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## Pricing a CDS Index Option

This example shows how to price CDS index options by using `cdsoptprice` with the forward spread adjustment. Unlike a single-name CDS, a CDS portfolio index contains multiple credits. When one or more of the credits default, the corresponding contingent payments are made to the protection buyer but the contract still continues with reduced coupon payments. Considering the fact that the CDS index option does not cancel when some of the underlying credits default before expiry, one might attempt to price CDS index options using the Black's model for non-knockout single-name CDS option. However, Black's model in this form is not appropriate for pricing CDS index options because it does not capture the exercise decision correctly when the strike spread (K) is very high, nor does it ensure put-call parity when (K) is not equal to the contractual spread (O'Kane, 2008).

However, with the appropriate modifications, Black's model for single-name CDS options used in `cdsoptprice` can provide a good approximation for CDS index options. While there are some variations in the way the Black's model is modified for CDS index options, they usually involve adjusting the forward spread F, the strike spread K, or both. Here we describe the approach of adjusting the forward spread only. In the Black's model for single-name CDS options, the forward spread F is defined as:

`$F=S\left(t,{t}_{E},T\right)=\frac{S\left(t,T\right)RPV01\left(t,T\right)-S\left(t,{t}_{E}\right)RPV01\left(t,{t}_{E}\right)}{RPV01\left(t,{t}_{E},T\right)}$`

where

S is the spread.

RPV01 is the risky present value of a basis point (see `cdsrpv01`).

t is the valuation date.

tE is the option expiry date.

T is the CDS maturity date.

To capture the exercise decision correctly for CDS index options, we use the knockout form of the Black's model and adjust the forward spread to incorporate the FEP as follows:

`${F}_{Adj}=F+\frac{FEP}{RPV01\left(t,{t}_{E},T\right)}$`

with FEP defined as

`$FEP=\left(1-R\right)Z\left(t,{t}_{E}\right)\left(1-Q\left(t,{t}_{E}\right)\right)$`

where

R is the recovery rate.

Z is the discount factor.

Q is the survival probability.

In `cdsoptprice`, forward spread adjustment can be made with the `AdjustedForwardSpread` parameter. When computing the adjusted forward spread, we can compute the spreads using `cdsspread` and the RPV01s using `cdsrpv01`.

Set up the data for the CDS index, its option, and zero curve. The underlying is a 5-year CDS index maturing on 20-Jun-2017 and the option expires on 20-Jun-2012. A flat index spread is assumed when bootstrapping the default probability curve.

```% CDS index and option data Recovery = .4; Basis = 2; Period = 4; CDSMaturity = datenum('20-Jun-2017'); ContractSpread = 100; IndexSpread = 140; BusDayConvention = 'follow'; Settle = datenum('13-Apr-2012'); OptionMaturity = datenum('20-Jun-2012'); OptionStrike = 140; SpreadVolatility = .69; % Zero curve data MM_Time = [1 2 3 6]'; MM_Rate = [0.004111 0.00563 0.00757 0.01053]'; MM_Dates = daysadd(Settle,30*MM_Time,1); Swap_Time = [1 2 3 4 5 6 7 8 9 10 12 15 20 30]'; Swap_Rate = [0.01387 0.01035 0.01145 0.01318 0.01508 0.01700 0.01868 ... 0.02012 0.02132 0.02237 0.02408 0.02564 0.02612 0.02524]'; Swap_Dates = daysadd(Settle,360*Swap_Time,1); InstTypes = [repmat({'deposit'},size(MM_Time));repmat({'swap'},size(Swap_Time))]; Instruments = [repmat(Settle,size(InstTypes)) [MM_Dates;Swap_Dates] [MM_Rate;Swap_Rate]]; ZeroCurve = IRDataCurve.bootstrap('zero',Settle,InstTypes,Instruments); % Bootstrap the default probability curve assuming a flat index spread. MarketData = [CDSMaturity IndexSpread]; ProbDates = datemnth(OptionMaturity,(0:5*12)'); ProbData = cdsbootstrap(ZeroCurve, MarketData, Settle, 'ProbDates', ProbDates); ```

Compute the spot and forward RPV01s, which will be used later in the computation of the adjusted forward spread. For this purpose, we can use `cdsrpv01`.

```% RPV01(t,T) RPV01_CDSMaturity = cdsrpv01(ZeroCurve,ProbData,Settle,CDSMaturity) % RPV01(t,t_E,T) RPV01_OptionExpiryForward = cdsrpv01(ZeroCurve,ProbData,Settle,CDSMaturity,... 'StartDate',OptionMaturity) % RPV01(t,t_E) = RPV01(t,T) - RPV01(t,t_E,T) RPV01_OptionExpiry = RPV01_CDSMaturity - RPV01_OptionExpiryForward ```
```RPV01_CDSMaturity = 4.7853 RPV01_OptionExpiryForward = 4.5971 RPV01_OptionExpiry = 0.1882```

Compute the spot spreads using `cdsspread`.

```% S(t,t_E) Spread_OptionExpiry = cdsspread(ZeroCurve,ProbData,Settle,OptionMaturity,... 'Period',Period,'Basis',Basis,'BusDayConvention',BusDayConvention,... 'PayAccruedPremium',true,'recoveryrate',Recovery) % S(t,T) Spread_CDSMaturity = cdsspread(ZeroCurve,ProbData,Settle,CDSMaturity,... 'Period',Period,'Basis',Basis,'BusDayConvention',BusDayConvention,... 'PayAccruedPremium',true,'recoveryrate',Recovery) ```
```Spread_OptionExpiry = 139.9006 Spread_CDSMaturity = 140.0000 ```

The spot spreads and RPV01s are then used to compute the forward spread.

```% F = S(t,t_E,T) ForwardSpread = (Spread_CDSMaturity.*RPV01_CDSMaturity - ... Spread_OptionExpiry.*RPV01_OptionExpiry)./RPV01_OptionExpiryForward ```
```ForwardSpread = 140.0040 ```

Compute the front-end protection (FEP).

```FEP = 10000*(1-Recovery)*ZeroCurve.getDiscountFactors(OptionMaturity)*ProbData(1,2) ```
```FEP = 26.3108```

Compute the adjusted forward spread.

`AdjustedForwardSpread = ForwardSpread + FEP./RPV01_OptionExpiryForward`
```AdjustedForwardSpread = 145.7273```

Compute the option prices using `cdsoptprice` with the adjusted forward spread. Note again that the `Knockout` parameter should be set to be `true` because the FEP was already incorporated into the adjusted forward spread.

```[Payer,Receiver] = cdsoptprice(ZeroCurve, ProbData, Settle, OptionMaturity, ... CDSMaturity, OptionStrike, SpreadVolatility,'Knockout',true,... 'AdjustedForwardSpread', AdjustedForwardSpread,'PayAccruedPremium',true); fprintf(' Payer: %.0f Receiver: %.0f \n',Payer,Receiver); ```
`Payer: 92 Receiver: 66 `

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