Lecture5.ipynb

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    "# Lecture 5: Deltas and Hedging\n",
    "\n",
    "Topics:\n",
    "\n",
    "* Derivatives, Deltas and Hedging \n",
    "* Risk transformation\n",
    "* Hedge Optimization\n",
    "* Brownian Bridge\n",
    "* Numerical integration"
   ]
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     "text": [
      "Populating the interactive namespace from numpy and matplotlib\n"
     ]
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   "source": [
    "%pylab inline\n",
    "\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "import inst \n",
    "import fmt\n",
    "\n",
    "lecture = 5"
   ]
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   "source": [
    "# Derivatives, Deltas and Hedging\n",
    "\n",
    "Judd Gregg: Derivatives are a huge, complex issue."
   ]
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    "## Finite difference\n",
    "\n",
    "The derivative of a smooth function $f(t)$ can be approximated numerically by finite difference schemes:\n",
    "\n",
    "* Forward difference: $\\frac{df(x)}{dx} \\approx \\frac{f(x+\\epsilon) - f(x)}{\\epsilon} + O(\\epsilon)$\n",
    "* Backward difference: $\\frac{df(x)}{dx} \\approx \\frac{f(x) - f(x-\\epsilon)}{\\epsilon} + O(\\epsilon)$\n",
    "* Central difference: $\\frac{df(x)}{dx} \\approx \\frac{f(x+\\epsilon) - f(x-\\epsilon)}{2\\epsilon} + O(\\epsilon^2)$\n",
    "* 2nd order derivative: $\\frac{d^2f(x)}{dx^2} \\approx \\frac{f(x+\\epsilon) - 2f(x) + f(x-\\epsilon)}{\\epsilon^2} + O(\\epsilon^2)$\n",
    "\n",
    "\n",
    "Central scheme has higher order of accuracy, requires two valuations of $f$.\n",
    "* The accuracy of finite difference schemes can be verified via Taylor expansions."
   ]
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    "## Generic pricing function\n",
    "\n",
    "A pricing function can be written very generically as: $\\renewcommand{bs}{\\boldsymbol} v = v(\\bs i, \\bs m, \\bs c, t)$, the inputs are:\n",
    "\n",
    "* $\\bs i$: the instrument itself, such as maturity, coupon rate etc\n",
    "* $\\bs m$: market environment, such as IR, CDS, FX rate etc\n",
    "* $\\bs c$: model parameters and assumptions, \n",
    "  * static parameters, e.g.: the tension spline parameter $\\lambda$ \n",
    "  * parameters calibrated to market inputs, e.g.: implied vol\n",
    "* $t$: time"
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   "source": [
    "The pricing function can be viewed as an interpolator."
   ]
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    "## Greeks\n",
    "\n",
    "The following terminologies are often used in practice\n",
    "* Delta: 1st order derivative to market variables: $\\bs \\delta = \\frac{\\partial v}{\\partial \\bs m}$\n",
    " * Delta is a vector of size $m$\n",
    " * Deltas are in the unit of currency\n",
    "* Gamma: 2nd order derivatives to market variables: $\\Gamma = \\frac{\\partial \\bs \\delta}{\\partial \\bs m}$\n",
    " * Gamma is a $m \\times m$ matrix that includes all cross gammas\n",
    "* Theta: $\\theta = \\frac{\\partial v}{\\partial t}$\n",
    "* Delta is also defined for certain common model parameters:\n",
    " * Vega: derivative to implied volatility"
   ]
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    "## How to compute Deltas\n",
    "\n",
    "Deltas are often computed by brute force\n",
    "  * bump, recalibrate (curves & other model parameters), revalue\n",
    "  * slow, but generic and robust\n",
    "  \n",
    "Advanced method of computing Delta\n",
    "  * algorithmic adjoint differentiation (AAD)\n",
    "  * Jacobian inversion"
   ]
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    "## Dynamic hedging\n",
    "\n",
    "The purpose of dynamic hedging is to remove the unwanted risks:\n",
    "\n",
    "* Compute Deltas to all relevant market risk factors\n",
    "* Aggregate the risk sensitivities to the portfolio level\n",
    "* Put on hedges that neutralize the portfolio's sensitivities to unwanted risk factors\n",
    "* The hedges are rebalanced frequently\n",
    "* Dynamic hedging allows a market maker to trade and risk manage illiquid derivatives\n",
    "* Risk managers impose limits on deltas to key risk factors"
   ]
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   "source": [
    "## PnL explain:\n",
    "\n",
    "* Attribute daily PnL to market risk factor changes\n",
    "* A critical step for a trading desk's operation in practice\n",
    "\n",
    "At a very high level, the PnL explain is the total derivative:\n",
    "\n",
    "$$\\renewcommand{bs}{\\boldsymbol}  d v = \\frac{\\partial v}{\\partial \\bs i} d \\bs i + \\frac{\\partial v}{\\partial \\bs m} d \\bs m + \\frac{\\partial v}{\\partial \\bs c} d \\bs c + \\frac{\\partial v}{\\partial t} dt $$\n",
    "\n",
    "* if the two sides differ a lot, the dynamic hedging is not working\n",
    "* on a normal day, the explained PnL should be more than 90%"
   ]
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   "source": [
    "## A Delta hedging example\n",
    "\n",
    "Given the following liquid benchmark CDS quotes of an issuer:"
   ]
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       "<center><font size=2><table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th>Maturity (Y)</th>\n",
       "      <th>0.25</th>\n",
       "      <th>0.5</th>\n",
       "      <th>1.0</th>\n",
       "      <th>2.0</th>\n",
       "      <th>3.0</th>\n",
       "      <th>5.0</th>\n",
       "      <th>7.0</th>\n",
       "      <th>10.0</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Coupon(bps)</th>\n",
       "      <td>100</td>\n",
       "      <td>100</td>\n",
       "      <td>100</td>\n",
       "      <td>100</td>\n",
       "      <td>100</td>\n",
       "      <td>100</td>\n",
       "      <td>100</td>\n",
       "      <td>100</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Quoted Spread(bps)</th>\n",
       "      <td>80</td>\n",
       "      <td>85</td>\n",
       "      <td>95</td>\n",
       "      <td>154</td>\n",
       "      <td>222</td>\n",
       "      <td>385</td>\n",
       "      <td>451</td>\n",
       "      <td>480</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>PV (% Points)</th>\n",
       "      <td>0.04965</td>\n",
       "      <td>0.0742</td>\n",
       "      <td>0.04909</td>\n",
       "      <td>-1.038</td>\n",
       "      <td>-3.42</td>\n",
       "      <td>-12.26</td>\n",
       "      <td>-19.57</td>\n",
       "      <td>-27.34</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table></font></center>"
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    "ir_rate = 0.02\n",
    "disc = inst.FlatCurve(ir_rate)\n",
    "\n",
    "terms = np.array([.25, .5, 1., 2, 3, 5, 7, 10])\n",
    "qs = np.array([80, 85, 95, 154, 222, 385, 451, 480])\n",
    "\n",
    "cps = np.ones(len(terms))*100\n",
    "insts = [inst.CDS(m, c*1e-4, .4) for m, c in zip(terms, cps)]\n",
    "\n",
    "#compute the upfront PV of the benchmark instruments\n",
    "ufr = np.array([i.pv01(disc, inst.FlatCurve(f*1e-4))*(c-f)*1e-4 for i, f, c in zip(insts, qs, cps)])\n",
    "\n",
    "mat_tag = 'Maturity (Y)'\n",
    "\n",
    "cds_data = pd.DataFrame({mat_tag:terms, 'Coupon(bps)':cps, 'Quoted Spread(bps)':qs, \n",
    "                         'PV (% Points)': ufr*100.}).set_index(mat_tag)\n",
    "fmt.displayDF(cds_data.T, \"4g\")\n",
    "benchmarks = dict(zip(insts, ufr))\n",
    "\n",
    "lbdm = 10"
   ]
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    "How to hedge the risk of a long $10MM, 4.5Y, 500bps coupon CDS position?\n",
    "\n",
    "We make the following assumptions:\n",
    "* the interest rate is always a flat curve\n",
    "  * there is a liquid 5Y zero coupon bond in the market\n",
    "* the CDS curve is built using a tension spline with $\\lambda = 10$"
   ]
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    "## Why spreads?\n",
    "\n",
    "The market trades in prices but market participants communicate in spreads:\n",
    "\n",
    "| Instruments | Quotes |\n",
    "| :---: | :----: |\n",
    "| Bond | yield, z-spread, BCDS, OAS |\n",
    "| Options | implied vol |\n",
    "| CDS | quoted spread, par spread |\n",
    "| ABS/MBS/CLO | discount margin |\n",
    "\n",
    "**PV01** : the ratio of PV changes in bps over 1bps spread change\n",
    "\n",
    "The advantages of spread over price includes:\n",
    "* free of the dirty/clean price ambiguity\n",
    "* comparable between different instruments\n",
    "* spreads are annualized, stable over time\n",
    "  * price of bonds will pull to par at maturity"
   ]
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    "## Replicating and hedging portfolio\n",
    "\n",
    "Given: \n",
    "\n",
    "* $v$ is the PV of the trade or portfolio to be hedged\n",
    "* $\\bs b$ is a vector of benchmark instruments' **PV**s \n",
    "  * $\\bs b$ is usually in unit notional, by convention\n",
    "* $\\bs q$ is the vector of benchmark quotes, usually in **spreads** \n",
    "  * e.g. the quoted spreads for CDS, par swaps for rates etc\n",
    "\n",
    "The replicating portfolio $\\bs h^T \\bs b$ is a benchmark portfolio that has identical first order risk to $v$: \n",
    "\n",
    "$$\\frac{\\partial v}{\\partial \\bs q}  = \\frac{\\partial (\\bs h^T \\bs b)}{\\partial \\bs q} = \\bs h^T  \\frac{\\partial \\bs b}{\\partial \\bs q} \\iff \\bs h^T = \\frac{\\partial v}{\\partial \\bs q} (\\frac{\\partial \\bs b}{\\partial \\bs q})^{-1} $$"
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    "Or more directly:\n",
    "\n",
    "$$\n",
    "\\bs h^T = \\frac{\\partial v}{\\partial \\bs b} = \\frac{\\partial v}{\\partial \\bs q}\\frac{\\partial \\bs q}{\\partial \\bs b} = \\frac{\\partial v}{\\partial \\bs q}(\\frac{\\partial \\bs b}{\\partial \\bs q})^{-1}\n",
    "$$\n",
    "\n",
    "* $\\frac{\\partial v}{\\partial \\bs q}$ are the Deltas of the bespoke instrument\n",
    "* $\\frac{\\partial \\bs b}{\\partial \\bs q}$ are the Deltas of the benchmark instruments.\n",
    "\n",
    "$-\\bs h^T \\bs b$ is the portfolio to neutralize all the first order risks, commonly referred as the hedging portfolio.\n",
    "\n",
    "* this equation is the core of a market maker's operation\n",
    "* we can trade illiquid instrument and hedge them using liquid benchmarks\n",
    "* economy of scale, the hedge is only on the portfolio level"
   ]
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   "source": [
    "## Benchmark Jacobian\n",
    "\n",
    "$\\frac{\\partial \\bs b}{\\partial \\bs q}$ is the Jacobian of benchmark instruments over benchmark quotes\n",
    "\n",
    "\n",
    "In this example, the benchmark instrument $\\bs b$ includes:\n",
    "\n",
    "* the standard CDS with 100bps coupon at benchmark tenors\n",
    "* the 5Y zero coupon bond\n",
    "\n",
    "The benchmark quotes $\\bs q$ includes:\n",
    "\n",
    "* quoted spread for all benchmark tenors\n",
    "* the flat interest rate"
   ]
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    "## Diagonality\n",
    "\n",
    "The benchmark Jacobian is diagonal if the dependency btw $\\bs b$ and $\\bs q$ is one to one\n",
    "  * Jacobian $\\frac{\\partial v}{\\partial \\bs q}$ requires other elements of $\\bs q$ remain constant when perturbing one of them\n",
    "  * other benchmark PVs remain unchanged because their $\\bs q$ are hold constant\n",
    "  \n",
    "The $\\bs b$ and $\\bs q$ are not always one-to-one\n",
    "  * eg, option price depends on both IR and implied vol\n",
    "  * the benchmark quote is not diagonal in this case\n",
    "    \n",
    "The benchmark Jacobian is diagonal in our example\n",
    "  * the diagonal elements of $\\frac{\\partial \\bs b}{\\partial \\bs q}$ are the PV01s"
   ]
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    "When computing benchmark Jacobian numerically, we **must** re-bootstrap all dependent curves for any perturbations:\n",
    "  * e.g., the CDS curve needs to be re-bootstrapped to reprice all benchmark instruments, even if we perturb IR\n",
    "  * survival probability may change, but PVs are kept the same for unperturbed tenors"
   ]
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       "<center><font size=2><table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>5Y ZCB</th>\n",
       "      <th>CDS @0.25Y</th>\n",
       "      <th>CDS @0.5Y</th>\n",
       "      <th>CDS @1Y</th>\n",
       "      <th>CDS @2Y</th>\n",
       "      <th>CDS @3Y</th>\n",
       "      <th>CDS @5Y</th>\n",
       "      <th>CDS @7Y</th>\n",
       "      <th>CDS @10Y</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>PV01</th>\n",
       "      <td>4.53</td>\n",
       "      <td>0.25</td>\n",
       "      <td>0.49</td>\n",
       "      <td>0.98</td>\n",
       "      <td>1.91</td>\n",
       "      <td>2.78</td>\n",
       "      <td>4.23</td>\n",
       "      <td>5.32</td>\n",
       "      <td>6.45</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table></font></center>"
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    " def calcDelta(bmks, ir, lbd, trades, pert=1e-4, fpert=inst.pert_bmk) : \n",
    "    mkt, pmkt, pkeys, cds = fpert(bmks, ir, lbd=lbd, its=3, pert=pert)\n",
    "    deltas, pv0 = inst.pv_deltas(mkt, pmkt, trades)\n",
    "    \n",
    "    return deltas/pert, pkeys, cds, pv0\n",
    "\n",
    "zcb = inst.ZeroCouponBond(maturity = 5.)\n",
    "bespoke = inst.CDS(maturity=4.5, coupon=0.05, recovery=0.4)\n",
    "bmks = [zcb] + insts\n",
    "notional = 10e6\n",
    "hi_names = ['5Y ZCB'] + ['CDS @%gY' % t for t in terms] # benchmark instruments\n",
    "\n",
    "deltas, pert_names, cds, _ = calcDelta(benchmarks, disc, lbdm, bmks + [bespoke])\n",
    "pv01m = deltas[:-1, :]\n",
    "pv01 = np.diag(pv01m)\n",
    "db = deltas[-1, :]\n",
    "\n",
    "fmt.displayDF(pd.DataFrame(pv01, index=hi_names, columns=['PV01']).T, \"2f\")"
   ]
  },
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   "source": [
    "## Hedging portfolio example"
   ]
  },
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   "source": [
    "Given the $\\frac{\\partial \\bs b}{\\partial \\bs q}$ is diagonal, the hedge notional is simply the bespoke instrument's risks divided by benchmark instruments' PV01:"
   ]
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       "<center><font size=2><table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>5Y ZCB</th>\n",
       "      <th>CDS @0.25Y</th>\n",
       "      <th>CDS @0.5Y</th>\n",
       "      <th>CDS @1Y</th>\n",
       "      <th>CDS @2Y</th>\n",
       "      <th>CDS @3Y</th>\n",
       "      <th>CDS @5Y</th>\n",
       "      <th>CDS @7Y</th>\n",
       "      <th>CDS @10Y</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>PV01 $= \\frac{\\partial b}{\\partial q}$</th>\n",
       "      <td>4.53</td>\n",
       "      <td>0.25</td>\n",
       "      <td>0.49</td>\n",
       "      <td>0.98</td>\n",
       "      <td>1.91</td>\n",
       "      <td>2.78</td>\n",
       "      <td>4.23</td>\n",
       "      <td>5.32</td>\n",
       "      <td>6.45</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>$\\delta = 10^{-4} \\frac{\\partial v}{\\partial q}$</th>\n",
       "      <td>390.43</td>\n",
       "      <td>3.38</td>\n",
       "      <td>9.88</td>\n",
       "      <td>45.11</td>\n",
       "      <td>69.88</td>\n",
       "      <td>871.55</td>\n",
       "      <td>3,744.18</td>\n",
       "      <td>-202.84</td>\n",
       "      <td>7.68</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Hedge Not (\\$MM)</th>\n",
       "      <td>-0.86</td>\n",
       "      <td>-0.14</td>\n",
       "      <td>-0.20</td>\n",
       "      <td>-0.46</td>\n",
       "      <td>-0.37</td>\n",
       "      <td>-3.14</td>\n",
       "      <td>-8.85</td>\n",
       "      <td>0.38</td>\n",
       "      <td>-0.01</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table></font></center>"
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    "hedge_items = ['PV01 $= \\\\frac{\\partial b}{\\partial q}$', \n",
    "               '$\\delta = 10^{-4} \\\\frac{\\partial v}{\\partial q}$', 'Hedge Not (\\$MM)']\n",
    "\n",
    "df_h_bmk = pd.DataFrame(np.array([pv01, db*notional*1e-4, -db/pv01*notional*1e-6]).T, \n",
    "                        index=hi_names, columns=hedge_items).T\n",
    "fmt.displayDF(df_h_bmk, \"2f\")"
   ]
  },
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   "source": [
    "* note that the deltas are usually communicated in PV changes for 1bps change in spread, thus the $10^{-4}$ factor."
   ]
  },
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   "source": [
    "## Compute deltas numerically\n",
    "\n",
    "Steps to compute $\\frac{\\partial v}{\\partial \\bs q}$ numerically:\n",
    "\n",
    "1. bump one tenor point $q_i$, re-bootstrap the whole CDS curve\n",
    "  1. for each tenor point ```j=1:maturity```, root search the knot value of hazard rate $h_j$\n",
    "    * for each trial value of $h_j$\n",
    "      * build tension spline interpolation on the whole hazard rate term structure $\\bs h$\n",
    "      * price the benchmark CDS instrument $b_j$, \n",
    "      * continue root search until $b_j$ matches the market price to a high precision\n",
    "  1. price the bespoke instrument $v$ using the bumped curve, and iterate if pricing error is large\n",
    "1. compute the delta using finite difference"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Tension and hedge notional\n",
    "\n",
    "The tension parameter has a noticeable effects on PV and hedge notionals.\n",
    "\n",
    "* all numbers below are in million USD\n",
    "* the leaks to longer maturities reduce with larger $\\lambda$ "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<center><font size=2><table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Bespoke PV</th>\n",
       "      <th>5Y ZCB</th>\n",
       "      <th>CDS @0.25Y</th>\n",
       "      <th>CDS @0.5Y</th>\n",
       "      <th>CDS @1Y</th>\n",
       "      <th>CDS @2Y</th>\n",
       "      <th>CDS @3Y</th>\n",
       "      <th>CDS @5Y</th>\n",
       "      <th>CDS @7Y</th>\n",
       "      <th>CDS @10Y</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>$\\lambda=0$</th>\n",
       "      <td>0.567</td>\n",
       "      <td>-0.840</td>\n",
       "      <td>-0.344</td>\n",
       "      <td>0.271</td>\n",
       "      <td>-1.095</td>\n",
       "      <td>1.011</td>\n",
       "      <td>-4.248</td>\n",
       "      <td>-9.625</td>\n",
       "      <td>1.441</td>\n",
       "      <td>-0.215</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>$\\lambda=2$</th>\n",
       "      <td>0.553</td>\n",
       "      <td>-0.846</td>\n",
       "      <td>-0.146</td>\n",
       "      <td>-0.140</td>\n",
       "      <td>-0.578</td>\n",
       "      <td>0.099</td>\n",
       "      <td>-3.507</td>\n",
       "      <td>-9.329</td>\n",
       "      <td>0.904</td>\n",
       "      <td>-0.077</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>$\\lambda=10$</th>\n",
       "      <td>0.541</td>\n",
       "      <td>-0.863</td>\n",
       "      <td>-0.136</td>\n",
       "      <td>-0.200</td>\n",
       "      <td>-0.461</td>\n",
       "      <td>-0.366</td>\n",
       "      <td>-3.136</td>\n",
       "      <td>-8.853</td>\n",
       "      <td>0.381</td>\n",
       "      <td>-0.012</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>$\\lambda=50$</th>\n",
       "      <td>0.534</td>\n",
       "      <td>-0.875</td>\n",
       "      <td>-0.140</td>\n",
       "      <td>-0.211</td>\n",
       "      <td>-0.440</td>\n",
       "      <td>-0.544</td>\n",
       "      <td>-3.014</td>\n",
       "      <td>-8.541</td>\n",
       "      <td>0.100</td>\n",
       "      <td>-0.001</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>$\\lambda=5000$</th>\n",
       "      <td>0.532</td>\n",
       "      <td>-0.880</td>\n",
       "      <td>-0.141</td>\n",
       "      <td>-0.214</td>\n",
       "      <td>-0.436</td>\n",
       "      <td>-0.600</td>\n",
       "      <td>-2.979</td>\n",
       "      <td>-8.424</td>\n",
       "      <td>0.001</td>\n",
       "      <td>0.000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table></font></center>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def hedgeRatio(lbdm) :\n",
    "    deltas, _, _, pv0 = calcDelta(benchmarks, disc, lbdm, bmks + [bespoke])\n",
    "    pv01 = np.diag(deltas[:-1, :])\n",
    "    db = deltas[-1, :]\n",
    "    return np.concatenate((pv0[-1:], -db/pv01))\n",
    "\n",
    "lbds = [0, 2, 10, 50, 5000]\n",
    "hn = [hedgeRatio(l)*notional for l in lbds]\n",
    "\n",
    "lkeys = [\"$\\lambda=%d$\" % l for l in lbds]\n",
    "df_hn = pd.DataFrame(np.array(hn)*1e-6, columns=np.concatenate((['Bespoke PV'], hi_names)), index=lkeys)\n",
    "\n",
    "fmt.displayDF(df_hn, \"3f\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Valuation and hedging ambiguity?\n",
    "\n",
    "The PV and hedge notionals depend on $\\lambda$\n",
    "  * two parties may not agree on the PVs and risks of the trades, because of model differences\n",
    "    * the quoted spreads are a few bps apart due to different $\\lambda$\n",
    "    * this contributes to a wider bid/offer for off-the-runs\n",
    "    * a common headache in operation, e.g. collateral posting\n",
    "  * the ambiguity is a reflection of incomplete market, not an arbitrage opportunity\n",
    "  \n",
    "Dynamic hedging in practice:\n",
    "  * can't possibly remove all risks, eg. jump-to-default risk\n",
    "  * the cost of hedging over time won't add up to the PV of the instruments"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Risk Transformation\n",
    "\n",
    "Warren Buffet: Risk comes from not knowing what you're doing."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## CDS big bang\n",
    "\n",
    "The big bang of CDS standardisation occurred in 2010:\n",
    "\n",
    "|  Era   | Liquid Instruments | Quoting Convention | At Trade Inception | Counterparty |\n",
    "| :---: | :----: | :----: | :-----: |\n",
    "| before |  par spread CDS | par spreads | $PV=0$, <br> no cash exchange | OTC |\n",
    "| after | standardised CDS with 100/500 cpn | quoted spreads, upfront | $PV \\ne 0$, <br> cash exchange | central clearing |\n",
    "\n",
    "At its core, it is simply a change in the liquid benchmark instruments.\n",
    "* the quoting convention change is a side product\n",
    "* ISDA publishes a one-day-lagged IR curve for the conversion of quoted spreads to PV, to ensure that the standardised CDS PV is solely determined by the corresponding quoted spread."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Real world hedging problem\n",
    "\n",
    "However, your bank's IT system hasn't been updated in time (it never does). \n",
    "\n",
    "In your bank:\n",
    "\n",
    "* end of day valuation still builds CDS curves from the par CDS spread\n",
    "* traders have to continue marking the par spreads, even though they all trade standardised CDS using quoted spreads\n",
    "* risks (Deltas) are all computed with regard to par CDS"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "From your bank's system point of view (contrary to the market): \n",
    "\n",
    "* the par CDS is still the benchmark instruments\n",
    "* the standardised CDS are the bespoke instruments you now have to hedge your positions with\n",
    "\n",
    "How do you continue to trade and hedge your book?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Par spread $\\leftrightarrow$ quoted spread\n",
    "\n",
    "the equivalent par CDS spreads ($\\bs p$) and quoted spreads ($\\bs q$) are shown below.\n",
    "* the conversion btw par spreads $\\bs p$ and quoted spreads $\\bs q$ is easy\n",
    "  * $\\bs q \\rightarrow$ CDS curve $\\rightarrow \\bs p \\rightarrow$ CDS Curve\n",
    "  * the last step is done by the end of day official IT system\n",
    "* CDS curves built from converted $\\bs q$ and $\\bs p$ are identical\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<center><font size=2><table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th>Maturity (Y)</th>\n",
       "      <th>0.25</th>\n",
       "      <th>0.5</th>\n",
       "      <th>1.0</th>\n",
       "      <th>2.0</th>\n",
       "      <th>3.0</th>\n",
       "      <th>5.0</th>\n",
       "      <th>7.0</th>\n",
       "      <th>10.0</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Coupon(bps)</th>\n",
       "      <td>100</td>\n",
       "      <td>100</td>\n",
       "      <td>100</td>\n",
       "      <td>100</td>\n",
       "      <td>100</td>\n",
       "      <td>100</td>\n",
       "      <td>100</td>\n",
       "      <td>100</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Quoted Spread(bps)</th>\n",
       "      <td>80</td>\n",
       "      <td>85</td>\n",
       "      <td>95</td>\n",
       "      <td>154</td>\n",
       "      <td>222</td>\n",
       "      <td>385</td>\n",
       "      <td>451</td>\n",
       "      <td>480</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>PV (% Points)</th>\n",
       "      <td>0.0497</td>\n",
       "      <td>0.0742</td>\n",
       "      <td>0.0491</td>\n",
       "      <td>-1.04</td>\n",
       "      <td>-3.42</td>\n",
       "      <td>-12.3</td>\n",
       "      <td>-19.6</td>\n",
       "      <td>-27.3</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Par Coupon (bps)</th>\n",
       "      <td>80</td>\n",
       "      <td>85</td>\n",
       "      <td>95</td>\n",
       "      <td>154</td>\n",
       "      <td>223</td>\n",
       "      <td>390</td>\n",
       "      <td>468</td>\n",
       "      <td>524</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table></font></center>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pars = np.array([i.par(disc, inst.zero2disc(cds)) for i in insts])\n",
    "cds_data['Par Coupon (bps)'] = pars*1e4\n",
    "fmt.displayDF(cds_data.T, \"3g\")\n",
    "bmk_par = {inst.CDS(i.maturity, s, .4) : 0 for i, s in zip(insts, pars)}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "where do the par spread and quoted spread diverge?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## What people often do\n",
    "\n",
    "Compute the hedge ratio as:\n",
    "\n",
    "* computes bespoke trade's deltas to par spreads: $\\frac{\\partial v}{\\partial p_i}$\n",
    "* compute the PV01 of standardised CDS to par spreads, i.e, $\\frac{\\partial b_i}{\\partial p_i}$\n",
    "* compute the hedge notional as the ratio of the bespoke delta and PV01s, both to par spread\n",
    "\n",
    "The following are the results:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<center><font size=2><table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>5Y ZCB</th>\n",
       "      <th>CDS @0.25Y</th>\n",
       "      <th>CDS @0.5Y</th>\n",
       "      <th>CDS @1Y</th>\n",
       "      <th>CDS @2Y</th>\n",
       "      <th>CDS @3Y</th>\n",
       "      <th>CDS @5Y</th>\n",
       "      <th>CDS @7Y</th>\n",
       "      <th>CDS @10Y</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>PV01 $= \\frac{\\partial b}{\\partial p}$</th>\n",
       "      <td>4.53</td>\n",
       "      <td>0.25</td>\n",
       "      <td>0.49</td>\n",
       "      <td>0.98</td>\n",
       "      <td>1.90</td>\n",
       "      <td>2.74</td>\n",
       "      <td>4.00</td>\n",
       "      <td>4.97</td>\n",
       "      <td>5.77</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>$\\delta = 10^{-4} \\frac{\\partial v}{\\partial p}$</th>\n",
       "      <td>124.94</td>\n",
       "      <td>1.17</td>\n",
       "      <td>3.38</td>\n",
       "      <td>16.16</td>\n",
       "      <td>-2.26</td>\n",
       "      <td>710.73</td>\n",
       "      <td>3,556.26</td>\n",
       "      <td>-181.90</td>\n",
       "      <td>6.28</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Hedge Not (\\$MM)</th>\n",
       "      <td>-0.28</td>\n",
       "      <td>-0.05</td>\n",
       "      <td>-0.07</td>\n",
       "      <td>-0.17</td>\n",
       "      <td>0.01</td>\n",
       "      <td>-2.59</td>\n",
       "      <td>-8.89</td>\n",
       "      <td>0.37</td>\n",
       "      <td>-0.01</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table></font></center>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "deltas_p, pert_names, _, _ = calcDelta(bmk_par, disc, lbdm, bmks + [bespoke])\n",
    "pv01m_p = deltas_p[:-1, :]\n",
    "pv01_p = np.diag(pv01m_p)\n",
    "db_p = deltas_p[-1, :]\n",
    "\n",
    "hedge_items_p = [l.replace(' q', ' p') for l in hedge_items]\n",
    "hn_p = pd.DataFrame(np.array([pv01_p, db_p*notional*1e-4, -db_p/pv01_p*notional*1e-6]).T, \n",
    "                        index=hi_names, columns=hedge_items_p).T\n",
    "\n",
    "fmt.displayDF(hn_p, \"2f\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## is incorrect\n",
    "\n",
    "There are serious errors in the resulting hedging portfolio\n",
    "\n",
    "* in particular the IR hedge amount is very wrong\n",
    "* leading to mishedges, confusions and headaches "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<center><font size=2><table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>5Y ZCB</th>\n",
       "      <th>CDS @0.25Y</th>\n",
       "      <th>CDS @0.5Y</th>\n",
       "      <th>CDS @1Y</th>\n",
       "      <th>CDS @2Y</th>\n",
       "      <th>CDS @3Y</th>\n",
       "      <th>CDS @5Y</th>\n",
       "      <th>CDS @7Y</th>\n",
       "      <th>CDS @10Y</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Par CDS $\\delta$/PV01</th>\n",
       "      <td>-0.2761</td>\n",
       "      <td>-0.0472</td>\n",
       "      <td>-0.0685</td>\n",
       "      <td>-0.1651</td>\n",
       "      <td>0.0119</td>\n",
       "      <td>-2.5896</td>\n",
       "      <td>-8.8937</td>\n",
       "      <td>0.3662</td>\n",
       "      <td>-0.0109</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Exact H/N</th>\n",
       "      <td>-0.8628</td>\n",
       "      <td>-0.1362</td>\n",
       "      <td>-0.2001</td>\n",
       "      <td>-0.4611</td>\n",
       "      <td>-0.3656</td>\n",
       "      <td>-3.1362</td>\n",
       "      <td>-8.8529</td>\n",
       "      <td>0.3814</td>\n",
       "      <td>-0.0119</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table></font></center>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ih = -db_p.dot(np.linalg.inv(pv01m_p))*notional\n",
    "hns = pd.DataFrame([-db_p/pv01_p*notional, -db/pv01*notional, ih], columns=hi_names, \n",
    "                    index=['Par CDS $\\\\delta$/PV01', 'Exact H/N', 'Inverted H/N'])\n",
    "fmt.displayDF(hns[:-1]*1e-6, \"4f\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "The root cause is that the $\\frac{\\partial \\bs b}{\\partial \\bs p}$ is not a diagonal matrix\n",
    "* $\\bs p$ is quotes for par CDS instruments, it is not one to one with $\\bs b$, the standardised CDS\n",
    "* this incorrect procedure ignores all off diagonal elements in $\\frac{\\partial \\bs b}{\\partial \\bs p}$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "The following is the full Jacobian matrix of the standardised CDS with 100bps coupon to par spread perturbations $\\frac{\\partial \\bs b}{\\partial \\bs p}$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<center><font size=2><table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>IR</th>\n",
       "      <th>CDS @0.25Y</th>\n",
       "      <th>CDS @0.50Y</th>\n",
       "      <th>CDS @1Y</th>\n",
       "      <th>CDS @2Y</th>\n",
       "      <th>CDS @3Y</th>\n",
       "      <th>CDS @5Y</th>\n",
       "      <th>CDS @7Y</th>\n",
       "      <th>CDS @10Y</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>5Y ZCB</th>\n",
       "      <td>4.5253</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>CDS @0.25Y</th>\n",
       "      <td>0.0001</td>\n",
       "      <td>0.2481</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>CDS @0.5Y</th>\n",
       "      <td>0.0003</td>\n",
       "      <td>0.0002</td>\n",
       "      <td>0.4940</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>CDS @1Y</th>\n",
       "      <td>0.0003</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0002</td>\n",
       "      <td>0.9788</td>\n",
       "      <td>-0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>-0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>-0.0000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>CDS @2Y</th>\n",
       "      <td>-0.0115</td>\n",
       "      <td>-0.0005</td>\n",
       "      <td>-0.0015</td>\n",
       "      <td>-0.0067</td>\n",
       "      <td>1.9003</td>\n",
       "      <td>0.0004</td>\n",
       "      <td>-0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>-0.0000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>CDS @3Y</th>\n",
       "      <td>-0.0543</td>\n",
       "      <td>-0.0011</td>\n",
       "      <td>-0.0032</td>\n",
       "      <td>-0.0140</td>\n",
       "      <td>-0.0377</td>\n",
       "      <td>2.7446</td>\n",
       "      <td>0.0005</td>\n",
       "      <td>-0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>CDS @5Y</th>\n",
       "      <td>-0.3067</td>\n",
       "      <td>-0.0022</td>\n",
       "      <td>-0.0065</td>\n",
       "      <td>-0.0287</td>\n",
       "      <td>-0.0708</td>\n",
       "      <td>-0.1771</td>\n",
       "      <td>3.9986</td>\n",
       "      <td>0.0094</td>\n",
       "      <td>-0.0003</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>CDS @7Y</th>\n",
       "      <td>-0.6523</td>\n",
       "      <td>-0.0023</td>\n",
       "      <td>-0.0068</td>\n",
       "      <td>-0.0303</td>\n",
       "      <td>-0.0750</td>\n",
       "      <td>-0.1833</td>\n",
       "      <td>-0.4337</td>\n",
       "      <td>4.9664</td>\n",
       "      <td>0.0077</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>CDS @10Y</th>\n",
       "      <td>-1.2097</td>\n",
       "      <td>-0.0020</td>\n",
       "      <td>-0.0058</td>\n",
       "      <td>-0.0257</td>\n",
       "      <td>-0.0635</td>\n",
       "      <td>-0.1556</td>\n",
       "      <td>-0.3520</td>\n",
       "      <td>-0.7054</td>\n",
       "      <td>5.7724</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table></font></center>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fmt.displayDF(pd.DataFrame(pv01m_p, index=hi_names, columns=pert_names), \"4f\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "For comparison, we also the full Jacobian matrix of stardardised CDS to quoted spreads $\\frac{\\partial \\bs b}{\\partial \\bs q}$, showing that it is indeed diagonal, so we can take the PV01 short cut."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<center><font size=2><table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>IR</th>\n",
       "      <th>CDS @0.25Y</th>\n",
       "      <th>CDS @0.50Y</th>\n",
       "      <th>CDS @1Y</th>\n",
       "      <th>CDS @2Y</th>\n",
       "      <th>CDS @3Y</th>\n",
       "      <th>CDS @5Y</th>\n",
       "      <th>CDS @7Y</th>\n",
       "      <th>CDS @10Y</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>5Y ZCB</th>\n",
       "      <td>4.5253</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>CDS @0.25Y</th>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.2479</td>\n",
       "      <td>-0.0000</td>\n",
       "      <td>-0.0000</td>\n",
       "      <td>-0.0000</td>\n",
       "      <td>-0.0000</td>\n",
       "      <td>-0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>CDS @0.5Y</th>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.4937</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>-0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>-0.0000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>CDS @1Y</th>\n",
       "      <td>-0.0000</td>\n",
       "      <td>-0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.9785</td>\n",
       "      <td>-0.0000</td>\n",
       "      <td>-0.0000</td>\n",
       "      <td>-0.0000</td>\n",
       "      <td>-0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>CDS @2Y</th>\n",
       "      <td>-0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>1.9111</td>\n",
       "      <td>-0.0000</td>\n",
       "      <td>-0.0000</td>\n",
       "      <td>-0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>CDS @3Y</th>\n",
       "      <td>-0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>2.7790</td>\n",
       "      <td>-0.0000</td>\n",
       "      <td>-0.0000</td>\n",
       "      <td>-0.0000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>CDS @5Y</th>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>-0.0000</td>\n",
       "      <td>-0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>-0.0000</td>\n",
       "      <td>4.2293</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>CDS @7Y</th>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>5.3189</td>\n",
       "      <td>0.0000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>CDS @10Y</th>\n",
       "      <td>-0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>-0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>-0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>6.4548</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table></font></center>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fmt.displayDF(pd.DataFrame(pv01m, index=hi_names, columns=pert_names), '4f')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Full inversion\n",
    "\n",
    "The following hedge ratio and notional are obtained by inverting the full Jacobian matrix of standardised CDS to par spreads: \n",
    "\n",
    "$$\\bs h^T = -  \\frac{\\partial v}{\\partial \\bs p} (\\frac{\\partial \\bs b}{\\partial \\bs p})^{-1} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "We did get the correct hedging portfolio."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<center><font size=2><table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>5Y ZCB</th>\n",
       "      <th>CDS @0.25Y</th>\n",
       "      <th>CDS @0.5Y</th>\n",
       "      <th>CDS @1Y</th>\n",
       "      <th>CDS @2Y</th>\n",
       "      <th>CDS @3Y</th>\n",
       "      <th>CDS @5Y</th>\n",
       "      <th>CDS @7Y</th>\n",
       "      <th>CDS @10Y</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Par CDS $\\delta$/PV01</th>\n",
       "      <td>-0.2761</td>\n",
       "      <td>-0.0472</td>\n",
       "      <td>-0.0685</td>\n",
       "      <td>-0.1651</td>\n",
       "      <td>0.0119</td>\n",
       "      <td>-2.5896</td>\n",
       "      <td>-8.8937</td>\n",
       "      <td>0.3662</td>\n",
       "      <td>-0.0109</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Exact H/N</th>\n",
       "      <td>-0.8628</td>\n",
       "      <td>-0.1362</td>\n",
       "      <td>-0.2001</td>\n",
       "      <td>-0.4611</td>\n",
       "      <td>-0.3656</td>\n",
       "      <td>-3.1362</td>\n",
       "      <td>-8.8529</td>\n",
       "      <td>0.3814</td>\n",
       "      <td>-0.0119</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Inverted H/N</th>\n",
       "      <td>-0.8627</td>\n",
       "      <td>-0.1362</td>\n",
       "      <td>-0.2001</td>\n",
       "      <td>-0.4610</td>\n",
       "      <td>-0.3656</td>\n",
       "      <td>-3.1361</td>\n",
       "      <td>-8.8530</td>\n",
       "      <td>0.3814</td>\n",
       "      <td>-0.0119</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table></font></center>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fmt.displayDF(hns*1e-6, \"4f\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Delta transformation\n",
    "\n",
    "Jacobian can transform Deltas to different market quotes:\n",
    "\n",
    "$$ \\frac{\\partial v}{\\partial{\\bs q}} = \\frac{\\partial v}{\\partial \\bs p} \\frac{\\partial \\bs p}{\\partial \\bs b} \\frac{\\partial \\bs b}{\\partial \\bs q} = \\frac{\\partial v}{\\partial \\bs p} \\left(\\frac{\\partial \\bs b}{\\partial \\bs p}\\right)^{-1} \\frac{\\partial \\bs b}{\\partial \\bs q} $$\n",
    "\n",
    "* simply a change of variables in multi-variate calculus\n",
    "* the right hand side can be computed without upgrading the system\n",
    "  * the first factor in the RHS is the only time-consuming part"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Be all inclusive\n",
    "\n",
    "It is important to include all risk factors in the Jacobian matrix\n",
    "\n",
    " * a common mistake is to leave out the IR from the Jacobian\n",
    " * missing any risk factor in the Jacobian will lead to slow PnL bleeding that is very difficult to detect\n",
    " * rigorous PnL explain process can help identify such problems"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Deltas to internal state variables\n",
    "\n",
    "\n",
    "The Jacobian matrix can also be used to speed up the delta calculation.\n",
    "\n",
    "* So far, we compute the deltas to market observables, such as quoted spreads or par spreads. \n",
    "* But this is not the only choice, we could compute deltas to model's internal state variables instead.\n",
    "\n",
    "Let consider computing all deltas of CDS to the cumulative hazard $g(t)$, which is a the natural state variable of the curve interpolation:\n",
    "\n",
    "$$ \\frac{\\partial v}{\\partial{\\bs q}} = \\frac{\\partial v}{\\partial \\bs g} \\frac{\\partial \\bs g}{\\partial \\bs b} \\frac{\\partial \\bs b}{\\partial \\bs q} = \\frac{\\partial v}{\\partial \\bs g} \\left(\\frac{\\partial \\bs b}{\\partial \\bs g}\\right)^{-1}\\frac{\\partial \\bs b}{\\partial \\bs q} $$\n",
    "\n",
    "* $\\bs g$ is the value of $g(t)$ at the benchmark tenors\n",
    "* the $\\frac{\\partial \\bs q}{\\partial \\bs b}$ is a diagonal matrix of 1/PV01"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Fast delta calculation\n",
    "\n",
    "$$ \\frac{\\partial v}{\\partial{\\bs q}}  = \\frac{\\partial v}{\\partial \\bs g} \\left(\\frac{\\partial \\bs b}{\\partial \\bs g}\\right)^{-1}\\frac{\\partial \\bs b}{\\partial \\bs q} $$\n",
    "\n",
    "* the right hand side is much easier to compute than $\\frac{\\partial v}{\\partial \\bs q}$\n",
    "* RHS no longer requires re-bootstrapping the curves\n",
    "\n",
    "The same Deltas are computed at a much lower cost that the brute-force bump and recalc."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<center><font size=2><table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>5Y ZCB</th>\n",
       "      <th>CDS @0.25Y</th>\n",
       "      <th>CDS @0.5Y</th>\n",
       "      <th>CDS @1Y</th>\n",
       "      <th>CDS @2Y</th>\n",
       "      <th>CDS @3Y</th>\n",
       "      <th>CDS @5Y</th>\n",
       "      <th>CDS @7Y</th>\n",
       "      <th>CDS @10Y</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>$\\delta$ to $g(t)$</th>\n",
       "      <td>23.29</td>\n",
       "      <td>14.79</td>\n",
       "      <td>21.39</td>\n",
       "      <td>46.59</td>\n",
       "      <td>43.09</td>\n",
       "      <td>185.00</td>\n",
       "      <td>353.27</td>\n",
       "      <td>-11.77</td>\n",
       "      <td>0.25</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>$\\delta$ to quoted spreads</th>\n",
       "      <td>390.43</td>\n",
       "      <td>3.38</td>\n",
       "      <td>9.88</td>\n",
       "      <td>45.11</td>\n",
       "      <td>69.88</td>\n",
       "      <td>871.55</td>\n",
       "      <td>3,744.18</td>\n",
       "      <td>-202.84</td>\n",
       "      <td>7.68</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Inverted $\\delta$ to q.s.</th>\n",
       "      <td>390.43</td>\n",
       "      <td>3.38</td>\n",
       "      <td>9.88</td>\n",
       "      <td>45.11</td>\n",
       "      <td>69.88</td>\n",
       "      <td>871.68</td>\n",
       "      <td>3,744.57</td>\n",
       "      <td>-203.01</td>\n",
       "      <td>7.69</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table></font></center>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def calcCurveDelta(ir, cds, lbd, trades, pert=1e-4) :\n",
    "    mkt, mktp, pkeys, _ = inst.pert_curve(ir, cds, pert) \n",
    "    deltas, _ = inst.pv_deltas(mkt, mktp, trades)\n",
    "\n",
    "    return deltas/pert, pkeys, cds\n",
    "\n",
    "deltas_c, _, _ = calcCurveDelta(disc, cds, lbdm, bmks + [bespoke])\n",
    "pv01m_c = deltas_c[:-1, :]\n",
    "pv01_c = np.diag(pv01m_c)\n",
    "db_c = deltas_c[-1, :]\n",
    "id_c = db_c.dot(np.linalg.inv(pv01m_c)).dot(pv01m)\n",
    "\n",
    "hn_c = pd.DataFrame(np.array([db_c, db, id_c]).T*1e-4*notional, index=hi_names,\n",
    "                      columns=['$\\delta$ to $g(t)$', '$\\delta$ to quoted spreads', \n",
    "                               'Inverted $\\delta$ to q.s.']).T\n",
    "\n",
    "fmt.displayDF(hn_c, \"2f\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Hedge Optimization\n",
    "\n",
    "Edmund North: there's a difference between a gamble and a calculated risk."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Cost of hedging\n",
    "\n",
    "Hedging are costly in practice\n",
    "\n",
    "* bid/ask, capital, financing, rebalancing  etc\n",
    "\n",
    "But not hedging is usually more costly\n",
    "\n",
    "We need to minimize the cost of hedging\n",
    " * desks/traders often choose not to zero out all unwanted risks\n",
    " * there are only a limited number of liquid and cost effective hedging instruments\n",
    "\n",
    "How to get the best results using a limited number of hedging instruments?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "define what is best first,"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Hedge optimization\n",
    "\n",
    "We can formulate the problem as an optimization problem:\n",
    "\n",
    "* a set of benchmark instruments $\\bs b$ with quotes $\\bs q$\n",
    "* limited set of hedging instruments $\\bs u$, whose deltas to $\\bs q$ is the Jacobian $J = \\frac{\\partial \\bs u}{\\partial \\bs q}$ \n",
    " * the dimension of $\\bs u$ is smaller than $\\bs q$, $J$ is rectangular\n",
    "* the portfolio PV is $v$ and its deltas are: $\\bs d = (\\frac{\\partial v}{\\partial \\bs q})^T$\n",
    "* we use $\\bs h$ to represent the best replicating portfolio, it has the same dimension as $\\bs u$\n",
    "  * the hedging portfolio is of course $-\\bs h$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Partial hedge\n",
    "\n",
    "Partial hedge is to take the subset of the full hedging portfolio, keeping notionals unchanged.\n",
    "\n",
    "* simple, but it is a starting point\n",
    "* we can use it as a baseline to evaluate other hedging strategies\n",
    "\n",
    "\n",
    "Can we apply PCA?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "* Yes, just zero out the PVs for the first few PCs of the covariance of changes.\n",
    "* People actually do this in practice"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Minimize the residual risk\n",
    "\n",
    "The residual risk is the remaining risk after including the hedges, we can find a replicating portfolio that minimizes its L-2 norm:\n",
    "\n",
    "$$\\min_{\\bs h} \\Vert J^T \\bs {h - d} \\Vert_2$$\n",
    "\n",
    "* a standard least square problem, the solution is:\n",
    "\n",
    "$$ \\bs h = (JJ^T)^{-1}J\\bs d $$\n",
    "\n",
    "* this approach is simple and useful in practice\n",
    "\n",
    "Ridge regression is an easy improvement, by adding a penalty of $\\lambda \\Vert W \\bs h \\Vert_2^2$:\n",
    "\n",
    "$$ \\bs h = (JJ^T + \\lambda W^T W)^{-1}J\\bs d $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Minimize the variance\n",
    "\n",
    "Given the covariance matrix $V$ of the risk factors $\\bs q$'s *daily changes*, the variance of the hedged portfolio is:\n",
    "\n",
    "$$\\text{var} = (J^T \\bs {h - d})^T V (J^T \\bs {h - d})$$\n",
    "\n",
    "The optimal replicating portfolio that minimizes the var is:\n",
    "\n",
    "$$\\small \\begin{array}\n",
    "\\\\ \\frac{\\partial \\text{var}}{\\partial \\bs h} &= 2(J^T \\bs {h - d})^T V \\frac{\\partial(J^T \\bs {h - d})}{\\partial \\bs h} = 2(J^T \\bs {h - d})^T V J^T = \\bs 0 \\\\\n",
    "\\bs h^T JVJ^T &= \\bs d^T V J^T \\iff\n",
    "JVJ^T \\bs h = J V \\bs d \\\\\n",
    "\\bs h &= (JVJ^T)^{-1} J V \\bs d\n",
    "\\end{array}$$\n",
    "\n",
    "* this approach takes into account the covariance of risk factors.\n",
    "\n",
    "The Ridge version is:\n",
    "\n",
    "$$\\bs h = (JVJ^T + \\lambda W^T W)^{-1} J V \\bs d $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Numerical example\n",
    "\n",
    "We use the previous example to illustrate the effects of different hedges. \n",
    "\n",
    "* Suppose only the ZCB and 5Y CDS are cost effective to trade\n",
    "  * not an uncommon situation in the credit market\n",
    "* Our bank's system only computes risks to par CDS spreads, but the hedging instruments are the standardised CDS\n",
    "* The bespoke trade is long $10MM, 4.5Y and 500bps\n",
    "\n",
    "The Jacobian $J = \\frac{\\partial \\bs u}{\\partial \\bs p}$ is:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<center><font size=2><table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>IR</th>\n",
       "      <th>CDS @0.25Y</th>\n",
       "      <th>CDS @0.50Y</th>\n",
       "      <th>CDS @1Y</th>\n",
       "      <th>CDS @2Y</th>\n",
       "      <th>CDS @3Y</th>\n",
       "      <th>CDS @5Y</th>\n",
       "      <th>CDS @7Y</th>\n",
       "      <th>CDS @10Y</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>ZCB</th>\n",
       "      <td>4.525</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5Y CDS</th>\n",
       "      <td>-0.3067</td>\n",
       "      <td>-0.002196</td>\n",
       "      <td>-0.006453</td>\n",
       "      <td>-0.02874</td>\n",
       "      <td>-0.07083</td>\n",
       "      <td>-0.1771</td>\n",
       "      <td>3.999</td>\n",
       "      <td>0.009446</td>\n",
       "      <td>-0.0003262</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table></font></center>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "hi = np.array([0, 6])\n",
    "hnm = ['ZCB', '5Y CDS']\n",
    "jac = np.array(pv01m_p)[hi, :].T\n",
    "lsqm = np.linalg.inv(jac.T.dot(jac)).dot(jac.T)\n",
    "fmt.displayDF(pd.DataFrame(jac.T, index = hnm, columns=pert_names), \"4g\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## A stylized market\n",
    "\n",
    "Correlation of daily rates changes:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<center><font size=2><table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>5Y ZCB</th>\n",
       "      <th>CDS @0.25Y</th>\n",
       "      <th>CDS @0.5Y</th>\n",
       "      <th>CDS @1Y</th>\n",
       "      <th>CDS @2Y</th>\n",
       "      <th>CDS @3Y</th>\n",
       "      <th>CDS @5Y</th>\n",
       "      <th>CDS @7Y</th>\n",
       "      <th>CDS @10Y</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>5Y ZCB</th>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>CDS @0.25Y</th>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>0.98</td>\n",
       "      <td>0.96</td>\n",
       "      <td>0.941</td>\n",
       "      <td>0.922</td>\n",
       "      <td>0.904</td>\n",
       "      <td>0.886</td>\n",
       "      <td>0.868</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>CDS @0.5Y</th>\n",
       "      <td>0</td>\n",
       "      <td>0.98</td>\n",
       "      <td>1</td>\n",
       "      <td>0.98</td>\n",
       "      <td>0.96</td>\n",
       "      <td>0.941</td>\n",
       "      <td>0.922</td>\n",
       "      <td>0.904</td>\n",
       "      <td>0.886</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>CDS @1Y</th>\n",
       "      <td>0</td>\n",
       "      <td>0.96</td>\n",
       "      <td>0.98</td>\n",
       "      <td>1</td>\n",
       "      <td>0.98</td>\n",
       "      <td>0.96</td>\n",
       "      <td>0.941</td>\n",
       "      <td>0.922</td>\n",
       "      <td>0.904</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>CDS @2Y</th>\n",
       "      <td>0</td>\n",
       "      <td>0.941</td>\n",
       "      <td>0.96</td>\n",
       "      <td>0.98</td>\n",
       "      <td>1</td>\n",
       "      <td>0.98</td>\n",
       "      <td>0.96</td>\n",
       "      <td>0.941</td>\n",
       "      <td>0.922</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>CDS @3Y</th>\n",
       "      <td>0</td>\n",
       "      <td>0.922</td>\n",
       "      <td>0.941</td>\n",
       "      <td>0.96</td>\n",
       "      <td>0.98</td>\n",
       "      <td>1</td>\n",
       "      <td>0.98</td>\n",
       "      <td>0.96</td>\n",
       "      <td>0.941</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>CDS @5Y</th>\n",
       "      <td>0</td>\n",
       "      <td>0.904</td>\n",
       "      <td>0.922</td>\n",
       "      <td>0.941</td>\n",
       "      <td>0.96</td>\n",
       "      <td>0.98</td>\n",
       "      <td>1</td>\n",
       "      <td>0.98</td>\n",
       "      <td>0.96</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>CDS @7Y</th>\n",
       "      <td>0</td>\n",
       "      <td>0.886</td>\n",
       "      <td>0.904</td>\n",
       "      <td>0.922</td>\n",
       "      <td>0.941</td>\n",
       "      <td>0.96</td>\n",
       "      <td>0.98</td>\n",
       "      <td>1</td>\n",
       "      <td>0.98</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>CDS @10Y</th>\n",
       "      <td>0</td>\n",
       "      <td>0.868</td>\n",
       "      <td>0.886</td>\n",
       "      <td>0.904</td>\n",
       "      <td>0.922</td>\n",
       "      <td>0.941</td>\n",
       "      <td>0.96</td>\n",
       "      <td>0.98</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table></font></center>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n = len(hi_names)\n",
    "\n",
    "corr = np.fromfunction(lambda i, j : .98**(abs(i-j)), (n, n))\n",
    "corr[0, 1:] = corr[1:, 0] = 0\n",
    "sig = np.eye(n)*15e-4\n",
    "sig[0, 0] = 5e-4\n",
    "cov = sig.dot(corr).dot(sig.T)\n",
    "fmt.displayDF(pd.DataFrame(corr, index=hi_names, columns=hi_names), \"3g\")\n",
    "\n",
    "varm = np.linalg.inv(jac.T.dot(cov).dot(jac)).dot(jac.T).dot(cov)\n",
    "opt_var = varm.dot(db_p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "Std dev of daily rates changes: \n",
    "* interest rate: 5bps \n",
    "* CDS spreads for all tenors: 15bps"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Optimal hedges\n",
    "\n",
    "* The optimal hedging portfolios (in \\$MM) using different methods for the bespoke trade:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<center><font size=2><table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Unhedged</th>\n",
       "      <th>Partial hedge</th>\n",
       "      <th>Min residual risk</th>\n",
       "      <th>Min var</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>ZCB</th>\n",
       "      <td>-0.000</td>\n",
       "      <td>-0.863</td>\n",
       "      <td>-0.872</td>\n",
       "      <td>-1.020</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5Y CDS</th>\n",
       "      <td>-0.000</td>\n",
       "      <td>-8.853</td>\n",
       "      <td>-8.793</td>\n",
       "      <td>-10.974</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table></font></center>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "opt_l2 = lsqm.dot(db_p).T\n",
    "opt_ph = (db/pv01)[hi]\n",
    "opt_pfs = [np.zeros(len(opt_l2)), opt_ph, opt_l2, opt_var]\n",
    "\n",
    "l2 = np.array([np.linalg.norm(jac.dot(np.array(p)) - db_p, 2) for p in opt_pfs])*1e-4\n",
    "vv = np.array([(jac.dot(np.array(p)) - db_p).T.dot(cov).dot((jac.dot(np.array(p)) - db_p)) \n",
    "               for p in opt_pfs])\n",
    "\n",
    "df_opt = pd.DataFrame(np.array(opt_pfs).T, index=hnm, \n",
    "                      columns=['Unhedged', 'Partial hedge', 'Min residual risk', 'Min var']).T\n",
    "\n",
    "df_opt['L2 norm of residual risk'] = l2\n",
    "df_opt['Std dev of daily PnL'] = np.sqrt(vv)\n",
    "\n",
    "fmt.displayDF(-df_opt.iloc[:, :-2].T*notional*1e-6, \"3f\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* The performances of these portfolios matches our expectations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<center><font size=2><table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Unhedged</th>\n",
       "      <th>Partial hedge</th>\n",
       "      <th>Min residual risk</th>\n",
       "      <th>Min var</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>L2 norm of residual risk</th>\n",
       "      <td>3,633.3</td>\n",
       "      <td>891.42</td>\n",
       "      <td>891.07</td>\n",
       "      <td>1,247.5</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Std dev of daily PnL</th>\n",
       "      <td>61,511</td>\n",
       "      <td>12,292</td>\n",
       "      <td>12,615</td>\n",
       "      <td>3,179.6</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table></font></center>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fmt.displayDF(df_opt.iloc[:, -2:].T*notional, \"5g\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* The variance minimization is an effective hedging strategy."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Theta\n",
    "\n",
    "In practice, we usually don't actively hedge Theta, instead we monitor it\n",
    "\n",
    "* Source of Theta includes reduction in maturity and accrual of coupon \n",
    "* The passing of time is not volatile, it is not considered a risk factor\n",
    "* Different hedges can lead to different theta profile\n",
    "* It is neither feasible nor necessary to maintain a flat theta\n",
    " \n",
    "Theta and convexity are often the trade off.\n",
    "* Positive theta usually means negative convexity, and vice versa"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Brownian Bridge"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Interpolate the Brownian motion\n",
    "\n",
    "Sometimes, we want to interpolate stochastically using Brownian motion:\n",
    "\n",
    "* Fill in missing historical data\n",
    "* Pricing model needs finer steps than what the simulation model produces\n",
    "* the exact stochastic path is costly to compute, we want to approximate the true path using larger sampling steps\n",
    "\n",
    "Suppose $\\bs w(t)$ is a n-dimensional brownian motion, and $\\bs v_T$ is its (known) terminal value, then:\n",
    "\n",
    "$$ \\bs w_b(t) = \\bs w(t) - \\frac{t}{T} (\\bs w(T) - \\bs v_T)$$ \n",
    "\n",
    "is the Brownian Bridge. Note that $\\bs w(t)$ can be correlated."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Examples of Brownian Bridge\n",
    "\n",
    "A one dimensional Brownian Bridge with $w(1) = 10$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x288 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "nt = 100\n",
    "w = np.cumsum(np.random.normal(size=[1000, nt]), 1)\n",
    "dt = arange(nt)/nt\n",
    "\n",
    "figure(figsize=[12, 4])\n",
    "subplot(1, 2, 1)\n",
    "plot(dt, w[:3, :].T, label=\"_nolegend_\")\n",
    "plot(dt, np.std(w, 0), 'k', linewidth=2)\n",
    "legend([\"std dev\"])\n",
    "title(\"$w(t)$\")\n",
    "xlabel(\"$t$\")\n",
    "\n",
    "ve = 10\n",
    "subplot(1, 2, 2)\n",
    "b = np.array([p - (p[-1] - ve)*dt for p in w])\n",
    "plot(dt, b[:3, :].T, label=\"_nolegend_\")\n",
    "plot(dt, np.std(b, 0), 'k', linewidth=2)\n",
    "legend([\"std dev\"])\n",
    "title(\"$b(t)$\")\n",
    "xlabel(\"$t$\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Numerical Integration"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Quadrature rules\n",
    "\n",
    "To compute the definite integral of a one dimensional function:\n",
    "\n",
    "$$ \\int_a^b f(x) dx $$\n",
    "\n",
    "* Rectangle rule: $ \\int_a^b f(x) dx \\approx (b-a)f(b) $\n",
    "* Trapezoidal rule:  $ \\int_a^b f(x) dx \\approx (b-a)\\frac{f(a) + f(b)}{2} $\n",
    "* Simpson's rule: $\\int_a^b f(x) dx \\approx \\frac{b-a}{6}[f(a) + 4 f(\\frac{a+b}{2}) + f(b)] $\n",
    "* Gaussian Quadrature: $ \\int_a^b f(x) dx \\approx \\sum_{i=1}^{N} w_i \\;f(x_i) $, where the nodes $x_i$ and weights $w_i$ are chosen so that the integral is exact for polynomials of degrees upto order $2N-1$.\n",
    "\n",
    "In practice, the Simpson's rule works well for reasonably smooth functions.\n",
    "\n",
    "* We sometime run into (nearly) discontinuous $f(x)$, where the simple Trapezoidal rule works very well.\n",
    "* Adaptive method can be used to achieve a desired level of accuracy"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Accuracy of numerical integration"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If the interval $[a, b]$ is divided into $N$ equal steps, then the numerical errors of the $\\int_a^b f(x) dx$ for smooth function $f(x)$ are:\n",
    "\n",
    "| Rectangle | Trapezoidal | Simpson |\n",
    "| :-----: | :------: | :------: |\n",
    "| $O(\\frac{1}{N})$ | $O(\\frac{1}{N^2})$ | $O(\\frac{1}{N^4})$ |"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x288 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import lin\n",
    "\n",
    "def plotIntegral(f, x, xs, lines) :\n",
    "    plot(xs, f(xs))\n",
    "    \n",
    "    for k, l in lines.items() :\n",
    "        l.build(x, f(x))\n",
    "        plot(xs, l(xs))\n",
    "        \n",
    "    legend(['$f(x)$'] + list(lines.keys()), loc='best')\n",
    "\n",
    "intRules = {\"Rect\":lin.PiecewiseFlat(), \"Trapozoidal\":lin.PiecewiseLinear(), \"Simpson\":lin.Quadratic()}\n",
    "\n",
    "f = np.sin\n",
    "x0 = 0.\n",
    "x1 = 4.\n",
    "dx = 2.\n",
    "\n",
    "x = np.arange(x0, x1 + 1e-10, dx/2)\n",
    "xs = np.arange(x0, x1 + 1e-10, .02)\n",
    "\n",
    "figure(figsize=[12, 4])\n",
    "subplot(1, 2, 1)\n",
    "plotIntegral(f, x, xs, intRules)\n",
    "plot(x, f(x), 'o');\n",
    "\n",
    "subplot(1, 2, 2)\n",
    "x = np.arange(x0, x1 + 1e-10, dx/4)\n",
    "plotIntegral(f, x, xs, intRules)\n",
    "plot(x, f(x), 'o');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Assignment\n",
    "\n",
    "Recommended Reading:\n",
    "\n",
    "* Andersen and Piterbarg: 6.4\n",
    "\n",
    "Homework:\n",
    "\n",
    "* Complete [homework set 5](http://yadongli.github.io/nyumath2048/)"
   ]
  }
 ],
 "metadata": {
  "celltoolbar": "Slideshow",
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}