FAST SIMULATION FOR COMPUTING CREDIT VALUE ADJUSTMENTS OF INTEREST RATE PORTFOLIOS

Transcription

1 FAST SIMULATION FOR COMPUTING CREDIT VALUE ADJUSTMENTS OF INTEREST RATE PORTFOLIOS Team members Jonatan Eriksson (Swedbank) Jonas Hallgren (KTH) Ola Hammarlid (Swedbank) Henrik Hult (KTH, team leader) Björn Löfdahl (KTH) Carl-Johan Rehn (Swedbank) 1. Project formulation Monte Carlo simulation is frequently used within the bank for the purpose of pricing and risk quantification. The proposed project is directed towards efficient Monte Carlo simulation for portfolios of interest rate products. The fixed-income portfolio of a typical bank consists of interest rate derivatives such as caps and floors, interest rate swaps, cross-currency swaps, swaptions and FX forwards. These instruments are being sold to customers and used for mitigating the bank s exposure on the fixed-income market. The portfolio consists of thousands of such derivatives. The valuation of such fixed income products is quite complicated in practice. In particular, the introduction of collateral agreements creates a significant computational overhead. The details of the collateral agreements as well as the counterparties respective credit worthiness must be taken into account and the theoretical risk neutral price must be corrected with the appropriate collateral value adjustment (CVA), default value adjustment (DVA), funding value adjustment (FVA), etc. In addition, in the aftermath of the financial crisis the fixed-income markets have become more complicated. The liquidity crisis transformed markets into multi-curve families where, for each currency, there is a yield-curve for each tenor (1m, 3m, 6m, etc). The stochastic evolution of the collection of yield-curves and exchange rates can be described by a high-dimensional stochastic process that is calibrated to be consistent with current market prices. For the purpose of pricing and risk management of the interest rate portfolio scenarios of the underlying stochastic process are generated and the value of the portfolio Date: This project is part of the Swedish Study Group Mathematics in Industry, Mittag-Leffler Institute, Djursholm, Aug,

2 2 FAST SIMULATION FOR INTEREST RATE PORTFOLIOS is computed under each scenario. As the number of contracts to value is high, as is the dimensionality of the underlying stochastic process, the computational cost is significant and even with a limited number of time steps the accuracy obtained by standard Monte Carlo is poor. The aim of this project is to develop a general framework for significantly reducing the computational time. For successful completion of the project it will be necessary to (1) apply appropriate dimension-reduction techniques to reduce the effective dimension of the underlying stochastic model and (2) develop appropriate variance-reduction techniques (e.g. stratification, antithetic variables, importance sampling) that are targeted towards large portfolios of interest rate derivatives. The project is successful if, at least in a simplified model, the appropriate dimension reduction techniques can be identified and variance reduction can be implemented that results in a reduction in computational time of a factor at least 10 over standard Monte Carlo Mathematical problem formulation. In this section the mathematical formulation of the project is presented briefly. It will be assumed that there is a d- dimensional stochastic process {X(t), t 0} representing, say, the relevant short rates and foreign exchange rates underlying the derivative portfolio. The no-artbitrage value at time t 0 of the derivative portfolio is assumed to be a nice function v(t, X(t)) that may be numerically costly to evaluate. The credit value adjustment may be represented as follows. If the counterpart defaults at one of the times t 1 < t 2 < < t n then the loss suffered at t k is proportional to max(v(t k, X(t k )), 0), where the proportionality constant depends on the default probability and the amount of recovery. The credit value adjustment can then be approximated by [ n ] E P c k max(v(t k, X(t k )), 0). k=1 By discretizing time the underlying stochastic process can be simulated at t 1,..., t n and a standard Monte Carlo simulation will converge to the true expectation. To obtain accurate results a large sample size may be needed and hence many function evaluations of v. Time constraints then considerably limits the level of accuracy that can be achieved. In this project we propose variance reduction techniques to improve accuracy. 2. Mathematical background In this section the main mathematical techniques used in the project is presented. The modelling of interest rates is based on a model for the short rate under the risk

3 FAST SIMULATION FOR INTEREST RATE PORTFOLIOS 3 neutral measure (here denoted by P). For practical purposes it is relevant to consider the multi-curve framework for modelling interest rates [1, 3, 4] in combination with multiple currencies. As a first step an elementary one-factor model is used to model the short rate and a single currency is considered The one-factor Hull-White model. The one-factor Hull-White model is a {X(t); t 0} is a Gaussian Markov process satisfying the stochastic differential equation dx(t) = (Θ(t) κx(t))dt + σ(t)db(t), X(0) = x 0, (2.1) where κ is the mean-reversion parameter, σ is the volatility and Θ is defined by a market discount factor D. With f(t) = t log D(t) being the instantaneous forward rate at time 0 for maturity t, Θ is given by Θ(t) = t f(t) + κf(t) + σ2 (t) 2κ (1 e 2κt ). The solution X to (2.1) is given by where X(t) = X(s)e κ(t s) + α(t) α(s)e (t s) + t s e κ(t u) σ(u)db(u), (2.2) α(t) = f(t) + σ2 (t) 2κ 2 (1 e κt ) 2. From (2.2) it follows that for every 0 s < t, conditional distribution X(t) X(s) = x is Gaussian with mean m(s, t, x) and variance v(s, t)), where m(s, t, x) = xe κ(t s) + α(t) α(s)e (t s), v(s, t) = t s e 2κ(t u) σ 2 (u)du Stochastic simulation with importance sampling. In this section the importance sampling methodology for computing expectations of diffusion processes will be described in some detail. Consider a one-dimensional diffusion processes {X(t); t [0, )} on a complete probability space with probability P and such that X is the unique strong solution to the stochastic differential equation dx(t) = b(t, X(t))dt + σ(t, X(t))dB(t), X(0) = x 0, (2.3) where B is a standard Brownian motion and b, σ are sufficiently nice so that a strong solution exists, see [2]. Let t 1 > 0 and h : R R be a measurable function with E P [ h(x(t 1 ) ] <. In this section importance sampling will be introduced with the aim of computing E P [h(x(t 1 ))].

4 4 FAST SIMULATION FOR INTEREST RATE PORTFOLIOS The importance sampling estimator of E P [h(x(t 1 ))] is constructed as the sampling average of independent copies of h(x(t 1 )) dp, where X is simulated under the sampling measure Q. It is implicitly assumed that P Q on the support of F so that, by the Radon-Nikodym theorem, the likelihood ratio exists. The estimator is unbiased because E Q [ h(x(t 1 )) dp ] = h(x(t 1 )) dp = EP [h(x(t 1 ))]. The performance of the estimator can be evaluated in terms of its variance: ( Var h(x(t 1 )) dp ) [ ( dp ) 2 ] [ = E Q h 2 (X(t 1 )) E Q h(x(t 1 )) dp ] 2 [ = E P h 2 (X(t 1 )) dp ] E P [h(x(t 1 ))] 2. The second term does not depend on the Q so it suffices to consider the second moment of the estimator. The idea behind importance sampling is to choose a practically implementable sampling measure Q that gives small second moment relative to E P [h(x(t 1 ))] 2. The optimal choice of sampling measure Q is given by the so-called zero-variance measure dp = h(x(t 1)) E P [h(x(t 1 ))], which, as the name indicates, gives zero variance. Since it requires knowledge of the quantity E[h(X(t 1 ))] it is obviously infeasible. Nevertheless, appropriate sampling measures Q are often constructed as approximations to the zero-variance measure. Importance sampling design as a stochastic control problem. It is convenient to consider sampling measures given by Girsanov s transformation. That is, given a viable Girsanov kernel θ, let Q be given by dp = exp { t1 0 θ(s, X(s))dB(s) 1 2 t1 0 } θ 2 (s, X(s))ds. The optimal choice of the Girsanov kernel can be interpreted as a stochastic control problem where the objective is to minimize the second moment of the importance sampling estimator: [ inf θ EP h 2 (X(t 1 )) exp { t1 0 θ(s, X(s))dB(s) t1 0 } ] θ 2 (s, X(s))ds

5 FAST SIMULATION FOR INTEREST RATE PORTFOLIOS 5 Let us denote the associated value function by W : [ { t1 W (t, x) = inf θ EP h 2 (X(t 1 )) exp θ(s, X(s))dB(s) t t1 t } ] θ 2 (s, X(s))ds X(t) = x, where 0 t t 1 and x R. The associated Hamilton-Jacobi-Bellman equation is given by } inf {θ 2 W θσw x + W t + bw x + σ2 θ 2 W xx = 0, W (t 1, x) = h 2 (x). By performing the minimization it follows that the optimal Girsanov kernel is given by θ = σ Wx and the Hamilton-Jacobi-Bellman equation may be rewritten as 2W W t + bw x + σ2 2 W xx σ2 W 2 x 4W = 0, W (t 1, x) = h 2 (x). (2.4) Since W is the minimal second moment a reasonable ansatz is that W (t, x) = V 2 (t, x) where V (t, x) = E P [h(x(t 1 )) X(t) = x]. Indeed, by Kolmogorov s backward equation V satisfies V t + bv x + σ2 2 V xx = 0, V (t 1, x) = h(x), and one readily verifies that V 2 satisfies (2.4). We conclude that the optimal Girsanov kernel is given by θ = σ W x 2W = σ V x V, and, in fact, the associated sampling measure Q is the zero-variance measure. 3. Fast CVA computations for the one-factor Hull-White model Let {X(t); t 0} be the Hull-White process. In this section an efficient importance sampling algorithm will be designed to compute the CVA of the form [ n ] E P c k max(v(t k, X(t k )), 0). k=1 In the sequel the following simple calculation will be used.

6 6 FAST SIMULATION FOR INTEREST RATE PORTFOLIOS Lemma 3.1. Let {X(t); t 0} be the one factor Hull-White process. a, b, R For each E P [max(ax(t) + b, 0) X(s) = x] ( ) am(s, t, x) + b = (am(s, t, x) + b)φ a + a ( ) am(s, t, x) + b v(s, t)ϕ v(s, t) a. v(s, t) The proposed importance sampling algorithm is designed as follows. Let us first linearize each h k (x) = v(t k, x) around x k = E P [X(t k )] and consider h lin k (x) = h k (x k ) + x h k (x k )(x x k ) = a k x + b k, where the coefficients are identified as a k = x h k (x k ) and b k = h k (x k ) x h k (x k )x k. By Lemma 3.1 the desired expectation when each h k is replaced by its linearization can be computed explicitly: h lin k V (k) (t, x) = E P [max(h lin k (X(t k )), 0) X(t) = x] ( ) a k m(t, t k, x) + b k = (a k m(t, t k, x) + b k )Φ a k v(t, tk ) ( ) a k m(t, t k, x) + b k + a k v(t, tk )ϕ, 0 t t k. a k v(t, tk ) [ n ] Consequently, the optimal sampling distribution for computing E P k=1 c k max(h lin k (X(t k)), 0) is given by a Girsanov transformation with kernel n k=1 θ(t, x) = σ(t) c kv x (k) (t, x) n k=1 c kv (k) (t, x) We [ propose to use this θ in the importance sampling implementation for computing n ] E P k=1 c k max(h k (t k, X(t k )), 0) Numerical experiments. In this section, we illustrate the results of some initial numerical experiments based on two hypothetical portfolios consisting of two swaps with contractual payments defined in Table 1. The Hull-White process X is given the parameters σ = and κ = 0.02, and the initial interest rate is 3%. Portfolio A has initial value close to zero. For this scenario, Monte Carlo should perform reasonably well, since many trajectories will result in positive portfolio values. Hence, the standard error should be reasonable. Portfolio B has a large negative initial value, and it is unlikely that any trajectory will result in a positive portfolio value. This will yield a large relative standard error.

7 FAST SIMULATION FOR INTEREST RATE PORTFOLIOS 7 Table 1. Swap contracts. Portfolio A Portfolio B Swap Type Payer Receiver Payer Receiver Maturity 5Y 4Y 5Y 4Y Nominal amt 1e6 5e5 1e6 5e5 Floating reference 6m 6m 6m 6m Coupons 3% 3% 7% 8% We implement an Euler simulation scheme and estimate the CVA for both portfolios using standard Monte Carlo and our proposed importance sampling scheme, respectively, with 1000 simulations. The results are displayed in Table 2. As we can see, the importance sampling scheme reduces the standard errors dramatically for both portfolios. For Portfolio A, which is an at the money portfolio, the variance reduction factor is 55, implying that the sample size needed for the importance sampling algorithm is 55 times smaller than for standard Monte Carlo. For portfolio B, which is a more extreme out of the money portfolio, the variance reduction factor is 32, 000. Table 2. Credit Value adjustment, standard error within brackets. MC IS Variance reduction Portfolio A 642(29.9) 621(4.03) 55 Portfolio B 2.59(1.79) 0.33(0.01) Conclusions and outlook In a simplified setting, using a one-factor Hull-White model for the short rate and a single currency, an efficient method for computing credit value adjustments have been constructed. The limited numerical experiments indicate and excellent performance and significant variance reduction, greater than expected. The results achieved thus far will be written in a joint paper that will be submitted, during fall 2015, to the Journal of Computational Finance. Future work may proceed in several directions. We have started to implement a sequential Monte Carlo version of the change of measure technique that is potentially useful in more complicated settings because it is generally more robust to misspecification of the change of measure than standard importance sampling. The current version of the algorithm may be improved further by considering a piecewise linear approximation of v(t k, ) based on the observations of

8 8 FAST SIMULATION FOR INTEREST RATE PORTFOLIOS v(t k, X(t k )). Then it can be argued that the convergence rate will be faster than N, where N is the sample size, which is quite remarkable It seems that the algorithm can be extended to a multi-dimensional (multicurve) setting to handle more realistic models. The Gaussianity assumption seems crucial, though, for the proposed method to work. References [1] D. Brigo and F. Mercurio. Interest rate models Theory and Practice. Springer, New York, [2] I. Karatzas and S.E. Shreve. Brownian motion and stochastic calculus. 2nd Edition. Springer, New York, [3] C. Kenyon. Post shock short-rate pricing. Risk Magazine, [4] J. Zhu. Multiple-curve valuation with one-factor hull-white model. SSRN Electronic Journal, June, 2012.