Volatility risk premium in the interest rate market: Evidence from delta-hedged gains on USD interest rate swaps

FINANA-00835; No of Pages 15 International Review of Financial Analysis xxx (2015) xxx–xxx

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International Review of Financial Analysis

Volatility risk premium in the interest rate market: Evidence from delta-hedged gains on USD interest rate swaps Suk Joon Byun 1, Ki Cheon Chang ⁎ Korea Advanced Institute of Science and Technology, Republic of Korea

a r t i c l e

i n f o

Article history: Received 14 September 2014 Received in revised form 8 March 2015 Accepted 30 March 2015 Available online xxxx JEL classification: G12 G13 Keywords: Volatility risk premium Interest rate swap Delta-hedged gain

a b s t r a c t This study examines whether the interest rate market compensates for volatility risk. It demonstrates that the delta-hedged gain (DHG) method introduced by Bakshi and Kapadia (2003) shows the existence and sign of DHG in the interest rate swap markets where they use measures different from what Bakshi and Kapadia assumed. This finding is applied to the USD interest rate swap and swaption market. The result shows that, over the short term, there is negative compensation for volatility risk premiums, akin to the equity or currency markets. Over the long term, the signs of compensation change and regression tests show the possibility that the volatility risk premium in the interest rate market can be different from those in other asset markets. However, this interpretation entails an overlapping data problem that is not easy to overcome especially for the long term DHG data. The difference in interest rate market may be due to the fact that the interest rate swap market is different from the equity or currency markets in that it is more driven by financial institutions and option traders than by individuals or directional traders. © 2015 Elsevier Inc. All rights reserved.

1. Introduction Why are traders willing to buy expensive options? According to existing research on the equity option and currency option markets, option buyers usually lose money because these markets compensate negatively for volatility risk. This implies that people tend to want to buy options or volatility rather than to sell. This study attempts to examine whether this phenomenon occurs in the interest swap markets as well. Regardless of their theories, practitioners trade options involving swaps (swaptions) and hedge their delta risks dynamically using interest rate instruments, usually interest rate swaps. In other words, they make or lose money by using dynamic delta hedging strategies. Academics refer to these gains or losses as delta-hedged gains (DHGs). If the volatility of swaptions is constant, these gains should average zero, but if it is stochastic and is compensated for by markets, the average will be different. This aspect is also analyzed in this study. Roughly 20 years have passed since Heston (1993) suggested the closed form formula for option pricing where volatility follows the square root process. Changes in volatility are generally accepted in interest rate markets; therefore, stochastic volatility models for interest

⁎ Corresponding author at: KAIST Business School, 85 Heogiro, Dongdaemoon-gu, Seoul 130-722, Republic of Korea. Tel.: +82 2 958 3968; fax: +82 2 958 3620. E-mail addresses: [email protected] (S.J. Byun), [email protected] (K.C. Chang). 1 KAIST Business School, 85 Heogiro, Dongdaemoon-gu, Seoul 130-722, Republic of Korea. Tel.: +82 2 958 3352; fax: +82 2 958 3620.

rate markets have been proposed in recent times. For example, Hagan, Kumar, Lesniewski, and Woodward (2002) proposed the stochastic alpha beta rho (SABR) model, while Wu and Zhang (2006) and Trolle and Schwartz (2009) suggested the stochastic bond price and volatility model. Apart from these modeling approaches, there was no empirical research on volatility risk premiums in interest rate markets prior to Fornari (2010). He studied volatility risk premiums in interest rate swap markets by using data for three currencies, USD, EUR, and GBP, from 1998 to 2006. He used a method based on the GARCH model to estimate realized volatility and volatility under a physical measure. To estimate volatility under a risk-neutral measure, he used swaptionimplied volatilities. He regarded the spread between the two volatilities as compensation for volatility risk. He concluded that interest rate volatility leads to negative compensation for volatility risk, which is in line with other studies focusing on different asset classes. He also finds that the process of selecting a particular model to calculate realized volatility can be arbitrary. To avoid this issue, Bollerslev, Tauchen, and Zhou (2009) used a model-free approach to calculate volatility spread. They applied their technique to stock markets because they required highfrequency intraday data that is seldom found in over-the-counter markets such as interest rate swap and swaption markets. Few existing studies have explored volatility risk premiums in interest rate markets. As a result, the methodology could be restricted. In order to address this gap, this study analyzes the interest rate swap market and uses the Bakshi and Kapadia (2003); (hereafter BK) method to estimate volatility risk premiums. Therefore our contribution is, on the

http://dx.doi.org/10.1016/j.irfa.2015.03.018 1057-5219/© 2015 Elsevier Inc. All rights reserved.

Please cite this article as: Byun, S.J., & Chang, K.C., Volatility risk premium in the interest rate market: Evidence from delta-hedged gains on USD interest rate swaps, International Review of Financial Analysis (2015), http://dx.doi.org/10.1016/j.irfa.2015.03.018

2

S.J. Byun, K.C. Chang / International Review of Financial Analysis xxx (2015) xxx–xxx

one hand, showing that the BK method can be applied to the market where the forward measure or annuity measure should be used. This procedure is also necessary from the fact that the interest rate market is different from the equity and currency markets in practical and theoretical ways. Applying this to extended market data of interest rate swap covering the financial crisis of 2008, on the other hand, it could contribute to the understanding of volatility risk premium in interest rate markets. The rest of this study is organized as follows. Section 2 introduces the instrument and market examined in this study. Section 3 demonstrates that the BK method is applicable to the interest rate swap markets and also explains the methodology involved. Section 4 presents the data used and the empirical findings of this study. Section 5 explains the robustness of our analysis. Finally, Section 6 provides the conclusion.

2.2. Swaptions and their market conventions An option on a swap is referred to as a swaption. There are two types of swaptions—European payer swaption and European receiver swaption. The former is an option giving the right but not the obligation to enter into a payer IRS at the swaption's maturity date. The latter is an option giving the right to enter into a receiver IRS at the swaption's maturity date. The length of the underlying swap Tβ − Tα is called the tenor of the swaption. Like traders in the currency options market, swaption market traders quote prices based on volatility and calculate option premiums by inputting the volatility into the following Black (1976) model:

PS

β
pffiffiffi  X  0; T; τ; N; K; σ ¼ NBl K; Sαβ ð0Þ; σ T ; 1 τi Bð0; T i Þ

Black 

Black 

2.1. Interest rate swap

RS

A number of terms related to interest rate markets that are used in this study are described as follows. Year fraction τ(t,T) is the number of days between date t and date T divided by the number of days in a year. Consider an investment of one unit of currency at date t and returns of R units at date T; the simply-compounded spot interest rates L(t,T) for this investment can be given as: Lðt; T Þ ¼

R−1 : τðt; T Þ

ð1Þ

An interest-rate swap (hereafter IRS or swap) is a contract that exchanges payments between two differently indexed legs on a prespecified set of dates Tα + 1, …, Tβ. On the payment date Ti (i = α + 1,…,β), the fixed leg pays out the amount: Nτ i K;

ð2Þ

corresponding to the fixed interest rate K, the nominal amount N, and a year fraction τi = τ(Ti − 1, Ti) between Ti − 1 and Ti. On the same date, the floating leg pays out the amount: Nτ i LðT i−1 ; T i Þ

ð3Þ

corresponding to the interest rate L(Ti − 1, Ti), reset at the previous payment date Ti − 1 for the maturity based on the current payment date Ti. Note that the floating-leg rate resets on dates Tα, …, Tβ − 1 and pays out on dates Tα + 1, …, Tβ. If the fixed leg is paid and the floating leg is received, the IRS is called a payer IRS (swap). If the floating leg is paid and the fixed leg is received, the IRS is called a receiver IRS (swap). The spot IRS or spot swap is one whose first reset date is on the usual spot date. For example, in the USD swap market, the spot date is two business days after the trade date. If a swap is traded for which the first reset date is later than the spot date, it is called a forward IRS or forward swap. The forward swap rate is the fixed rate of the relevant swap contract that makes the value of the forward swap zero. When discount factors exist, the forward swap rate can be easily calculated using its definition. The time t forward swap rate Sαβ(t) with the first rest date tα and the last payment date tβ is
 Bðt; t α Þ−B t; t β Sαβ ðt Þ ¼ Xβ Bðt; t i Þτi i¼αþ1

ð5Þ

i¼αþ1

2. Instruments and markets

ð4Þ

where B(t,T) is the time t price of the zero coupon bond that pays the unit currency at T.

β
X pffiffiffi  0; T; τ; N; K; σ ¼ NBl K; Sαβ ð0Þ; σ T ; −1 τi Bð0; T i Þ

ð6Þ

i¼αþ1

where: PSBlack Payer swaption price using the Black model RSBlack Receiver swaption price using the Black model N Notional amount Bl(K, F, v, w) = FwΦ(wd1) − KwΦ(wd2) Non-discounted Black price d1 ¼ ln ð F=KvÞþv =2 ; d2 ¼ ln ð F=KvÞ−v =2 ω 1 if payer swaption, otherwise −1 Φ(x) Cumulative standard normal density function   T ¼ T α ; …; T β Set of dates (reset dates and payment dates) τ = {τα + 1, …, τβ} Set of year fractions B(0,T) Present value of discount bonds that pay one unit of currency on date T T Option expiry Tα First reset date, the effective date of swap Tβ Last payment date Sα,β(t) Forward swap rate at time t, for a swap with first reset date tα and last payment date tβ. σ Volatility of forward swap rates K Strike of swaption This volatility σ is called Black volatility. The pricing formula presented above can be applied by assuming that the forward swap rate dynamics are as in Eq. (7), and by using the results of the lognormal forward-swap model (LSM or swap market model) (Brigo & Mercurio, 2007; Jamshidian, 1997) where the forward swap rates are martingales. 2

2

dSt P ¼ μdt þ σ t dW t St

ð7Þ

where St is the abbreviation for the forward swap rate Sα,β(t). Wt is the standard Wienner process, while superscript P denotes the dynamics defined by a physical measure. In recent times, practitioners have often quoted swaption prices by using the premium itself rather than volatility. However, the premium can be easily converted into Black volatility and information providers such as Reuters and Bloomberg provide both premium and Black volatility. As such, recent changes in quotation methods do not affect this research. 3. Delta-hedged gain and interest rate swap markets This section provides an overview of an interest rate model with stochastic volatility, and reviews the meaning of DHG. In this section, forward swap rate dynamics are derived using various measures, such

Please cite this article as: Byun, S.J., & Chang, K.C., Volatility risk premium in the interest rate market: Evidence from delta-hedged gains on USD interest rate swaps, International Review of Financial Analysis (2015), http://dx.doi.org/10.1016/j.irfa.2015.03.018

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3

as risk-neutral measures and annuity measure. This shows how the volatility risk premium arises in interest rate settings. This study follows Wu and Zhang (2006) and Trolle and Schwartz (2009) in setting up a swap market model with stochastic volatility. For simplicity, two factor models are used, one for the underlying bond prices and the other for the volatility process. Like Wu and Zhang (2006), dynamics of zero coupon bond prices B(t,T) under riskneutral measures are assumed to be as follows:

Let κA = κ + η∑iβ= α + 1wiσi. Since κP = κ + λ2η, the relationship between κA and κP is

pffiffiffiffiffi dBðt; T Þ Q ¼ r ðt Þdt þ σ ðt; T Þ vt dW t Bðt; T Þ

ð8Þ

pffiffiffiffiffi Q vt ¼ κvt þ η vt dZ t

ð9Þ

Proposition 1. Volatility dynamics under different measures: If it is assumed that the bond price under a risk-neutral measure follows Eqs. (8) and (9), volatility processes will be pffiffiffiffiffi (i) dvt ¼ ðκ þ λ2 ηÞvt dt þ η vt dZ Pt under a physical measure, and
 pffiffiffiffiffi β (ii) dvt ¼ κ þ ρη∑i¼αþ1 wi σ i ðt Þ vt dt þ η vt dZ At under an annuity

Q where dW Q t and dZ t are Wiener processes under risk neutral measure. Their correlation is ρ. The volatility of the zero coupon bond B(t, Ti) is denoted as σ(t, Ti) = σi(t) = σi for brevity when there is no worry for confusion. Based on Ito's lemma and the fact that the forward swap rate is the function of zero coupon bonds, as in Eq. (4), the dynamics of the forward swap rate under the risk-neutral Q-measure is expressed as follows:

pffiffiffiffiffi Q Q dSαβ ðt Þ ¼ μ dt þ σ S ðt Þ vt dW t

ð10Þ

where: Q

μ ¼

β X ∂Sαβ ðt Þ i¼α

∂Bi

r ðt ÞBi þ

2 β β 1 X X ∂ Sαβ ðt Þ σ σ BB v 2 i¼α j¼α ∂Bi ∂B j i j i j t

β X ∂Sαβ ðt Þ pffiffiffiffiffi σ S ðt Þ ¼ Bi vi vt ∂B i i¼α

ð11Þ

pffiffiffiffiffi P Q dZ t ¼ dZ t −λ2 vt dt:

ð12Þ

Then the forward swap rate under physical measure P is ð13Þ

pffiffiffiffiffi P where μ P(t) = μQ(t) + λ1σS(t)vt and dvt ¼ ðκ þ λ2 ηÞvt dt þ η vt dZ t . P Let κ = κ + λ2η. Note that λ2 is regarded as the volatility risk premium relative to the risk-neutral measure. Since this study aims to model the forward swap rate directly, it is most convenient to involve a measure A under which every asset price divided by an annuity is a martingale. In this case, A is called as annuity measure. Here, the annuity Aαβ(t) is defined as ∑βi = α + 1B(t, ti)τi. In this measure, the forward swap rate will move in accordance with the following stochastic differential equations (for derivations, see Appendix A): ð14Þ

and
 Xβ pffiffiffiffiffi A dvt ¼ κ þ ρη i¼αþ1 wi σ i vt dt þ η vt dZ t where wi = Bi(t)/A(t)τi.

measure. Using this proposition, the DHG and its properties can be derived. Before considering the DHG in swap markets, it is important to examine the meaning of the delta hedge in the swap market model because it differs when the underlying asset is spot, as in the case of the usual equity or currency option. Let P(t) be the value of a derivative contract on a forward swap rate. After trading on this derivative, its delta risk can be hedged by using its underlying forward swap contract. In order to concretely identify the underlying instrument of the forward swap, let us set the forward swap as a payer swap with the notional amount N, fixed rate K, first reset date Tα, and last payment date Tβ. The value of the forward swap F is then: β X

τ i Bðt; T i Þ:

ð17Þ

i¼αþ1

pffiffiffiffiffi P Q dWt ¼ dW t −λ1 vt dt

pffiffiffiffiffi A dSðt Þ ¼ σ S ðt Þ vt dW t

ð16Þ

Let ξ = λ2 − ∑βi = α + 1wiσi. Here, ξ is regarded as the volatility risk premium of the swap market, relative to the annuity measure. These results are summarized by the following proposition:

F t ¼ NðSt −K Þ

and Bi is the short form for B(t, ti). In order to describe the dynamics under physical measure P, the market prices of risks λ can be expressed as follows:

pffiffiffiffiffi P P dSαβ ðt Þ ¼ μ dt þ σ S ðt Þ vt dW t


 Xβ P A κ ¼ κ þ λ2 − i¼αþ1 wi σ i η:

ð15Þ

The delta of the forward swap is: β X ∂F t ¼N τ i Bðt; T i Þ: ∂St i¼αþ1

ð18Þ

The number of underlying assets needed to hedge the option's delta risk can be calculated by dividing the delta of the options by the delta of the underlying assets. If the option on equity is considered, the delta of the underlying asset ∂St/∂St is 1. Thus, the number of underlying assets Nunder for hedging the delta risk of the option is equal to the delta of the option itself. Nunder ¼

∂P t ∂St

,

∂St P ¼ Δt ∂St

ð19Þ

where Pt is the value of options on equity and St is the value of equity. However, in the case of swap markets, the delta of the underlying asset ∂Ft/∂St is not 1, as in Eq. (18). Thus, the number of underlying contracts Nswap for hedging the delta risk of derivatives should be changed as follows: Nswap

∂P ¼ t ∂St

,

∂F t ΔPt ¼ F ∂St Δt

ð19Þ

where ΔFt = ∂Ft/∂St is the delta of the underlying contract. This point is made clearer by considering the practical swap market. Practitioners use a delta that is calculated using a numerical approach (difference method) rather than an analytical one (differential method). This is called the basis point value (BPV) and it expresses the change in the value of an instrument if the underlying interest rate changes by one basis point (bp).

Please cite this article as: Byun, S.J., & Chang, K.C., Volatility risk premium in the interest rate market: Evidence from delta-hedged gains on USD interest rate swaps, International Review of Financial Analysis (2015), http://dx.doi.org/10.1016/j.irfa.2015.03.018

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S.J. Byun, K.C. Chang / International Review of Financial Analysis xxx (2015) xxx–xxx

For example, consider USD payer swap with a notional amount of USD 10,000,000 and a maturity of three years. This swap has a BPV of roughly USD 2800. This means if the three-year interest rate increases by 1 bp (1/10,000), the value of the swap increase by USD 2800. If the BPV of the derivatives is USD 50,000, then 17.86 units of receiver swap with notional amount of USD 10,000,000 are required to hedge the delta risk of the derivatives. With this understanding of delta hedging in swap markets and following the BK model, it is possible to derive the formula for DHG. First, the case of constant volatility is considered. In this case, the forward swap rate dynamics is set as follows: dSαβ ðt Þ P P ¼ μ dt þ σ S dW t Sαβ ðt Þ

ð20Þ

The DHG will then be changed as in the following proposition to reflect that the dynamic hedging will be accomplished using a forward instrument: Proposition 2. The DHG, D, under the swap market model when volatility is constant is given as: Dt;tþτ

Aðt þ τÞ−Aðt Þ ¼ P ðt þ τÞ−P ðt Þ− P ðt Þ−Aðt þ τÞ Aðt Þ

Z

tþτ t

¼

Xβ Xβi¼αþ1

pffiffiffiffiffi P P dvt ¼ κ vt dt þ η vt dZ t :

ð27Þ

By combining this with the similar argument in Proposition 2 and following the BK's Proposition 2, the relationship between the DHG and volatility risk premium can be derived. Let P^ t ¼ P t =At . Using Ito's lemma, the dynamics of P^ t under P-measure can be expressed as follows:

ð21Þ

ð28Þ

The term A(t + τ)/A(t) in Eq. (21) can be regarded as a kind of compounding factor or, inversely, A(t)/A(t + τ) can be regarded as a discount factor for the period from t to t + τ. Adjusting the definition of the annuity as follows clearly demonstrates this point: Aαβ ðt Þ ¼

ð26Þ

! 2^ 2^ 2 ^ ∂P^ t ∂P^ t ∂P 1∂ P ∂ P ∂ P^ t t t 2 t ^ þ dP t ¼ dS þ dv þ σ v þ 2 ηvt þ ρσ S ηvt dt 2 ∂S2t S t ∂St t ∂vt t ∂t ∂St ∂vt ∂vt

Price of derivative contract Aαβ(t) = ∑βi = α + 1τiB(t, Ti) Value of forward swap with fixed rate K, first reset date tα, and last payment date tβ ΔPu/ΔFu. ∂P(t)/∂St ∂Ft(St)/∂St

Nswap ΔPt ΔFt

pffiffiffiffiffi P P dSαβ ðt Þ ¼ μ dt þ σ S ðt Þ vt dW t

1 ∂F u ðSu Þ dSu N AðuÞ swap ∂Su

where: P(t) A(t) Ft(K)

If the underlying asset is a spot asset, the delta hedge trade will cost cash. In this case, cash has to be paid (received) to buy (sell) the underlying asset. Thus, the cost or interest on the cash will be included in the DHG. However, if the underlying asset is forward, as in our case, the delta-hedged trade does not cost cash, and a contract is made without cash on the trade date. In this case, the last term in Eq. (25) will disappear in our DHG in Proposition 2. Next, the DHG under stochastic volatility is considered. In the preceding section, it has been shown that the dynamics of the forward swap rate under the P-measure can be set as follows:

Bðt; T i Þτ i ð22Þ

Bðt; t α ÞBðt α ; T i Þτi i¼αþ1 Xβ ¼ Bðt; t α Þ i¼αþ1 Bðt α ; T i Þτi

By using the martingale property of the ratio of any asset price to the numeraire Aαβ(t) under the A-measure, the following PDE can be derived: ∂P^ t ∂P^ 1 ∂2 P^ t 2 ∂2 P^ ∂2 P^ t A σ S vt þ 2t ηvt þ ρσ S ηvt ¼ 0 þ κ vt t þ 2 ∂t ∂vt 2 ∂St ∂St ∂vt ∂vt Combining the two results yields dP^ t ¼


 ∂P^ ∂P^ t ∂P^ P dSt þ t dvt − κ −ξη vt t dt ∂St ∂vt ∂vt

pffiffiffiffiffi On substituting dvt with κ P vt dt þ η vt dZ Pt and rearranging: dP^ t ¼

∂P^ t ∂P^ ∂P^ pffiffiffiffiffi P dSt þ ξηvt t dt þ t η vt dZ t : ∂St ∂vt ∂vt

P^ tþτ ¼ P^ t þ

Aðt Þ Bðt i ; t Þ ¼ ¼ Bðt; t þ τÞ: Aðt þ τ Þ Bðt; t þ τ Þ

ð23Þ

Thus, the third part of the RHS in Eq. (21) can be interpreted as the money cost of the option premium, while the last part can be regarded as the compounded delta gains. Comparing this to the results of the BK model (Eq. (6)) makes it possible to understand the difference between the DHGs of the (spot) equity market and the (forward) swap market as follows: N−1

ð30Þ

ð31Þ

Integrating leads to:

Therefore,



ð29Þ



N1




Dt;tþτ ¼ P ðt þ τ Þ−P ðt Þ−∑i¼0 Δi Siþ1 −Si −∑n¼0 r P ðt Þ−Δtn Stn

τ

Z

tþτ t

∂P^ u dS þ ∂Su u

Z

tþτ t

ξηvu

Z tþτ ^ ∂P^ u ∂P u pffiffiffiffiffi P duþ η vu dZ u ∂vu ∂vu t

ð32Þ

Substituting P^ t and P^ tþτ with Pt/At and Pt + τ/At + τ respectively in the original definition gives: P tþτ P t ¼ þ Atþτ At

Z

tþτ t

∂P^ u dS þ ∂Su u

Z

tþτ t

ξηvu

Z tþτ ^ ∂P^ u ∂P u pffiffiffiffiffi P duþ η vu dZ u ∂vu ∂vu t

ð33Þ

Multiplying At + τ on both sides and rearranging gives: P tþτ ¼ P t þ

dAt P þ Atþτ At t

Z

tþτ t

∂P^ u dS þ ∂Su u

Z

tþτ t

N ð24Þ

ξηvu

! Z tþτ ^ ∂P^ u ∂P u pffiffiffiffiffi P duþ η vu dZ u ∂vu ∂vu t

ð34Þ

or

From Proposition 2 and assuming no volatility risk, it is apparent that:

 τ N−1  N−1 Dt;tþτ ¼ P ðt þ τ Þ−P ðt Þ−rP ðt Þ−∑i¼0 Δi Siþ1 −Si −∑n¼0 rΔt n St n : N ð25Þ

P tþτ −P t −

dAt P −Atþτ At t

Z

tþτ t

∂P^ u dS ¼ 0 ∂Su u

ð35Þ

Please cite this article as: Byun, S.J., & Chang, K.C., Volatility risk premium in the interest rate market: Evidence from delta-hedged gains on USD interest rate swaps, International Review of Financial Analysis (2015), http://dx.doi.org/10.1016/j.irfa.2015.03.018

S.J. Byun, K.C. Chang / International Review of Financial Analysis xxx (2015) xxx–xxx

By plugging the above equation into Eq. (35), the following relationship between DHG and volatility risk premium is derived: Proposition 3. The DHG, D, in the swap market model when volatility is stochastic is given as: Z Dt;tþτ ¼ Atþτ

tþτ t

! Z tþτ ^ ∂P^ u ∂P u pffiffiffiffiffi P ξηvu duþ η vu dZ u ∂vu ∂vu t

ð36Þ

Based on the martingale property of the Ito integral, P Et

h

i

Dt;tþτ ¼

Z

tþτ t

" P Et

# Atþτ ∂P^ u ξηvu du: Au ∂vu

ð37Þ

From this proposition, it is seen that the volatility risk premium and expected DHG share the same sign. Thus, the sign of the volatility risk premium can be inferred from the DHG. 4. Empirical analysis 4.1. Data For this empirical study, data for USD swaptions and swaps were obtained from Bloomberg. The USD interest rate market involving the US treasury is the largest interest rate market. Therefore, the USD interest rate swap and swaption markets are the most liquid. The data in this study covers the period from May 10, 2000 to June 20, 2012. For interest rate data, the term structures of interest rates were collected for each business day in the following manner. For periods that were less than or equal to one year, LIBOR was used. For periods that were longer than one year, swap rates were used. LIBOR tenors are for 3, 6, 9, or 12 months, while swap tenors are annually spaced over periods ranging from two years to 15 years. To bootstrap discount factors from interest rates, swap rates were linearly interpolated into semi-annually spaced ones. Thus, the number of interest rates for each day increased from 18 to 30. For swaption data, at-the-money forward (ATMF) Black volatilities were used. Simultaneously buying or selling an ATMF straddle (a package of call and put options with the same strikes and expiries) along with hedging delta risk dynamically is often called gamma trading. The gains from gamma trading can be practically regarded as DHGs from academic perspective. For gamma trading, options with short expiries are used since a strategy that requires more than one year to determine profits or losses is not regarded as appropriate from a trading perspective. Following this practice, options with expiries of 1, 3, 6, and 12 months were used in this study. Swaptions with tenors (the maturity of the underlying interest rate swap) of 1, 2, 3, 4, 5, 7, and 10 years were used. This process yielded 28 different options for determining DHGs. Table 1 presents the general statistics for the ATMF Black volatilities of the 28 swaptions. Dates with missing data such as volatility and interest rates that are required to calculate option premiums or their Greeks were excluded. The sample size is 3060. Average ATMF volatilities range from 24.35% to 45.20%. Generally, the longer the total maturity (sum of swaption expiry and underlying swap tenor), the lower the volatility. Standard deviations of ATMF volatilities range from 8.88% to 30.99%. An overview of volatility and swap rates is presented below. Fig. 1 presents time series graphs for four selected ATMF volatilities. The right side in Fig. 1 shows implied volatilities of swaptions for one month into two years, one month into 10 years, 12 months into one year, and 12 months into 10 years, while the left side displays their underlying interest rates. The swap rates declined between 2000 and 2004, while slope of the term structures of the swap rates steepened. The swap rates then bounced back until 2008, before declining again until 2012. Fig. 1 shows that volatilities are inversely related to swap

5

Table 1 General statistics on ATMF black volatilities for 28 swaptions. This sample consists of 28 swaptions with four different expiries (1, 3, 6, and 12 months) and seven tenors (1, 2, 3, 4, 5, 7, and 10 years). Numbers in [] are option expiries. For example ‘1 m’ means that the option will expire in one month. The numbers in the headings in the table (1, 2, 3, 4, 5, 7, and 10 years) indicate the tenors of the underlying swaps. In addition, size is the number of samples, mean is the sample average (in %), stdev is the standard deviation, and kurt is the kurtosis of the samples. [1 m]

1 years

2 years

3 years

4 years

5 years

7 years

10 years

size mean stdev skew kurt median

3060 45.20 30.99 0.59 −0.72 40.72

3060 42.79 24.91 0.33 −1.01 41.60

3060 38.49 21.23 0.49 −0.55 37.79

3060 35.25 18.12 0.50 −0.46 34.48

3060 33.24 16.53 0.59 −0.05 32.00

3060 29.56 14.18 0.92 1.10 28.10

3060 26.75 13.19 1.37 2.73 24.30

[3 m] size mean stdev skew kurt median

3060 44.06 28.31 0.42 −1.10 41.00

3060 40.99 22.23 0.17 −1.32 39.65

3060 37.09 18.86 0.27 −1.06 36.00

3060 34.14 16.26 0.29 −0.93 33.23

3060 32.36 14.88 0.38 −0.63 31.63

3060 28.95 12.73 0.65 0.16 28.00

3060 26.34 11.68 1.05 1.29 24.60

[6 m] size mean stdev skew kurt median

3060 43.80 26.48 0.33 −1.28 40.00

3060 39.29 20.42 0.18 −1.39 36.36

3060 35.42 17.00 0.23 −1.21 33.25

3060 32.69 14.62 0.24 −1.11 31.31

3060 31.03 13.35 0.30 −0.93 29.83

3060 28.00 11.42 0.52 −0.35 26.85

3060 25.63 10.42 0.85 0.45 23.90

[12 m] size mean stdev skew kurt median

3060 40.53 22.24 0.36 −1.26 35.00

3060 35.63 17.11 0.31 −1.24 32.38

3060 32.29 14.12 0.29 −1.20 30.30

3060 30.07 12.25 0.29 −1.13 28.66

3060 28.65 11.20 0.34 −0.99 27.40

3060 26.33 9.74 0.50 −0.60 25.00

3060 24.35 8.88 0.76 −0.03 22.60

rates. This is seen, for instance, in the one month into two year swaption (marked as “swaption 1m2y”) and its underlying swap rate. In 2004, when the swap rate was low, the volatility was high; in 2007, when the rate was high, the volatility was low. This pattern is common among all swaptions, although this tendency is weaker among options with longer expiries and tenors, as can be seen in the bottom row of Fig. 1. 4.2. Delta-hedged gains of 28 swaptions To determine whether volatility risk premium exists, the DHG was first calculated according to Eq. (22) using an option notional amount of USD 10,000,000, which is considered a tradable amount in the swap market. Based on Proposition 3, if traders forecast future volatility without systemic bias and care about volatility risk, it can be expected that the DHG will be systemically different from zero. Three types of DHGs are reported in this study: the DHG itself, the DHG divided by the premium, and the DHG divided by the underlying swap rate. This has been done to prevent selection bias that could result from selecting the preferred result among the available data. In Table 2, the average DHGs for different swaptions have different signs. Short-expiry swaptions have negative DHGs, while long-expiry swaptions have positive DHGs. Swaptions with expiries of one month and three months have negative average DHGs, whereas swaptions with expiries of six and 12 months have positive DHGs. This result shows that the volatility risk premium (estimated from the expected DHG based on historical data) is not necessarily negative in the interest rate swap market. This is inconsistent with the findings of previous studies on equity and currency markets where negative DHGs have been consistently found. This result can be interpreted in several ways. One possible explanation is that the swaption market is sellerdriven, especially in the case of long-expiry swaptions. When sellers

Please cite this article as: Byun, S.J., & Chang, K.C., Volatility risk premium in the interest rate market: Evidence from delta-hedged gains on USD interest rate swaps, International Review of Financial Analysis (2015), http://dx.doi.org/10.1016/j.irfa.2015.03.018

6

S.J. Byun, K.C. Chang / International Review of Financial Analysis xxx (2015) xxx–xxx

Fig. 1. 4 selected time series graphs for four selected ATM volatilities (on the right side) and their underlying interest rates (on the left side) are presented. Swaption AmBy means a swaption with an expiry of A month(s) and a tenor of B year(s). For example, swaption 1m2y means a swaption with maturity of one month and tenor of two years. Therefore the underlying interest swap notated as swap AmBy is a forward start swap with forward period of A month and swap maturity of B year(s).

exceed buyers, the swaption premium clearly decreases, resulting in higher DHGs. Another explanation is that volatility traders are dominant in the swaption market. If volatility traders are dominant, implied volatility and, thus, the option premium may be lower than in other situations (Cherian & Vila, 1997). Different characteristics of different markets are even more apparent when their participants are taken into consideration. A relatively large number of participants in the equity option market are individual investors, whereas corporate and central banks are the main players in the currency option market. These are directional traders who are concerned with the direction of the underlying assets, even if they trade options. However, the interest rate market is mostly governed by financial institutions that are professional option traders and care about the volatility rather than direction of underlying asset. Therefore, the swaption market is likely to involve more volatility traders than the other markets. DHGs are analyzed from an alternate perspective as well. Fig. 2 presents the time trends in DHGs. It should be noted that the DHGs of long-expiry swaptions fell sharply in 2008. For example, the 12 month into 10 year swaption (28th swaption) experienced a significant drop in DHGs from October 21, 2008 to November 20, 2008. This occurred because these options became ATM (the underlying swap rate eventually returned to the strike) near expiry, while the realized volatility (the movement of the underlying swap rate) decreased simultaneously. This led to the sharp decline in DHGs. The overall levels in Fig. 2 are not different from previous results. Despite fluctuations, DHGs from short-maturity swaptions are generally

below zero, while those from long-maturity swaptions are over zero. These patterns confirm our previous findings and provide more insights into DHGs among swaptions. Short-maturity swaptions show noisy random walk patterns, whereas long-maturity swaptions show trending patterns. This is because long-expiry swaptions have more overlapping periods than short-expiry swaptions. The same effect can be seen in the fact that the moving average with a long window is smoother than the moving average with a short window. Fig. 2 also shows that positive DHGs for swaptions with longer expiries were observed over 2007–2008. This was the period of the financial crisis; since the crisis was not reflected in longer swaptions, these swaptions may have been traded at relatively low premiums. This indicates that the volatility risk premium for longer options does not usually reflect economywide crises or risks. 4.3. Delta-hedged gains and realized volatility Out-of-the money (OTM) option data is needed to analyze the crosssectional aspect of volatility risk premiums. However, unlike the equity option and currency option markets, the swaption market lacks OTM option data because of its unique characteristics. Therefore, to obtain complete information on volatility in the swaption market, it is necessary to create a volatility “cube” of three-dimensional information on option expiries, swap tenors, and moneyness or strikes. However, this additional dimension of volatility information makes it difficult to obtain reliable OTM option data since it is practically difficult to quote all

Please cite this article as: Byun, S.J., & Chang, K.C., Volatility risk premium in the interest rate market: Evidence from delta-hedged gains on USD interest rate swaps, International Review of Financial Analysis (2015), http://dx.doi.org/10.1016/j.irfa.2015.03.018

−64,120 4,108,193 2678 0.0257 0.2816 2678 28 6353 187,204 2678 0.1136 0.2665 2678 22 9170 22,909 2678 mean stdev size

300,993 746,132 2678

23 19,170 45,718 2678

0.1190 0.2710 2678

547,720 1,281,142 2678

24 22,472 66,368 2678

0.0977 0.2685 2678

599,802 1,750,722 2678

25 24,732 86,392 2678

0.0838 0.2697 2678

585,194 2,168,022 2678

26 19,068 106,078 2678

0.0584 0.2723 2678

362,565 2,552,091 2678

27 9880 139,299 2678

0.0316 0.2706 2678

67,922 3,188,688 2678

−678,364 3,013,098 2864 −0.0389 0.2387 2864 21 −27,066 127,501 2864 −417,398 2,245,086 2864 0.0016 0.2245 2864 15 707 12,153 2864 mean stdev size

−82,749 571,939 2864

16 6390 26,385 2864

0.0545 0.2222 2864

177,400 976,453 2864

17 6550 41,359 2864

0.0396 0.2314 2864

169,988 1,287,028 2864

18 2836 54,265 2864

0.0157 0.2261 2864

67,782 1,546,838 2864

19 −3582 66,056 2864

−0.0064 0.2219 2864

−124,986 1,792,023 2864

20 −15,720 89,346 2864

−0.0319 0.2237 2864

−914,766 2,270,815 2957 −0.0851 0.2333 2957 14 −35,445 89,826 2957 −640,748 1,681,573 2957 −0.0285 0.2532 2957 8 −608 8839 2957 mean stdev size

−147,376 469,222 2957

9 945 19,987 2957

0.0144 0.2626 2957

−42,724 840,554 2957

10 −1104 29,156 2957

−0.0061 0.2306 2957

−99,614 1,080,734 2957

11 −6621 37,524 2957

−0.0343 0.2170 2957

−226,369 1,284,442 2957

12 −12,503 45,703 2957

−0.0530 0.2084 2957

−374,172 1,403,625 2957

13 −23,462 62,369 2957

−0.0769 0.2119 2957

−564,924 1,423,206 3019 −0.0895 0.2362 3019 7 −21,299 52,297 3019 −397,137 1,109,174 3019 −0.0822 0.3361 3019 1 −1265 6647 3019 mean stdev size

−153,958 378,325 3019

2 −1389 13,465 3019

−0.0359 0.2875 3019

−133,453 598,275 3019

3 −2252 19,996 3019

−0.0295 0.2768 3019

−141,144 792,292 3019

4 −5049 24,193 3019

−0.0477 0.2514 3019

−188,618 874,635 3019

5 −7857 29,256 3019

−0.0579 0.2379 3019

−250,763 945,224 3019

6 −13,979 37,947 3019

−0.0804 0.2306 3019

D/rate D/p DHG D/rate D/p DHG D/rate D/p DHG D/rate D/p DHG D/rate D/p DHG D/rate D/p DHG D/rate D/p DHG

Table 2 Delta-hedged gains. D/p is DHG divided by swaption premium on trade date, while D/r is DHG divided by forward swap rate. In addition, size indicates the number of items in the sample. The numbers 1 to 28 in the headline of the table represent swaptions that are combinations of four expiries (1, 3, 6, and 12 months) and seven swap tenors (1, 2, 3, 4, 5, 7, and 10 years). For example, 1 is for swaptions with 1 month expiry and 1 year swap tenor, 2 is for swaptions with 1 month expiry and 2 year swap tenor, and so on.

S.J. Byun, K.C. Chang / International Review of Financial Analysis xxx (2015) xxx–xxx

7

swaption prices in terms of three dimensions. Considering this, this study focuses on time series analysis of volatility risk premiums for the swaptions market. Bakshi and Kapadia (2003) have shown that DHG is negatively related to realized volatility if the volatility risk premium is negative. This study has, thus, tested regression in a manner that is similar to that by Bakshi and Kapadia (2003). Proposition 2 of the BK model for the ATMF options explains that if the volatility risk is priced (λ = 0 in this proposition), the parameter β1 in the following equation should be different from zero: DHGt ¼ β0 þ β1 VOLt þ β2 DHGt−1 þ εt ;

ð38Þ

where VOLt is the estimate of historical volatility. Therefore, it is possible to run a regression under the null hypothesis H0 : β1 = 0 to determine whether volatility risk is priced or not. However, since DHGs overlap greatly, it is not easy to analyze the results of such a regression statistically. The fact that the DHG is calculated not by simply summing up data but by summing up data resulting from an option pricing formula makes statistical interpretation more difficult. In this section, the regression results for two different data frequencies are presented with the aim of demonstrating that the overlapping problem cannot be resolved by simply changing the data frequency. If daily DHGs are used for a regression, the data will be overlapping almost until the expiry of the swaption. This overlapping results in bias in the regression. In order to avoid overlapping, data that has been sampled based on non-overlapping periods should be used. This will reduce the number of samples in a regression. In this case, for a one year swaption, the number of data items available is only 12 (note that data for 12 years is used for this study). This sample is too small to provide reliable results in a regression. Therefore, only daily and monthly data is used in this study. Table 3 presents the results of a regression using daily data. In this table, all estimates of β1 are positive, but are inconsistent in terms of statistical significance, with some being significant and the others not. As mentioned earlier, due to the overlapping problem with daily data, it is not possible to rely on statistical significance that is based on p-value. Therefore, it is uncertain whether the volatility risk premium is not negative based on the estimates for β1 . In spite of this, the regression using daily data shows that volatility risk premiums in the interest rate swap market are different from those in the equity or currency markets. Further analysis based on regression results using monthly data is conducted (see Table 4). The approximate number of business days in one month is 22. All non-business days are excluded from the data in this study because there are no data available for calculating option premiums and Greeks on non-business days. Even when monthly data is used to relax the overlapping problem, swaptions with maturities that are longer than one month are affected by overlapping. Despite this result, this approach is the best way to mitigate the overlapping problem. In monthly data regression, almost all parameters for β1 are negative, but are still insignificant. This shows that adjusting data frequency does not fundamentally remedy this problem, as predicted. The above findings make it clear that running regressions with highly overlapped data does not lead to any reliable statistical result. Therefore, the focus is now on an analysis of correlation patterns by investigating scattergrams for DHGs and realized volatility. In this study, DHGs were plotted against realized volatility, as shown in Fig. 3. In Fig. 3, no positive or negative relationship or correlation between DHGs and realized volatility is indicated. Instead, the relationship between DHGs and realized volatility is cone-shaped, which means that as realized volatility increases, DHGs largely deviate from zero. For a detailed analysis, the data is divided into two periods: pre-crisis and post-crisis. The pre-crisis period covers the time from the first date in the study to December 31,2007, and the post-crisis period lasts from January 1, 2008 until the last date in the study data. In Fig. 3, the

Please cite this article as: Byun, S.J., & Chang, K.C., Volatility risk premium in the interest rate market: Evidence from delta-hedged gains on USD interest rate swaps, International Review of Financial Analysis (2015), http://dx.doi.org/10.1016/j.irfa.2015.03.018

8 S.J. Byun, K.C. Chang / International Review of Financial Analysis xxx (2015) xxx–xxx

Please cite this article as: Byun, S.J., & Chang, K.C., Volatility risk premium in the interest rate market: Evidence from delta-hedged gains on USD interest rate swaps, International Review of Financial Analysis (2015), http://dx.doi.org/10.1016/j.irfa.2015.03.018

Fig. 2. The time series graphs for DHGs divided by the initial premia of 28 swaptions are presented. The numbers 1 to 28 in the headline of the table represent swaptions that are combinations of four expiries (1, 3, 6, and 12 months) and seven swap tenors (1, 2, 3, 4, 5, 7, and 10 years). For example swaption 1 is for swaptions with 1 month expiry and 1 year swap tenor, swaption 2 is for swaptions with 1 month expiry and 2 year swap tenor, and so on.

DHGt ¼ β0 þ β 1 VOLt þ β2 DHGt−1 þ εt DHGt is DHG using ATMF swaption straddle on date t and dynamic delta-hedging from date t to the expiry of the swaption. DHGt-1 is DHG using ATMF swaption straddle on date t-1 and dynamic delta-hedging from date t-1 to the expiry of the swaption. VOLt is the historical volatility from date t to the expiry of the swaption. Numbers in line with βi (i = 0,1,2) are estimates, and the numbers below the estimates are p-values. Further, adjR2 means adjusted R-square. The numbers 1 to 28 in the headline of the table represent swaptions that are combinations of four expiries (1, 3, 6, and 12 months) and seven swap tenors (1, 2, 3, 4, 5, 7, and 10 years). For example, 1 is for swaptions with 1 month expiry and 1 year swap tenor, 2 is for swaptions with 1 month expiry and 2 year swap tenor, and so on. Finally, df means degree of freedom.

β0 β1 β2 R2 adjR2 df

1

2

3

4

5

6

7

8

9

10

11

12

13

14

−0.0128 0.0058 0.0156 0.1185 0.9165 0 0.8415 0.8414 2985

−0.0139 0.0028 0.0260 0.0088 0.8948 0 0.8085 0.8084 2985

−0.0172 0.0003 0.0387 0.0006 0.8834 0 0.7939 0.7937 2985

−0.0202 0.0000 0.0452 0.0001 0.8844 0 0.8001 0.8000 2985

−0.0224 0.0000 0.0497 0.0001 0.8745 0 0.7820 0.7819 2985

−0.0256 0.0000 0.0574 0.0001 0.8727 0 0.7780 0.7778 2985

−0.0233 0.0000 0.0511 0.0021 0.8776 0 0.7811 0.7810 2985

0.0000 0.9975 −0.0019 0.7088 0.9669 0 0.9348 0.9348 2883

−0.0016 0.5860 0.0058 0.3607 0.9586 0 0.9208 0.9207 2883

−0.0033 0.2081 0.0083 0.1837 0.9618 0 0.9286 0.9285 2883

−0.0045 0.0770 0.0095 0.1555 0.9618 0 0.9294 0.9294 2883

−0.0047 0.0665 0.0084 0.2378 0.9626 0 0.9299 0.9299 2883

−0.0049 0.0629 0.0084 0.3176 0.9635 0 0.9320 0.9319 2883

−0.0038 0.1484 0.0061 0.5090 0.9698 0 0.9432 0.9431 2883

Table 3 (continued)

β0 β1 β2 R2 adjR2 df

15

16

17

18

19

20

21

22

23

24

25

26

27

28

−0.0005 0.7993 0.0013 0.7102 0.9722 0 0.9457 0.9457 2790

−0.0031 0.1929 0.0129 0.0149 0.9639 0 0.9392 0.9391 2725

−0.0034 0.1492 0.0130 0.0305 0.9707 0 0.9512 0.9511 2725

−0.0039 0.0915 0.0134 0.0386 0.9735 0 0.9563 0.9562 2725

−0.0032 0.1558 0.0102 0.1313 0.9774 0 0.9609 0.9608 2725

−0.0027 0.2178 0.0079 0.2832 0.9809 0 0.9658 0.9658 2725

−0.0020 0.3472 0.0057 0.4737 0.9847 0 0.9716 0.9715 2725

−0.0011 0.5906 0.0055 0.1728 0.9869 0 0.9775 0.9774 2539

−0.0006 0.7679 0.0054 0.2865 0.9885 0 0.9805 0.9805 2539

−0.0006 0.7862 0.0048 0.4673 0.9896 0 0.9826 0.9826 2414

0.0005 0.8185 0.0005 0.9345 0.9924 0 0.9854 0.9854 2414

0.0007 0.7386 −0.0010 0.8926 0.9929 0 0.9856 0.9855 2414

0.0013 0.4810 −0.0043 0.5311 0.9947 0 0.9879 0.9879 2414

0.0008 0.6834 −0.0027 0.7398 0.9943 0 0.9877 0.9877 2414

S.J. Byun, K.C. Chang / International Review of Financial Analysis xxx (2015) xxx–xxx

Please cite this article as: Byun, S.J., & Chang, K.C., Volatility risk premium in the interest rate market: Evidence from delta-hedged gains on USD interest rate swaps, International Review of Financial Analysis (2015), http://dx.doi.org/10.1016/j.irfa.2015.03.018

Table 3 Regression results using daily data. Regression equation:

9

10

DHGt ¼ β0 þ β 1 VOLt þ β2 DHGt−22 þ εt DHGt is DHG using ATMF swaption straddle on date t and dynamic delta-hedging from date t to the expiry of the swaption. DHGt-22 is DHG using ATMF swaption straddle on date t-22 and dynamic delta-hedging from date t-22 to the expiry of the swaption. VOLt is the historical volatility from date t to the expiry of the swaption. Numbers in line with βt (i = 0,1,2) are estimate numbers and the numbers below the estimates are p-values. In addition, adjR2 means adjusted R-square. The numbers 1 to 28 in the headline of the table represent swaptions that are combinations of four expiries (1, 3, 6, and 12 months) and seven swap tenors (1, 2, 3, 4, 5, 7, and 10 years). For example, 1 is for swaptions with 1 month expiry and 1 year swap tenor, 2 is for swaptions with 1 month expiry and 2 year swap tenor, and so on. Finally, df means degree of freedom.

β0 β1 β2 R2 adjR2 df

1

2

3

4

5

6

7

8

9

10

11

12

13

14

0.0116 0.8917 −0.4759 0.3839 0.1738 0.0445 0.0375 0.0229 132

0.0855 0.4963 −1.0164 0.3324 0.0265 0.7620 0.0083 −0.0067 132

0.0250 0.8164 −0.4416 0.6293 0.1619 0.0605 0.0288 0.0141 132

−0.0232 0.8249 −0.1339 0.8820 0.0878 0.3105 0.0080 −0.0070 132

−0.0507 0.6475 0.0355 0.9693 0.0974 0.2619 0.0095 −0.0055 132

−0.1233 0.2726 0.5058 0.5853 0.1421 0.1006 0.0235 0.0087 132

−0.1322 0.0678 0.5331 0.3230 0.1999 0.0208 0.0525 0.0381 132

0.0138 0.8837 −0.2018 0.7385 0.5318 0.0000 0.2836 0.2724 128

−0.0021 0.9888 0.0204 0.9863 0.4039 0.0000 0.1626 0.1496 128

0.0499 0.7230 −0.4892 0.6718 0.5392 0.0000 0.2938 0.2828 128

0.0578 0.6943 −0.6450 0.5985 0.5627 0.0000 0.3224 0.3118 128

0.1101 0.4664 −1.1336 0.3623 0.5503 0.0000 0.3200 0.3094 128

0.0137 0.9178 −0.3877 0.7189 0.5471 0.0000 0.3065 0.2957 128

−0.0427 0.6590 0.0485 0.9460 0.5335 0.0000 0.2906 0.2795 128

15

16

17

18

19

20

21

22

23

24

25

26

27

28

−0.0142 0.8782 0.0682 0.9043 0.6943 0.0000 0.4771 0.4686 123

0.2664 0.1893 −1.9644 0.2200 0.6613 0.0000 0.4864 0.4779 121

0.2971 0.2084 −2.3385 0.2295 0.6655 0.0000 0.4912 0.4828 121

0.3048 0.2070 −2.4898 0.2173 0.6440 0.0000 0.4619 0.4530 121

0.2185 0.3106 −1.8457 0.3074 0.6575 0.0000 0.4647 0.4558 121

0.1396 0.5330 −1.2460 0.5016 0.6839 0.0000 0.4824 0.4738 121

0.0517 0.7694 −0.4784 0.7237 0.7277 0.0000 0.5315 0.5237 121

0.1418 0.6960 −0.6722 0.7258 0.8606 0.0000 0.7537 0.7493 112

0.0841 0.7585 −0.4870 0.8024 0.8760 0.0000 0.7848 0.7810 112

−0.0412 0.7653 0.3375 0.7043 0.8855 0.0000 0.8048 0.8011 106

−0.0156 0.9193 0.1605 0.8703 0.8909 0.0000 0.8037 0.8000 106

0.0306 0.8467 −0.1603 0.8775 0.9033 0.0000 0.8164 0.8129 106

0.0240 0.8943 −0.1329 0.9126 0.8915 0.0000 0.7959 0.7920 106

0.0078 0.9760 −0.0274 0.9875 0.8678 0.0000 0.7529 0.7483 106

Table 4 (continued)

β0 β1 β2 R2 adjR2 df

S.J. Byun, K.C. Chang / International Review of Financial Analysis xxx (2015) xxx–xxx

Please cite this article as: Byun, S.J., & Chang, K.C., Volatility risk premium in the interest rate market: Evidence from delta-hedged gains on USD interest rate swaps, International Review of Financial Analysis (2015), http://dx.doi.org/10.1016/j.irfa.2015.03.018

Table 4 Regression results using 22-day frequency data. Regression equation:

S.J. Byun, K.C. Chang / International Review of Financial Analysis xxx (2015) xxx–xxx

pre-crisis data is marked “o” and post-crisis data is marked “x”. These two data sets demonstrate very different patterns, but do not demonstrate the existence of any correlation. For example, the plot for the one month into one year swaption (Opt_1 in Fig. 3) for the pre-crisis period is distributed symmetrically around zero, while the plot for the post-crisis period has an upward slope. In addition, the overall levels in the pre-crisis period are higher than those in the post-crisis period, although they all seem to be negative. This explains the results presented in Table 2. Even though the DHGs are negative on an average, especially for swaptions with short expiries, the fact that there is no relationship between DHGs and realized volatility suggests that the volatility risk premium in the swaption market may be different from the premiums in the equity or currency markets. 5. Robustness To assess the robustness of this study's findings, two cases are considered. In the first case, a swaption is valuated and delta-hedged using a normal formula instead of the Black formula. The normal formula is an option pricing formula that assumes that the underlying processes are normal. The potential pitfall of this formula is that the underlying value can be negative, which implies arbitrage in the economy. But this model is also accepted in the interest rate swap market because it seems that interest rates change in absolute terms rather than relative terms. For example, the US Federal Reserve Board of Governors (FRB) usually changes its base rate (the FED rate) by 25 bps regardless of the current level of interest rate. Thus, it is considerable to use a normal formula to calculate DHGs in order to include different market views. The other case used in this study to assess robustness is that of the volatility spread, in terms of whether it is consistent with the study results. The volatility spread, defined as implied volatility minus realized volatility, is a simple measure of how the market perceives volatility risk. The simplest way to calculate realized volatility is to use historical volatility, which is measured as the annualized standard deviation of historical log returns of underlying values. This method is appropriate for checking the robustness of this study's results and is, thus, used to test the volatility spread. 5.1. Normal case The normal formula for a payer swaption price PSNormal is: PS

Normal

Xβ pffiffiffi ¼ Nσ N T ðnðdÞ þ dΦðdÞÞ i¼αþ1 τi Bð0; T i Þ;

ð39Þ

while that for the receiver swaption price RSNormal is: Normal

RS

Xβ pffiffiffi ¼ Nσ N T ðnð−dÞ−dΦð−dÞÞ i¼αþ1 τi Bð0; T i Þ;

ð40Þ

where T is the swaption expiry, n(x) is the standard normal density function, Φ(x) is the cumulative standard normal density function, d ¼ pffiffiffi ð F−K Þ=σ N T , F is the forward swap rate, σN is the normal volatility, and the other terms are the same as in Eqs. (5) and (6). The results of this formula for DHGs are presented in Table 5. To save space, the results for the DHG/swap rate and DHG/premium are omitted; however, they demonstrate patterns that are similar to the results for the DHGs themselves. The overall result is consistent with the results arrived at using the Black formula. The DHGs are negative for short-

11

expiry swaptions and positive for long-expiry swaptions. This further confirms this study's previous results. 5.2. Volatility spread To confirm the robustness of this study's results from an alternate perspective, the spread between realized volatility and risk-neutral volatility is calculated. This spread is clearly related to the volatility risk premium. The results of this calculation are presented in Table 6. Similar to the results described above, these results show that the volatility spreads are positive (implied volatility is higher than realized volatility) for short-expiry swaptions and negative for long-expiry swaptions. All the spreads are statistically significant given the standard errors, defined as standard deviation divided by the number of samples. The results of these two tests demonstrate that our results are valid and quite stable. 6. Conclusion This study investigates whether the interest rate swap market compensates for volatility risk. It shows that the DHG model proposed by Bakshi and Kapadia (2003) can be applied to the interest rate swap market where market participants usually price and hedge options using the Black (1976) formula. The findings of the application of this approach to USD swap and swaption data suggest that the interest rate swap market compensates for volatility risk differently across swaptions. The signs of the expected DHGs of swaptions change with different maturities and tenors. Swaptions with short maturities lead to negative expected DHGs, while the contrary is true for swaptions with long expiries. This is tested by checking the relationship between realized volatility and DHG. However, no consistent relationship was found. This implies that the volatility risk premium has no consistent sign, therefore the volatility risk is compensated differently from other assets such as equity and currency. To check the robustness of this study's findings, DHGs from a normal model and the volatility spread are tested. These two tests also support the study's previous results. This study's findings may result from the characteristic of interest rate market different from other assets. One possible explanation is that interest rate market is one where option sellers exceed buyers especially in long term swaption market. If it is the case, the option premium decreases, leading to an increase in DHGs. Another explanation is that option traders are dominant in the interest rate market. If volatility traders are dominant, the implied volatility in the market may be lower than otherwise, and traders do not seek compensation for bearing volatility risk. This can be related to the fact that in general, the equity and currency option markets are more buyer-driven and dominated by individual investors, corporate and central banks whereas the interest rate market is governed mostly by financial institutions who are professional option traders. Since reliable OTM swaption data could not be obtained for this study, the analysis of the cross-sectional aspect of volatility risk premium in the interest rate swap market was restricted. In addition, due to the problems involved in statistical analysis using overlapping data, various regressions on the DHG data were not attempted. The regression results seemingly show contradicting results, suggesting that volatility risk is not compensated, but this interpretation may result from statistical error due to overlapping data. Therefore, in order to obtain more

Fig. 3. The scatter grams between DHG and realized volatility for eight selected swaptions (swaption 1: 1 month into 1 year swaption, swaption 5: 1 month into 5 year swaption, swaption 9: 3 month into 2 year swaption, swaption 13: 3 month into 7 year swaption, swaption 17: 6 month into 4 year swaption, swaption 21: 6 month into 10 year swaption, swaption 25: 12 month into 4 year swaption, and swaption 28: 12 month into 10 year swaption) are presented. The horizontal axis is realized volatility and the vertical axis is DHG in USD. Black dots with o-mark denotes data prior to the financial crisis of 2008 and gray dots with x-mark denotes data after the financial crisis.

Please cite this article as: Byun, S.J., & Chang, K.C., Volatility risk premium in the interest rate market: Evidence from delta-hedged gains on USD interest rate swaps, International Review of Financial Analysis (2015), http://dx.doi.org/10.1016/j.irfa.2015.03.018

12

S.J. Byun, K.C. Chang / International Review of Financial Analysis xxx (2015) xxx–xxx

Please cite this article as: Byun, S.J., & Chang, K.C., Volatility risk premium in the interest rate market: Evidence from delta-hedged gains on USD interest rate swaps, International Review of Financial Analysis (2015), http://dx.doi.org/10.1016/j.irfa.2015.03.018

Mean Stdev Size

1

2

3

4

5

6

7

8

9

10

11

12

13

14

−0.0929 0.3346 3019

−0.0438 0.2861 3019

−0.0359 0.2755 3019

−0.0535 0.2501 3019

−0.0632 0.2366 3019

−0.0849 0.2290 3019

−0.0931 0.2344 3019

−0.0543 0.2488 2957

−0.0064 0.2634 2957

−0.0227 0.2265 2957

−0.0483 0.2112 2957

−0.0655 0.2022 2957

−0.0869 0.2061 2957

−0.0929 0.2269 2957

15

16

17

18

19

20

21

22

23

24

25

26

27

28

−0.0237 0.2246 2864

0.0308 0.2295 2864

0.0206 0.2307 2864

−0.0002 0.2208 2864

−0.0208 0.2135 2864

−0.0439 0.2128 2864

−0.0491 0.2260 2864

0.0508 0.2589 2678

0.0731 0.2564 2678

0.0604 0.2535 2678

0.0504 0.2516 2678

0.0282 0.2505 2678

0.0066 0.2465 2678

0.0047 0.2570 2678

Table 5 (continued)

Mean Stdev Size

S.J. Byun, K.C. Chang / International Review of Financial Analysis xxx (2015) xxx–xxx 13

Please cite this article as: Byun, S.J., & Chang, K.C., Volatility risk premium in the interest rate market: Evidence from delta-hedged gains on USD interest rate swaps, International Review of Financial Analysis (2015), http://dx.doi.org/10.1016/j.irfa.2015.03.018

Table 5 Average DHG/premium with normal forward swap rate. The numbers 1 to 28 in the headline of the table represent swaptions that are combinations of four expiries (1, 3, 6, 12 months) and seven swap tenors (1, 2, 3, 4, 5, 7, 10 years). For example, 1 is for swaptions with 1 month expiry and 1 year swap tenor, 2 is for swaptions with 1 month expiry and 2 year swap tenor, and so on. In addition, size is the number of items in the sample.

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mean stdev size

1

2

3

4

5

6

7

8

9

10

11

12

13

14

−0.0666 0.1762 2988

−0.0246 0.1324 2988

−0.0159 0.1142 2988

−0.0158 0.0946 2988

−0.0176 0.0823 2988

−0.0209 0.0705 2988

−0.0201 0.0678 2988

−0.0046 0.1533 2886

0.0154 0.1165 2886

0.0116 0.0992 2886

0.0067 0.0856 2886

−0.0005 0.0769 2886

−0.0050 0.0699 2886

−0.0057 0.0711 2886

15

16

17

18

19

20

21

22

23

24

25

26

27

28

0.0503 0.1451 2793

0.0535 0.1203 2728

0.0388 0.1038 2728

0.0289 0.0939 2728

0.0166 0.0859 2728

0.0081 0.0791 2728

0.0041 0.0792 2728

0.1272 0.1509 2542

0.0704 0.1143 2542

0.0619 0.1068 2417

0.0524 0.0977 2417

0.0370 0.0922 2417

0.0291 0.0880 2417

0.0233 0.0855 2417

Table 6 (continued)

mean stdev size

S.J. Byun, K.C. Chang / International Review of Financial Analysis xxx (2015) xxx–xxx

Please cite this article as: Byun, S.J., & Chang, K.C., Volatility risk premium in the interest rate market: Evidence from delta-hedged gains on USD interest rate swaps, International Review of Financial Analysis (2015), http://dx.doi.org/10.1016/j.irfa.2015.03.018

Table 6 Volatility spread. Volatility spread is defined as realized volatility minus risk neutral volatility. The numbers 1 to 28 in the headline of the table stand for swaptions that are combinations of four expiries (1, 3, 6, 12 months) and seven swap tenors (1, 2, 3, 4, 5, 7, 10 years). For example, 1 is for swaptions with 1 month expiry and 1 year swap tenor, 2 is for swaptions with 1 month expiry and 2 year swap tenor, and so on. In addition, stdev is the standard deviation and size is the number of items in the sample.

S.J. Byun, K.C. Chang / International Review of Financial Analysis xxx (2015) xxx–xxx

information on the volatility risk premium, these problems should be overcome in future studies. Appendix A. Derivation of forward swap rate dynamics under annuity measure A

15

while the dynamics of P^ t under the P-measure are: dP^ t ¼

∂P^ t ∂P^ 1 ∂2 P^ t 2 2 σ S dt þ t dSt þ 2 ∂S2t S t ∂t ∂St

On substituting the PDE into this equation: Radon–Nikodym derivatives are needed, from measure Q to measure A. Radon–Nikodym derivatives can be expressed as the ratios of two numeraire assets, as follows: dA Aðt ÞMð0Þ ¼ dQ Að0ÞM ðt Þ

dA ¼ dQ

B ðt Þτi i¼αþ1 i Að0Þ

Bð0; t Þ 1 Xβ ¼ B ðt Þτi i¼αþ1 i Bð0Þ Að0Þ

From the dynamics of bonds, the following can be obtained: Z t

Z t pffiffiffiffiffi 1 2 Q Bi ðt Þ ¼ Bi ð0Þ exp r ðuÞ− σ i ðuÞ vu du þ σ i ðuÞ vu dW u : 2 0 0 Therefore, using this result with the definition of Radon–Nikodym derivatives implies: Z t

Z t pffiffiffiffiffi 1 2 Q r ðuÞ− σ i ðuÞ vu du þ σ i ðuÞ vu dW u β Bi ð0Þ exp X dA 1 2 0 Z t 0 τi ¼ dQ Að0Þ i¼αþ1 r ðuÞdu exp 0  Z t Z t β pffiffiffiffiffi 1 X 1 2 Q ¼ Bi ð0Þ exp − σ i ðuÞ vu dW u τi : σ i ðuÞ vu du þ Að0Þ i¼αþ1 0 2 0

Let dA/dQ = m(t). The following can then be obtained: dmðt Þ ¼

 Z t Z t β pffiffiffiffiffi pffiffiffiffiffi 1 X 1 2 Q Q B ð0Þ exp − σ i ðuÞ vu dW u τ i σ i ðt Þ vt dW t : σ i ðuÞ vu du þ Að0Þ i¼αþ1 i 2 0 0

β pffiffiffiffiffi 1 X Q ¼ B ðt Þτ i σ i ðt Þ vt dW t Að0Þ i¼αþ1 i

¼

∂P^ t dS ∂St t

The integral form of this result is:

where M(t) is the value of the money market account at time t, with an initial value of M(0) = 1. It should be noted that M(t) = exp{∫0tr(u)du} = 1/B(0, t). Xβ

dP^ t ¼

β X pffiffiffiffiffi Aðt Þ Bi ðt Þ Q τ σ ðt Þ vt dW t Að0ÞBð0; t Þ i¼αþ1 Aðt Þ i i

β X pffiffiffiffiffi Bi ðt Þ Q ¼ mðt Þ τ σ ðt Þ vt dW t Aðt Þ i i i¼αþ1

Thus,

pffiffiffiffiffiXβ Bi ðt Þ A Q Q dmðt Þ Q ¼ dZ t −ρ vt τ σ ðt Þdt: dZ t ¼ dZ t −Cov dZ t ; i¼αþ1 Aðt Þ i i mðt Þ Using this result in the volatility process produces the desired result. Appendix B. Proof of Proposition 2 The swap market model is a model for pricing derivative assets, in which underlying assets are the forward swap rates and derivative values solely depend on the forward swap rates and on time. In the swap market, when a measure that uses the annuity Aαβ(t) as the numeraire is considered, the forward swap rate is a martingale (Brigo & Mercurio, 2007). Thus the measure is called a forward swap measure. Let P^ t ¼ P t =Aαβ ðt Þ. Under the forward swap measure, P^ t is a martin-

P^ tþτ −P^ t ¼

Z

tþτ t

∂P^ u dS : ∂Su u

After carrying out some manipulations, the following is derived: P ðt þ τÞ ¼ P ðt Þ þ

Z tþτ Aðt þ τÞ−Aðt Þ 1 ∂P u P ðt Þ þ Aðt þ τÞ dS : Aðt Þ AðuÞ ∂Su u t

To clarify the meaning of the delta (as mentioned above, the delta of derivatives does not give the number of underlying assets), more manipulations are needed. Let Ft(K) be the value of an underlying forward swap contract which has a predefined notional amount, direction (payer or receiver), and fixed rate K, with the same length as a forward swap underlied from derivatives. Substituting ΔPu/ΔFu ⋅ ∂Fu/∂Su into ∂Pu/∂Su will produce: P ðt þ τ Þ ¼ P ðt Þ þ

Z tþτ P Aðt þ τ Þ−Aðt Þ 1 Δu ∂F u ðSu Þ P ðt Þ þ Aðt þ τ Þ dSu ; Aðt Þ AðuÞ ΔuF ∂Su t

where ΔPu = ∂Pu/∂Su and ΔFu = ∂Ft(Su)/∂Su. The above derivation shows that a swaption can be replicated with the annuity A(t) and underlying forward swap Ft(K). Note that here, ΔPu/ΔFu is the number of underlying assets. Thus, if a swaption is purchased at t and replicated through dynamic hedging, the gain from this swaption or the DHG will be as follows: Dt;tþτ ¼ P ðt þ τÞ−P ðt Þ−

Z tþτ P Aðt þ τ Þ−Aðt Þ 1 Δu ∂F u ðSu Þ P ðt Þ−Aðt þ τ Þ dSu ð Þ Aðt Þ A u ΔuF ∂Su t

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gale and satisfies the following PDE: 2 ∂P^ t 1 ∂ P^ t 2 2 σ S ¼0 þ 2 ∂S2t S t ∂t

Please cite this article as: Byun, S.J., & Chang, K.C., Volatility risk premium in the interest rate market: Evidence from delta-hedged gains on USD interest rate swaps, International Review of Financial Analysis (2015), http://dx.doi.org/10.1016/j.irfa.2015.03.018