A Hybrid Commodity and Interest Rate Market Model
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1 A Hybrid Commodity and Interest Rate Market Model University of Technology, Sydney June 1
2 Literature A Hybrid Market Model Recall: The basic LIBOR Market Model The cross currency LIBOR Market Model LIBOR Market Model (LMM): Miltersen/Sandmann/Sondermann (1997), Brace/Gatarek/Musiela (1997), Jamshidian (1997), Musiela/Rutkowski (1997) Multicurrency LIBOR Market Model: Schlögl () LMM calibration: Pedersen (1998) Integrating commodity risk, interest rate risk, and stochastic convenience yields: Gibson/Schwartz (199), Miltersen/Schwartz (1998), Miltersen (3)
3 A model of forward LIBOR Recall: The basic LIBOR Market Model The cross currency LIBOR Market Model The forward LIBOR L(t, T ) is defined in terms of zero coupon bond prices by L(t, T ) := 1 ( ) B(t, T ) δ B(t, T + δ) 1 Note that irrespective of the model we choose, L(t, T ) is a martingale under IP T +δ. Therefore, assuming deterministic volatility for L(t, T ) means that it is lognormally distributed under IP T +δ.
4 Setup A Hybrid Market Model Recall: The basic LIBOR Market Model The cross currency LIBOR Market Model Discrete tenor lognormal forward LIBOR model (as in Musiela/Rutkowski (1997)) Horizon date T N for some N IN, finite number of maturities T i = T N (N i)δ, i {,..., N} Dynamics of (domestic) forward LIBORs dl(t, T i ) = L(t, T i )λ(t, T i )dw Ti+1 (t) where λ(, ) is a deterministic function of its arguments W Ti+1 ( ) is a Brownian motion under the time T i+1 forward measure Note that lognormality in this model is a measure dependent property.
5 Recall: The basic LIBOR Market Model The cross currency LIBOR Market Model Links between domestic forward measures By Ito s lemma ( ) B(t, T ) d = B(t, T ) δl(t, T ) B(t, T + δ) B(t, T + δ) 1 + δl(t, T ) λ(t, T )dw T +δ(t) Setting γ(t, T, T + δ) = δl(t, T ) 1 + δl(t, T ) λ(t, T ) we can write dp Ti dp = B(t, T ( i) B(, T i+1 ) = E t Ti+1 B(t, T F t i+1 ) B(, T i ) ) γ(u, T i, T i+1 ) dw Ti+1 (u) Thus dw Ti (t) = dw Ti+1 (t) γ(t, T i, T i+1 )dt
6 Adding a foreign economy, case 1 Recall: The basic LIBOR Market Model The cross currency LIBOR Market Model Assume lognormal forward LIBOR dynamics in the foreign economy as well d L(t, T i ) = L(t, T i ) λ(t, T i )d W Ti+1 (t) Then the foreign forward measures are linked in a manner analogous to the domestic forward measures. This leaves us with the freedom of specifying one further link (only) between a domestic and a foreign forward measure.
7 Recall: The basic LIBOR Market Model The cross currency LIBOR Market Model Measure Links 1 Measure Links Domestic Foreign Domestic Foreign T forward measure T forward measure T forward measure T forward measure T 1 forward measure T 1 forward measure T 1 forward measure T 1 forward measure T forward measure T forward measure T forward measure T forward measure T i forward measure T i forward measure T i forward measure T i forward measure T N forward measure T N forward measure T N forward measure T N forward measure
8 Recall: The basic LIBOR Market Model The cross currency LIBOR Market Model Linking domestic & foreign forward measures X(t): spot exchange rate in units of domestic currency per unit of foreign currency Time T i forward exchange rate: X(t, T i ) = B(t, T i )X(t) B(t, T i ) This is a martingale under P Ti. Conversely is a martingale under P Ti. So we can write 1 X(t, T i ) = B(t, T 1 i) B(t, T i ) X(t) dx(t, T N ) = X(t, T N )σ X (t, T N ) dw TN (t)
9 Recall: The basic LIBOR Market Model The cross currency LIBOR Market Model Domestic vs. foreign forward measures d P TN = X(T N) B(T N, T N )B(, T N ) dp TN X() B(, T N )B(T N, T N ) = X(T N, T N ) X(, T N ) resp. restricting P TN, P TN to the information given at time t: d P TN = X(t, T N) dp TN X(, T N ) Ft By the dynamics assumed forx(t, T N ), ( d P ) TN = E t σ X (u, T N )dw TN (u) dp TN Thus d W TN (t) = dw TN (t) σ X (t, T N )dt P TN a.s.
10 Forward exchange rate volatilities Recall: The basic LIBOR Market Model The cross currency LIBOR Market Model Note that all measure relationships and therefore all volatilities are now fixed. To determine the remaining forward exchange rate volatilities, inductively make use of the relationship X(t, T i ) X(t, T i+1 ) = B(t, T i+1) B(t, T i ) B(t, T i ) B(t, T i+1 )
11 Recall: The basic LIBOR Market Model The cross currency LIBOR Market Model For ease of notation, consider just the first step of the induction. Writing all processes under the domestic time T N 1 forward measure and applying Ito s lemma then yields dx(t, T N 1 ) = X(t, T N 1 ) ( ( γ(t, TN 1, T N ) γ(t, T N 1, T N ) + σ X (t, T N )) dw TN 1 (t)) Thus we must set σ X (t, T N 1 ) = γ(t, T N 1, T N ) γ(t, T N 1, T N ) + σ X (t, T N ) i.e. we can choose only one σ X (t, T i ) to be a deterministic function of its arguments. So for FX options we can have a Black/Scholes type formula for only one maturity, as all other forward exchange rates are not lognormal.
12 Adding a foreign economy, case Recall: The basic LIBOR Market Model The cross currency LIBOR Market Model Assume lognormal forward LIBOR dynamics in the domestic economy only; assume lognormal forward exchange rates dx(t, T i ) = X(t, T i )σ X (t, T i )dw Ti (t) for σ X a deterministic function of its arguments. Thus for all maturities T i d W Ti (t) = dw Ti (t) σ X (t, T i )dt Since the derivation of the links between forward exchange rate volatilities did not depend on the lognormality assumptions, it is valid in the present context as well and therefore γ(t, T i 1, T i ) = σ X (t, T i 1 ) σ X (t, T i ) + γ(t, T i 1, T i )
13 A commodity as foreign currency Recall: The basic LIBOR Market Model The cross currency LIBOR Market Model The commodity market can naturally be considered as a foreign interest rate market. The currency is the physical commodity itself. The zero coupon bond prices C(t, T ) quote (as seen at time t) the amount of the commodity that has to be invested at time t to physically receive one unit of the commodity at time T. Thus the yield of C(t, T ) is the convenience yield (adjusted for storage costs, if applicable). Since convenience yields are implicit rather than explicitly quoted in the market, Case of the multicurrency model is applicable.
14 Pedersen (1998) Calibration Calibration to market prices of caps (or caplets) and swaptions. Calibration of a non-parametric volatility function λ(, ), piecewise constant on a discretisation of both time to maturity and calendar time. Unconstrained non-linear optimisation of weighted sum of quality of fit and smoothness criteria. Correlation is exogenous to the calibration procedure: Assumed to be constant in time and estimated from historical data. Reduction of dimension via principal components analysis.
15 The nonparametric approach Suppose we have n fac factors (the dimension of the driving Brownian motion) and discretise process time into n cal segments, and forward time (maturities) into n fwd segments. The i-th component of (1 i n fac ) of the volatility function λ(t, T ) will be given by λ i (t, x) = λ ijk, t [t j 1, t j ), x [x k 1, x k ) where x = T t is the forward tenor, t j, j >, and x k, k >, are the chosen process and forward times, respectively. For convenience set t = x =.
16 Objective function A Hybrid Market Model w caps QOF caps + w swaptions QOF swaptions + smooth Quality of fit QOF = 1 N N i=1 ( ) PV i 1 PV i smooth = scale fwd smooth fwd + scale cal smooth cal smooth fwd = smooth cal = n cal n fwd j=1 j= n cal n fwd j= j=1 ( ) voli,j 1 vol i,j 1 ( ) voli,j 1 vol i 1,j
17 Reducing the dimesionality of the problem Original dimensionality: n fac n cal n fwd Separate volatility levels and correlation: Volatility levels given by volatility grid vol i,j, 1 i n cal, 1 j n fwd where vol i,j is the volatility as seen at time t i 1 (assumed constant until t i ) of the basic period rate L(, t i 1 + x j ) for the forward period beginning at time t i 1 + x j. This is the object which will be calibrated.
18 Covariance and correlation Principal components representation Let vol be the vector of basic period forward rate volatilities as seen on time t j 1. Let Corr be the corresponding correlation matrix. The covariance matrix is then computed as Cov = vol T Corr vol Let Γ be the diagonal matrix containing the eigenvalues of Cov and V be the corresponding matrix of eigenvectors, i.e. we have the eigenvalue/eigenvector decomposition of Cov Cov = V T ΓV
19 As Cov is positive semidefinite, all entries γ k on the diagonal of Γ will be non-negative and we have where Cov = W T W w ik = γ k v ik We can then extract the stepwise constant volatility function for forward LIBORs as λ ijk = w ik W will provide values for as many factors as the rank of the covariance matrix. For a given n fac, we only use the rows of W corresponding to the n fac largest eigenvalues.
20 Spot measure dynamics Brownian motion under the rolling spot LIBOR measure Q is related to BM under the T i forward measure by dw Ti (t) = φ(t, T i )dt + dw Q (t) with φ(, T i ) defined recursively as φ(t, T i ) φ(t, T i 1 ) = γ(t, T i 1, T i ) = δl(t, T i 1) 1 + δl(t, T i 1 ) λ(t, T i 1) Under an appropriate extension of the discrete tenor LMM to continuous tenor, Q coincides with the spot risk neutral measure and the futures price corresponds to the expected future spot price under this measure.
21 Futures vs. forward A Hybrid Market Model Thus for the futures price G(t, T ) observed at time t for maturity T, we have G(t, T ) = E Q [X(T, T ) F t ] { T = X(t, T )E Q [exp σ X (u, T )dw Q (u) t 1 T ] T σx (u, T )du + σ X (u, T )φ(u, T )du} t t F t { } T X(t, T ) exp σ X (u, T )φ(u, T )du where φ is the frozen coefficient approximation for φ. t
22 Step 1: Calibrate LMM for interest rates using Pedersen approach. An output of this is the matrix W (I). Step : Calibrate the volatility of forward commodity prices to the market using an appropriately modified Pedersen approach. An output of this is the matrix W (C).
23 Step 3: Suppose we have an exogenously given covariance matrix Σ CI of all forward LIBORSs and commodity prices. In order to achieve an approximate fit to this covariance matrix, we exploit the property that multivariate normally distributed random variables are invariant under orthonormal rotations. We seek a square matrix Q, which minimises and Σ CI W (C) Q(W (I) ) QQ I We then replace W (C) by W (C) Q when determining the volatility functions for forward commodity prices.
24 Notes A Hybrid Market Model The dimension of Q is the total number of factors, which may be greater than or equal to the greater of the number of factors in W (I) and W (C). W (I) and W (C) are padded with zeroes where needed. Due to the dependence of the convexity adjustment on interest rate volatilities, steps and 3 need to be repeated iteratively.
25 The commodity and interest rate market on the calibration date 5 May 8 16 Crude Oil Nearest Futures 1 Crude Oil Futures Curve.5 3M Forward Rates USD USD Decimal Points Years Maturity Times Reset Times Left: The WTI Crude Oil nearest futures between 5 and end of 8. The circle indicates the calibration date. Middle: The futures curve as seen at calibration date with maturities up to five years. Right: The 3 month USD forward rates for reset dates (expiries) between 3 months and 4 years and 9 months.
26 Historically estimated interest rate correlation matrix Interest Forward Rate Correlation Matrix Forward Time Forward Time 5 6
27 Calibrated interest rate volatility matrix Interest Forward Rate Volatility Calendar Time Forward Time
28 Market prices vs. model prices 1. x 1 3 Caplets 8 x 1 3 Caps. Swaptions Caplet Expiries Last Caplet Expiries 1 3 Swaption Expiries
29 Historically estimated commodity correlation matrix WTI Crude Oil Futures Correlation Future Maturity Future Maturity 3
30 Calibrated commodity volatility matrix WTI Crude Oil Forward Volatility Forward Time Calendar Time
31 Commodity futures vs. forwards & call option prices Futures and Forwards Time to Maturity Fit of Call Prices Time to Maturity
32 Target & model cross correlations Cross Correlations Absolute Commodity Forwards Interest ForwardsCommodity Forward
33 Six-factor correlation fitting errors s Absolute Error of Cross Correlation Fit Interest ForwardsCommodity Forwards Interest Forwards
34 Factors before & after rotation.6 Originally Calibrated Volatility Factors.35 Cross Transformed Volatility Factors Forward Time Forward Time
35 Target & model cross correlations Cross Correlations Absolute Er Commodity Forwards Commodity Forwards 1 1 Interest Forwards
36 Twelve-factor correlation fitting errors lations Absolute Error of Cross Correlation Fit Interest Forwards.6 3 Commodity Forwards Interest Forwards
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