p\\ \end{cases} \end{equation} Finally, it is useful to note that drifts $dm_1^j(t), .., dm_n^j(t)$ in \ref{LIBORtransition_1} cancel out thus we can, using \ref{mainFormula_1} and \ref{mainFormula_2} and fixing some measure (let it be w.l.o.g $Q^{T_1}$), rewrite \ref{LIBORrateDynamicsVectorNotation_2} as \begin{equation} \label{mainFormula_3} \left[ {\begin{array}{c} dW_1^{Q_{T_1}}(t) \\ \vdots \\ dW_j^{Q_{T_1}}(t) \\ \vdots \\ dW_n^{Q_{T_1}}(t) \\ \end{array} } \right] = \left[ {\begin{array}{c} 0 \\ \vdots \\ \sum\limits_{k=1}^{j-1} \frac{\delta L_k(t)}{1 + \delta L_k(t)} {\sigma}_k(t) dt \\ \vdots \\ \sum\limits_{k=1}^{n-1} \frac{\delta L_k(t)}{1 + \delta L_k(t)} {\sigma}_k(t) dt \\ \end{array} } \right] + \left[ {\begin{array}{ccc} h_{11}(t) & \ldots & 0 \\ \vdots & \ddots & \vdots \\ h_{1n}(t) & \ldots & h_{nn}(t) \\ \end{array} } \right] \left[ {\begin{array}{c} dB_1^{Q_{T_1}}(t) \\ \vdots \\ dB_j^{Q_{T_1}}(t) \\ \vdots \\ dB_n^{Q_{T_1}}(t) ) \\ \end{array} } \right] \end{equation} The formulae \ref{mainFormula_1} , \ref{mainFormula_2} and \ref{mainFormula_3} are the main result of this section. Of course it would be also interesting to prove that all bond prices, discounted with the $T_j$-bond as numeraire are indeed martingales under $Q^{T_j}$. But it is mainly technical question and we can spare it (or postpone till the next tutorial). \\ But the last task we have to do is to derive the dynamics of LIBOR rate under the spot martingale measure $Q$, i.e. the measure associated with the bank account as the numeraire. For this we need to introduce the idea of a [discrete] bank account. It is as follows: assume $T_{j-1} < t < T_j$, i.e. $L_{j-1}$ (the interest rate from $T_{j-1}$ to $T_j$) is already fixed and no more random. We define the rolling over bank account at time $t$ as follows: invest 1\euro $ $ at $T_0$ and reinvest on each reset date until $T_{j-1}$. At time $t$ the present value of such bank account is \begin{equation} \label{LIBORbankAccount} B(t) = P(t, T_j) \prod\limits_{k=0}^{j-1}(1 + \delta L_k) \end{equation} We derive the dynamics of $W^Q$ via the dynamics of $W^{Q_{T_{j+1}}}$. Inverting \ref{RadonNikodymProcess_1} yields \begin{equation} \label{SpotMeasureRadonNykodimLIBORbankAccount} {\eta}(t) = \frac{dQ}{dQ^{T{j+1}}} = \frac{B(t) P(0, T_{j+1})}{P(t, T_{j+1})} = \frac{P(t, T_j) \overbrace{\prod\limits_{k=0}^{j-1}(1 + \delta L_k) P(0, T_{j+1})}^{:=C}}{P(t, T_{j+1})} \end{equation} Analogous to \ref{RadonNykodimProcessTwoTForwardMeasures} we obtain \begin{eqnarray} \label{LIBORbankAccountGirsanovKernel} d[{\eta}(t)] = C d[P_{j+1}(t)] = C d[1 + \delta L_{j+1}(t)] = C \delta {\sigma}_{j+1}(t) L_{j+1}(t) dW^{Q_{T_{j+1}}} \nonumber \\ = C \delta {\sigma}_{j+1}(t) L_{j+1}(t) \frac{P(t, T_j)}{P(t, T_{j+1})} \frac{P(t, T_{j+1})}{P(t, T_{j})} dW^{Q_{T_{j+1}}} = {\eta}(t) \frac{\delta {\sigma}_{j+1}(t) L_{j+1}(t)}{1 + \delta L_{j+1}(t)}dW^{Q_{T_{j+1}}} \end{eqnarray} So $\frac{\delta {\sigma}_{j+1}(t) L_{j+1}(t)}{1 + \delta L_{j+1}(t)}$ is the Girsanov kernel and you know how proceed further. \section{What's next?} We did big (and hopefully good) job, so that now you are well prepared to your exam in interest rate modeling. However, this theory is only the top of iceberg so for your job [in order both to find and to do it] you may need to know the following things:\\ \\ 1). The LIBOR market model, which we have considered is actually the \emph{forward} LIBOR model. It is not compatible with the \emph{swap} LIBOR model (just like Black-76 for caps is not compatible with Black'76 for swaptions). However, there is a surprisingly good approximation for the swaption prices in the forward model. \\ \\ 2). Implied volatilities and implied correlations. As I steadily pointed out, the LIBOR market model is consistent with Black-76. Unfortunately the market prices of caps and floors are not! This phenomenon is known as volatility smile (analogously correlation smile in case of swaptions). So if you put the historical volatility into Black-76, you will hardly get the current market price of a cap or a floor. But you can proceed other way around and find an \emph{implied} volatility, which makes Black-76 consistent with current price. This is known as ``plugging the wrong value into the wrong formula to get the right price''. Implied volatility can be time dependent and even strike dependent. It reflects the market expectation of the future volatility. There is nothing unnatural in the dependence on strike. Obviously, if the price of an asset goes too high or too low there will be a big hype and the volatility will grow. \\ Analogously the implied correlations can be extracted from swaptions. Swaptions and Caps(Floors) are very liquid derivatives, so one can consider them as ``primary'' traded assets as well. We use them to calibrate our models and then to price not so liquid derivatives.\\ \\ 3). Practically, it is difficult to extract the implied correlations from the market, so one introduce a parametrization. ${\rho}_{jk} = \exp(-\beta |T_j - T_k|)$ is a popular choice.\\ \\ 4). There are many models for stochastic implied volatility and the most popular is probably the SABR model. There is a whole book on the LIBOR with SABR (see References). \\ \\ 5). We have derived the dynamics of the LIBOR rates under different risk neutral measures. Due to pretty complicated drifts we should carefully choose the measure under which to simulate - in order to achieve good numerical accuracy.\\ \\ 6). The LIBOR model is essentially discrete but how about derivatives, whose maturities are not the multiples of the tenor $\delta$? In this case one has to interpolate the LIBOR dynamics but this is in no way trivial task if we want to stay arbitrage free and keep the positivity of all rates. \\ \\ 7). So far we have assumed that the LIBOR rates are driven by a multidimensional Wiener Process. To model a defaultable term structure and, in general, for better reality approximation we might need to add jump processes.\\ \\ 8). In our model the number of Brownian Motions is equal to the number of LIBOR rates. This is indeed too many. The \emph{Principle Component Analysis} shows that 3 random factors are enough, they are interpreted as parallel shifts, tilting(affects the slope) and flex(affect the curvature) of the forward curve.\\ \\ 9). After the Crisis'2008 the world of interest rates became multicurve. In particular, it means that the LIBOR or EURIBOR rates are still used as a reference for IR Swaps but for discounting one takes another curve, see Bianchetti and Carlicchi(2012). Moreover, the swaps are usually collateralized, which makes even a pricing of plain-vanilla swaps very challenging, see Brigo(2012). \newpage \section{References} Fischer Black (1976). The pricing of commodity contracts, Journal of Financial Economics, 3, 167-179.\\ \\ Oldrich Vasicek (1977). An Equilibrium Characterisation of the Term Structure. Journal of Financial Economics 5 (2): 177–188\\ \\ John C. Hull - Options, Futures and Other Derivatives, 7th Ed. - Prentice Hall, 2008\\ \\ Damiano Brigo and Fabio Mercurio - Interest Rate Models - Theory and Practice: With Smile, Inflation and Credit, 2nd Ed. - Springer, 2006\\ \\ Damiano Brigo - Counterparty Risk FAQ: Credit VaR, PFE, CVA, DVA, Closeout, Netting, Collateral, Re-hypothecation, WWR, Basel, Funding, CCDS and Margin Lending, 2012 - http://arxiv.org/pdf/1111.1331v3.pdf\\ \\ Marco Bianchetti and Mattia Carlicchi - Interest Rates After The Credit Crunch: Multiple-Curve Vanilla Derivatives and SABR, 2012\\ http://arxiv.org/ftp/arxiv/papers/1103/1103.2567.pdf\\ \\ Damir Filipovic - Term Structure Models: A Graduate Course - Springer, 2009\\ \\ Riccardo Rebonato, Kenneth McKay and Richard White - The SABR/LIBOR Market Model: Pricing, Calibration and Hedging for Complex Interest Rate Derivatives - Wiley, 2009\\ \\ Dariusz Gatarek, Przemyslaw Bachert and Robert Maksymiuk - The LIBOR Market Model in Practice - Wiley, 2012\\ \\ Thomas Bj\"ork - Arbitrage Theory in Continuous Time, 3rd Ed. - Oxford Finance, 2009\\ \\ Steven E. Shreve - Stochastic Calculus for Finance II: Continuous-Time Models, 2nd Ed. - Springer, 2004\\ \\ James Jardine - An Examination and Implementation of the Libor Market Model (Master Thesis), 2006 - http://www.jimme.net/Thesis.pdf\\ \newpage \section{Solutions to Exercises} \emph{Exercise} \ref{CholeskyExercise1} $$ cov[dW_1(t), dW_1(t)] = VAR[dW_1(t)] = {\sigma}_1^2 VAR[dB_1(t)] = {\sigma}_1^2 dt $$ $$ cov[dW_2(t), dW_2(t)] = VAR[dW_2(t)] \overset{(\star)}{=} {\rho}^2 {\sigma}_2^2 VAR[dB_1(t)] + (1-{\rho}^2) {\sigma}_2^2 VAR[dB_2(t)] = {\sigma}_2^2 dt $$ $$ cov[dW_1(t), dW_2(t)] = \mathbb{E}[dW_1(t) \cdot dW_2(t)] - \underbrace{\mathbb{E}[dW_1(t)]\mathbb{E}[dW_2(t)]}_{=0} $$ $$ = \mathbb{E}[\sigma_1 \sigma_2 \rho dB_1(t) dB_1(t) + \sigma_1 \sigma_2 \sqrt{1 - \rho^2} \underbrace{dB_1(t) dB_2(t)}_{=0 (\star)} ] = \sigma_1 \sigma_2 \rho dt $$ where in $(\star)$ we used the independence of $B_1(t)$ and $B_2(t)$\\ \\ \emph{Exercise} \ref{CholeskyExercise2}\\ $$ \left[ {\begin{array}{c} dB^Q_1(t) \\ dB^Q_2(t) \\ \end{array} } \right] = \left[ {\begin{array}{c} \theta_1 dt + dB^P_1(t) \\ \theta_2 dt + dB^P_2(t) \\ \end{array} } \right] $$ Thus $$ \left[ {\begin{array}{cc} \sigma_1 & 0 \\ \sigma_2 \rho & \sigma_2\sqrt{1 - \rho^2} \\ \end{array} } \right] \left[ {\begin{array}{c} dB^Q_1(t) \\ dB^Q_2(t) \\ \end{array} } \right] = \left[ {\begin{array}{c} \sigma_1 dB^Q_1(t) \\ \sigma_2 \rho dB^Q_1(t) + \sigma_2 \sqrt{1-\rho^2} dB^Q_2(t) \\ \end{array} } \right] $$ $$ = \left[ {\begin{array}{c} dW^Q_1(t) \\ dW^Q_2(t) \\ \end{array} } \right] = \left[ {\begin{array}{c} \sigma_1 [\theta_1 dt + dB^P_1(t)] \\ \sigma_2 \rho [\theta_1 dt + dB^P_1(t)] + \sigma_2 \sqrt{1-\rho^2} [\theta_2 dt + dB^P_2(t)] \\ \end{array} } \right] $$ $$ = \left[ {\begin{array}{c} \sigma_1 \theta_1 dt + dW^P_1(t) \\ \sigma_2[ \rho \theta_1 + \sqrt{1-\rho^2} \theta_2 ] dt + dW^P_2(t)] \\ \end{array} } \right] =:\left[ {\begin{array}{c} \mu_1 dt + dW^P_1(t) \\ \mu_2 dt + dW^P_2(t)] \\ \end{array} } \right] $$ Respectively, if the drifts coefficients $\mu_1$ and $\mu_2$ are given and you are going to kill the drifts by $W^P_1(t)$ and $W^P_1(t)$, you have to choose $ \theta_1 = -\frac{\mu_1}{\sigma_1} \\ $ and $ \theta_2 = -\left( \frac{\mu_2}{\sigma_2} - \rho \theta_1 \right) / \sqrt{1 - \rho^2 } $ \\ Looking at this solution you are probably trying to reconcile it with section 5.4.3 of Shreve's book. In this case, do not be confused with Example 5.4.4, note that in this example there are \emph{two} stocks but only \emph{one} Brownian Motion.\\ \\ \emph{Exercise} \ref{ChangeOfNumeraireExercise1}\\ \\ Recall: $dB_t = rB_t dt$ und $dS_t = S_t(\mu dt + \sigma dW_t)$ $$ d\left(\frac{B_t}{S_t}\right) = \frac{dB_t}{S_t} + B_t d\left(\frac{1}{S_t}\right) =\frac{r B_t dt}{S_t} + B_t \left( -\frac{1}{S^2_t} dS_t + \frac{1}{2} \frac{2S_t}{S^4_t}d^2S_t \right) $$ $$ =\frac{B_t}{S_t} \left( [r - \mu + \sigma^2] dt - \sigma dW_t \right) =\frac{B_t}{S_t} \left( [r - \mu + \sigma^2] dt + \sigma dW_t \right) $$ since $W(t)$ is statistically indistinguishable from $-W(t)$. We kill the drift of $B_t$ under numeraire $S_t$ as follows: $$ \left( [r - \mu + \sigma^2] dt + \sigma dW_t \right) = \sigma \left( [\frac{r - \mu}{\sigma} + \sigma] dt + dW_t \right) =: \sigma dW^{Q_S}_t $$ Now consider $$ d \left(\ln \frac{B_t}{S_t} \right) = \frac{S_t}{B_t} d\left(\frac{B_t}{S_t}\right) - \frac{1}{2} \frac{S^2_t}{B^2_t} d^2\left(\frac{B_t}{S_t}\right) = \left( [r - \mu + \frac{\sigma^2}{2}] dt + \sigma dW_t \right) $$ And finally $$ \frac{B_t}{S_t} = \frac{B_0}{S_0} \exp \left( [r - \mu + \frac{\sigma^2}{2}] t + \sigma W_t \right) = \frac{1}{S_0} \exp \left( [r - \mu + \frac{\sigma^2}{2}] t + \sigma W_t \right) $$ \\ \emph{Exercise} \ref{ChangeOfNumeraireExercise2}\\ \\ Here to keep the notation simpler I omit $(t)$, writing e.g. $S_1$ instead of $S_1(t)$. $$ d\left( \frac{S_2}{S_1} \right) = S_2 d\left( \frac{1}{S_1} \right) + \frac{1}{S_1} dS_2 + dS_2 d\left( \frac{1}{S_1} \right) $$ $$ =S_2 \left[ -\frac{1}{S^2_1} S_1 \left( rdt + \sigma_1 dB_1 \right) + \frac{1}{S_1} {\sigma}_1^2 dt \right] +\frac{S_2}{S_1} \left( rdt + \sigma_2 \rho dB_1 + \sigma_2 \sqrt{1 - \rho^2}dB_2 \right) $$ $$ +\frac{S_2}{S_1} \underbrace{( -rdt -\sigma_1 dB_1 + \sigma_1^2 dt ) ( rdt + \sigma_2 \rho dB_1 + \sigma_2 \sqrt{1 - \rho^2}dB_2 )}_{\sigma_1 \sigma_2 \rho dB_1 dB_1\text{remains only}} $$ $$ =\frac{S_2}{S_1} \left[ (\sigma_1^2 - \sigma_1 \sigma_2 \rho)dt + (\sigma_2 \rho - \sigma_1) dB_1 + \sigma_2 \sigma \sqrt{1 - \rho^2} dB_2 \right] $$ The dynamics of $\frac{S_0}{S_1}$ follows from the exercise \ref{ChangeOfNumeraireExercise1} setting $r = \mu$, i.e. $$ d\left(\frac{S_0}{S_1}\right) = \frac{S_0}{S_1} \sigma_1 \left( \sigma_1 dt + dB_1 \right) = \frac{S_0}{S_1} \sigma_1 dB^{Q_{S_1}}_1 $$ Respectively (recalling that we have already replaced $-dB_1$ with $dB_1$ in the previous equation and thus must do it in the following too) we obtain $$ \frac{S_2}{S_1} \left[ (\sigma_1^2 - \sigma_1 \sigma_2 \rho)dt + (\sigma_2 \rho - \sigma_1) dB_1 + \sigma_2 \sigma \sqrt{1 - \rho^2} dB_2 \right] $$ $$ =\frac{S_2}{S_1} \left[ (\sigma_1^2 - \sigma_1 \sigma_2 \rho)dt + (\sigma_1 - \sigma_2 \rho) (\sigma_1 dt + dB_1) - (\sigma_1 - \sigma_2 \rho)\sigma_1 dt + \sigma_2 \sigma \sqrt{1 - \rho^2} dB_2 \right] $$ $$ =\frac{S_2}{S_1} \left[ (\sigma_1 - \sigma_2 \rho) (\sigma_1 dt + dB_1) + \sigma_2 \sigma \sqrt{1 - \rho^2} dB_2 \right] =\frac{S_2}{S_1} \left[ (\sigma_1 - \sigma_2 \rho) dB_1^{Q_{S_1}} + \sigma_2 \sigma \sqrt{1 - \rho^2} dB_2^{Q_{S_1}} \right] $$ Note that the drift under the neu numeraire $S_1$ was completely absorbed with $dB_1$. This is because we have already started with the risk-neutral dynamics under the numeraire $S_0$. Had we started under the real world measure like in Exercise \ref{ChangeOfNumeraireExercise1} it would not be the case and a part of the drift would be absorbed with $dB_2$. \end{document}