function λ ( t , ω ) = h ( t , X ( t , ω )), where X ( t , ω ) is a random process. The random
processes of this type are called Cox processes. In paper [7] the default was presented by using Cox
processes. We wish to highlight a complexity of the consistency formal mathematic and financial essence
of the problem. It was supposed that conditional probability of the event that default does not occur before
date t can be presented by the formula
23. 23
P { η > t | X ( u ) , 0 ≤ u ≤ t } = exp –
t
0
h ( u , X ( u , ω )) d u = Secu19()(19)
Following [7] define filtration of the ζ-algebras :
G t = ζ { X ( u , ω ) , 0 ≤ u ≤ t }
H t = ζ { χ ( η ≤ u ) , 0 ≤ u ≤ t }
F t = G t H t
In applications of Cox processes to default problem
*) the random process X is interpreted [7] as the state variable process X , for which the spot rate
r t = R ( X t ) at time t. In other words X characterizes the company that issued bond;
**) events from H t shows whether default observed until the moment t or nor;
***) events from F t described simultaneously default and company state until the moment t
Note that formula (19) shows that default distribution is also depend on the process X as far as one
inexplicitly assumed in (19) that η = η ( X ( )). Nevertheless the formula (19) can be interpreted as
following. For each constant value X ( ) = v there uniquely defined default time η = η ( v ) of the firm
in the state v for which
P { η ( v ) > t } = exp –
t
0
h ( u , v ) d u
Then if the state of the firm is a random process X ( t , ) that independent on η ( v ) then the formula
(19) holds. Next, one should explicitly define the random process X ( t , ). The formula
r t = R ( X ( t , ) )
can be used as a definition of the process X if the function R ( x ) is formally defined. These facts were
probably missed in [7]. This is a way that stochastic intensity can be studied. Nevertheless it is not clear
whether function R can be formally defined and whether or not X will be independent on η ( v ).
At the end let us highlight the fact that introduced here risky bond pricing approach differs from the
reduced form standard though both approaches relates to the reduced form interpretation of the default. In
simplified case when default can be observed only at maturity the main distinction is that based on prices
observations we calculate probability of default along with recovery rate of the bond. The standard reduce
form approach assumes that recovery rate of the corporate bond to be known. It is stipulated by the fact
that in the single price format we deal with expected value of the equation of the type (14). Therefore it is
impossible to derive two unknown Δ , Р ( D ). Our approach separates market price that depends on
market scenario and spot price. This approach makes it possible to present equations for higher moments
of the market price (16) that helps to present two unknown in the form (16′). It is clear that different
25. 25
Bibliography.
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http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1964307.
2. I. Gikhman. FX Basic Notions and Randomization.
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1971373.
3. I. Gikhman. О построении цены производных инструментов., http://www.slideshare.net/list2do
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systems. Lap Lambert Academic Publishing, 2011, p.252.
6. J. Hull. Options, Futures, and other Derivatives, 7 ed., 814.
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1998.
8. R. Jarrow, S Turnbull. Derivatives Securities, South-Western College Publishing, 2ed. 2000, 684.