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An Analytic Framework for Computing Value-at-Risk in Incomplete Markets with Credit Risk Mårten Grebäck Abstract This master thesis presents a framework for analytic computation of Valueat-Risk for a portfolio of arbitrary derivatives. The underlying securities are modelled by jump-diffusion processes, where the number of jumps are Poisson distributed and the diffusion processes are driven by Brownian motions. The method is based on a delta-gamma approximation of the portfolio value and Fourier inversion of the characteristic function. In the second half of the thesis credit risk is introduced by a default event indicator variable. In case of a default, a certain fraction of the defaulted option’s value is subtracted from the portfolio value. Several examples are presented illustrating the effects on Value-at-Risk with this extension. Acknowledgements I would like to thank my supervisor Henrik Hult, at the institution of mathematical statistics, KTH, for always giving me guidance when needed. Mr Hult’s remarks and help has been invaluable. I would also like to thank Kaj Nyström and Jimmy Skoglund at the financial risk control group of Swedbank/Föreningssparbanken for helping me with the subject of my thesis. I am also most grateful for the motivating company and useful opinions of my fellow-students. Contents Introduction 7 1 Preliminaries 9 1.1 Value-at-Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Credit Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3 Incomplete Markets . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4 Delta-Gamma Approximation . . . . . . . . . . . . . . . . . . . . 14 2 The Framework 17 2.1 Portfolio Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Delta-Gamma Approximation . . . . . . . . . . . . . . . . . . . . 18 2.3 Characteristic Function . . . . . . . . . . . . . . . . . . . . . . . 18 2.4 Fourier Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.5 Conditional Value-at-Risk . . . . . . . . . . . . . . . . . . . . . . 21 2.6 Delta-Gamma-Vega Approximation . . . . . . . . . . . . . . . . . 22 2.7 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3 Credit Risk 3.1 Portfolio Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 29 3.2 Stochastic Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3 Delta-Gamma Approximation . . . . . . . . . . . . . . . . . . . . 31 3.4 Default Component . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4 Conclusions 37 A Examples: Market Risk 39 B Examples: Credit Risk 45 Bibliography 51 Introduction Sensitivity measures such as delta, gamma and vega1 describes different aspects of a portfolio of financial derivatives. Every night, banks (and other financial institutions) calculate these measures, which results in an enormous amount of data. These data are valuable for each specific trader in managing their allocated risk. But the Greeks are not sufficient for senior management in measuring the risk in a bigger perspective, interesting for the bank as a whole. Value-at-Risk tries to provide such a measure. The Value-at-Risk (VaR) framework is not only being used by financial institutions, but by central bank regulators as well. They use it in determining how much capital a bank must keep to protect themselves, and their customers, from future losses. Simulations of VaR, for a large portfolio, often takes hours to run and demands a great deal of computational power. An analytic framework for computing VaR would make life easier for many traders and risk managers. Duffie and Pan (2001) presents such a framework. In this master thesis we implement this approach and investigate its pros and cons. Extensions are made for managing vega risk and the computation of conditional Value-at-Risk. The underlying securities are modelled by jump-diffusion processes, where the number of jumps are Poisson distributed and the diffusion processes are driven by Brownian motions. The size of the jumps are assumed to be Gaussian. The portfolio value (or return) at time T , where T is the time horizon for the VaR computation, is approximated by a second order Taylor expansion. This approximation is called the “delta-gamma” approximation. From this form we are able to reach the characteristic function, which after Fourier inversion gives us the cumulative distribution function. This is carried out in Chapter 2. 1 Delta (∆), vega (ν), theta (Θ), rho (ρ), and gamma (Γ) are usually referred to as the Greeks. They are defined as the derivative of the portfolio value w.r.t., respectively, the underlying price, the volatility, the time, and the interest rate. Γ is defined as the second derivative of the portfolio value w.r.t. the underlying price. 7 Market risk as well as credit risk is managed within the framework. In Chapter 3 we introduce credit risk. In case of a default the market value of the portfolio is decreased by some exogenously given fraction of the time T value of the defaulted contract. Individual, as well as common, default intensities are modelled with Cox-Ingersoll-Ross processes. Basically, the same procedure as in Chapter 2 is repeated, but with some modifications for incorporation of credit risk. Examples can be found at the end of each chapter, and in the appendices. 8 CHAPTER 1 Preliminaries 1.1 Value-at-Risk In the mid-nineties Value-at-Risk became a widely used measure for managing market risk. What made VaR exceptional was that it provided an easily understood, single number, measure for the risk of a portfolio. The definition of VaR is: Given a time horizon T and confidence level p, Value-at-Risk is the value x such that the probability of loosing more than x, during the time interval [0, T ], is less than or equal to 1 − p. In other words it gives you a lower limit of what you, with some given probability, can expect your portfolio to be worth at some future time T . Commonly used time horizons and confidence levels are, respectively, 1 day or 2 weeks and 1 or 5 percent. Two portfolios with exactly the same VaR may have different potential losses beyond the quantile of interest. Artzner et al. (1999) suggests a, by their definitions, coherent risk measure called conditional Value-at-Risk 1 to cover this shortage. Conditional Value-at-Risk (C-VaR) tells us how much we can expect to loose, given that we are in the far end of the tail (e.g. beyond the 5% quantile). There are two main approaches for calculating VaR. The historical simulationand the model building approach. In the historical simulation approach past data is used in the following manner. Suppose that we want to calculate the one day 99%-VaR and that we have access to data for the last two years, or 500 days.2 We start by identifying market variables that affect the portfolio, e.g. equity prices and interest rates, and note their movements. This gives 1 Other 2 One names that are used are Tail Conditional Expectation and Tail VaR. year has approximately 250 trading days. 9 1.1. VALUE-AT-RISK CHAPTER 1. PRELIMINARIES Probability density function 99% Value− at−Risk 0 Figure 1.1: 99% Value-at-Risk over some time horizon T . us 500 scenarios for how the price of our portfolio may change from today till tomorrow. For each scenario we calculate tomorrow’s portfolio value. The 1% quantile, or the fifth worst scenario, tells us the 99%-VaR. For an analytical model the historical approach is of no interest. The biggest concern using the model building approach is the derivation of the probability density function of the portfolio. For an approximation of the density function certain assumptions have to be made about the movements of the underlying risk factors, and how these movements will affect the value of the portfolio (i.e. the pricing model). In most models it is assumed that the underlying risk factors, in one way or another, are normally distributed. It is known that the “normal” assumption often leads to under-estimation of potential losses. To come to terms with that problem the usage of fat tailed distributions have arisen. Fat tailed distributions implies larger VaR at high confidence levels. Student’s-t distribution (or just the “t-distribution”) is one example of a distribution with fat tails. The probability density function, ft (x), of the t-distribution is defined by ft (x) = Tn x2 1+ n − n+1 2 , −∞ < x < ∞ , where n is the degrees of freedom (d.f.) and Tn is a constant depending on the exogenously given variable n. As d.f. goes to infinity, student’s-t approaches the standard normal distribution. Figure 1.2 illustrates the difference between the probability density function of a normally distributed random variable and a t-distributed random variable. 10 CHAPTER 1. PRELIMINARIES 1.2. CREDIT RISK Normal probability density function "Fat tailed" probability density function 0 Figure 1.2: The probability density functions of a normally distributed random variable and of a t-distributed random variable. Another way of increasing the risk for large losses is by including jumps in the underlying price process. In the jump-diffusion models this is done by adding a compound Poisson process to the diffusion process with some exogenously given intensity λ. The extreme losses in the jump-diffusion model shows up much further out in the tail than the 1% quantile. Duffie and Pan (1997) makes a comparison between a jump-diffusion and a pure diffusion model3 . This shows that with an expected frequency of roughly once every 140 years, one will lose overnight at least one quarter of the value of one’s position. In the plain vanilla model, one would have to wait far longer than the age of the universe for this to happen. They also bring up figures which shows that between 1986 and 1996 there was numerous daily returns, in many markets, of at least 5 standard deviations in size. In the plain vanilla model this is expected to be the case less than once per million days. Jaschke (2002) brings up some other aspects about the drawbacks of VaR. 1.2 Credit Risk While market risk means the risk of unexpected changes in prices and rates, credit risk concerns the risk that you will suffer a financial loss because your counterparty not is able to meet his, or hers, obligations. We will call such an event a default event. A default event may be triggered by a number of reasons, for example by a decline in the credit worthiness of a counterparty or by the fact that a firm not is able to pay back its debts. The risk taker requires compensation for this risk. In bond markets, for example, riskier issues have to promise a higher yield than the not so risky issues. 3 Both models have IID shocks, constant mean return, and a constant annual volatility of 15%. The intensity λ is one per year and the jump standard deviation is set to 10 percent. 11 1.2. CREDIT RISK CHAPTER 1. PRELIMINARIES In the pricing of credit risk and credit risk derivatives the structural models and the reduced-form models are most common. Structural models (or the firm’s value models) are concerned with modelling and pricing credit risk that is specific to a particular corporate obliger (a firm). The way in which a default event occurs is by the movement of the firm’s value. Several default triggers are possible, e.g.: • As soon as the firm’s value hits a barrier the default event is triggered. The barrier can be either constant or time-dependent. For example the default can occur as soon as the firm’s value falls below the discounted value of its outstanding debts. • A default can only occur at the maturity of an outstanding debt. The firm continues to operate until this happens. The main issue in this framework is the construction of a meaningful process describing the movements of the firm’s value. There is an extensive amount of literature covering this field, e.g. Merton (1974), Black and Cox (1976), Longstaff and Schwartz (1995) and Ericsson (2000). In the reduced-form models (or the intensity-based models) a jump process, whose first jump determines the default, defines the default process. The probability of a default in a specific time interval is given by the intensity of the jump process. Poisson processes are common to use when modelling rare events. Two properties of the Poisson process, which makes it an especially appropriate process to choose when modelling defaults, are: • The Poisson process has no memory, i.e. it is not more likely for a jump to occur only because it has been a long time since the last jump. • The probability of two or more jumps at exactly the same time is of small order. This may, a bit sloppy, be expressed as P (#jumps [0, t] > 1) = O(t2 ), where t is small. A jump in the Poisson process says that a default event has occurred, but it does not tell the actual loss. The fractional loss, or the loss-given-default (LGD), is often assumed to be exogenously given. In case of a default, lets say at time t, the value of the portfolio is decreased by the LGD multiplied with the positions value at time t. Literature concerning the intensity-based approach is, just to name a few, Jarrow and Turnbull (1995), Duffie et al. (1996) and Schönbucher (1996, 1998a, 1998b). Two commercial products for quantifying credit risk are CreditMetricsTM , developed by J.P. Morgan, and CreditRisk+TM , developed by Credit Suisse First Boston. Cossin and Pirotte (2001) gives an overview of the existing frameworks for managing credit risk. 12 CHAPTER 1. PRELIMINARIES 1.3 1.3. INCOMPLETE MARKETS Incomplete Markets Imagine a market with n tradable assets whose prices are driven by some stochastic processes S1 , . . . , Sn . Under the assumption of absence of arbitrage, there exists an equivalent probability measure P under which the discounted price processes S1 /B, . . . , Sn /B are martingales. When the market is complete, all possible cash flows can be replicated by a self-financing trading strategy.4 If the market is complete there exists an unique equivalent probability measure under which pricing is carried out. If all possible cash flows can not be replicated by a self-financing trading strategy the market is incomplete. The market may still be free of arbitrage, in the sense that there exists equivalent martingale measures,5 but there is no unique equivalent probability measure. Björk (1998) explains completeness, and absence of arbitrage, by this meta-theorem: Meta-theorem Let M denote the number of underlying traded assets in the model excluding the risk free asset, and let R denote the number of random sources. Generically we then have the following relations. • The model is arbitrage free if and only if M ≤ R. • The model is complete if and only if M ≥ R. • The model is complete and arbitrage free if and only if M = R. For an incomplete market the problem comes down to which of the existing martingale measures that should be chosen for a valuation of the portfolio. If the underlying asset, Sk , is following a jump-diffusion model a natural martingale measure, Q, can be derived by changing the drift of the driving Brownian motion. This is done to obtain the drift under the new measure. Conditioning on the number of jumps, j, gives us a log-normal distribution under Q. Hence, to compute a price of a derivative we take expectation under Q and since the conditional distribution is log-normal we can apply the Black-Scholes formula (j) to obtain the price, C(Sk ), associated with j jumps. Hence the unconditional value of a portfolio consisting of n European options would be EQ [C(S)] = n ∞ (j) C(Sk )pj k=1 j=0 where S = (S1 , . . . , Sn ) and pj is the probability of j jumps. (Infinity is “chosen” depending on the demands of accuracy. For a Poisson distributed number of jumps it can be pointed out that the probability of 10 jumps in one year, with an annual jump intensity of 2, is approximately 3.82 · 10−5 .) 4 By “self-financing” a trading strategy where no additional means are provided, or withdrawn, after t = 0 is intended. 5 A probability measure, P ∗ , and its associated expectation operator, E∗ , forms a martingale measure. 13 1.4. D-G APPROXIMATION 1.4 CHAPTER 1. PRELIMINARIES Delta-Gamma Approximation The delta-gamma approximation is based on a second order Taylor expansion of the portfolio value. What makes the delta-gamma model especially appropriate is that the characteristic function of the quadratic form is known analytically. The value of a portfolio consisting of n different positions can be represented by 1 V = A + ∆ X + X ΓX 2 where A is a scalar, ∆ a vector of the portfolio deltas in Rn , Γ is a matrix of the gammas for the different assets with dimension n × n and X is normally distributed with mean 0 and covariance matrix Σ. Solve the eigenvalue problem D ΛD = 12 (Σch ) ΓΣch , where Σch denotes the Cholesky factorisation of the positive definite matrix Σ, Λ is a diagonal matrix with the eigenvalues, λ1 , . . . , λn , on its diagonal and D is a matrix with the corresponding eigenvectors as rows. Let X = Σch Y and α = ∆ Σch , where Y is standard normal. Then V can be expressed on a quadratic form, as a linear combination of independent random variables, as V = A+ n (αk Yk + λk Yk2 ) k=1 n αk = A+ λk Yk + 2λk k=1 n αk2 = A+ λk Zk − 4λk 2 α2 − k 4λk k=1 where Zk is a non-central χ2 distributed random variable with 1-degree of freeα2 dom and non-centrality parameter δk2 = 4λk2 . Now the characteristic function of k V , ϕ(t) = E[ eitV ], can be calculated as (see, for example, Davies (1973)), E[ eitV ] = n exp it k=1 λk δk2 /(1 − 2itλk ) − Πnk=1 (1 − 2itλk )1/2 α2k 4λk + itA . Further examining the behaviour of the delta-gamma approach we take a look at the delta approximation. Suppose that we only got one underlying asset, x, and that we use the Black-Scholes pricing formula to price a European call option on that asset. Since ∆ > 0 and Γ > 0 the option price, f (x), is an increasing, convex, function of x, where f ! C 1 (i.e. the space of functions, f , with continuous first derivatives). For small changes in the underlying security’s price, we know that the first order Taylor expansion is a reasonably good approximation of the function value, i.e. f (x + a) ≈ f (x) + ∆ · a. The delta approximation over-estimates the loss of a long position, and underestimates the loss of a short position, when the price of the underlying asset increases. This is illustrated in Figure 1.3. 14 CHAPTER 1. PRELIMINARIES 1.4. D-G APPROXIMATION For the second order Taylor expansion, or the delta-gamma approximation, we have, for f ! C 2 , 1 f (x + a) ≈ f (x) + ∆ · a + Γ · a2 . 2 Option Price f(x) Figure 1.4 shows that the delta-gamma approximation under-estimates the loss of a long position, and over-estimates the loss of a short position. f′(x)a a x x+a Figure 1.3: The delta approximation over-estimates the loss of a long position, and Option Price f(x) under-estimates the loss of a short position. a 2 f′(x)a + 0.5f′′(x)a x x+a Figure 1.4: The delta-gamma approximation under-estimates the loss of a long position, and over-estimates the loss of a short position. 15 1.4. D-G APPROXIMATION CHAPTER 1. PRELIMINARIES 16 CHAPTER 2 The Framework In this chapter we follow the model set-up by Duffie and Pan (2001). 2.1 Portfolio Dynamics Consider a portfolio consisting of m derivatives, C1 , . . . , Cm , written on n underlying assets. Each of the price processes, S1 , . . . , Sn , for the underlying securities are modelled by a jump-diffusion process. The total value of the portfolio then m becomes C(R) = k=1 Ck (R), where R = (log(S1 ), . . . , log(Sn )) denotes the log-price process. Suppose that1 d RT = R0 + (M − λµ) · T + √ N (T ) T Σ X0 + ch (µ + Qch Xj ), (2.1) j=1 where M , in Rn , is the mean-return vector, Xj is standard normal in Rn , for j = 0, 1, 2 . . ., and Xj and Xk are independent for j = k. Further, Σ and Q are symmetric, positive semi-definite matrices in Rn×n and N (T ) is a Poisson process, independent of X1 , X2 , . . ., with constant arrival intensity λ. Thus, conditional on j jumps before T , we have a multi-variate normally distributed random variable RT with mean R0 + (M − λµ)T + jµ and covariance ΣT + jQ. The purpose of the model is to compute P (C(RT ) ≤ cq ), where C(RT ) is the portfolio value at time T and cq is the q-th quantile portfolio value. For this we need the cumulative distribution function (CDF) of C(RT ). We intend to reach the CDF by Fourier inversion of the characteristic function. From a quadratic form of independent normal random variables it is shown that the characteristic function may be readily calculated. 1 Let Ωch denote the Cholesky factorisation of some matrix Ω. That is Ω = Ωch (Ωch ) . 17 2.2. D-G APPROXIMATION 2.2 CHAPTER 2. THE FRAMEWORK Delta-Gamma Approximation A Taylor expansion around initial value R0 gives the delta-gamma approximation of the portfolio value as C(RT ) ≈ C ∆,Γ (RT ) = C(R0 ) + ∆ (RT − R0 ) 1 + (RT − R0 ) Γ(RT − R0 ) 2 where m ∂Ci (R) ∆k = ∂Rk i=1 , Γk,l m ∂ 2 Ci (R) = ∂Rk ∂Rl i=1 R=R0 , k, l = 1, . . . , n. R=R0 In the case of European options delta and gamma are easily computed. Conditional on j jumps, compute the Black-Scholes delta and gamma (under the measure derived in the way laid out in Section 1.3). Note that the underlying assets in Black-Scholes’ framework follow a diffusion process, and that we use the logarithm of that process. Let ∆(BS) denote ∂C ∂S . This yields ∆(R) = and Γ (R) ∂C ∂S ∂C ∂eR = · = ∆(BS) · = ∆(BS) eR ∂R ∂S ∂R ∂R ∂C ∂S ∂ 2 S ∂C ∂2C ∂ ∂ 2 C ∂S · · + = = · = ∂R2 ∂R ∂S ∂R ∂R∂S ∂R ∂R2 ∂S ∂ ∂C ∂ 2 S ∂C ∂S ∂S = · + · = ... ∂S ∂S ∂R ∂R ∂S ∂R2 = Γ(BS) e2R + ∆(BS) eR ˜ where J˜ gives reasonable accuracy, weighted Add up for j ranging from 0 to J, by the probability, pj , of j jumps. Since the jump process is Poisson distributed, the probability of N (T ) = j jumps before T is pj = e−λT (λT )j . j! We chose “reasonable accuracy” to be J˜ = 30 jumps.2 This gives us ∆= 30 (R) pj ∆j , Γ = j=0 2.3 30 (R) p j Γj . j=0 Characteristic Function For the computation of P (C ∆,Γ (RT ) ≤ cq ) we need the distribution function of C ∆,Γ (RT ). One way of getting this is via the characteristic function, ϕ(t) = ∆,Γ E[ eitCT ]. Throughout this section j is fixed and the portfolio is assumed to consist of n contracts. 2 With an annual jump arrival intensity of 4, the probability of 30 jumps during one year is 7.96 · 10−17 . 18 CHAPTER 2. THE FRAMEWORK 2.3. CHAR. FUNCTION ch We define ηj = (ηj1 , . . . , ηjn ) as the eigenvalues3 of 12 (Θch j ) Γ Θj , i.e. 1 ηj 1 ch ch .. Dj Dj = (Θj ) Γ Θj , . 2 ηjn where Dj is an orthogonal, full matrix whose rows are the corresponding eigenvectors to (ηj1 , . . . , ηjn ). Denote the k-th eigenvector and eigenvalue by Djk and ηjk , respectively. Given j jumps before T , RT has mean-value Mj = R0 + (M − λµ)T + jµ and covariance matrix Θj = ΣT + jQ. Let Z be standard normal, i.e. Z ∼ N (0, I) where I is the identity matrix. This enables us to 4 represent RT as Θch j Z + Mj . With these preparations at hand, and using the fact that Θj is symmetric, we can express C ∆,Γ (RT ), conditional on j jumps, as 1 C ∆,Γ (RT ) = A + B RT + RT ΓRT 2 1 ch ch = A + B (Θch j Z + Mj ) + (Θj Z + Mj ) Γ(Θj Z + Mj ) = . . . 2 1 = A + B Mj + Mj ΓMj + Z (Θch j ) (B + ΓMj ) 2 + (Dj Z) Λj (Dj Z), (2.2) where 1 A = C(R0 ) − ∆ R0 + R0 ΓR0 , 2 B = ∆ − ΓR0 , Λj = diag(ηj1 , . . . , ηjn ). Remembering that the goal of our manipulations is to express C ∆,Γ (RT ) on a quadratic form of independent normal variables (see Section 1.4), we define δj = (δj1 , . . . , δjn ) by δjk = (Djk ) aj 2ηjk and 1 γj = A + B Mj + Mj ΓMj − ηjk (δjk )2 , 2 n k=1 where aj = (Θch j ) (B C ∆,Γ (RT ) = γj + = γj + = γj + + ΓMj ). Equation (2.2) may now be re-written as n k=1 n k=1 n ηjk (δjk )2 + n Zk ak + k=1 n (Djk Zk ) ηjk (Djk Zk ) = . . . k=1 k Zk ak − 2ηjk Djk Zk δj + n ηjk (Djk Zk + δjk )2 k=1 ηjk (Djk Zk + δjk )2 . k=1 3 All eigenvalues are assumed to be non-zero. ch ch ch ch = T ] = E[Θj Z + Mj ] = Mj and E[(Θj Z)(Θj Z) ] = Θj E[ZZ ](Θj ch) ch ch Θj I(Θj ) = Θj . 4 E[R 19 2.4. FOURIER INVERSION CHAPTER 2. THE FRAMEWORK Since Dj is an orthogonal matrix, Dj Z is standard normal, and C ∆,Γ (RT ) is finally on a quadratic form. Davies (1973) gives us the characteristic function by ϕj (u) = E[exp iuC ∆,Γ (RT ) NT = j] n exp iu k=1 ηjk (δjk )2 /(1 − 2iuηjk ) + iuγj = . Πnk=1 (1 − 2iuηjk )1/2 The unconditional characteristic function is acquired by adding up for j ranging from 0 to a point of reasonable accuracy, weighted by the probability, pj , of j jumps. That is ϕ(u) = J˜ pj ϕj (u) , j=0 where cutting of at J˜ jumps gives “reasonable accuracy”. 2.4 Fourier Inversion The inversion theorem proves that the distribution function is uniquely given by the characteristic function by (see e.g. Gut (1995)) ∞ 1 − e−itx 1 ϕ(t) dt. F (x) − F (0) = 2π −∞ it We need an explicit expression for the distribution function, hence the inversion theorem is not satisfactory because of the constant term F (0). Gil-Pelaez (1951) derived an explicit formula for F (x), as 1 ∞ Im[e−itx ϕ(t)] 1 dt. F (x) = − 2 π 0 t Applying this to our problem yields FC ∆,Γ (cq ) = P (C ∆,Γ T ∞ ∞ 1 Ij (u, cq ) 1 du (RT ) ≤ cq ) = − pj 2 π j=0 u 0 where, for u ! R, Ij (u, cq ) = Im[e−iucq ϕj (u)] = Φj (u)Ψj (u, cq ) where Φj (u) = and Ψj (u, cq ) = sin n exp −2u2 k=1 (ηjk δjk )2 /(1 + 4(uηjk )2 ) Πnk=1 (1 + 4(uηjk )2 )1/4 n n k k 2 uη (δ ) 1 j j arctan(2uηjk ) + + u(γj − cq ) . 2 1 + 4(uηjk )2 k=1 k=1 20 (2.3) CHAPTER 2. THE FRAMEWORK 2.5 2.5. CONDITIONAL VAR Conditional Value-at-Risk Given that the time T value of the portfolio, C(RT ), is less than the q-th quantile value, cq , conditional VaR is defined as the mean of C(RT ), i.e. C-V aRC(RT ) (cq ) = E [C(RT ) | C(RT ) < cq ] cq 1 = xfC (x)dx, FC (cq ) −∞ (2.4) where fC (c) is the probability density function of C(RT ). The framework being used in this thesis only gives the cumulative distribution function, FC (c), between c0 and cq . c0 ∆c cq Figure 2.1: The cumulative distribution function between c0 and cq . The CDF is approximated from n data points and the step size is ∆c. Splitting Equation (2.4) in two, yields one integral over the known section [c0 , cq ] and one over the unknown section (−∞, c0 ], 1 FC (cq ) cq c0 1 xfC (x)dx = xfC (x)dx + xfC (x)dx . FC (cq ) −∞ −∞ c 0 cq I1 I2 For the calculation of I2 , let ∆c be the step size and n the number of data points (x) yields, on [c0 , cq ]. The standard differential approximation f (x) ≈ F (x+∆x)−F ∆x n−1 FC (c0 + (k + 1)∆c) − FC (c0 + k∆c) · ∆c I2 = (c0 + k∆c) ∆c k=0 = n−1 [(c0 + k∆c) {FC (c0 + (k + 1)∆c) − FC (c0 + k∆c)}] . k=0 21 2.6. D-G-V APPROXIMATION CHAPTER 2. THE FRAMEWORK All the parameters are known, so I2 may be readily calculated. For the (−∞, c0 ] interval, some kind of approximation of FC (c) has to be done. At least, far out in the tail FC (c) = δ · (−c)−β is a reasonable approximation, where β is a constant and δ is given by FC (c0 ) = P (C(RT ) ≤ c0 ) = q0 ⇒ δ · (−c0 )−β = q0 ⇔ δ = q0 · (−c0 )β . Hence, given β δ is known. It should be pointed out that FC (c) is taken, almost, out of the blue. Many other approximations may work just as well. The justification for this particular choice comes from regularly varying distributions. These models may be appropriate for modelling financial data (see e.g. Embrechts et al. (1997)). For the estimation of β the least square method is used. That is, let FCβ (c) = q0 · (−c0 )β (−c)−β and solve the minimisation problem min β n 2 FC (ck ) − FCβ (ck ) . k=1 With β at hand, I1 can be calculated by c0 I1 = xfC (x) dx = FC (x) = fC (x) = δ · (−β)(−x)−β−1 (−1) −∞ c0 c0 = −x · δ(−β)(−x)−β−1 dx = −δ β(−x)−β dx −∞ β (−c0 )1−β , if β > 1. = −δ 1−β −∞ Dividing the sum of I1 and I2 with the probability that C(RT ) is less than cq gives us the conditional VaR. 2.6 Delta-Gamma-Vega Approximation In the delta-gamma approximation, as well as in the Black-Scholes framework for pricing European options, the volatility, σ, is assumed to be constant. If a variable volatility is desirable, vega risk can be introduced to the model. Vega, ν, is defined by ν = ∂C ∂σ , i.e. vega tells how sensitive the portfolio is against changes in the volatility. (The Black-Scholes framework is applicable under the assumption that σ is constant between today and the time horizon, T , for the VaR computation and between T and the time of maturity for the options.) A second order Taylor expansion around initial values R0 and σ0 , neglecting the second order derivatives with respect to σ and w.r.t. σ and R, yields the “delta-gamma-vega” approximation C(RT , σT ) ≈ C ∆,Γ,ν (RT , σT ) = C(R0 , σ0 ) + ∆ (RT − R0 ) 1 +ν (σT − σ0 ) + (RT − R0 ) Γ(RT − R0 ), 2 22 (2.5) CHAPTER 2. THE FRAMEWORK 2.7. ERROR ANALYSIS where ∆ and Γ are as before, and m ∂Ci (R) νk = ∂σk i=1 , k = 1, . . . , n R=R0 for a portfolio consisting of m derivatives written on n underlying assets. The vega term is not affected by the fact that the logarithm of the underlying assets’ diffusion processes are being used, as the delta and the gamma terms were. (In the same way as in Section 2.2, ν is achieved by adding up for j ranging from ˜ where J˜ gives reasonable accuracy, weighted by the probability of j 0 to J, jumps.) Given j jumps before T , RT has the mean-value Mj = R0 + (M − λµ)T + jµ and covariance matrix Θj = ΣT + jQ. The diagonal of Θj is the volatility, i.e. the mean-value of σT is Mjσ = diag(Θj ). Define R̃T in the following way R̃T = σT RT ∼N Mjσ Mj σ Θj Covj , , Covj Θj where Covj = Cov(σT , RT ) < 0 and Θσ is the covarians matrix for the volatilities. Equation (2.5) may now be re-expressed as ˜ (R̃T − R̃0 ) + 1 (R̃T − R̃0 ) Γ̃(R̃T − R̃0 ), C ∆,Γ,ν (R̃T ) = C(R̃0 ) + ∆ 2 where ˜ = ∆ and Γ̃ = ν ∆ 0n×n 0n×n 0n×n Γ . ˜ is a vector in Rn and Γ̃ is a matrix in R2n×2n .) The approximation of the (∆ portfolio value is now on the same form as in Section 2.2, and the remainder of the computations may be carried out in the same way. 2.7 Error Analysis For the evaluation of Equation (2.3) numerical integration is used in the following way, Kj J 1 I ((k + 1/2)h , c ) 1 j j q . pj − P (C ∆,Γ (RT ) ≤ cq ) ≈ 2 π k + 1/2 j=0 k=0 For this numerical method three different types of errors will have to be handled. The truncation errors introduced by J, Kj < ∞, and the discretisation error introduced by hj > 0. Assign an error tolerance, α, for the total error. This gives each of the J + 1 terms an assigned error tolerance of α/(J + 1) divided 23 2.7. ERROR ANALYSIS CHAPTER 2. THE FRAMEWORK by pj . This tolerance is then split in two, half for the error introduced by hj and the other half for the error introduced by Kj . The truncation error ∞ introduced by J < ∞ is the easiest to handle. Simply choose J so that j=J+1 pj is less than the desired accuracy. Moving on to the error introduced by Kj , Davies (1980) handles it in the following manner. If |ϕ(u)| ≤ B(u), and B(u) is a monotonically decreasing function in u (for u ≥ U, U = (Kj + 1/2)hj ), the truncation error is bounded by 1 π ∞ k=Kj +1 Ij ((k + 1/2)hj , cq ) ≤ k + 1/2 ∞ u=U B(u) du . πu Davies considers three different bounds on I, resulting in (1) Bj (U ) 2 Φj (U ) Π4(U ηjk )2 >1 = πJ (2) Bj (U ) = Φj (U ) , πU 2 βj2 where B (3) only is valid if n k=1 (3) Bj (U ) = 1 + 4(U ηjk )2 1/4 4(U ηjk )2 , 2.5 Φj (U ), π ln(1 + 4(U ηjk )2 ) + 2β 2 U 2 ≥ 1. The truncation (1) (2) (3) error is bounded by the minimum of (Bj , Bj , Bj ). For managing the discretisation error the following expression, derived by Davies (1973), is examined. P (C ∆,Γ ≤ cq ) + ∞ (−1)n {P (C ∆,Γ < cq − 2πn/hj ) − P (C ∆,Γ > cq + 2πn/hj )} n=1 = ∞ 1 1 − Im[ϕj ((k + 1/2)hj )e−i(k+1/2)hj cq ]/(k + 1/2). 2 π k=0 A minimisation of the summation over n is desirable. This can be done by choosing hj in a way that makes max{P (C ∆,Γ < cq − 2πn/hj ), P (C ∆,Γ > cq + 2πn/hj )} (2.6) less than the allocated error. To find bounds on Equation (2.6), consider the ∆,Γ moment generating function, E[euC ], of C ∆,Γ . Let ξ(u) denote its logarithm. Conditional of j jumps and for C ∆,Γ ≥ x the following holds, E [IC ∆,Γ >x − eu(C ∆,Γ −x) | N (T ) = j] ∆,Γ = P (C (RT ) > x | N (T ) = j) − eξj (u) e−ux ≤ 0 ⇔ P (C ∆,Γ (RT ) > x | N (T ) = j) ≤ eξj (u) e−ux , where IX>x is the indicator variable of X > x. Let x = ξj (u). This gives us the desired bounds on Equation (2.6), P (C ∆,Γ (RT ) > x | N (T ) = j) ≤ eξj (u)−uξj (u) . 24 CHAPTER 2. THE FRAMEWORK 2.8. EXAMPLES The logarithm of the moment generating function is computed by (see e.g. Johnson et al. (1994)) ! " ∆,Γ ξj (u) = ln E[ euC | N (T ) = j ] = n uηjk (δjk )2 k=1 2.8 1 − 2uηjk 1 ln(1 − 2uηjk ). 2 n + uγj − k=1 Examples In the following example the 1-day Value-at-Risk for a portfolio of 17, at-themoney, European call-options is reviewed. They are written on 17 different underlying assets, whose price processes follows Equation (2.1). Today’s value, S0 , is set to 100 units for all contracts, the annual jump arrival intensity, λ, is set to 4 and the risk free interest rate is assumed to be 5 percent. The expected jump amplitude is set so that µj + Qj,j /2 = 0, for j = 1, . . . , 17, M = 0 and let Σ = 2λQ = Cov, where the covariance matrix, Cov, can be found in Appendix A. Further, the error tolerance is set to 0.00001. In Figure 2.2 the “actual” and the delta-gamma approximated Value-at-Risk are plotted. In the “actual” approach a simulation of the underlying securities’ time T prices is done and the portfolio value is evaluated with the Black-Scholes formula. In the delta-gamma approach the time T value of the portfolio is simulated, using the delta-gamma approximation of the portfolio value. In both cases 100 000 simulations were made. In Figure 2.3 the analytical and the delta-gamma Value-at-Risk are plotted. Using the method laid out in Section 2.5 conditional VaR is computed to be 32 units, with a β-value of 4.67. 1-day Analytic solution Delta-Gamma simulation 1% 33.3367 33.3777 5% 22.4067 22.4106 1% 96.4767 96.4418 5% 65.8667 65.7183 10-days Analytic solution Delta-Gamma simulation Table 2.1: A comparison between the analytical and the simulated delta-gamma VaR for a long position in the portfolio (the results are taken from 100 000 simulations). An example where the price process is driven by pure diffusion is also presented. That is, the annual jump arrival intensity, λ, is zero. The results can be seen in Figures 2.4 and 2.5. Figure 2.6 shows the analytical 5-day5 VaR with annual jump intensities λ = 1, 2, 4. Otherwise the same data as above is being used. More examples can be found in Appendix A. 5 5-days, as in 5 trading days. That is one “ordinary” week. 25 2.8. EXAMPLES CHAPTER 2. THE FRAMEWORK 5 4.5 4 Probability (%) 3.5 3 2.5 2 1.5 1 0.5 0 160 140 120 100 80 60 40 20 Value−at−Risk Figure 2.2: 1-day VaR with an annual jump arrival intensity of 4. The dotted line is the simulation of the “actual” VaR and the solid line the simulation of the delta-gamma approximation. 5 4.5 4 Probability (%) 3.5 3 2.5 2 1.5 1 0.5 0 120 110 100 90 80 70 60 50 40 30 20 Value−at−Risk Figure 2.3: 1-day VaR with an annual jump arrival intensity of 4. The dashed line is the analytical VaR and the solid line the simulation of the delta-gamma approximation. 26 CHAPTER 2. THE FRAMEWORK 2.8. EXAMPLES 5 4.5 4 Probability (%) 3.5 3 2.5 2 1.5 1 0.5 0 40 38 36 34 32 30 28 Value−at−Risk 26 24 22 20 Figure 2.4: 1-day VaR with pure diffusion (no jump component). The dotted line is the simulation of the “actual” VaR and the solid line the simulation of the delta-gamma approximation. 5 4.5 4 Probability (%) 3.5 3 2.5 2 1.5 1 0.5 0 40 38 36 34 32 30 28 Value−at−Risk 26 24 22 20 Figure 2.5: 1-day VaR with pure diffusion (no jump component). The dashed line is the analytical VaR and the solid line the simulation of the delta-gamma approximation. 27 2.8. EXAMPLES CHAPTER 2. THE FRAMEWORK 5 4.5 4 Probability (%) 3.5 3 2.5 2 1.5 1 0.5 0 95 90 85 80 75 70 65 Value−at−Risk 60 55 50 45 Figure 2.6: 5-day VaR with annual jump intensities λ = 1, 2, 4 (respectively, dotted-, dash-dotted-, and solid line). 28 CHAPTER 3 Credit Risk Credit risk concerns the risk of a financial loss because your counterparty not is able to meet his, or hers, obligations. Such an event will be referred to as a default event. A default event may be triggered by a number of reasons, some stated in Section 1.2. The risk taker requires compensation for this risk. Financial institutions devote considerable resources to the pricing and management of credit risk. 3.1 Portfolio Value For the incorporation of credit risk the portfolio value is divided into two parts, the value component and the default component. The value component is the market value, conditional on no default within the time horizon, T , of the VaR computation. All default-dependent payoffs can be priced by discounting them with the short rate process r̄ = r + Lλ, where r is the deterministic short rate, L the loss-given-default (LGD) and λ the intensity of the default process (see e.g. Schönbucher (1997)). In the case of a portfolio of European options the market value of option i with maturity at Ti is # $ Ti λi (s) ds . Vi (Rt , λt ) = Ci (Rt ) · E exp −Li t The default component is the total loss from defaults. In case of a default affecting firm i at τi , 0 < τi < T < Ti , the value of the portfolio is reduced by the LGD multiplied with the time T value of that contract. Thus, the total value of a portfolio consisting of n contracts at time T is expressed by WT = n i=1 Vi (RT , λT ) − n i=1 29 Li Di (T )Vi (RT , λT )+ , 3.2. STOCHASTIC INTENSITY CHAPTER 3. CREDIT RISK where Di (T ) is the indicator of a default event and where it will be assumed that Li = L, ∀i. Let Vi (RT , λT )+ denote the positive part of Vi (RT , λT ) and, for analytical tractability, approximate Vi (RT , λT )+ by Vi (R0 , λ0 )+ . Further, the dependency between Di (T ) and λT is ignored.1 These approximations eliminates the dependency between the value component and the default component, which enables us to write the characteristic function as ϕ(u) ≈ ϕV (u)ϕD (u). Once the characteristic function is known, the Fourier inversion method laid out in Section 2.4 is applicable. 3.2 Stochastic Intensity Default events are modelled as the first jumps of a series of Cox processes2 with intensities λ1 (t), . . . , λn (t). Each firm’s default intensity process is defined as λi = λIi + pi λC where λIi is the intensity of a default specific to firm i and λC is the intensity of a common default which will affect firm i with probability pi . The individual and common default intensities are defined as Cox-Ingersoll-Ross processes, % dλIi (t) = κIi (λ̄Ii − λIi (t))dt + σiI λIi (t) dBiI (t) % dλC (t) = κC (λ̄C − λC (t))dt + σ C λC (t) dB C (t) where B C , B1I , . . . , BnI are standard Brownian motions and λ̄C , λ̄I1 , . . . , λ̄In denotes the long-run means. Assuming that pi = p, κIi = κC = κ, and that √ σiI = pσ C = σ, λi is also a CIR process. Let λ̄i = λ̄Ii + pλ̄C , and Bi be a Brownian motion, then λi is defined by & dλi (t) = κ(λ̄i − λi (t))dt + σ λi (t) dBi (t). The probability of survival for each contract i to time T > 0 may now be computed by (see Cox et al. (1985) for details) # $ T λi (s)ds Ft P (τi > T ) = E exp −L t = H1i (T − t, L)e−H2 (T −t,L)Lλi (3.1) where # H1i (T, L) = 1 2γe 2 (γ+κ)T (γ + κ)(eγT − 1) + 2γ $2κλ̄i /σ2 2(eγT − 1) (γ + κ)(eγT − 1) + 2γ & γ = κ + 2Lσ 2 . H2 (T, L) = Both the individual and the common default intensities are initiated at their long-run means, i.e. λIi (0) = λ̄Ii and λC (0) = λ̄C . 1 Duffie and Pan (2001) shows that the covariance between DT and λT is of small order, over short time horizons. 2 Roughly, a Cox process is a Poisson process with stochastic intensity. 30 CHAPTER 3. CREDIT RISK 3.3 3.3. D-G APPROXIMATION Delta-Gamma Approximation For the delta-gamma approximation we Taylor expand the value component, n V (RT , λT ) = i=1 Vi (RT , λT ), around its initial values. Let YT = (RT , λT ) , then V (YT ) can be approximated in exactly the same way as in Section 2.2. Delta and gamma are expressed as ∆= ∂2V ∂R12 ... .. . ... . . . ∂2V ∂Rn ∂R1 Γ= 2 ∂ V ... ∂λ1 ∂R1 . .. . . . ∂2V . . . ∂R1 ∂λn ∂V ∂V , ∂Ri ∂λi , ∂2V ∂R1 ∂Rn ∂2V ∂R1 ∂λ1 ∂2V 2 ∂Rn ∂2V ∂λ1 ∂Rn .. . .. . .. . .. . ∂2V ∂λ1 ∂Rn ∂2V ∂λ21 ∂2V ∂λn ∂Rn ∂2V ∂λn ∂λ1 ... .. . ... ∂2V ∂R1 ∂λn 2 ∂ V ∂Rn ∂λn . 2 V . . . ∂λ∂1 ∂λ n .. .. . . ∂2V . . . ∂λ2 .. . n For analytical tractability we make the, quite considerable, approximation that λT is Gaussian.3 This gives us V (RT , λT ) on a normally distributed form with mean M and covariance matrix Θ. Further, assuming independency between λ and R yields M= MR Mλ , Θ= ΘR 0n×n . 0n×n Θλ The characteristic function may now be computed, with parameters Miλ = E0 (λi (T )) = λi (0) + (1 − exp(−κT ))(λ̄Ii + pλ̄C − λi (0)) Θλi,i = Cov0 (λi (T ), λi (T )) (1 − exp(−κT ))2 2 I 1 − exp(−κT ) 2 σ λi (0) + σ (λ̄i + pλ̄C ) κ 2κ = Cov0 (λi (T ), λj (T )) (1 − exp(−κT ))2 C 1 − exp(−κT ) C λ (0) + = pσ 2 exp(−κT ) , λ̄ κ 2κ = exp(−κT ) Θλi,j in the same way as in Section 2.3. For an incorporation of vega risk, let Vt depend on Rt , λt and σt . Proceeding as in Section 2.6 will give the delta-gamma-vega approximation. 3 By modelling λt with an Ornstein-Uhlenbeck process, which is Gaussian, we could have got around this approximation. However the usage of the Ornstein-Uhlenbeck process would have given a positive probability of λt < 0. 31 3.4. DEFAULT COMPONENT 3.4 CHAPTER 3. CREDIT RISK Default Component A firm can default either from a common or an individual default event. In case of a common default the firm has some given probability of survival. Let the indicator variable ξi denote if firm i has survived a common default or not (where zero denotes survival). This probability is already defined as P (ξi = 1) = p. The default indicator can be expressed as Di (T ) = DiI (T ) + ξi DC (T ). The risk of double counting defaults is obvious, but the probability is of small order and is shown by Duffie and Pan (2001) to have negligible effect. The individual and common default indicators are triggered with the probability defined by Equation (3.1). That is # $ T pIi (T ) = P (DiI (T ) = 1) = 1 − E exp − # C C p (T ) = P (D (T ) = 1) = 1 − E exp − 0 0 T λIi (s)ds $ λ (s)ds C . Letting K(T ) denote the default component, we get the characteristic function as ϕD (u) = E e−iuK(T ) ≈ pC (T )E e−iuK(T ) DC (T ) = 1 +(1 − pC (T ))E e−iuK(T ) DC (T ) = 0 n ! n " ' ' 1 − p + pe−iuLVi (R0 ,λ0 ) ki , ≈ 1 − pC (T ) + pC (T ) i=1 i=1 where ki = 1 − pIi (T ) + pIi (T )e−iuLVi (R0 ,λ0 ) . When the characteristic function, ϕ(u), is known Value-at-Risk can be calculated in the same way as in Section 2.4. 3.5 Examples The 1-day VaR for the same portfolio as in Section 2.8 on page 25 is reviewed, but we introduce credit risk this time. Each of the 17 default intensities are assumed to have the same long-run means, λ̄i = 0.03, ∀i, and λ̄I /λ̄ = 0.2. Further, let κ = 0.25, the loss-given-default, L, is assumed to be 50%, the volatility is 0.22 and the probability for a common default to affect firm i is 80%. That is pi = p = 0.8, ∀i. In Figure 3.1 the “actual” and the delta-gamma approximated Value-at-Risk are plotted. In the “actual” approach a simulation4 of the underlying securities’ 4 100 000 simulations of the price at time T were made. 32 CHAPTER 3. CREDIT RISK 3.5. EXAMPLES time T prices is done and the portfolio value is evaluated with the Black-Scholes formula. In the delta-gamma approach the time T value of the portfolio is simulated 100 000 times, using the delta-gamma approximation of the portfolio value. In Figure 3.2 the analytical and the delta-gamma Value-at-Risk are plotted. Using the method laid out in Section 2.5 conditional VaR is computed to be 31 units, with a β-value of 3.21. 1-day Analytic solution Delta-Gamma simulation 1% 32.9319 32.9790 5% 22.0819 22.0407 1% 94.9819 94.9311 5% 64.8219 64.5547 10-days Analytic solution Delta-Gamma simulation Table 3.1: A comparison between the analytic and the simulated delta-gamma VaR for a long position in the portfolio (the results are taken from 100 000 simulations). An example where the price processes are driven by pure diffusion is also presented. That is, the annual jump arrival intensity, λ, is zero. The results can be seen in Figures 3.3 and 3.4. Figure 3.5 shows the analytical 5-day5 VaR with annual jump intensity λ = 4, and default intensities λ̄ = 0.003, 0.03, 0.06. Otherwise the same data as above is used. The options portfolio shows to be fairly insensitive towards credit risk. More examples can be found in Appendix B. 5 5-days, as in 5 trading days. That is one “ordinary” week. 33 3.5. EXAMPLES CHAPTER 3. CREDIT RISK 5 4.5 4 Probability (%) 3.5 3 2.5 2 1.5 1 0.5 0 160 140 120 100 80 Value−at−Risk 60 40 20 Figure 3.1: 1-day VaR. The dotted line is the simulation of the “actual” VaR and the solid line the simulation of the delta-gamma approximation. 5 4.5 4 Probability (%) 3.5 3 2.5 2 1.5 1 0.5 0 110 100 90 80 70 60 Value−at−Risk 50 40 30 20 Figure 3.2: 1-day VaR. The dashed line is the analytical VaR and the solid line the simulation of the delta-gamma approximation. 34 CHAPTER 3. CREDIT RISK 3.5. EXAMPLES 5 4.5 4 Probability (%) 3.5 3 2.5 2 1.5 1 0.5 0 40 38 36 34 32 30 28 Value−at−Risk 26 24 22 20 Figure 3.3: 1-day VaR with pure diffusion (no jump component). The dotted line is the simulation of the “actual” VaR and the solid line the simulation of the delta-gamma approximation. 5 4.5 4 Probability (%) 3.5 3 2.5 2 1.5 1 0.5 0 45 40 35 30 Value−at−Risk 25 20 Figure 3.4: 1-day VaR with pure diffusion (no jump component). The dashed line is the analytical VaR and the solid line the simulation of the delta-gamma approximation. 35 3.5. EXAMPLES CHAPTER 3. CREDIT RISK 5 4.5 4 Probability (%) 3.5 3 2.5 2 1.5 1 0.5 0 120 110 100 90 80 70 Value−at−Risk 60 50 40 Figure 3.5: 5-day VaR with annual jump intensity λ = 4, and default intensities λ̄ = 0.003, 0.03, 0.06 (respectively, dotted-, dash-dotted-, and solid line). 36 CHAPTER 4 Conclusions We have presented an analytic framework for computing Value-at-Risk for a portfolio exposed to market risk as well as credit risk. The price processes for the underlying securities are modelled by jump-diffusions. The number of jumps are Poisson distributed and the diffusion processes are driven by Brownian motions. Credit risk is introduced by a default event indicator variable. In case of a default a certain fraction of the defaulted option’s value is subtracted from the portfolio value. The cumulative distribution function for the market value of the portfolio is reached via Fourier inversion of the characteristic function. It is shown that the characteristic function is easily computed once the expression for the portfolio value is on a quadratic form of independent normal random variables. As an example, we look at a portfolio of European call options. In the preliminaries it is shown that the delta-gamma approximation under-estimates the loss of a long position. This characteristic is clearly seen in the examples presented. When the underlying is driven by a pure diffusion process the under-estimation is not as rough as in the jump-diffusion case. Through the examples we can also observe how insensitive a portfolio consisting of European call options is towards credit risk. Value-at-Risk stays at, approximately, the same level although the long-run default intensity is increased 20 times. A considerably higher sensitivity is observed towards market risk. The analytic delta-gamma approach gives results which deviates approximately ±0.4% from the simulated delta-gamma VaR. This accuracy is satisfactory, but the massive under-estimation made by the delta-gamma approximation when the underlying securities are modelled by jump-diffusions is not. The focus in this thesis has been on the accuracy of the analytic approximation. More effort can be made to improve the accuracy of the delta-gamma approximation. 37 CHAPTER 4. CONCLUSIONS 38 APPENDIX A Examples: Market Risk Below Value-at-Risk for a portfolio of 17, at-the-money, European call options is presented. The 5 and 10-day VaR, using annual jump intensities 2 and 4, is computed. The expected jump amplitude is set so that µj + Qj,j /2 = 0, for j = 1, . . . , 17, and Σ = 2λQ = Cov where the covariance matrix, Cov, can be found in Appendix A on page 44. 100 000 simulations were made for each example. For the cause of simplicity, the following input data were used for all options. Option’s time to maturity (T ) Underlying’s value today (S0 ) Risk free interest rate (r) 1 year 100 5% 39 Error tolerance (α) Mean return (M̄ ) 0.00001 0 APPENDIX A. EXAMPLES: MARKET RISK 5 4.5 4 Probability (%) 3.5 3 2.5 2 1.5 1 0.5 0 200 180 160 140 120 100 Value−at−Risk 80 60 40 Figure A.1: 5-day VaR with λ = 2. The dotted line is the simulation of the “actual” VaR and the solid line the simulation of the delta-gamma approximation. 5 4.5 4 Probability (%) 3.5 3 2.5 2 1.5 1 0.5 0 130 120 110 100 90 80 Value−at−Risk 70 60 50 40 Figure A.2: 5-day VaR with λ = 2. The dashed line is the analytical VaR and the solid line the simulation of the delta-gamma approximation. 40 APPENDIX A. EXAMPLES: MARKET RISK 5 4.5 4 Probability (%) 3.5 3 2.5 2 1.5 1 0.5 0 200 180 160 140 120 100 Value−at−Risk 80 60 40 Figure A.3: 5-day VaR with λ = 4. The dotted line is the simulation of the “actual” VaR and the solid line the simulation of the delta-gamma approximation. 5 4.5 4 Probability (%) 3.5 3 2.5 2 1.5 1 0.5 0 130 120 110 100 90 80 Value−at−Risk 70 60 50 40 Figure A.4: 5-day VaR with λ = 4. The dashed line is the analytical VaR and the solid line the simulation of the delta-gamma approximation. 41 APPENDIX A. EXAMPLES: MARKET RISK 5 4.5 4 Probability (%) 3.5 3 2.5 2 1.5 1 0.5 0 220 200 180 160 140 120 Value−at−Risk 100 80 60 Figure A.5: 10-day VaR with λ = 2. The dotted line is the simulation of the “actual” VaR and the solid line the simulation of the delta-gamma approximation. 5 4.5 4 Probability (%) 3.5 3 2.5 2 1.5 1 0.5 0 140 130 120 110 100 90 Value−at−Risk 80 70 60 Figure A.6: 10-day VaR with λ = 2. The dashed line is the analytical VaR and the solid line the simulation of the delta-gamma approximation. 42 APPENDIX A. EXAMPLES: MARKET RISK 5 4.5 4 Probability (%) 3.5 3 2.5 2 1.5 1 0.5 0 220 200 180 160 140 120 100 80 60 Value−at−Risk Figure A.7: 10-day VaR with λ = 4. The dotted line is the simulation of the “actual” VaR and the solid line the simulation of the delta-gamma approximation. 5 4.5 4 Probability (%) 3.5 3 2.5 2 1.5 1 0.5 0 140 130 120 110 100 90 80 70 60 Value−at−Risk Figure A.8: 10-day VaR with λ = 4. The dashed line is the analytical VaR and the solid line the simulation of the delta-gamma approximation. 43 Covariance matrix for contracts 1–17 (A–Q): A 0.33640 0.07888 0.01856 0.06264 0.09396 0.08816 0.09280 0.05046 0.08584 0.12470 0.10904 0.10440 0.06728 0.05220 0.03248 0.05394 0.09976 B 0.11560 0.02176 0.05508 0.03672 0.07752 0.09520 0.05916 0.01258 0.10234 0.11186 0.08568 0.06902 0.06120 0.03808 0.06324 0.08772 C 0.02560 0.01728 0.00864 0.01824 0.02560 0.01856 0.01184 0.02064 0.03008 0.01728 0.01856 0.03600 0.02688 0.02480 0.02064 D 0.07290 0.03645 0.08208 0.08640 0.04698 0.02997 0.08127 0.08883 0.07776 0.06264 0.04860 0.03024 0.05859 0.08127 E 0.07290 0.06156 0.05400 0.03132 0.01998 0.06966 0.06345 0.05832 0.03915 0.03645 0.01512 0.04185 0.05805 Contract M N O P Q M 0.08410 0.06525 0.03248 0.06293 0.09976 N 0.20250 0.10080 0.05580 0.07740 O 0.31360 0.03472 0.04816 P 0.09610 0.09331 Q 0.18490 F 0.14440 0.12160 0.06612 0.02812 0.13072 0.14288 0.12312 0.09918 0.06840 0.02128 0.08246 0.13072 G 0.16000 0.08120 0.01480 0.13760 0.15040 0.11520 0.09280 0.07200 0.04480 0.08680 0.13760 H 0.08410 0.01073 0.08729 0.09541 0.07308 0.05887 0.05220 0.01624 0.05394 0.08729 I 0.13690 0.04773 0.03478 0.02664 0.02146 0 0.04144 0.02294 0.03182 J 0.18490 0.14147 0.12384 0.08729 0.07740 0.07224 0.07998 0.14792 K 0.22090 0.13536 0.10904 0.10575 0.05264 0.11656 0.16168 L 0.12960 0.09396 0.06480 0.04032 0.07812 0.13932 APPENDIX A. EXAMPLES: MARKET RISK 44 Contract A B C D E F G H I J K L M N O P Q APPENDIX B Examples: Credit Risk Below Value-at-Risk for a portfolio of 17, at-the-money, European call options exposed to market risk as well as credit risk is presented. Value-at-Risk is computed with time horizons 5 and 10 days, and annual jump intensities 2 and 4. Let the mean return vector, M̄ , be the null vector, and Σ = 2λQ = Cov where the covariance matrix, Cov, can be found in Appendix A on the facing page. 80% of the defaults are individual, i.e. λ̄I /λ̄ = 0.8. 100 000 simulations were made for each example. For the cause of simplicity, the following input data were used for all options. Underlying’s value today (S0 ) Risk free interest rate (r) Option’s time to maturity (T ) Error tolerance (α) Vol. def. intensities (σλ ) 100 5% 1 year 0.00001 0.22 45 Common def. prob. (p) Mean reversion (κ) Loss-given-default (L) Long-run mean (λ̄) 0.2 0.25 0.5 0.03 APPENDIX B. EXAMPLES: CREDIT RISK 5 4.5 4 Probability (%) 3.5 3 2.5 2 1.5 1 0.5 0 200 180 160 140 120 100 Value−at−Risk 80 60 40 Figure B.1: 5-day VaR with λ = 2. The dotted line is the simulation of the “actual” VaR and the solid line the simulation of the delta-gamma approximation. 5 4.5 4 Probability (%) 3.5 3 2.5 2 1.5 1 0.5 0 130 120 110 100 90 80 Value−at−Risk 70 60 50 40 Figure B.2: 5-day VaR with λ = 2. The dashed line is the analytical VaR and the solid line the simulation of the delta-gamma approximation. 46 APPENDIX B. EXAMPLES: CREDIT RISK 5 4.5 4 Probability (%) 3.5 3 2.5 2 1.5 1 0.5 0 180 160 140 120 100 Value−at−Risk 80 60 40 Figure B.3: 5-day VaR with λ = 4. The dotted line is the simulation of the “actual” VaR and the solid line the simulation of the delta-gamma approximation. 5 4.5 4 Probability (%) 3.5 3 2.5 2 1.5 1 0.5 0 120 110 100 90 80 70 Value−at−Risk 60 50 40 Figure B.4: 5-day VaR with λ = 4. The dashed line is the analytical VaR and the solid line the simulation of the delta-gamma approximation. 47 APPENDIX B. EXAMPLES: CREDIT RISK 5 4.5 4 Probability (%) 3.5 3 2.5 2 1.5 1 0.5 0 220 200 180 160 140 120 Value−at−Risk 100 80 60 Figure B.5: 10-day VaR with λ = 2. The dotted line is the simulation of the “actual” VaR and the solid line the simulation of the delta-gamma approximation. 5 4.5 4 Probability (%) 3.5 3 2.5 2 1.5 1 0.5 0 140 130 120 110 100 90 Value−at−Risk 80 70 60 Figure B.6: 10-day VaR with λ = 2. The dashed line is the analytical VaR and the solid line the simulation of the delta-gamma approximation. 48 APPENDIX B. EXAMPLES: CREDIT RISK 5 4.5 4 Probability (%) 3.5 3 2.5 2 1.5 1 0.5 0 200 180 160 140 120 Value−at−Risk 100 80 60 Figure B.7: 10-day VaR with λ = 4. 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