Modelos estocásticos para la valoración de riesgo financiero

1. MODELOS ESTOCÁSTICOS PARA LA VALORACIÓN DE RIESGO FINANCIERO Viswanathan (ARUN) Arunachalam Departamento de Estadística Universidad Nacional de Colombia Bogotá, Colombia varunachalam@unal.edu.co XXIII Seminario financiero: Las finanzas cuantitativas -UNAB Bucaramanca 20 de Septiembre 2019 Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

2. AGENDA 1 FINANCIAL RISK 2 MARKET RISK -OPTION PRICING 3 THE SKEW NORMAL MIXTURES 4 CREDIT RISK 5 OPERATIONAL RISK Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

3. Financial Risk FINANCIAL RISK Financial risk is one of the high-priority risk types for every business. Risk can be referred to like the chances of having an unexpected or negative outcome. Any action or activity that leads to loss of any type can be termed as risk. There are different types of risks that a firm might face and needs to overcome. Widely, risks can be classified into three types: Business Risk, Non-Business Risk, and Financial Risk. Financial Risk: Financial Risk as the term suggests is the risk that involves financial loss to firms. Financial risk generally arises due to instability and losses in the financial market caused by movements in stock prices, currencies, interest rates and more. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

4. Financial Risk FINANCIAL RISK Financial risk is one of the high-priority risk types for every business. Risk can be referred to like the chances of having an unexpected or negative outcome. Any action or activity that leads to loss of any type can be termed as risk. There are different types of risks that a firm might face and needs to overcome. Widely, risks can be classified into three types: Business Risk, Non-Business Risk, and Financial Risk. Financial Risk: Financial Risk as the term suggests is the risk that involves financial loss to firms. Financial risk generally arises due to instability and losses in the financial market caused by movements in stock prices, currencies, interest rates and more. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

5. Financial Risk TYPES OF FINANCIAL RISKS Financial risk is caused due to market movements and market movements can include a host of factors. Based on this, financial risk can be classified into various types such as Market Risk : This type of risk arises due to the movement in prices of financial instrument. Market risk can be classified as Directional Risk and Non-Directional Risk. Directional risk is caused due to movement in stock price, interest rates and more. Non-Directional risk, on the other hand, can be volatility risks. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

9. Financial Risk TYPES OF FINANCIAL RISKS Financial risk is caused due to market movements and market movements can include a host of factors. Based on this, financial risk can be classified into various types such as Credit Risk: This type of risk arises when one fails to fulfill their obligations towards their counterparties. Credit risk can be classified into Sovereign Risk and Settlement Risk. Sovereign risk usually arises due to difficult foreign exchange policies. Settlement risk, on the other hand, arises when one party makes the payment while the other party fails to fulfill the obligations. Operational Risk : This type of risk arises out of operational failures such as mismanagement or technical failures. Operational risk can be classified into Fraud Risk and Model Risk. Fraud risk arises due to the lack of controls and Model risk arises due to incorrect model application. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

10. Financial Risk TYPES OF FINANCIAL RISKS Financial risk is caused due to market movements and market movements can include a host of factors. Based on this, financial risk can be classified into various types such as Credit Risk: This type of risk arises when one fails to fulfill their obligations towards their counterparties. Credit risk can be classified into Sovereign Risk and Settlement Risk. Sovereign risk usually arises due to difficult foreign exchange policies. Settlement risk, on the other hand, arises when one party makes the payment while the other party fails to fulfill the obligations. Operational Risk : This type of risk arises out of operational failures such as mismanagement or technical failures. Operational risk can be classified into Fraud Risk and Model Risk. Fraud risk arises due to the lack of controls and Model risk arises due to incorrect model application. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

11. Financial Risk STOCHASTIC MODELING Stochastic modeling is a form of a financial model that is used to help make investment decisions. This type of modeling forecasts the probability of various outcomes under different conditions, using random variables. Deterministic modeling produces constant results Stochastic modeling produces distribution(various) of results. A stochastic model incorporates random variables to produce many different outcomes under diverse conditions. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

14. Market Risk -Option Pricing OPTION PRICING There is empirical evidence that financial stock returns are not normally distributed but are characterized by skewness, leptokurticity, heavy-tailedness and other non-Gaussian properties. The skewness and kurtosis of the empirical distribution function (EDF) of stock returns contribute significantly to the phenomenon of volatility smile. In recent years, there have been considerable efforts to report that the unconditional probability distributions of returns on financial stocks are not normally distributed. Specifically, these distributions tend to have heavier tails (leptokurtic) and asymmetry. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

18. Market Risk -Option Pricing LOG-SKEW-NORMAL DISTRIBUTION- OPTION PRICING By assuming that the stock distribution follows a Log-Skew-Normal mixture (LSNMIX) distribution, we calculate an explicit formula for option valuation for both European call and put options and Greek measures. We also show that some of the well-known models are obtained as special cases from the proposed model An example from past S&P500 daily returns to price is presented to illustrate the model. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

21. Market Risk -Option Pricing Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

22. Market Risk -Option Pricing LOG-SKEW-NORMAL DISTRIBUTION A random variable Y has a LSN distribution with asymmetry parameter λ ∈ R, denoted as Y ∼ LSN (Λ1) , if its pdf is of the form fY y; Λ1 = 2 σy ϕ ln y − µ σ Φ λ ln y − µ σ = 1 y φSN ln y; Λ1 , y ∈ R+ , (1) where Λ1 = µ, σ, λ , σ > 0, φSN(·) denotes the pdf of the skew-Normal (SN) distribution, and ϕ (·) and Φ(·) denote the pdf and cdf of a standard univariate normal variable, respectively. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

23. Market Risk -Option Pricing LOG-SKEW-NORMAL DISTRIBUTION If µ = 0 and σ = 1 then Y is said to have a (standard) LSN distribution, i.e., Y ∼ LSN (λ) . The parameter λ controls the skewness, which is positive when λ > 0 and negative when λ < 0. The cdf of (1) is given by FY (y; Λ1) = Φ ln y − µ σ − 2T ln y − µ σ ; 0, λ , (2) where the function T (z; α, λ) with α ≥ 0 is given as T (z; α, λ) =sign (λ) arctan (|λ|) 2π − z α |λ|x 0 ϕ(x, α, 1)ϕ (y)dydx , and sign(·) is the signum function. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

24. Market Risk -Option Pricing −4 −3 −2 −1 0 1 2 3 4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Probability Density Functions SN(µ,σ,λ), µ= 0, σ= 1 x SN(µ,σ,λ) λ= 0 λ= 2 λ= 5 FIGURE: Comparison of the SN pdf with λ > 0. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

25. Market Risk -Option Pricing −4 −3 −2 −1 0 1 2 3 4 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Probability Density Functions SN(µ,σ,λ), µ= 0, σ= 1 x SN(µ,σ,λ) λ= 0 λ= −2 λ= −5 FIGURE: Comparison of the SN pdf with λ < 0. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

26. The Skew Normal Mixtures MIXTURE we consider the finite mixture model: We assume that if Y is SNMIX distributed then the transformation X = exp {Y} is distributed as a LSNMIX. Let us assume that fY(y) is the weighted sum of m-component SNMIX densities, that is, fY (y; ˜) = m j=1 ωjφSN (y; µj, σj, λj). (3) We use the notation Y ∼ SNMIX (˜) , where ˜ = (ξ1, . . . , ξm) , and ξj = (ωj, µj, σj, λj) is the parameter vector that defines the j-th component and probability weights, ωj, satisfy the conditions m j=1 ωj =1, 0 < ωj < 1, for each j. (4) Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

27. The Skew Normal Mixtures The choice of a finite mixture is attractive from the application view point because of its flexibility and allows us to consider different kinds of shaped distributions. For instance, the two component SNMIX model has the advantage of numerical tractability, because it has only seven parameters. Assuming ξj = (ωj, µj, σj, λj) , with µ1 = −1, µ2 = 1, σ1 = σ2 = 1, λ1 = .5 and λ2 = −2. Figure 3 shows the pdf shape of the SNMIX for three values of ω. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

28. The Skew Normal Mixtures −5 −4 −3 −2 −1 0 1 2 3 4 5 0 0.1 0.2 0.3 0.4 0.5 Probability Density Functions SNMix(µ1 ,σ1 ,λ1 ,µ2 ,σ2 ,λ2 ,ω) µ1 = −1, σ1 = 1, λ1 = 0.5, µ2 = 1, σ2 = 1, λ2 = −2 x SNMix(µ1 ,σ1 ,λ1 ,µ2 ,σ2 ,λ2 ,ω) ω= 0.06 ω= 0.46 ω= 0.91 Normal FIGURE: Comparison of the pdf of the SNMIX with varying ω. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

29. The Skew Normal Mixtures OPTION PRICING FORMULA Let St be the price of the underlying stock at time t and C(t, τ; K) the price of the call option with strike price K and maturity date of T = t + τ. It is assumed that r is the annual risk-free rate. In the absence of arbitration, the price of European call options can be written as follows: C(t, τ; K) = E e−rτ max{ST − K, 0} = e−rτ E [max{ST − K, 0}] Ct(K) = e−rτ ∞ K (ST − K) f (ST) dST. Here E[.] is the expected value conditional (risk neutral) on any information that is available at time t, f (ST) is the risk-neutral pdf (risk-neutral distribution, RND) for the underlying at maturity. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

30. The Skew Normal Mixtures In an arbitrage-free economy, the stock price discounted by the risk free rate becomes martingale, that is: E[e−rτ ST] =St, τ > 0 (5) where the standard deviation of ln (St) is σ √ τ. The net premium of the derivative, which is the expected value of the option at maturity in a risk-neutral world, E [max{ST − K, 0}] is calculated using statistical or numerical methods. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

31. The Skew Normal Mixtures OPTION PRICE USING THE LSNMIX We define the distribution of the logarithm of the stock price ST using its location and scale parameters A and B, respectively, and also ˜, the parameter of the SNMIX. These parameters satisfy the following relationship: ln [ST] =A + BY, with Y ∼SNMIX (˜) . (6) Then, the pdf of ST is a LSNMIX. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

32. The Skew Normal Mixtures OPTION PRICE USING THE LSNMIX PROPOSITION The price of a European call option is given by erτ Ct K; Λ = m j=1 2ωjE [ST] Υj (˜, B) ∞ −δ1j ϕ (z) Φ [λjzj]dz − m j=1 ωjK [1 − FY (−δ2j; λj)] , (7) where zj = z + Bσj and FY(·) is given in (2), δ1j = δ2j + Bσj, δ2j = − κ − µj σj , κ = ln K − A B , (8) δ2j = 1 Bσj ln E [ST] K − ln [Υj (˜, B)] − 1 2 Bσj, Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

33. The Skew Normal Mixtures OPTION PRICE USING THE LSNMIX PROPOSITION The LSNMIX European put option price is given by erτ Pt K; Λ = m j=1 ωjKFY (−δ2j; λj) − m j=1 2E [ST] ωj Υj (˜, B) −δ1j −∞ ϕ (z) Φ [λjzj]dz, (10) where δ1j and δ2j are given in (8) and Υj (˜, B) is given in (9). Using the option valuation formula, we can obtain the put-call parity relationship by subtracting expression (7) from (10) to obtain the following equality erτ Ct K; Λ − Pt K; Λ =E [XT] − K. (11) In our case, the pdf is a RND and therefore will not violate the put-call parity relationship. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

34. The Skew Normal Mixtures BLACK-SCHOLES Assuming ξj = 1 m , µ, σ, 0 for all j in (3), substituting in expressions (7) and (10) yields, respectively, erτ Ct K; Λ =E [ST] Φ 1 Bσ ln E [ST] K + 1 2 Bσ − KΦ 1 Bσ ln E [ST] K − 1 2 =E [ST] Φ (d1) − KΦ (d2) , (12) where d2 = 1 Bσ ln E [ST] K − 1 2 Bσ, (13) and d1 = d2 + Bσ. Note that when B = √ τ, these expressions coincide with the option pricing formula. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

35. The Skew Normal Mixtures BAHRA(1997) When ξj = ωj, µ∗ j , σj, 0 for all j, where µ∗ j = B µj − 1 2σ2 j in (3), substituting in expressions (7) and (10) yields, respectively, erτ Ct K; Λ = m j=1 ωjE [ST] Υj (˜, B) Φ (δ1j) − K m j=1 ωjΦ (δ2j) , (14) where δ1j and δ2j are given in (8) and Υj (˜, B) = m l=1 ωl exp B2 (µl − µj) . (15) Note that when B = √ τ, these expressions coincide with the option pricing formula given in [?]. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

36. The Skew Normal Mixtures CORNS & SATCHELL (2007) -SKEW BROWNIAN MOTION When ξj = 1 m , µ, σ, λ for all j in (3), substituting in expressions (7) and (10) yields, respectively, erτ Ct K; Λ = E [ST] Φ (ρσB) ∞ −δ1 ϕ (z) Φ [λ (z + Bσ)]dz − K [1 − FY (−δ2; λ)] , (16) where δ2 = 1 Bσ ln E [ST] 2K Φ (ρσB) − 1 2 Bσ, (17) and δ1 = δ2 + Bσ. Note that when B = √ τ, these expressions coincide with the option pricing formula (Corns & Satchell). Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

37. The Skew Normal Mixtures NUMERICAL RESULTS We now present an example from real market data to model the distribution of the stock prices and compare the numerical values of a European option under the assumption that the stock movement follow a LSNMIX distribution. The market data of the S&P500 index for the daily prices were considered from January 4, 2010 to October 13, 2014. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

38. The Skew Normal Mixtures Statistics Values Mean 0.0004 Stan. Dev. 0.0102 Minimum -0.0690 Maximum 0.0463 Skewness -0.4854 Kurtosis 7.6325 JB test 1120.1201 TABLE: Summary of the Descriptive Statistics Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

39. The Skew Normal Mixtures NUMERICAL RESULTS Estimate MME MLE µ1 -0.0039 -0.0025 µ2 0.0036 0.0045 σ1 0.0056 0.0056 σ2 0.0137 0.0132 λ1 1.4683 1.4686 λ2 -0.6444 -0.6457 ω 0.6158 0.6158 TABLE: Estimates for adjusting the SNMIX (˜) Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

40. The Skew Normal Mixtures NUMERICAL RESULTS −0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0 50 100 150 200 250 Density functions adjusted for daily returns on S&P 500 Index Daily returns Frequency Histogram Normal Empirical NMix SNMix FIGURE: Returns vs. normal distribution and SNMIX. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

41. The Skew Normal Mixtures Strike Maturity (K) τ = 0.25 τ = 0.5 τ = 0.75 τ = 1.0 1300 122,7236 146,2137 168,1804 188,9434 1350 80,8813 106,9346 130,0796 151,5625 1400 47,3226 73,8094 96,9744 118,4221 1450 24,1159 47,8415 69,5369 90,0320 1500 10,5793 29,0350 47,9058 66,5672 TABLE: Comparison prices of call option Black Scholes. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

42. The Skew Normal Mixtures NUMERICAL RESULTS Strike Maturity (K) τ = 0.25 τ = 0.5 τ = 0.75 τ = 1.0 1300 123,0441 143,9754 164,5054 184,3572 1350 78,9742 103,5501 125,8430 146,8139 1400 44,7446 70,3303 92,9945 114,1851 1450 22,2624 45,2095 66,5457 86,9097 1500 10,7043 28,0958 46,5603 65,1000 TABLE: Comparison prices of call option Corrado & Su. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

43. The Skew Normal Mixtures NUMERICAL RESULTS Strike Maturities (K) τ = 0.25 τ = 0.5 τ = 0.75 τ = 1.0 1300 121,8372 143,5546 164,1304 183,8455 1350 76,7124 99,9953 121,4675 141,8595 1400 35,8423 59,5374 81,2789 101,9446 1450 8,3470 25,7672 45,2634 65,0156 1500 1,6477 8,0504 19,5020 34,6045 TABLE: Comparison prices of call option SNMIX. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

44. The Skew Normal Mixtures 88 90 92 94 96 98 100 102 104 106 −10 0 10 20 30 40 50 60 70 Moneyness (%) Calloption BS CS SNMIX FIGURE: Call option for different strikes. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

45. The Skew Normal Mixtures 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 105 110 115 120 125 130 135 140 145 150 Maturities Calloption BS CS SNMIX FIGURE: Call option for different maturities. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

46. The Skew Normal Mixtures NUMERICAL RESULTS 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 Strike Impliedvolatility BS SNMIX FIGURE: Implied volatility. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

47. Credit Risk CREDIT RISK The credit risk problem is one of the most important issues in finance. It is the risk of counter parties not fulfilling their obligations in full on the due date. Losses can result from both default and decline in the market value due to deterioration of credit quality of an issuer or counter party. For example, a corporate bond issued by a company has a credit risk associated with it. Credit risk analysis basically consists of finding the likelihood of default of an obligor going into debt. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

51. Credit Risk Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

52. Credit Risk CREDIT RISK Recent financial crisis has stressed the importance of the study of the correlations in the financial market. In this regard, the study of the risk of default of the counter party, in any financial contract, has become crucial in the credit risk. It is the most intensely studied topic in modern financial world. There are many parameters associated with bond issuer or bond itself, which quantify the credit risk associated with it. Credit rating is one of the important parameters. Credit rating of a credit risky bond, issued by a company, is an indicator of its creditworthiness. Better the credit rating of a bond, safer it is. Credit rating plays a very important role in evaluating the probability of default and thus quantifying the credit risk. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

56. Credit Risk CREDIT RISK After the Basel II accord in 2004 (now Basel III), these ratings became instrumental in evaluating the risk of a bond or loan and hence became an increasingly important instrument in Credit Risk, as they allow banks to base their capital requirements on internal as well as external rating systems. The level of rating changes from time to time because of random credit risk and thus can be modeled by an appropriate stochastic process. In 1997, Jarrow et al. first time applied Markov processes to capture the time evolution of credit ratings. These Markov models are called "migration models". Other papers deals with the problem of suitability of Markov process for the description of accurate rating dynamics was addressed. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

60. Credit Risk Main issues regarding the suitability of Markov processes are 1 Non-Markov behavior: It was proved that the new rating of a company depends not only on its last rating but on all the previous ones, the effect called rating momentum. 2 Ageing or time spent in a rating: In the credit risk problem, a complete knowledge of the duration inside the states is of fundamental importance. The credit migration probability depends on the time spent by a company in a particular rating. 3 Time dependence: It means that in general, transition probabilities tends to vary with the state of the economy, being high during recession and low during periods of economic expansion. Rating evaluation at two different time points is different. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

63. Credit Risk CREDIT RISK Empirical studies document that prior rating changes may carry predictive power for the direction of future rating changes, the property called rating momentum. If a firm gets a lower rating than its previous rating, the probability that its next rating will also be lower than the present one is high. Therefore, the Markov property fails, as the current rating does not fully determine the transition intensities. In the case of upward movement, this phenomenon does not hold. In the credit risk migration model, the rating organisations that gives the rating, estimates the reliability of the company that issued bonds. In their paper, they applied time-homogeneous semi-Markov processes (SMP) to solve the first issue. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

67. Credit Risk CREDIT RISK - MOTIVATION We will solve first and second issues, described above, by the use of Markov regenerative process (MRGP) with embedded semi-Markov process. Since rating momentum exists only in downward moving rating, we can say Markov property is satisfied only when there is a migration from a given rating to a better rating. This behavior of the ratings can be modeled by MRGP. Also, the time spent in a state is a general distribution in MRGP. Hence, MRGP models both the randomness of time in the transition between two states and non-Markovian behavior of the states. The fact that in the proposed MRGP model, the embedded process is SMP, makes it unique. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

70. Credit Risk MARKOV REGENERATIVE PROCESS (MRGP) Firstly, we give the definition of Markov renewal sequence (MRS). A sequence of random variables {(Xn, Tn), n = 0, 1, . . .} is called a Markov renewal sequence if 1 T0 = 0, Tn+1 ≥ Tn; Xn ∈ Ω = {0, 1, 2, . . .} 2 ∀ n ≥ 0, P{Xn+1 = j, Tn+1 − Tn ≤ t | Xn = i, Tn, Xn−1, Tn−1, . . . , X0, T0} = P{Xn+1 = j, Tn+1 − Tn ≤ t | Xn = i} (Markov property) = P{X1 = j, T1 − T0 ≤ t | X0 = i} (time homogenity) Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

71. Credit Risk MARKOV REGENERATIVE PROCESS (MRGP) Firstly, we give the definition of Markov renewal sequence (MRS). A sequence of random variables {(Xn, Tn), n = 0, 1, . . .} is called a Markov renewal sequence if 1 T0 = 0, Tn+1 ≥ Tn; Xn ∈ Ω = {0, 1, 2, . . .} 2 ∀ n ≥ 0, P{Xn+1 = j, Tn+1 − Tn ≤ t | Xn = i, Tn, Xn−1, Tn−1, . . . , X0, T0} = P{Xn+1 = j, Tn+1 − Tn ≤ t | Xn = i} (Markov property) = P{X1 = j, T1 − T0 ≤ t | X0 = i} (time homogenity) Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

72. Credit Risk MARKOV REGENERATIVE PROCESS (MRGP) A stochastic process {Z(t), t ≥ 0} on Ω is called an MRGP if there exists a Markov renewal sequence {(Xn, Tn), n = 0, 1, . . .} such that all conditional finite dimensional distributions of {Z(Tn + t), t ≥ 0} given {Z(u), 0 ≤ u ≤ Tn, Xn = i} are same as those of {Z(t), t ≥ 0} given X0 = i, i ∈ Ω ⊂ Ω. This implies that in this case {Z(T+ n ), n = 0, 1, . . .} or {Z(T− n ), n = 0, 1, . . .} is an embedded discrete time Markov chain (DTMC) and also that Tn’s are regeneration points of {Z(t), t ≥ 0}. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

73. Credit Risk CREDIT RISK The main difference between semi-Markov processes and Markov regenerative process is that in semi-Markov processes, all transition points are regeneration points but in MRGP not all transition points are regeneration points. In MRGP the time spent inside a state can be any distribution unlike Markov environment, where it has to be negative exponential. Therefore, by using MRGP in credit risk problem, we can address the issue of the duration of the rating inside a class called ageing problem. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

76. Credit Risk THE GLOBAL AND LOCAL KERNEL The global kernel K(t) = [Kij(t)]i,j∈Ω associated with the process is defined as Kij(t) = P{Z(T1) = j, T1 ≤ t|Z(0) = i}, i, j ∈ Ω , t ≥ 0 and it follows that pij = lim t→∞ Kij(t), i, j ∈ Ω where P = [pij]i,j∈Ω is the one-step transition probability matrix of the embedded Markov chain with state space Ω . Also, the local kernel E(t) = [Eij(t)]i∈Ω ,j∈Ω given by Eij(t) = P{Z(t) = j, T1 > t|Z0 = i}, i ∈ Ω , j ∈ Ω , t ≥ 0 describes the behaviour of the process between two regeneration epochs of the embedded Markov chain. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

77. Credit Risk TRANSTION PROBABILITIES AND RENEWAL EQUATION Now, the transition probabilities for {Z(t), t ≥ 0} are defined by Vij(t) = P{Z(t) = j | Z(0) = i}, i ∈ Ω , j ∈ Ω, t ≥ 0. They can be obtained by solving the generalised Markov renewal equation V(t) = E(t) + K(t) ∗ V(t) or Vij(t) = Eij(t) + γ∈Ω t 0 Vγj(t − y)dKiγ(y), i ∈ Ω , j ∈ Ω. In the credit risk environment, the first part of above equation can be interpreted as the probability that firm has a lower rating j at time t before the next regeneration time point that is before getting better rating given the firm was at state i at time 0. In second part of above equation, Kiγ(y) represents the probability that firm will get a better rating γ in time y and then firm will migrate to rating j in time (t − y) following one of the possible paths. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

78. Credit Risk CREDIT RATING MODEL If we represent Laplace transform of any matrix A by Ã, then the solution of above equation in transform domain is given by ˜V(s) = [I − ˜K(s)]−1 × ˜E(s). Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

79. Credit Risk CREDIT RATING MODEL The rating issued to a firm by a rating agency gives its degree of reliability. For example, there are the following ratings given by Standard and Poor’s, Ω = {AAA, AA, A, BBB, BB, B, CCC, D}. The bonds having rating above BB are investment grade bonds whereas those having BB or below BB are speculative bonds and state D corresponds to default. Let us assume that these ratings are Ω = {1, 2, . . . , 8} in the same order as above. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

80. Credit Risk DETERMINING THE LOCAL KERNAL Now, we need to determine Eij(t), i ∈ Ω and j ∈ Ω, the probability of migrating from i to j without any regeneration (upward movement). Clearly, we have Eij(t) = 0 if i > j, t ≥ 0, i ∈ Ω , j ∈ Ω. For i ≤ j, let Ω(i) for each i ∈ Ω be the set of all states reachable from state i by downward movement that is Ω(i) = {i, i + 1, ..., 8}. Let {M(i)(t); t ≥ 0}, i ∈ Ω be the subordinated semi-Markov process (since sojourn time in each state is general distribution in which there are only downward movements from state i but no upward movement. Here, by choosing parameters such that probability of downward movement is high, the problem of rating momentum can be addressed. The transition probabilities of this semi-Markov process for each i ∈ Ω will give us Eij(t) ∀j, local kernel of MRGP. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

81. Credit Risk DETERMINING THE LOCAL KERNAL Let (Yn, Sn) be the Markov renewal sequence where Yn represents the state at nth transition and Sn, n ≥ 1 with state space equal to R+ represents the time of nth transition. The kernel Q(t) = [Qjk(t)], j, k ∈ Ω of subordinated SMP is defined by Qjk(t) = P(Yn+1 = k; Sn+1 − Sn ≤ t | Yn = j), j, k ∈ Ω and it follows that qjk = lim t→∞ Qjk(t), j, k ∈ Ω is the transition probabilities of the embedded Markov chain in the process. Therefore, for j, k ∈ Ω Gjk(t) = P(Sn+1 − Sn ≤ t | Yn = j, Yn+1 = k) = Qjk(t) qjk if qjk = 0 1 if qjk = 0 . Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

82. Credit Risk DETERMINING THE LOCAL KERNAL Let Fij(t); i ∈ {1, 2, . . . , 7} and j ∈ Ω represents the distribution of migrating from i to j. It basically represents the distribution function of the waiting time in each state i, given that next state is known. Clearly, Gjk(t) is same as Fjk(t). Furthermore, probability that the process will be in state j at time t is given by Hj(t) = P(Sn+1 − Sn ≤ t | Yn = j), j ∈ Ω. We can observe that Hj(t) = 8 k=j Qjk(t). Now, the transition probabilities of the SMP {M(t), t ≥ 0} are defined by φjk(t) = P(M(t) = k | M(0) = j), j, k ∈ Ω and are obtained by solving the following renewal equation k tArunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

83. Credit Risk DETERMINING THE LOCAL KERNAL In the matrix form, it can be written as φ(t) = H(t) + Q(t) ∗ φ(t). where H(t) is the diagonal matrix with jth diagonal entry 1 − Hj(t). This φ(t) will be upper triangular matrix. In the Laplace domain, solution of above equation is given by ˜φ(s) = [I − ˜Q(s)]−1 ˜H(s). Therefore, for each i ∈ Ω , this φij(t), j ∈ Ω will give Eij(t), j ∈ Ω of the MRGP and hence describes the behavior of rating evolution between two regeneration epochs that is how the rating moves to lower rating before going to upper rating. Hence, by taking Fij(t) with different intensities, increasing or decreasing, we can take care of downward momentum using local kernel. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

84. Credit Risk DETERMINING THE GLOBAL KERNAL Let pij be the one-step transition probabilities for the embedded Markov chain. Then, for i, j ∈ Ω pij = lim t→∞ Kij(t). The global kernel K(t) = [Kij(t)]i,j∈Ω is given by Kij(t) = P(Z(T1) = j; T1 ≤ t | Z(0) = i) = Fij(t)pij if i > j since we need only one regeneration epoch before time t. In case if i ≤ j, we need the probability of going to j by a regeneration which is possible from states greater than j. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

85. Credit Risk DETERMINING THE GLOBAL KERNAL Therefore, when i ≤ j, we have Kij(t) = 6 l=j+1 t 0 Klj(t − s)dEil(s) where Eil is element of local kernel E(t) given in earlier. Thus, Kij(t) = Fij(t)pij if i > j 6 l=j+1 t 0 Klj(t − s)dEil(s) if i ≤ j . Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

86. Credit Risk TIME-DEPENDENT SOLUTION The time-dependent solution of the proposed model is given by Vij(t) = Eij(t) + γ∈Ω t 0 Vγj(t − y)dKiγ(y); i ∈ Ω , j ∈ Ω where E(t) and K(t) are given earlier. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

87. Credit Risk TIME-DEPENDENT SOLUTION This equation can be solved numerically using discretization. Let h be the step size of discretization, then we have the countable linear system given by Vh ij(kh) = Eij(kh) + γ∈Ω k τ=1 diγ(τh)Vh γj((k − τ)h), k = 0, 1, . . . where dij(kh) = Kij(kh) − Kij((k − 1)h) if k > 0 0 if k = 0. . Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

88. Credit Risk TIME-DEPENDENT SOLUTION Eij(kh) is given by numerical solution of SMP, Eij(kh) = gij(kh) + 8 l=i k τ=1 qil(τh)Elj((k − τ)h) where qij(kh) = Qij(kh) − Qij((k − 1)h) if k > 0 0 if k = 0 and gij(kh) = 1 − Hi(kh) if i = j 0 if i = j . Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

89. Credit Risk TIME-DEPENDENT SOLUTION In the Laplace transform domain the solution can be obtained as ˜V(s) = [I − ˜K(s)]−1 ˜E(s) where ˜E(s) is given by transform solution of subordinated SMP, ˜E(s) = [I − ˜Q(s)]−1 ˜H(s) where Q(t) and H(t) are described in subsection 3.1. On solving the above system of equations, the following measures can be obtained: Vij(t), which represents the probability of being in state j after a time t given that initially it was in state i at time 0. This takes into account the time spent in a particular state (ageing). 7 j=1 Vij(t), which represents the probability that the system will never go into the default state in time t. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

90. Credit Risk TIME-DEPENDENT SOLUTION In the Laplace transform domain the solution can be obtained as ˜V(s) = [I − ˜K(s)]−1 ˜E(s) where ˜E(s) is given by transform solution of subordinated SMP, ˜E(s) = [I − ˜Q(s)]−1 ˜H(s) where Q(t) and H(t) are described in subsection 3.1. On solving the above system of equations, the following measures can be obtained: Vij(t), which represents the probability of being in state j after a time t given that initially it was in state i at time 0. This takes into account the time spent in a particular state (ageing). 7 j=1 Vij(t), which represents the probability that the system will never go into the default state in time t. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

91. Credit Risk NUMERICAL EXAMPLE we illustrate the applicability of the proposed model with the help of a numerical example. Since real data is not available, we start with the transition matrix given in Jarrow, Lando and Turnbull (1997) to show how the proposed model works. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

92. Credit Risk Table 1. 1 year transition probability matrix P             AAA AA A BBB BB B CCC D AAA 0.8910 0.0963 0.0078 0.0019 0.0030 0.0000 0.0000 0.0000 AA 0.0086 0.9010 0.0747 0.0099 0.0029 0.0029 0.0000 0.000 A 0.0009 0.0291 0.8896 0.0649 0.0101 0.0045 0.0000 0.0009 BBB 0.0006 0.0043 0.0656 0.8428 0.0644 0.0160 0.0018 0.0045 BB 0.0004 0.0022 0.0079 0.0719 0.7765 0.1043 0.0127 0.0241 B 0.0000 0.0019 0.0031 0.0066 0.0517 0.8247 0.0435 0.0685 CCC 0.0000 0.0000 0.0116 0.0116 0.0203 0.0754 0.6492 0.2319 D 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000             Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

93. Credit Risk NUMERICAL EXAMPLE Assuming Weibull distribution for Fi,j(t), i, j ∈ Ω with parameters as given in Carty and Fons (1994) [1], we have generated the rating migration matrices at different time instants. The transition matrices constructed by the proposed model at times 5, 12 and 15 years are given in Table 2,3 and 4 respectively. For example, the element 0.0659 in row BBB and in column BB represents the probability that a firm will have a rating BB at time 12 given that it initially started with rating BBB. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

94. Credit Risk NUMERICAL EXAMPLE Table 2. 5 year transition probability matrix, Vij(5)         AAA AA A BBB BB B CCC D AAA 0.9997 0.0002 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 AA 0.0001 0.9967 0.0026 0.0004 0.0001 0.0001 0.0000 0.0000 A 0.0001 0.0002 0.9943 0.0034 0.0008 0.0005 0.0003 0.0004 BBB 0.0001 0.0002 0.0006 0.9801 0.0102 0.0041 0.0021 0.0026 BB 0.0002 0.0002 0.0002 0.0006 0.9717 0.0161 0.0043 0.0062 B 0.0001 0.0001 0.0002 0.0002 0.0005 0.9710 0.0114 0.0168         Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

97. Credit Risk Table 5. Probabilities of not going into default in a time horizon of 15 years given initial rating                           AAA AA A BBB BB B 1 1.0000 1.0000 1.0000 1.0000 0.9999 0.9994 2 1.0000 1.0000 1.0000 0.9998 0.9993 0.9975 3 1.0000 1.0000 0.9999 0.9993 0.9982 0.9941 4 1.0000 1.0000 0.9998 0.9985 0.9964 0.9892 5 1.0000 1.0000 0.9996 0.9974 0.9938 0.9830 6 1.0000 0.9999 0.9993 0.9958 0.9906 0.9755 7 1.0000 0.9999 0.9987 0.9939 0.9866 0.9669 8 1.0000 0.9998 0.9980 0.9916 0.9819 0.9573 9 1.0000 0.9997 0.9971 0.9891 0.9768 0.9471 10 1.0000 0.9995 0.9960 0.9864 0.9712 0.9363 11 0.9999 0.9993 0.9948 0.9835 0.9653 0.9252 12 0.9999 0.9992 0.9936 0.9805 0.9592 0.9139 13 0.9998 0.9990 0.9924 0.9774 0.9530 0.9024 14 0.9998 0.9987 0.9913 0.9743 0.9467 0.8908 15 0.9998 0.9985 0.9902 0.9711 0.9403 0.8793                           Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

98. Credit Risk Table 6. Discrete time distribution function of first time of default in a time horizon of 15 years given rating at time 0.         1 2 3 4 5 6 7 8 AAA 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 AA 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0001 0.0002 A 0.0000 0.0000 0.0001 0.0002 0.0004 0.0007 0.0013 0.0020 BBB 0.0000 0.0002 0.0007 0.0015 0.0026 0.0042 0.0061 0.0084 BB 0.0001 0.0007 0.0018 0.0036 0.0062 0.0094 0.0134 0.0181 B 0.0006 0.0025 0.0059 0.0108 0.0170 0.0245 0.0331 0.0427         Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

99. Credit Risk Table 6. Discrete time distribution function of first time of default in a time horizon of 15 years given rating at time 0.         9 10 11 12 13 14 15 AAA 0.0000 0.0000 0.0001 0.0001 0.0002 0.0002 0.0002 AA 0.0003 0.0005 0.0007 0.0008 0.0010 0.0013 0.0015 A 0.0029 0.0040 0.0052 0.0064 0.0076 0.0087 0.0098 BBB 0.0109 0.0136 0.0165 0.0195 0.0226 0.0257 0.0289 BB 0.0232 0.0288 0.0347 0.0408 0.0470 0.0533 0.0597 B 0.0529 0.0637 0.0748 0.0861 0.0976 0.1092 0.1207         Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

100. Operational Risk INTRODUCTION The Operational risk (OpRisk) management has become an increasingly important component in financial institutions and plays a key role in the development of integrated risk management. OpRisk frequently arises due to other type of risks, and the size of an OpRisk may be intensely impacted by market or credit risk factors. The definition of OpRisk is defined by Basel II as “The risk of direct or indirect loss resulting from inadequate or failed internal processes, people and systems or from external events”. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

103. Operational Risk In recent years, considerable importance has been given to model the Oprisk in the financial world. As a result of its relevance, the Basel Committee on Banking Supervision (BCBS) has included a specific section on Oprisk in the document “International Convergence of measures and capital standards: a Revised Framework”, known as Basel II, a specific section on OpRisk. OpRisk measurement focuses on the calculation of capital for OpRisk and Basel II offers three methods for calculating OpRisk. One of the three OpRisk methods, the “advanced measurement approach (AMA)” is the most frequently used method. The Loss distribution approach (LDA) in the banking industry is one of the paradigms of the AMA in the banking sector and actuarial mathematics, in particular insurance sector. An understanding of loss processes is a crucial element in the determination of OpRisk. Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

106. Operational Risk Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

107. Operational Risk THANK YOU VERY MUCH Arunachalam (UNAL) Modelos estocásticos para la valoración de riesgo financieroXXIII Seminario financiero: Las finanzas cu

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