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Regularized Pricing & Risk Models

https://www.youtube.com/watch?v=aga-Tak3c3M&list=PLUl4u3cNGP63ctJIEC1UnZ0btsphnnoHR

Bond Duration

Sensitivity of bond price \((\ln P)\) to bond yield \(y\)

  • Duration gives the “weighted time”
  • Duration of zero coupon bond = maturity
  • Duration of regular coupon bond < maturity
  • As there is only a fixed \(y\) for all payment dates, the duration is a sensitivity to “parallel” move

Good measure for price changes for small variation in yield

Second derivative required for large changes in yield

$$ \begin{aligned} P &= \sum \limits_{i=1}^n e^{-y t_i} C_i \ P_y &= \dfrac{\partial P}{\partial y}= - \sum \limits_{i=1}^n t_i e^{-y t_i} C_i \ \implies d &= \dfrac{P_y}{P} \ c &= \dfrac{\partial^2 P}{\partial y^2} = \sum \limits_{i=1}^n t_i^2 e^{-y t_i} C_i \ \end{aligned} $$ where

  • \(d=\) Bond Duration
  • \(c =\) Bond convexity: Always positive
  • \(P=\) price of bond
  • \(y=\) yield of bond
  • \(C_i = i\)th cashflow

Swaps

Valuing fixed and float legs of the swap

Swap can be hedged with bond $$ \begin{aligned} \text{PV}\text{fixed} &= \sum_i C \delta_i \Alpha_i = C \sum_i w_i \ \text{PV}\text{float} &= \sum_i C r_i \delta_i \Alpha_i = \sum_i r_i w_i \ \text{PV}\text{fixed} &= \text{PV}\text{float} \ \implies C &= \dfrac{\sum_i r_i w_i}{\sum_i w_i} \end{aligned} $$ where

  • \(c =\) swap rate (fixed leg coupon)
  • Weighted sum of forward rates (assuming same frequency of payments of fixed & floating legs)
  • \(\Alpha_i =\) discount factor for payment date \(i\)
  • \(\delta_i =\) day count fraction
  • \(r_i =\) forward rate (floating rate of future payment)

Yield Curve

  1. Select input instruments
  2. Choose interpolation
  3. Interpolation space (daily forward rates, zero rates, etc)
  4. Spline (piece-wise constant, linear, tension spline, etc)
  5. Knot points and model parameters
  6. Calibrate
  7. Solve for spline parameters such that input instruments are re-priced at par

Bond Spread

\[ P= \sum_{i=1}^n e^{-s t_i} \Alpha_i C_i \]

where

  • \(\Alpha_i =\) discount factor for payment date \(i\) computed from curve
  • \(s=\) bond spread
  • \(t_i =\) future time of payment in years
  • \(C_i = i\)th cashflow

If the model is available for typical movements of the curve embedded in \(\Alpha_i\) we can build more effective risk model for bond, rather than using single “parallel” shift mode (bond duration)

Hedging

\[ x = \arg \min {\left \vert \left \vert F^T (r + Hx ) \right \vert \right \vert}^2 \]

where

  • \(r =\) portfolio risk
  • \(H =\) hedging portfolio risks
  • \(x =\) weights of hedging instruments
  • \(F =\) market scenarios (factors)

PCA

Use SVD to decompose market movements data \(D\) into principal comments \(P\) and corresponding uncorrelated market dynamics \(U\) with weights \(S\) $$ D = U \cdot S \cdot P^T $$ Use few SVD components with largest singular values - low rank approximation of market data $$ P^T (r + Hx) = 0 $$

PCA Risk Model

“Formally” tuned to historical data

Hedge coefficients are unstable, especially if historical window is short

Costly to re-hedge when PC factors change

Instability is coming from PCs corresponding to small singular values

Over-fitting to historical data

NO assumptions of shape of yield curve

Regularized Risk Models

Assumption: Forward rates move smoothly $$ H^T R = I \ {\vert \vert L \cdot J \cdot R \vert \vert}^2 \to \min \ R \sim \Big(HH^T + \lambda^2 (L \cdot J)^T \cdot L \cdot J \Big)^{-1} $$ where

  • \(J =\) Jacobean matrix translating shifts of yield curve inputs to movements of forward rates
  • \(L=\) Smoothness regularity matrix
  • \(\lambda =\) regularization parameter

Pricing Model

image-20240225131906900

HJM Heath-Jarrow-Morton Model

Evolution of forward rates $$ {df}{t, s} = \mu^\beta V(t, s) \rho (t, s) \cdot dB_t^Q $$ where} dt + f_{t, s

  • \(f =\) forward rate
  • \(\mu=\) drift
  • \(\beta=\) model skew factor
  • \(\rho=\) Correlation/factor structure
  • \(V(t, s)=\) parametric volatility surface
  • \(d B_t^Q =\) Brownian motion

Regularized Volatility Surface

image-20240225132738320

Challenges

  • High dimensionality
  • Need to calibrate many elements
  • Large memory requirement to store matrix
  • Relatively small number of calibration instruments
  • Under-determined problem
  • Sensitivity areas of calibration instruments overlap significantly
  • Ill-posed inverse problem
  • Unstable, noisy solution
  • Need regularity conrtaints
  • Has to be smooth to produce realistic prices for similar instruments

IDK

Represent volatility surface as linear combination of \(n\) basis functions $$ v = v_0 + \beta x $$ where

  • \(v =\) vector of volatility grid elements
  • \(\beta=\) matrix corresponding to basis functions
  • \(x=\) vector of weights

Make \(n\) equivalent to number of calibration instruments \(M\)

“Formally” unambiguous

Make basis functions piecewise constant matching sensitivity of calibration instruments, 0 otherwise

Sensitivities

\[ \begin{aligned} J_{ij} &= \dfrac{\partial q_i}{\partial x_j} \\ q &= J \cdot x \\ &= \ln \dfrac{q_{mdl}}{q_0} \\ q_\text{in} &= \ln \dfrac{q_\text{market}}{q_0} \end{aligned} \]

where

  • \(q_{mdl} =\) model price
  • \(q_\text{market} =\) market price
  • \(q_0 =\) base price
  • \(x=\) vector basis functions coefficients
Last Updated: 2024-05-14 ; Contributors: AhmedThahir

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