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<p>1</p>
<p>Credit Index Options in CDSOBjorn Flesaker Madhu Nayakkankuppam December 14, 2011 Igor Shkurko</p>
<p>11.1</p>
<p>IntroductionBackground</p>
<p>Current standard models for credit index swap options have been developed assuming lognormal spread dynamics under a pricing measure that takes the value of the index premium leg (risky annuity) as the numeraire. The sole rationale for this modeling approach is to obtain a closed form valuation formula of the Black-Scholes kind. The primary deciencies of this approach are that it does not properly handle: Strike adjustment A contractual cash ow upon exercise to compensate for the difference between the option strike and the coupon of the underlying swap; Loss settlement A contractual cash ow upon exercise to compensate for realized losses on the underlying index at the time of valuation; and Price-based options Options where the strike and spot are specied in price terms rather than in CDS spread terms, hence requiring appropriate price-to-spread conversions before the model can be used. Two further undesirable aspects of this modeling approach are: Armageddon scenario The rare (but theoretically possible) event of all names in the underlying index defaulting prior to option expiry, resulting in a vanishing numeraire; and Volatility estimate The inability to obtain unbiased estimates of implied volatility from historical volatility even under idealized conditions, owing to the strictly positive quadratic variation of the risky annuity numeraire process.</p>
<p>Originally published on September 4, 2009</p>
<p>2</p>
<p>1.2</p>
<p>Summary</p>
<p>The new Bloomberg model makes a small enhancement to the standard Black model. In this new model, we retain the essential assumption of lognormal spreads, but not the goal of a closed form solution, thus eliminating the need for the risky annuity numeraire. Specically, we assume that the options are valued as if the following conditions are satised:</p>
<p> Any defaults that have taken place before the valuation date are accommodated by adjusting theswaption notional amount by the index factor, and by adding a default settlement amount (based on actual or assumed recovery on the defaulted names) to the cash ows settled upon exercise.</p>
<p> The underlying swap is valued with the usual market convention of a at spread to maturity.1 The forward index spread to the option expiration date is computed as the replacement spreadon a forward starting swap with no-knockout to account for defaults taking place between the valuation date and option expiry. This essentially amounts to nding a replacement spread from a CDS with a default leg starting immediately and a premium leg starting at the option expiration date, with both legs running until the maturity date of the underlying swap.</p>
<p> Aside from their impact on the forward spread (through the no-knockout assumption), potentialdefaults between the valuation date and option expiry are not considered.</p>
<p> The realized index spread at option expiry is assumed to be lognormally distributed under theusual risk-neutral probability measure, i.e. using the default-free money market account as the numeraire asset.</p>
<p> For a given index spread volatility, the mean of the realized index spread is calibrated so thatthe risk neutral expectation of the terminal swap value matches the forward price quoted at the valuation time.</p>
<p> The option cash ow upon exercise is modeled precisely, including the strike adjustment amountand compensation for known defaults.</p>
<p> The expectations involved in the forward and the option valuation calculations are carried outthrough numerical quadrature. The new Bloomberg model recties several shortcomings of the Black model by</p>
<p> correctly handling strike adjustment and settlement of realized losses; directly handling price-based indices; and avoiding the technical pitfalls of a vanishing numeraire.1 We note that the assumption of a stochastic, at spread curve can be problematic, but will not attempt to address this currently.</p>
<p>3</p>
<p>Hence, the Black model will be decommissioned.2 Clients should migrate to the new Bloomberg model, noting that any changes in pricing, hedging and P&L are the result of a more precise valuation. Further details of both the original Black model and the new Bloomberg model are given in Section 2, along with numerical comparisons in Section 3.</p>
<p>2</p>
<p>Details</p>
<p>While both models assume an adjusted index spread process that is lognormal in the risk-neutral measure, there are differences in the calibration of the parameters of the spread process, in the option payoff function, and in the choice of the numeraire asset. To see these differences clearly, consider the valuation at time t = 0 of a (receiver or payer) swaption with strike K and expiration time te 0 on an underlying spread-based index with maturity tm te , contractual coupon c, and notional amount N (adjusted with a weighting factor for any defaults since inception of the index). For the underlying forward swap from te to tm , denote by St the clean forward index spread adjusted for no-knockout, by Lt the corresponding clean forward risky annuity, and by Vt the corresponding clean principal value at time t (0 t te ). As a result, the following relation holds:</p>
<p>Vt = N (St c)Lt ,where is an indicator variable that differentiates between payer swaps ( = 1) and receiver swaps ( = 1). Finally, let Pt denote the risk-free discount factor to time t with P0 = 1, and let Et [] denote the risk-neutral expectation conditional on information up to time t.</p>
<p>2.1</p>
<p>The Black Model</p>
<p>This model [3] assumes a lognormal process for the adjusted index spread</p>
<p>St = S0 e 2 </p>
<p>1</p>
<p>2 t+</p>
<p>t</p>
<p>where is the index spread volatility, is a standard normal random variable, and S0 is the adjusted forward index spread at t = 0. The terminal value of the swap is then given by</p>
<p>Vte = N (Ste c) Lte ,</p>
<p>(1)</p>
<p>and the terminal option payoff is taken to be its positive part (Vte )+ . By using the risky annuity Lt as the pricing numeraire, the option price O is obtained from the classical, Black formula:</p>
<p>O = N L0 [S0 (d+ ) K (d )] ,2</p>
<p>The Black model was decommissioned around mid 2010 from the Bloomberg terminal.</p>
<p>4</p>
<p>where</p>
<p>d = </p>
<p>1 ln (S0 /K) 2 2 te te</p>
<p>and () is the cumulative distribution function of a standard normal variate. While this approach leads to an appealing, analytic pricing formula, it also suffers from the following shortcomings. Payoff The option payoff incorrectly handles two important components that inuence the exercise decision. 1. When the strike K differs from the index contractual coupon c, the contract calls for a strike adjustment amount equal to N (c K)A(K) to be settled upon exercise. Here A(K) denotes the deterministic risky annuity as of te for a swap starting at te and maturing at tm , using a at quoted spread equal to K . By splitting the option payoff as</p>
<p>[Vte ]+ = [N {(Ste c) + (c K)} Lte ]+ ,the Black model can be seen to employ a strike adjustment term of the form N (c K)Lte , i.e. with a risky annuity computed with the random spread at expiry Ste , not with the xed strike spread K . 2. The contract requires defaults that have occurred since index inception to be settled upon option exercise. While defaults between the valuation time t = 0 and option expiry te are accommodated with an adjusted forward spread and defaults prior to t = 0 alter the notional via a weighting factor, settlement upon exercise of realized index losses as of the valuation time do not play a role in the exercise decision. By incorrectly accounting for these terms in the option payoff, the model delivers an accurate price only for options struck near the contractual coupon of an underlying index that is lossless. Numeraire Since the numeraire Lt can vanish with positive probability (the "Armageddon scenario") for an event where the option payoff is nonzero, the option price must strictly be viewed as being conditional upon not all names in the index defaulting prior to option expiry. However, this is more a technical issue than a practical one. For realistic index spreads and option maturities, the negligible probability of the Armageddon scenario has no material impact on valuation; see also [1]. Following the general approach3 of [2], the Bloomberg model recties these shortcomings.</p>
<p>2.2</p>
<p>The Bloomberg Model</p>
<p>The Bloomberg model too assumes a lognormal adjusted index spread process of the form</p>
<p>St = me 2 3</p>
<p>1</p>
<p>2 t+</p>
<p>t</p>
<p>,</p>
<p>(2)</p>
<p>We thank Claus Pedersen for bringing his earlier work to our attention.</p>
<p>5</p>
<p>where is the index spread volatility and is a standard normal random variable, and m is chosen to satisfy the nonlinear equation F0 = E0 [Vte ] . (3) Here F0 denotes the current clean adjusted forward price for delivery at time t = 0 of the underlying swap, i.e. the parameter m of the lognormal spread process is calibrated to match the market quoted clean forward price at t = 0, adjusted for no-knockout. Note that, as a result of such a calibration, the value of m will generally depend on , the assumed spread volatility. This is in contrast with the Black model where S0 is simply the adjusted forward spread at t = 0. Denote the strike adjustment amount by</p>
<p>H(K) = N (c K)A(K),</p>
<p>(4)</p>
<p>where A(K) denotes the clean forward risky annuity for a at, quoted spread of K . Further, at the valuation time t = 0, let l 0 denote the fractional index loss relative to the original, unadjusted notional N0 , so that the default settlement upon exercise is D = N0 l. The option payoff is then taken to be (Vte + H(K) + D)+ , and the option value O is given by</p>
<p>O = Pte E0 (Vte + H(K) + D)+ .</p>
<p>(5)</p>
<p>Note that both the strike adjustment H(K) and the realized index loss D are deterministic payoff corrections fully known at the valuation time. The expectations in the calibration equation (3) and the pricing equation (5) are computed by numerical integration, hence the integrands need to be evaluated at the nodes of the chosen quadrature scheme. In the interest of a fast implementation of the calibration and the pricing procedures within CDSO , we make three simplifying approximations in the implementation. In order to compute Vte and A(K) in (3) and (5), we need to strip a at credit curve and infer a hazard rate corresponding to an adjusted index spot spread Ste or strike level K . To speed up these computations, we assume that</p>
<p> the interest rate up to swap expiry is a constant; the index coupon is paid continuously (rather than quarterly); and Ste and K are spot spreads, rather than forward spreads adjusted for the knockout provision.These assumptions yield a well known analytic formula for as the ratio between the spread level and the index loss given default, preempting a numerical root nding procedure. Note that the last assumption applies only to the calculation of ; given this , the terminal swap value Vte is then correctly calculated as the forward swap value adjusted for no-knockout. All three assumptions are consistently used in both the calibration and the pricing procedures. The latter two assumptions affect in opposing ways leading to some error cancellation. The impact of any residual approximation error in is mitigated by calibrating the mean of the spread process to the no-knockout forward price which is calculated exactly.</p>
<p>6</p>
<p>2.3</p>
<p>Put-Call Parity</p>
<p>In practice, it is sufcient for the option pricer to handle only one of payer swaptions (call options on index spread, = 1) or receiver swaptions (put options on index spread, = 1), as the other may be inferred from an appropriate put-call parity relationship. Since the contractual payoff as shown in (5) includes a strike adjustment term H(K) and a loss settlement term D, this put-call parity is given by</p>
<p>OC = OP + Pte (F0 + H(K) + D) ,where OC is the call premium, OP the put premium, and F0 , H(K), D are valued for payer swaps with = 1.</p>
<p>2.4</p>
<p>Related Calculations</p>
<p>Both models require a volatility input, for which CDSO provides an estimate of the 90 calendarday historical volatility of the 5-year spread of the underlying swap. When the underlying swap is spread-based, this estimate is exactly computed as the sample volatility of the closing market quotes. When the underlying is price-based, an approximate replacement spread is computed using the same three assumptions mentioned above. The sample volatility of this approximate replacement spread is then used as an estimate of historical spread volatility. This estimate is acceptable for a wide range of price quotes for the underlying, but becomes noticeably biased if prices deviate signicantly from par. In any case, the estimate is only an indicative wake-up value, which the user is expected to override with a suitable implied volatility. Both models provide a full set of risk measures (delta, gamma, vega, spread DV01, and additionally for the Bloomberg model only, theta and interest rate DV01) using a simple bump-and-reprice approach, as well as an implied volatility for a given option premium. Delta is ratio of the change in option premium to the change in the principal value of the underlying swap when the spot spread is bumped up by a basis point. Gamma is the change in delta for a 10 bp shift in the spot spread. Vega is the difference in option premiums for a 1 vol-point increase in implied volatility. Theta is the difference in option premiums for a one calendar day decrease in option maturity, i.e. using the terminal distribution obtained by shortening the exercise time te by 1/365.25 in (2), but retaining the same forward price F0 and forward spread m. Spread DV01 is the change in the option premium when the spot spread is increased by 1 bp, while IR DV01 is the change in the option premium for a parallel 1 bp upward shift of the benchmark yield curve.</p>
<p>2.5</p>
<p>Price-Based Underlying</p>
<p>For price-based indices (e.g. high yield and emerging market indices), the Black model requires the spot quote and the strike to rst be converted (for example, using the default swap pricer CDSW ) to an equivalent at spread, appropriately adjusted for the knockout feature of the option. The Bloomberg model directly handles this case by setting the strike adjustment amount in (4) as H(K) = N (K 1), where K is the strike expressed as a fraction of par.</p>
<p>7</p>
<p>3</p>
<p>Comparisons</p>
<p>We present, via a series of plots, numerical comparisons between the Black and the Bloomberg models for valuing a payer swaption ( = 1) expiring in te = 0.1222 years on an underlying swap maturing in tm = 5.3222 years, with an input volatility of = 30%, a at yield curve at 5%, and unit notional N = 1. (Pricing and risks for a receiver swaption, which may be obtained by put-call parity, show similar patterns, and hence are not included here.) Figure 1 corresponds to a spread-based payer swap with a contractual coupon of c = 100 basis points, and Figure 2 corresponds to a payer sw...</p>