III. Valuation Framework for CDS options

Transcription

1 III. Valuation Fraework for CDS options In siulation, the underlying asset price is the ost iportant variable. The suitable dynaics is selected to describe the underlying spreads. The relevant paraeters such as volatilities and correlations in the dynaics are calibrated with arket data. The siulated paths of the asset prices are obtained by repeatedly drawing rando nubers. Finally, the option values can be easily calculated by averaging the siulated values of the paths. As for a CDS option, its underlying asset is a CDS contract and the priary variable to deterine the value of a CDS contract is CDS spread. To get the value of a CDS option, we need to siulate the CDS spread. However, instead of directly siulating CDS spreads, we siulate forward one-period CDS spreads which can be stripped out fro regular quotes for CDS contracts. The dynaics to be used in this paper are fro the one-period forward CDS spread odel presented by Brigo (2005). The ain idea of this odel is to convert the arket quotes for CDS contracts to forward one-period CDS spreads, and siulate these spreads with specific dynaics. To get siulated ulti-period CDS spread, we need only to substitute the siulated one-period CDS spreads with certain forula which will be introduced later. To get the siulated value of a European CDS option, we only need to find out the option prices at the aturity with the siulated CDS spreads, and discount these prices to obtain the option value. For an Aerican option, the procedure ay be a little ore 13

2 coplicated. We have to copute the values of the Aericans option along each path by using least-squares ethod. The priary advantage of this odel is that it is siilar with LIBOR arket odel in interest rate theories. The one-period CDS spread odel uses forward spreads as ain variables, and assues the forward one-period CDS spread is a artingale and log-noral distributed under respective probability easure. These concepts are the sae as those in LIBOR arket odel. As long as the basic ideas about LIBOR arket odel are realized, path-dependent CDS-related products can be easily priced. In this chapter, we first introduce forward one-period CDS spread and the one-period CDS spread odel. Secondly we present how we apply this odel to a European CDS option. Finally, we detail how we price an Aerican CDS option with this odel Dynaics of one-period forward CDS spreads The definition of one-period forward CDS spreads is siilar with that of forward interest rates. A one-period forward CDS spread which is alive for three onth in one onth fro now can be denoted by S(0;1,3). The first nuber zero in the bracket represents the current tie, and the 1 and 3 ean the starting and ending tie of a one-period forward CDS spread repeatedly. The starting and ending tie of the spread do not change with the passage of tie. Once the 1 is reached, the forward CDS spread one onth ago now becoes a spot CDS spread. 14

3 The only difference between forward interest rates and one-period forward CDS spreads is that arket quotes for one-period forward CDS spreads are still not universal. The prevalent quotes for CDS contracts are usually spot spreads. To obtain one-period forward CDS spreads, we have to strip these arket CDS quotes with certain forula. An exaple is given to explain how to reach one-period CDS spreads. The one-period forward CDS spreads fro now (tie 0) to tie T for a reference copany are denoted by S 1 0, S 2 0, S M 0, where T 1 < T 2 < < T M. 2 The k th spread can be coputed by 3 S k 0 = 1 R P 0, 1 P 0, P 0, 1 P 0, 1 P 0,. (3.1) As Equation (3.1) shows, the one-period CDS spread is ainly coposed of risk-free and corporate zero coupon bonds. The risk-free zero coupon bonds can be obtained by zero curve which is usually stripped fro interest rate swaps. As for the corporate zero coupon bonds in Equation (3.1), we do not replace the with bond prices quoted in the arket. This is priarily because not all corporate bonds for the reference copany are liquid enough to reflect the true value of bonds. The bias ay arise when we estiate the zero curve for the reference copany. Therefore, inserting such data to our odel ay produce unrealistic results. To deal with this proble, we first recall fro the following equation 2 For clarity, we siplify the notation S 0; T 0, T 1, S 0; T 1, T 2, S 0; T M 1, T M as S 1 0, S 2 0, S M 0. 3 For details, please see Brigo (2006). 15

4 P 0, = E Q D 0, 1 τ> G 0. Under the assuption of independence of interest rates and default, this can be P 0, = E Q D 0, G 0 E Q 1 τ> G 0 = P 0, Q τ > G 0. (3.2) The survival probabilities can be calculated fro arket quotes for CDS contracts on the reference copany. 4 Given the zero curve for interest rates and survival probabilities for the reference copany, the values of corporate zero coupon bonds can be calculated. Therefore, we can obtain the set of one-period forward CDS spreads with Equation (3.1). Since one-period forward CDS spreads have been reached, we now discuss how the dynaics are derived. Changed to one-period CDS spread expression, Equation (2.15) can be rewritten as follows S k 0 = E Q D 0, 1 1 <τ F 0 1 R C k 0, (3.3) where C k 0 eans C k 1,k 0. Following the concept in the section 2.2.2, we set C k as the nueraire. The dynaics for the one-period CDS spread under the probability easure Q k can be 4 In this paper, we adopt the reduced for odel to atch the survival probability, and the default intensity function is assued to be piece-wise constant. 16

5 assued to be ds k t S k t = ς k dw k k t (3.4) where ς k is the instantaneous volatility for the k th one-period CDS spread, W k is the Brownian otion under the easure Q k. Equation (3.4) eans that the dynaics for each one-period CDS forward spread are log-noral distributed under their respective probability easures. However, these dynaics are eaningful to valuation only when all the probability easures are changed to identical easure. Thus Equation (3.4) has to be further derived. Suppose now there is a CDS-related product involving two one-period CDS spreads, S 1 t and S 2 t. Both dynaics for S 1 t and S 2 t are artingales and log-noral distributed under the respective easures Q 1 and Q 2. For eaningful valuation, the probability easures have to be consistent. With the forula for change of nueraire by Brigo (2006), we can overcoe the proble of different probability easures. The Brownian otion for S 1 t under the easure Q 2 can be expressed as dw 1 1 t = dw 1 2 t + ρ 1,2 dln S 1 t dln C 1 t C 2 t (3.5) where ρ 1,2 is the instantaneous correlation between S 1 t and S 2 t. Therefore, the dynaics for S 1 t under the easure Q 2 thus is ds 1 t S 1 t = ς ρ 1,2 ς 2 S 2 t T 2 T 1 1 S 2 t T 2 T R dt + ς 2 2 dw 1 t (3.6) 17

6 In practice, there are usually ore than two dynaics when a CDS-related product is valued. This eans Equation (3.6) has to be further generalized for valuation. Suppose we split a particular tie spanning over the aturity of the CDS-related product into periods, t 0, t 1, t 2,, t. With the sae procedure described above, the k th dynaics for one-period spread under the easure Q is ds i t S i t = ς i k=i+1 ρ i,k ς k S k t 1 S k t R dt + ς i dw i t (3.7) Now that we have reached a consistent probability easure for the dynaics of the one-period CDS spreads, the next step is to derive a feasible forula for siulation. By Ito-Doeblin forula 5, Equation (3.4) can be arranged as dln S i t = ς i k=i+1 ρ i,k ς k S k t 1 S k t R 1 2 ς i 2 dt + ς i dw i t (3.8) With a little algebra, Equation (3.8) becoes ln S i t + t = ln S i 0t + ς i k=i+1 ρ i,k ς k S k t 1 S k t R 1 2 ς i 2 t + ς i W i n t + t W i n t (3.9) 5 For ore details, please see Steven E. Shreve (2000). 18

7 Equation (3.9) is the forula dealing with Monte Carlo siulation for one-period CDS spreads. This forula is alost the sae with the forula for LIBOR arket odel in interest rate theory except that the denoinator in Equation (3.9) appears a recovery rate ter. Another difference behind the forula is that the arket quotes for forward CDS spreads are not prevalent. To apply this odel to the valuation, we have to strip out one-period forward spreads fro arket quotes for CDS contracts. After inserting these estiated spreads, we can start a Monte Carlo siulation Valuation fraework for European CDS options Before siulating Aerican CDS options, we have to notice that all siulated spreads resulting fro Equation (3.9) are of one-period length. For holders of Aerican CDS options, they exercise options on the basis of whether the CDS spot spread is greater than the exercise price. In other words, the holders decisions priarily depend on ulti-period spreads instead of one-period spreads. Therefore, this eans that the siulated one-period spreads have to be converted to ulti-period spreads so that the valuation is consistent with the arket convention. Brigo (2006) derives a forula relating these two types of CDS spreads as S 0, t i=1 w i S i t (3.10) where 19

8 w i = t i t i 1 P 0, t i t i t i 1 P 0, t i. i=1 The ter w i in Equation (3.10) can be seen as the weight of i th one-period spread and is coposed of a set of corporate zero coupon bonds observed at the initial tie. Now that Equation (3.10) has provided us an idea of how to convert one-period siulated CDS spreads to ulti-period siulated CDS spreads, we start to calculate the value of a European CDS option under this siulation fraework. Suppose a European CDS option atures at tie T 0, and the underlying forward CDS contract starts fro T 0 to T n, in which the protection preius are paid at T 1, T 2, T n-1, T n. Recalled fro Equation (2.19), the option value under the easure Q 0,n is expressed as V 0 = n i=1 T i T i 1 P 0, T i E Q 0,n ax S 0,n T 0 K, 0 F 0 (3.11) For siulation purpose, we split the contract life fro T 0 to T n into periods, t 0 = T 0, t 1, t 2,, t = T n, and set C as the valuation nueraire. Thus the option value under the easure Q is 6 V 0 = P 0, T E Q n i=1 T i T i 1 P T 0, T i P T 0, T ax S 0,n T 0 K, 0 F 0 (3.12) 6 For ore details, please see the appendix. 20

9 If we shift fro the initial tie to the option aturity T 0, the option value appears to be n V T 0 = T i T i 1 P T 0, T i ax S 0,n T 0 K, 0 i=1 (3.13) Apparently, Equation (3.13) represents the option value at T 0. The suation ter explains that there are n cash flows of protection payents on the CDS contract once the option is exercised. In credit odels, one of the ost difficult parts is to describe a default event in atheatics. Asset values always draatically change with the occurrence of a default. In spite of this, Equation (3.12) provides an easy way to value a European CDS option. We siulate CDS spreads at the option aturity, insert these spreads in Equation (3.13) to obtain the iediate exercise value, and reach the initial option values with Equation (3.12). It is unnecessary to consider the default event in siulation because this is iplicitly included in the probability easure Q. Intuitively speaking, the default inforation is actually contained in the corporate zero coupon bonds in Equation (3.12). Consequently, we only need to focus on the variations of the CDS spreads in siulating European CDS options Valuation fraework for Aerican CDS options with least-squares Monte Carlo siulation Although the basic concepts of valuing Aerican CDS options are uch the sae 21

10 with those of valuing European CDS options, there are still soe differences between the. The clearest distinction is that the life of underlying CDS contract for an Aerican CDS option varies as the holder exercises at different oents. For exaple, suppose the option aturity and the protection aturity for an Aerican CDS option are 1 and 2 years respectively. When the option is exercised at the end of the six onths, the holder will receive a CDS contract of which the life is four and a half years. This eans the holder has to ake periodical payents for four and a half years. Analogously, the life of the CDS contract is 4 years when the option is exercised at the end of one year. Consequently, the underlying CDS contract depends ainly on the tie at which the holder exercises. To deal with this situation, let us further take an exaple to show how we apply the one-period spread odel. Suppose a Berudan CDS option with option aturity T 0 and protection aturity T n is exercisable at T 1, T 2 T n = T 0. As entioned above, the underlying CDS life is fro T 1 to T n when this option is exercised at T 1. Because Berudan CDS option has the early-exercise characteristic, we have to start fro the oent that the Berudan CDS option atures and recursively calculate the siulated value. Suppose we siulate h paths of one-period CDS spreads for the underlying CDS contract. With Equation (3.10), the siulated ulti-period CDS spreads at the option aturity, S 1 T n,t n Tn, S 2 T n,t n Tn S h T n,t n Tn, can be calculated fro these siulated one-period CDS spreads. Using these siulated ulti-period CDS spreads, we can get the corresponding iediate exercise values, EV 1 T n, EV 2 T n, EV h T n, with Equation (3.13). 22

11 Only these iediate exercise values are not enough for us to find the value of the Berudan option. We have to further look for the relationship of option values aong exercisable oents so that the option value can be recursively calculated. The following forula relates the option value at with that at T 7 k 1. V 1 = P 1, T E V Q P T k, T F (3.14) With Equation (3.14), the option values at T n 1, V 1 T n 1, V 2 T n 1, V h T n 1, can be calculated fro the corresponding iediate exercise values, EV 1 T n, EV 2 T n, EV h T n. Let s further cobine the above procedure with least-squares regression. We refer to the siulated ulti-period spreads at T n 1 as the independent variable, and the option values at T n 1 are set as dependent variable. Then we regress the dependent variable on the independent variable according to the following regression odel. V i = β 0 L 0 S i,tn + β 1 L 1 S i,tn + β 2 L 2 S i,tn for i = 1,2 ; k=1,2 n-1 (3.15) where L 0 S i,tn = exp S i,tn 2 7 This forula provides an idea of discount siilar with the oney arket account in risk-neutral easure. The difference is that we use corporate zero coupon bonds as our discount factor here. 23

12 L 1 S i,tn = exp S i,tn 2 1 S i,tn L 2 S i,tn = exp S i,tn 2 1 2S i,tn + S i,tn 2 Thus a set of estiated option values at T n 1, V 1 T n 1, V 2 T n 1, V h T n 1, are obtained. The iediate exercise value on each path, EV 1 T n 1, EV 2 T n 1, EV h T n 1, is then copared with the corresponding estiated option value. Once the exercise value is greater, the option is exercised at T n 1. Consequently, repeating these steps recursively until the initial tie, we can deterine the optial exercise tie on each path. With Equation (3.14), the Berudan option thus can be valued by discounting these cash flows of which the option is optially exercised. 8 For an Aerican CDS option, we only need increase the nuber of the exercisable oents. The ore the exercisable oents are, the ore accurate the value of an Aerican CDS option is. Now that a coplete siulation of Aerican CDS options can be ipleented, let us specify our procedure to price Aerican CDS options. 1. Strip out the iplied intensity fro arket quotes for CDS contracts, and calculate survival probabilities during protection aturity of an option so that corporate zero coupon bonds can be reached. 8 This ethod is known as least-squares approach. For ore details, please see Longstaff and Schwartz (2001). 24

13 2. Strip out one-period CDS spreads by Equation (3.1), and calculate historical volatilities and correlations with these one-period CDS spreads to calibrate for siulation Siulate one-period CDS spreads with Equation (3.9), and convert the to desired ulti-period CDS spreads. 4. Calculate iediate exercise values during the option aturity with Equation (3.13). 5. Calculate recursively initial values of Aerican CDS options with least-squares approach. 9 We use historical volatilities and correlations as our proxy because the arket for European CDS options is not prevalent. Brigo (2006) provides a forula dealing with calibration for correlation of one-period CDS spreads under the situation that the arket is prevalent. 25