bloomberg model[1]

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1 Credit Index Options in CDSO Bjorn Flesaker Madhu Nayakkankuppam Igor Shkurko December 14, 2011 * 1 Introduction 1.1 Background 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 for- mula of the Black-Scholes kind. The primary deficiencies of this approach are that it does not properly handle: Strike adjustment A contractual cash flow upon exercise to compensate for the difference between the option strike and the coupon of the underlying swap; Loss settlement A contractual cash flow upon exercise to compensate for realized losses on the un- derlying index at the time of valuation; and Price-based options Options where the strike and spot are specified 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 volatil- ity even under idealized conditions, owing to the strictly positive quadratic variation of the risky annuity numeraire process. * Originally published on September 4, 2009

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Page 1: Bloomberg Model[1]

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Credit Index Options in CDSO

Bjorn Flesaker Madhu Nayakkankuppam Igor Shkurko

December 14, 2011∗

1 Introduction

1.1 Background

Current standard models for credit index swap options have been developed assuming lognormalspread 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 for-mula of the Black-Scholes kind. The primary deficiencies of this approach are that it does not properlyhandle:

Strike adjustment A contractual cash flow upon exercise to compensate for the difference betweenthe option strike and the coupon of the underlying swap;

Loss settlement A contractual cash flow upon exercise to compensate for realized losses on the un-derlying index at the time of valuation; and

Price-based options Options where the strike and spot are specified in price terms rather than in CDSspread terms, hence requiring appropriate price-to-spread conversions before the model can beused.

Two further undesirable aspects of this modeling approach are:

Armageddon scenario The rare (but theoretically possible) event of all names in the underlying indexdefaulting prior to option expiry, resulting in a vanishing numeraire; and

Volatility estimate The inability to obtain unbiased estimates of implied volatility from historical volatil-ity even under idealized conditions, owing to the strictly positive quadratic variation of the riskyannuity numeraire process.

∗Originally published on September 4, 2009

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1.2 Summary

The new Bloomberg model makes a small enhancement to the standard Black model. In this newmodel, we retain the essential assumption of lognormal spreads, but not the goal of a closed formsolution, thus eliminating the need for the risky annuity numeraire. Specifically, we assume that theoptions are valued as if the following conditions are satisfied:

• 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 (basedon actual or assumed recovery on the defaulted names) to the cash flows settled upon exercise.

• The underlying swap is valued with the usual market convention of a flat 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 thevaluation date and option expiry. This essentially amounts to finding a replacement spread froma CDS with a default leg starting immediately and a premium leg starting at the option expirationdate, with both legs running until the maturity date of the underlying swap.

• Aside from their impact on the forward spread (through the no-knockout assumption), potentialdefaults between the valuation date and option expiry are not considered.

• 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 thenumeraire asset.

• 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 thevaluation time.

• The option cash flow upon exercise is modeled precisely, including the strike adjustment amountand compensation for known defaults.

• The expectations involved in the forward and the option valuation calculations are carried outthrough numerical quadrature.

The new Bloomberg model rectifies several shortcomings of the Black model by

• correctly handling strike adjustment and settlement of realized losses;

• directly handling price-based indices; and

• avoiding the technical pitfalls of a vanishing numeraire.

1We note that the assumption of a stochastic, flat spread curve can be problematic, but will not attempt to address thiscurrently.

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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.

2 Details

While both models assume an adjusted index spread process that is lognormal in the risk-neutral mea-sure, there are differences in the calibration of the parameters of the spread process, in the optionpayoff 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) swaptionwith 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 sinceinception of the index). For the underlying forward swap from te to tm, denote by St the clean forwardindex spread adjusted for no-knockout, by Lt the corresponding clean forward risky annuity, and by Vtthe corresponding clean principal value at time t (0 ≤ t ≤ te). As a result, the following relation holds:

Vt = ξN(St − c)Lt,

where ξ is an indicator variable that differentiates between payer swaps (ξ = 1) and receiver swaps(ξ = −1). Finally, let Pt denote the risk-free discount factor to time t with P0 = 1, and let Et [·] denotethe risk-neutral expectation conditional on information up to time t.

2.1 The Black Model

This model [3] assumes a lognormal process for the adjusted index spread

St = S0e− 1

2σ2t+σ

√tε

where σ is the index spread volatility, ε is a standard normal random variable, and S0 is the adjustedforward index spread at t = 0. The terminal value of the swap is then given by

Vte = ξN (Ste − c)Lte , (1)

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:

O = ξN L0 [S0Φ (d+)−KΦ (d−)] ,

2The Black model was decommissioned around mid 2010 from the Bloomberg terminal.

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where

d± = ξ

(ln (S0/K)± 1

2σ2te

σ√te

)and Φ (·) 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 followingshortcomings.

Payoff The option payoff incorrectly handles two important components that influence the exercisedecision.

1. When the strike K differs from the index contractual coupon c, the contract calls for a strikeadjustment amount equal to ξN(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 flat quoted spread equal to K. By splitting the option payoff as

[Vte ]+ = [ξN {(Ste − c) + (c−K)}Lte ]

+ ,

the Black model can be seen to employ a strike adjustment term of the form ξN(c−K)Lte ,i.e. with a risky annuity computed with the random spread at expiry Ste , not with the fixedstrike spread K.

2. The contract requires defaults that have occurred since index inception to be settled uponoption exercise. While defaults between the valuation time t = 0 and option expiry te areaccommodated with an adjusted forward spread and defaults prior to t = 0 alter the notionalvia a weighting factor, settlement upon exercise of realized index losses as of the valuationtime do not play a role in the exercise decision.

By incorrectly accounting for these terms in the option payoff, the model delivers an accurateprice 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 beingconditional upon not all names in the index defaulting prior to option expiry. However, this ismore 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; seealso [1].

Following the general approach3 of [2], the Bloomberg model rectifies these shortcomings.

2.2 The Bloomberg Model

The Bloomberg model too assumes a lognormal adjusted index spread process of the form

St = me−12σ2t+σ

√tε, (2)

3We thank Claus Pedersen for bringing his earlier work to our attention.

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where σ is the index spread volatility and ε is a standard normal random variable, and m is chosen tosatisfy the nonlinear equation

F0 = E0 [Vte ] . (3)

Here F0 denotes the current clean adjusted forward price for delivery at time t = 0 of the underlyingswap, i.e. the parameter m of the lognormal spread process is calibrated to match the market quotedclean forward price at t = 0, adjusted for no-knockout. Note that, as a result of such a calibration, thevalue of m will generally depend on σ, the assumed spread volatility. This is in contrast with the Blackmodel where S0 is simply the adjusted forward spread at t = 0.

Denote the strike adjustment amount by

H(K) = ξN(c−K)A(K), (4)

where A(K) denotes the clean forward risky annuity for a flat, quoted spread of K. Further, at thevaluation time t = 0, let l ≥ 0 denote the fractional index loss relative to the original, unadjustednotional N0, so that the default settlement upon exercise is D = ξN0l. The option payoff is then takento be (Vte +H(K) +D)+, and the option value O is given by

O = Pte E0

[(Vte +H(K) +D)+

]. (5)

Note that both the strike adjustment H(K) and the realized index loss D are deterministic payoff cor-rections fully known at the valuation time.

The expectations in the calibration equation (3) and the pricing equation (5) are computed by numericalintegration, 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 <GO>,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 flat credit curve and infer a hazard rate λ corresponding to an adjustedindex spot spread Ste or strike level K. To speed up these computations, we assume that

• 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 λ as the ratio between the spread leveland the index loss given default, preempting a numerical root finding procedure. Note that the lastassumption applies only to the calculation of λ; given this λ, the terminal swap value Vte is then correctlycalculated as the forward swap value adjusted for no-knockout. All three assumptions are consistentlyused in both the calibration and the pricing procedures. The latter two assumptions affect λ in opposingways leading to some error cancellation. The impact of any residual approximation error in λ is mitigatedby calibrating the mean of the spread process to the no-knockout forward price which is calculatedexactly.

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2.3 Put-Call Parity

In practice, it is sufficient for the option pricer to handle only one of payer swaptions (call options onindex spread, ξ = 1) or receiver swaptions (put options on index spread, ξ = −1), as the other may beinferred 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

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 withξ = 1.

2.4 Related Calculations

Both models require a volatility input, for which CDSO <GO> provides an estimate of the 90 calendar-day historical volatility of the 5-year spread of the underlying swap. When the underlying swap isspread-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 samethree assumptions mentioned above. The sample volatility of this approximate replacement spread isthen used as an estimate of historical spread volatility. This estimate is acceptable for a wide range ofprice quotes for the underlying, but becomes noticeably biased if prices deviate significantly from par.In any case, the estimate is only an indicative wake-up value, which the user is expected to overridewith a suitable implied volatility.

Both models provide a full set of risk measures (delta, gamma, vega, spread DV01, and additionally forthe Bloomberg model only, theta and interest rate DV01) using a simple bump-and-reprice approach, aswell as an implied volatility for a given option premium. Delta is ratio of the change in option premium tothe change in the principal value of the underlying swap when the spot spread is bumped up by a basispoint. Gamma is the change in delta for a 10 bp shift in the spot spread. Vega is the difference in optionpremiums for a 1 vol-point increase in implied volatility. Theta is the difference in option premiums for aone calendar day decrease in option maturity, i.e. using the terminal distribution obtained by shorteningthe 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 IRDV01 is the change in the option premium for a parallel 1 bp upward shift of the benchmark yield curve.

2.5 Price-Based Underlying

For price-based indices (e.g. high yield and emerging market indices), the Black model requires the spotquote and the strike to first be converted (for example, using the default swap pricer CDSW <GO>) toan equivalent flat spread, appropriately adjusted for the knockout feature of the option. The Bloombergmodel directly handles this case by setting the strike adjustment amount in (4) as H(K) = Nξ(K−1),where K is the strike expressed as a fraction of par.

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3 Comparisons

We present, via a series of plots, numerical comparisons between the Black and the Bloomberg modelsfor valuing a payer swaption (ξ = 1) expiring in te = 0.1222 years on an underlying swap maturing intm = 5.3222 years, with an input volatility of σ = 30%, a flat yield curve at 5%, and unit notionalN = 1. (Pricing and risks for a receiver swaption, which may be obtained by put-call parity, showsimilar 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 swaption on a price-based swap with a contractual coupon of500 basis points. Each figure contains four plots showing option premium, delta, gamma and vegaas a function of spot spread. The underlying swap is assumed to be lossless, and the option strike ischosen so that no strike adjustment is necessary. Hence, the only differences between the models arisefrom the calibration in (3) to the forward price, the choice of numeraire, and the numerical quadratureschemes used. The plots show close agreement between the two models on option premium and allrisk measures.

In Figure 3, we show the effect of the differences arising from strike adjustment. Although it may benot be evident in this figure, strike adjustment can lead to appreciable differences in option premium forstrikes far away from the money. Similarly, the impact of loss settlement (not graphed) can be significantin indices that have sustained multiple unsettled defaults with low expected recoveries.

The last two figures (Figure 4 and Figure 5) show the additional convexity impact of the volatility depen-dent calibration procedure in (3). At a modest 30% volatility, the differences begin to be visible in thegamma and the vega plots in Figure 1 and Figure 2. At elevated levels of volatility, the differences canbe more significant as shown in Figure 4 and Figure 5.

In conclusion, the Bloomberg model provides a fast numerical valuation of spread- and price-basedcredit index options, circumventing technical problems associated with the risky annuity numeraire. Bycorrectly accounting for strike adjustment and loss settlement in the option payoff, the Bloomberg modeloffers a more precise valuation than the Black model does.

References

[1] Damiano Brigo and Massimo Morini. Arbitrage-Free Pricing of Credit Index Options: The No-Armageddon Pricing Measure and the Role of Correlation after the Subprime Crisis. SSRN eLi-brary, 2007. http://ssrn.com/paper=1080064.

[2] Claus M. Pedersen. Valuation of portfolio credit default swaptions. Technical report, Lehman Broth-ers, New York, 2003.

[3] Phillip Schönbucher. Credit Derivatives Pricing Models: Models, Pricing, Implementation. WileyFinance, New York, 2003.

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