Valuation of Bermudan-DB-Underpin Option

Valuation of Bermudan-DB-Underpin Option Mary, Hardy 1, David, Saunders 1 and Xiaobai, Zhu 1 1 Department of Statistis and Atuarial Siene, University of Waterloo Marh 31, 2017 Abstrat The study of embedded options has grown importane in pension design, with many novel forms of hybrid plan being proposed to meet the needs of employees and sponsors. In 2002, the State of Florida introdued temporarily) a swithing hybrid plan, under whih employees were given the option to onvert their defined ontribution DC) plans to defined benefit DB) plans. The ost of suh a swith is alulated in terms of the aumulated benefit obligation ABO), whih is the present value of the arued benefit. If the ABO is greater than the DC aount, the employee is assumed to fund the differene. In this work we reonsider the swithing hybrid plan, with additional downside protetion for the employees. The new option is similar to the DB- Underpin hybrid design, also knowns as the floor-offset plan, but with a Bermudan-style exerise feature. We adopt an arbitrage-free priing methodology to value the option, and speify the situation where early swith is not optimal. Department of Statistis and Atuarial Siene SAS), Mathematis 3 M3), University of Waterloo, ON, Canada, N2L 3G1. Email: x32zhu@uwaterloo.a 1

1 Two hybrid pendion plans: DB underpin and the Florida option 1.1 Introdution Rising interest in risk sharing pension plans suggest the potential for new hybrid style pension designs. Hybrid pension plans ombine aspets of both Defined Benefit DB) and Defined Contribution DC) plans, and may be designed to provide a more seure retirement benefit to employees ompared with DC plans, while reduing the risks to employers ompared with DB plans. In this setion, we will give a brief desriptions of two existing hybrid pension designs, the DB-underpin plan and the Florida s seond eletion option. In both plans, as well as the new design explored in subsequent setions of this paper, we are onsidering true hybrid plans where employees have some ombination of DC and DB benefits. 1.2 Notation and assumptions We introdue some preliminary notation and assumptions. denotes the annual ontribution rate as proportion of salary) into the DC plan. We assume that ontributions are paid annually. We also assume that all ontributions are paid by the plan sponsor/employer, although this is easily relaxed to allow for employee ontributions. L t denotes the salary from t to t1 for t = 0, 1,..., T 1, where t denotes years of servie, and T denotes servie at retirement. We assume that salaries inrease deterministially at a rate µ L per year, ontinuously onvertible, so that L t = L 0 e µ L t r denotes the risk free rate of interest, onvertible ontinuously. b denotes the arual rate in the DB plan. ät ) denotes the atuarial value at retirement of a pension of 1 per year. 2

denotes the stohasti prie index proess of the funds in the DC aount. We assume follows a Geometri Brownian Motion, so that dst) St) = rdt σ SdZ Q t 1.3 DB-Underpin Plan The DB Underpin pension plan, also known as the floor-offset plan in the North Amerian, provides a guaranteed defined benefit DB) minimum whih underpins a DC plan. Plan sponsors make periodi ontributions into the member s DC aount, and separately ontribute to the fund whih overs the guarantee. Employees usually have limited investment options to make the guarantee value more preditable. At the retirement date, after T years of servie, if the member s DC balane is higher than the guaranteed minimum defined benefit underpin, the plan sponsor will not inur any extra ost. However, if the DC benefit is smaller, the plan sponsor will over the differene. Using arbitrage-free priing, we an alulate the present value of the umulative ost of DB-underpin plan at time t = 0 as follows. We assume more for larity than neessity) that DB benefits are based on the final 1-year s salary, and we ignore all exits before retirement. [ T 1 ] [ ) ] T 1 E Q e rt L t E Q e rt S T bt ät )L T 1 L t S t=0 t=0 t }{{} }{{} PV of Contribution to DC aount Value of DB underpin option Notie that the option value does not have an expliit solution, but an be determined using Monte Carlo simulation. See Chen and Hardy 2009) for details on the valuation and funding of the DB underpin option. 1.4 Florida Seond Eletion Option In 2002, publi employees of the State of Florida were given an option to swith from their DC plan to DB plan anytime before their retirement date. The ost of partiipating in the DB plan is alulated by the aumulated benefit obligation ABO), whih is the present value of the arued benefit, based on urrent servie and pensionable salary. 3

We denote the ABO of the DB benefit for an employee with t years of servie as K t, so that rt t) K t = b L t 1 t ät ) e Under the Florida Option hybrid plan, suppose the employee hooses to swith from DC to DB at time τ. If the ABO at τ is more than the DC aount balane, the employee needs to fund the differene from her own resoures. If the DC aount is more than the ABO, at τ, then the employee retains the differene in a separate DC top-up aount whih an be withdrawn at retirement. Mathematially, the present value of the total DC and DB benefit ost at ineption is [ τ 1 E Q e rt L t e ) ] rτ K T e rt τ) K τ sup 0 τ T t=0 where the first term, as in the previous setion, is the present value of the DC ontributions, and the seond term is the additional funding required for the DB benefit, offset by the ABO at transition, whih is funded from the DC ontributions. The sup indiates that the valuation assumes the swith from DC to DB is made at the optimal time to maximize the value of the benefits to the employee. Milevsky and Promislow 2004) studied the prie and optimal swithing time of the Florida option with deterministi assumptions. 2 Bermudan-style DB underpin plan The Florida Option design has the advantage that it provides employees with the flexibility to hoose their plan type based on their hanging risk appetite. However, employees retain the investment risk through the DC period of membership, and also have the additional risk of a suboptimal hoie of the swithing time. Moreover, when the DC investment falls below the ABO of the DB plan, the employee may be unable to make up the differene. Inspired by the idea of ombining the DB-underpin plan with the Florida option, we investigate a new hybrid design, whih adds a guarantee at the time of the swith from DC to DB. If the employee deides to swith at time τ, and her DC aount is below the ABO at that time, then the plan sponsor will over the differene. 4

2.1 Problem Formulation We assume that ontribution are made annually into the DC aount until the employee swithes to DB. Suppose we are valuing the benefits at a ontribution time t. We let w t denote the known aount balane in the DC plan at t, immediately before the ontribution at that date. Then at t τ, τ = 1, 2,..., we have stohasti proess for future values of the DC aount, τ τ 1 W tτ F t = w t where is the prie index proess of the DC funds. Assume that the employee has not swithed from DC to DB before time t, and that the DC aount is w t at t. Then the present value at time t of the ost of future benefits past and future servie, DC and DB), whih is denoted by the funtion Ct, w t ), an be expressed in three parts: 1. The present value of the future ontribution into the DC aount before the member swithes to DB plan. 2. The present value of the ost of the DB benefit, offset by the ABO at the time of the swith u=t τ S u L u 3. The differene between the ABO and the DC balane, if positive. We take the maximum value of the sum of these three parts, maximizing over all the possible swithing dates, as follows. Ct, w) = sup E Q t 0 τ T t [ τ 1 e ru L u e ) rτ K T e rt t τ) K tτ ] e τr K tτ W tτ ) W t = w Notie that the swithing time τ is involved in all three parts, whih makes the analysis more omplex. However we an transform our problem into the simpler ase given in equation 5

1), using a put-all parity approah. Details are given in the appendix. Ct, w) = [ E Q t KT e rt t)] sup E [e Q τr W τt K tτ e T t τ)r) ] W t = w w 1) 0 τ T t The new formulation also onsists of three terms: The PV of the DB plan benefits at time t A Bermudan type all option, with underlying W t and strike K t Offset by the available DC balane at t. The first and third terms do not depending on the swithing time, and the first part does not depend on the DC balane. To study the optimal exerising strategy, we omit the first and third part and define our value funtion as [ vt, w) = sup E Q t e τr W tτ K tτ e T t τ)r) ] W t = w 0 τ T t [ τ 1 = sup E Q t e τr 0 τ T t τ u L tu w τ T τ t)r bt τ)ät )L τt 1 e 2) ) ] We assume that the exerising dates are at the beginning of eah year, before the ontribution is made into the DC aount, so the admissible exerising dates are t = 0, 1, T. At time t, given that the DC aount balane is w, we define the exerising value of the option, denoted v e t, w), as the value if the member deided to swith at that date, and the holding value, denoted v h t, w) as the value if the member deides not to swith. Then v e t, w) = w K t ) 3) = w tbl t 1 ät )e T t)r) 4) [ v h t, w) = E Q t e r v t 1, w L t ) S )] t1 5) The option is similar to a Bermudan all option, but with an underlying proess following a generalized geometri Brownian motion. The option is not analytially tratable. In the next setion will present some of the properties of the value funtion. 6

2.2 Charateristis of the value funtion 2.2.1 The value funtion at T 1, v T 1 Sine the option value at time T 1 depends only on the stok performane in the period [T 1, T ], it an be solved using the Blak-Sholes Formula for the holding value [ v h T 1, w) = E Q e r w L T 1 ) S ) ] T bl T 1 T ät ) S T 1 = Nd 1 )w L T 1 ) Nd 2 )bl T 1 T ät )e r where d 1 = 1 [ ) )] w LT 1 ln r σ2 S σ S bt L T 1 ät ) 2 d 2 = d 1 σ S thus vt 1, w) = max w bt 1)L T 2 ät )e r, v h T 1, w) ) 2.2.2 The Exerise Frontier The following proposition gives the general onvexity and monotoniity of the value funtion Proposition 1. At eah observation date 0 t T, the value funtion vt, w) is a ontinuous, stritly positive, non-dereasing and onvex funtion of w. The proof is given in Appendix. Like other Bermudan- and Amerian-type options, there exists a ontinuation region C and a stopping region or exerising region) D. When the time and DC aount value pair, t, w) is in the ontinuation region, it is optimal for the member to stay in the DC plan. Where t, w) is in the stopping region, it is optimal for the member to swith to the DB plan. The option to swith is exerised when t, w) moves into the stopping region. Mathematially, the ontinuation and stopping regions are defined as v h t, w) > v e t, w) t, w) C v h t, w) v e t, w) t, w) D 7

In order to value the option to swith we need to identify the ontinuation and stopping regions. Consider first the ase when w < K t at time t. We have v e t, w) = w K t ), so w < K t v e t, w) = 0 The value funtion, given in equation 2), is the expeted value of a funtion bounded below by zero, and whih has a positive probability of being greater than zero, whih means that the expeted value is stritly greater than zero. The holding funtion is the expeted disounted value of the 1-year ahead value funtion assuming the option is not exerised immediately), so it too must be stritly greater than 0. So whenever w < K t, we have v h t, w) > v e t, w). Thus if w K t, it annot be optimal to exerise. Therefore to explore the exerise frontier, we need only onsider ases when w > K t. When w > K t we have v e t, w) = w K t t, w) D vw, t) = v e t, w) vw, t) = w K t and from Appendix B.0.3, vt, w) h vt, w h) for any w R and h > 0, whih means that v h t, w) h v h t, w h) also. First, assume that w > K t and that t, w) D, t, w) D = v e t, w) v h t, w) = h > 0 v e t, w h) = v e t, w) h v h t, w) h v h t, w h) = t, w h) D Next, assume that w > K t and that t, w) C. Note that v h t, w) h v h t, w h) for all h > 0 implies that v h t, w h) v h t, w) h t, w) C = v h t, w) < v e t, w) = h > 0 v h t, w h) v h t, w) h > v e t, w) h = v e t, w h) = t, w h) C These results show that it is optimal for the employee to swith to the DB plan only when his/her DC aount balane is above a ertain threshold at eah possible swithing time. Some 8

numerial illustrations on the exerise boundary under a ontinuous setting will be presented in next setion. We let the funtion ϕt) denote the boundary between the ontinuation and exerise regions. The results above an be summarized in the following proposition. Proposition 2. There exists a funtion ϕt) suh that { v e t, w) if w ϕt) vt, w) = v h t, w) if w < ϕt) Notie, it is possible, under ertain parameters, that ϕt) = for some t < T, whih means that it is not optimal to swith regardless of the DC aount value. The next proposition will speify the situations where ϕt) <. Proposition 3. The behavior of exerise boundary ϕt) depending on the ratio i. If bät )e rt < 1, then ϕt) <, t [0, T ] ii. If bät )e rt ϕt) <, t > t. bät )e rt 1, there exists a t [0, 1,, T 1], suh that ϕt) =, t t, and a) If > bät ) 1 e µ L ) T e µ L ) e r, t = T 1, whih means ϕt) =, b) If t < T = 1, t bät )e rt = 0. t is determined as t, where t satisfies t 1)bL t ät )e rt t ) t bl t 1äT )e rt t ) L t = 0 The proof is given in the Appendix C. This proposition learly demonstrate that the ratio between ontribution rate to DC aount and arual rate of DB benefit will onstrut the shape of the exerise boundary. In extreme ase, when ontribution rate is muh higher than the DB arual rate, it is optimal for employee to wait until the retirement date, and the Bermudan-DBunderpin option simplifies to a DB-underpin plan. 9

3 Numerial Examples Sine the Bermudan-type DB swithing option is not analytially tratable, numerial methods must be adopted. In this setion, we evaluate the option using the least square method proposed by Longstaff and Shwartz 2001), and ompare the ost of Bermudan-style DC/DB option with the floor offset from Chen and Hardy 2009), whih is a European-style DC/DB option, and also with the Florida swithing option. The parameters are adopted from Milevsky and Promislow 2004) and Larrabee et al. 2016). b = 0.016, perentage value for people retire at age 65. = 0.125, as adopted by Chen and Hardy 2009). µ L = 0.0459, assumed salary growth in Larrabee et al. 2016). σ S = 0.15, ät ) = 14.75, L 0 = 1, t = 0, W 0 = 0 and r = 0.04. 3.1 Cost Table 1 display the present value of eah pension option). Time to Retirement DB DC v0,0) Seond Eletion DB-Underpin 10yr 2.3911 1.2838 0.0089 0.0001) 0 0.0031 0.0012) 15yr 3.6941 1.9547 0.0409 0.0003) 0 0.0186 0.0021) 20yr 5.0729 2.6457 0.1078 0.0006) 0.0287 0.0385 0.0031) 30yr 8.0718 4.0903 0.3562 0.0013) 0.2368 0.1062 0.0055) 40yr 11.4165 5.6227 0.7460 0.0024) 0.6095 0.2300 0.0083) Table 1: Cost of eah pension plan The values of three options, our Bermudan-DB-underpin, Florida s seond eletion option and the DB-underpin plan, are all expressed as the value added to the DB plan. For example, the value of the Bermudan-DB underpin option is equal to v0, 0) = C0, 0) DB. From the table, we an observe that 10

For short term, all three options ost relatively insignifiant ompare to the DB plan, and Florida s seond eletion has zero ost. For long term, Bermudan-DB-underpin and seond eletion option ost muh heavier than DB underpin, but at most 6.5% of the DB plan. The ost of Bermudan-DB-underpin an be greater than the sum of seond-eletion option and DB underpin option. Inreases in time to retirement greatly impat the value of the option. For example, the option pries for Bermudan-DB-underpin on a 40-year plan doubles a 30-year plan. 3.2 Sensitivity Tests In this setion, we present the sensitivity tests over five fators:, µ L, r, σ S,b; and we set the time horizon to be 30 years. Details are displayed in Table 2 below. 11

Fator Sensitivity Tests -0.04-0.03-0.02-0.01 0.125 0.01 0.02 0.03 0.04 DC Value 2.7814 3.1086 3.4358 3.7630 4.0903 4.4175 4.7447 5.0719 5.3991 v0, 0) 0.0857 0.1355 0.1989 0.2703 0.3562 0.4503 0.5547 0.6687 0.7914 2 nd Eletion 0.0228 0.0571 0.1042 0.1641 0.2368 0.3206 0.4148 0.5190 0.6325 µ L -0.04-0.03-0.02-0.01 0.459 0.01 0.02 0.03 0.04 DB Value 2.5304 3.3817 4.5194 6.0398 8.0718 10.7873 14.4165 19.2666 25.7484 v0, 0) 0.5296 0.4811 0.4339 0.3892 0.3587 0.3243 0.2977 0.2732 0.2543 2 nd Eletion 0.3433 0.3058 0.2770 0.2547 0.2368 0.2224 0.2081 0.1986 0.1885 DB underpin 0.4602 0.3522 0.2571 0.1769 0.1062 0.0771 0.0430 0.0319 0.0102 r -0.04-0.03-0.02-0.01 0.04 0.01 0.02 0.03 0.04 DB Value 26.7993 19.8534 14.7077 10.8958 8.0718 5.9797 4.4299 3.2817 2.4312 v0, 0) 0.0157 0.0443 0.1074 0.2112 0.3568 0.5106 0.6603 0.7935 0.8918 2 nd Eletion 0 0 0 0.0717 0.2368 0.4313 0.6156 0.7685 0.8859 σ S -0.04-0.03-0.02-0.01 0.15 0.01 0.02 0.03 0.04 v0, 0) 0.2732 0.2895 0.3088 0.3303 0.3557 0.3833 0.4149 0.4440 0.4790 DB underpin 0.0225 0.0382 0.0635 0.0875 0.1062 0.1504 0.1709 0.2067 0.2645 b -0.004-0.003-0.002-0.001 0.016 0.001 0.002 0.003 0.004 DB Value 6.0538 6.5583 7.0628 7.5673 8.0718 8.5763 9.0808 9.5852 10.0897 v0, 0) 0.6091 0.5301 0.4641 0.4085 0.3564 0.3110 0.2720 0.2369 0.2064 2 nd Eletion 0.4899 0.4111 0.3439 0.2861 0.2368 0.1944 0.1576 0.1253 0.0986 Table 2: Sensitivity Test over, µ L, r, σ S and b) The impat of eah fator are summarized as below: Inrease in ontribution rate would inrease the value funtion as the employer is spending more money into DC aount. Inrease in salary growth would derease the value funtion, but less signifiantly. The inreases in ABO prie is offset by the inrease in DC ontributions. Inrease in market volatility would inrease the prie of the options. Inrease in DB arual rate b, would derease the value funtion, sine the fast aumulation of DB benefit would disourage employee from entering into DC aount. 12

Inrease in risk-free rate r would inrease the value funtion. It is interesting to notie that the Bermudan-DB-underpin ost less than 10% of DB plan in most ases exept for two. The first is when the salary growth is small, and the seond is when the interest rate is high. However, in either ase, the total ost that the employer will inur sum of the DB benefit and the option value) is indeed smaller. The inreases in the option value is offset by the dereases in the ost of DB benefit. One thing we want to point out is when the assumed interest rate is high, as adopted in Milevsky and Promislow 2004), the Bermudan- DB-underpin value is very lose to the seond-eletion option value. This is due to the fat that when the DC aount is aumulating fast with higher interest rate, the guarantee will beome ostless omparing to the ost of the seond eletion. The variability in the option values shown in our sensitivity test, strongly suggest a need to adopt more sophistiated models for example, interest rate model), and to onsider more realisti assumptions for example, different disounting rate for ABO alulations, employee ontributions). 4 Conlusion and Future Work In this paper, we disuss a new pension design, whih ombines the Florida s seond eletion option and the DB-underpin option, to form a Bermudan-type DB-underpin plan. We summarize some key harateristis of the option, suh as onvexity and monotoniity. Also, we provide illustration on the behavior of early exerising region, and speifially inlude the situation that the Bermudan-DB-underpin will simplify into DB-underpin plan. Our numerial illustration demonstrate that although the Bermudan-type DB underpin ost muh more than the traditional DB-underpin plan or Florida s seond eletion plan, it does not ost more than 10% of the DB plan in general. In ases when the ost of the option is high ompare to DB benefit, we argue that the total osts inurred by the employer is indeed dereasing. Here is a list of what we will study in the future Similar to DB-underpin in pratie, the underlying DB plan is often smaller than a regular 13

DB benefit. Thus, we may set our guarantee appliable only to a ertain proportion of the DB benefit e.g. 90% of ABO is guaranteed, 0.9K t ). Hybrid pension plan in ontinuous setting. Florida s Seond Eletion option uses Arued Benefit Obligation to alulate the prie of swith, it maybe interesting to investigate using Projeted Benefit Obligation, whih we would believe to have large differene in both the shape of the exerising boundary as well as the option values. We may onsider to adopt the atuarial valuation on Arued Benefit Obligation, whih often sets the disount rate different from market risk-free rate. Sine the valuation is under arbitrage-free and omplete market assumption, it may also interesting to assume salary and interest rate to be stohasti where salary is assumed to be hedgable). Inorporate employee ontribution into wealth proess maybe a more reasonable risksharing approah. It keeps the same ontribution ost for the employer, but redue the hane that employee will swith to DB plan with defiit, thus redue the option value. The Bermudan-type DB-underpin maybe undesirable in the sense that the employers are bearing all the risk. It maybe interesting to study a similar option where employees are still guaranteed for the swith from DC to DB, but s/he must give up all his/her DC aount regardless of its balane. Referenes and Notes Chen, K. and Hardy, M. 2009). The DB underpin hybrid pension plan. North Amerian Atuarial Journal, 134):407 424. Hatem, B.-A., Mihèle, B., and L Euyer, P. 2002). A dynami programming proedure for priing amerian-style asian option. Management Siene, 485):625 643. Larrabee, M., Wade, D., and Hunter, K. 2016). Florida retirement system pension plan - atuarial valuation as of July 1, 2016. 14

Longstaff, Franis, A. and Shwartz, Eduardo, S. 2001). Valuing amerian options by simulation: A simple least-squares approah. The Review of Finanial Studies, 141):113 147. Milevsky, Moshe, A. and Promislow, S. 2004). Florida s pension eletion: From DB to DC and bak. The Journal of Risk and Insurane, 713):381 404. A Cost Funtion Here is the derivation of equation 1): Ct, w) = sup 0 τ T t [ τ 1 E Q e ru L tu e rτ K T e rt τ t) T K tτ e t τ)r) e ) τr K tτ e T τ t)r W t,w τt F t [ τ 1 = sup E Q e ru L tu e rτ K T e rt τ t) T K tτ e t τ)r) 0 τ T t K tτ e T t)r e τr W t,w τt e τr W t,w τt K tτ e T t τ)r) F t [ τ 1 = sup E Q e ru L tu K T e rt t) e τr W t,w τt e τr Wτ t,w K tτ e T t τ)r) F t 0 τ T t ] ] ] To further redue our equation, we need optional sampling theorem. First, observe that E [ ) ] ] Q e t s)r W s,w t F s = E Q [e t s)r w S t 1 t L u F s S s S u Define a new proess X t as X t = e rt W 0,w t u=s t 1 t 1 = w e t s)r e t u)r L u = w e s u)r L u = e tr w S t 1 t w S 0 u=s ) t 1 w e ur L u e tr S u e ur u=s ) L u 15

and it is easy to verify that X t is a martingale: E Q [X t F s ] = e tr w S s S 0 e t s)r w = e sr w S s 1 s w S 0 = X s s e tr S ) s e t s)r e ur L u S u e sr S s S u e ur ) L u t 1 u=s1 Let τ [0, T t] be any stopping time, by optional sampling theorem, we have e tr e t u)r e ur) L u E Q [X τt F t ] = X t = E Q e τt)r w τ 1 tτ e tτ)r τ L u F t = E [w Q S 0 S u }{{} =e τt)r W t,w t tτ tτ 1 e ur L u F t ] X t [ ] τ 1 ] E [e Q τr W t,wt tτ F t = E Q e ur L ut F t W t Substitute the last line into ost funtion, we have { [ τ 1 ] Ct, w) = sup E Q e ru L tu E [ ] Q e τr Wτt F [ t,w t E Q rt K T e t)] 0 τ T t E [e Q τr W τ K tτ e T t τ)r) = E [ [ Q rt K T e t)] sup E Q e τr W t,w τt K tτ e T t τ)r) ] }{{} F t w 0 τ T t The ABO of DB plan at t }{{} Prie of the Option ]} B Charateristis of Value Funtion Here is the proof of Proposition 1) 16

B.0.1 Value funtion is non-dereasing in w For h > 0, we have x k) x h k) 0, for all x, therefore, vt, w) vt, w h) sup E Q 0 τ T t 0 e rτ { W t,w tτ K tτ e T t τ)r) W t,w tτ h S ) } tτ T t τ)r K tτ e }{{} 0 sine h τ is stritly positive a.s.. Notie, although value funtion is inreasing in the initial DC balane, the ost funtion Ct, w) is the opposite. Ct, w h) Ct, w) [ { sup E Q e rτ W t,w tτ h τ 0 τ T [ { sup E Q e rτ h τ 0 τ T = 0 ) T t τ)r K tτ e W t,w tτ K tτ e T t τ)r) }] h }] W t,w tτ K tτ e T t τ)r) W t,w tτ K tτ e T t τ)r) h B.0.2 Holding value funtion and exerise funtion is non-dereasing in w For x > y, v h t, x) v h t, y) = e r E Q v t 1, x L t ) S ) t1 v t 1, y L t ) S ) t1 S }{{ t } 0 from previous subsetion 0 Thus, v h is inreasing in DC aount balane. It is also easy to see that v e t, x) v e t, y) = x K t e T t)r ) y K t e T t)r ) 0 17

B.0.3 Continuity of value funtion in w For x > y, using the fat that sup[x] sup[y ] sup[x Y ] and x k) y k) x y), we have vt, x) vt, y) [ sup E Q e τr x S τ tτ 0 τ T t e τr y S τ tτ [ sup E Q e τr x y) S )] tτ 0 τ T t x y) Thus, v is ontinuous in w and learly τ L ut S ut τ L ut S ut T τ t)r K tτ e T τ t)r K tτ e ) ) ] F t vt, w h) vt, w) h 6) B.0.4 Convexity of value funtion We follow similar to Hatem et al. 2002), and prove the onvexity by indution. For any w 1 > 0 and w 2 > 0, and 0 λ 1 [ v h T 1, λw 1 1 λ)w 2 ) = E Q e r λw 1 1 λ)w 2 L T 1 ) S ) T bl T 1 T ät ) S T 1 λv h T 1, w 1 ) 1 λ)v h T 1, w 2 ) Thus, v h satisfy the onvexity at time T-1, and similarly v e at time T-1. Sine v is also onvex at time T-1. vt 1, w) = max v e T 1, w), v h T 1, w) ) ] 18

We now assume that result hold for time t 1, where 0 t T 2, then holding value at time t is v h t, λw 1 1 λ)w 2 ) = E [e Q r v t 1, λw 1 1 λ)w 2 L t ) S )] t1 S t E [e Q r λv t 1, w 1 L t ) 1 = λv h t, w 1 ) 1 λ)v e t, w 2 ) )] 1 λ)e Q [ e r λv t 1, w 2 L t ) S )] t1 and sine v e holds the onvexity for all t, thus, v h is a onvex funtion at t, whih proves the onvexity of v. C Properties of ϕt) This setion provides proof for Proposition 3. First, notie that if ϕt) <, then for suffiient large w [ vt, w) = v e t, w) v h t, w) = E e r v t 1, w L t ) S )] t1 S t E [e r v e t 1, w L t ) S )] t1 = w L t ) N d 1,t,w ) t 1)bL t ät )e rt t 1) e r Nd 2,t,w ) where d 1,t,w = 1 [ ) w L t ln σ S t 1)L t bät )e T t 1)r )] r σ2 S 2 d 2,t,w = d 1,t,w σ S 19

whih is the Blak-Sholes Formula, with initial stok prie wl t and strike value t 1)bL t ät )e rt t 1). Here we define ft) = lim w ve t, w) E [e r v e t 1, w L t ) S )] t1 = lim w ve t, w) w L t ) N d 1,t,w ) t 1)bL t ät )e rt t 1) e r Nd 2,t,w ) ) = lim w w L t ) N d 1,t,w ) t 1)bL t ät )e rt t) Nd 2,t,w ) ) rt t) tbl t 1 ät )e w = lim t 1)bL t ät )e rt t) N d 2,t,w ) w L t ) N d 1,t,w ) ) w L t t 1)bL t ät )e rt t) rt t) tbl t 1 ät )e = t 1)bL t ät )e rt t) tbl t 1 ät )e rt t) L t Where seond line to third is based on Put-Call Parity, and the third to fourth lines is based on the fats that lim w N d 1,t,w ) = 0, lim w N d 2,t,w ) = 0 and lim w wn d 1,t,w ) = 0. Clearly, whenever ϕt) <, we have ft) 0. I. If < 1, we prove ϕt) <, t [0, T ] by indution. bät )e rt At time T, ϕt ) = T bl T 1 ät ) <. At time t, assume ϕt 1) <, first we observe that for suffiient large w <, we have vt 1, w) = v e t 1, w). Next, we an show [e r v e t 1, w L t ) 1 lim w vh t, w) E [e r v = lim w E =0 t 1, w L t ) 1 The last line is due to the fat that if w ϕt 1) and for w < ϕt 1) ) F t ] ) v e t 1, w L t ) 1 v t 1, w) v e t 1, w) = 0 < ϕt 1) < v t 1, w) v e t 1, w) v e t 1, ϕt 1)) ϕt 1) < )) F t ] sine vt 1, w) is an inreasing funtion of w Appendix B.0.1). Thus, the differene is bounded by ϕt 1) < and we are able to apply the dominating onvergene theorem. 20

Next, lim w ve t, w) v h t, w) = lim w ve t, w) E rt t) = t 1)bL t äe > t 1)bL t äe > 0 T rt t) T [e r v e w L t ) S )] t1 tbät )e rt t) L t 1 L t tbät )e rt t) L t 1 L t bät )e rt Whih implies ϕt) < otherwise if ϕt) =, lim w v e t, w) v h t, w) 0). Thus ϕt) <, t [0, T ] II. For 1, we split the proof into three parts: the first is when > 1, the bät )e rt bät )e rt seond part and third part onsider speial ases when > bät ) 1 e µ L ) T e µ L ) e r, and when = 1. bät )e rt a) If > 1, we first prove that there exists a t bät )e rt suh that ϕt) =, t t, then prove that ϕt) <, t > t by indution from time T to t 1 as in part I). i. Immediately we have f0) < 0 that lim w ve 0, w) v h 0, w) f0) < 0 and thus ϕ0) =. Also, notie we an write ft) in the form of ft) = e µ Lt ht) where ht) is a strit inreasing funtion of time t. By the fat that )) log bät )e f0) < 0 and f rt > 0 where log > 0 by assumption. Then, we an find t suh that Here we set t = t, and we have and for t = t, first notie that r ft) < 0, t < t ft) = 0, t = t ft) > 0, t > t ) bät )e rt r lim w ve t, w) v h t, w) ft) < 0, t < t ft ) 0 = bät )e T r t 1 t e µ L) e rt 21

Next, for any finite w > t L t 1bäT )e rt t ), we have v h t, w) = E [e Q r v t 1, w L t ) S )] t 1 S [ t > E Q e r w L t ) S ) t 1 T t 1)r t 1)L t bät )e max T 0, w L t t 1)L t bät )e t )r) max 0, w t L t 1bäT )e T r e t r) = v e t, w) The seond to third line follows from Jensen s Inequality that [ E Q e r w L t ) S ) ] t 1 T t 1)r t 1)L t bät )e max 0, E [e Q r w L t ) S )]) t 1 T t 1)r t 1)L t bät )e = max 0, w L t t 1)L t bät )e T t )r) Thus, we have v h t, w) > v e t, w), w <, and lim t v e t, w) v h t, w) ft ) 0, whih implies ϕt ) =. ] ii. At time T, again we have ϕt ) <. At time t > t, assume ϕt 1) <. Exatly same as part I), we have Thus ϕt) <, t > t. lim w ve t, w) v h t, w) = ft) > 0, t > t b) If > bät ) 1 e µ L ) T e µ L ) e r, immediately we notie that > bät ) ) ) 1 e µ L T e µ L e r = bät )e ) r T T 1)e µ L > bät )e r bät )e T r Thus, we know there exist t as defined previously, and for time T 1, lim w ve T 1, w) v h T 1, w) = T bl T 1 ät )e r T 1)bL T 2 ät )e r L T 1 < T bl T 1 ät )e r T 1)bL T 2 ät )e r bät ) 1 e µ L) T e µ L ) e r L T 1 = 0 Thus, ϕt) =, t T 1, and the option simplifies to the DB-underpin option. 22

) When bät )e rt ϕ0) =. = 1, we have f0) = 0. Thus t = t = 0, immediately we have 23