QF5203 Lecture 7
Interest Rate Options and their Risk Measures Part 2
1. References
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2. Interest Rate Swaption
3. Swaption Volatility
4. Example –
5. SABR Model
6. Example – SABR Swaption
7. Swaption Risk Sensitivities
8. CMS Swaps
9. Example – CMS Swap
10. Bermudan Swaption
1. References
• Options, Future and Other Derivatives,
• Interest Rate Option Models,
• The Volatility Surface: A Practitioner’s Guide, J. Gatherall
• Managing Smile Risk, Hagan, P., Kumar, D., Lesniewski, A., and Woodward, D.
, September, 84–108 (2002)
• Convexity Conundrums, Pricing CMS Swaps, Caps and Floors,
• QuantLib Python Cookbook, , Luigi Ballabio
• https://www.quantlib.org/quantlibxl/
2. Interest Rate Swaption
• An interest rate payer or receiver swaption is an over the counter (OTC) interest rate derivatives which are intended to provide the owner with protection against a rise (fall) in future swap rates.
• A European payer swaption gives the holder the right, but not the obligation, to enter into a pre-specified payer swap at some time 𝑇 in the future at a pre-specified fixed rate of 𝐾.
• A European receiver swaption gives the holder the right, but not the obligation, to enter into a pre-specified receiver swap at some time 𝑇 in the future at a pre-specified fixed rate of 𝐾.
• The payout of the payer and receiver swaptions is given by:
𝑉 𝑇 =𝑁𝐵 𝑇;𝑇,𝑇 𝑀𝑎𝑥 0,𝐹 𝑇;𝑇,𝑇 −𝐾 forapayerswaption
𝑉(𝑇)=𝑁𝑀𝑎𝑥0,𝐾−𝐹𝑇;𝑇,𝑇 forarecieiverswaption 𝑅12
• In the above, 𝑁 is the notional, 𝐾 is the strike, F 𝑇; 𝑇 , 𝑇 is the (forward) swap rate 𝑠𝑒
observed at time 𝑇 with corresponding start and end dates of 𝑇 and 𝑇 respectively, 𝑠𝑒
and B 𝑇;𝑇 ,𝑇 = σ 𝛼 𝑡 ,𝑡 𝑍(𝑡 ) is the annuity associated with the fixed leg. 𝑠 𝑒 𝑛 𝑛−1 𝑛 𝑛
2. Interest Rate Swaption
• Interest rate swaptions are quoted in terms of the number of months or years from today to expiry, followed by the number of months or years from the expiry date to the maturity date of the underlying swap.
• For example, a 5 year into 5 year (or 5 year 5 year) 2% payer swaption gives the owner the right to enter into a 5 year payer swap in 5 years time at a rate of 2%.
• If in 5 years time the then prevailing 5 year swap rate was 3%, then the swaption holder will (rationally) exercise the swaption and enter into a swap where he or she pays 2%.
• Note that since a swap rate can be thought of as a (weighted average) basket of LIBOR rates, an interest rate swaption is an option on a portfolio (of LIBOR forwards).
• Therefore, whereas a cap (floor) is a portfolio of options on LIBOR forwards, a payer (receiver) swaption is an option on a portfolio of LIBOR forwards.
• Note that Cap/Floors and Payer/Receiver swaptions are not completely independent, and traders will often seek to arbitrage one against the other (e.g. 5×10 cap versus the 5y into 5y payer swaption).
2. Swaption Valuation
• If the forward swap rate is assumed to be lognormally distributed, then the valuation of the payer (or receiver) swaption can again be obtained using the Black formula give by:
𝑑𝑁𝐾− 𝑑𝑁 𝑇,𝑇;𝑡𝐹)𝑇,𝑇;𝑡(𝐵𝑁=)𝑡(𝑉 𝑃𝑠𝑒𝑠𝑒12
𝑑−𝑁 𝑇,𝑇;𝑡𝐹− 𝑑−𝑁𝐾)𝑇,𝑇;𝑡(𝐵𝑁=)𝑡(𝑉 𝑅𝑠𝑒2𝑠𝑒1
for a payer and receiver swaption, respectively, and where
ln 𝐹(𝑡;𝑇𝑠,𝑇𝑒) +1𝜎2(𝑇 −𝑡) ln 𝐹(𝑡;𝑇𝑠,𝑇𝑒) −1𝜎2(𝑇 −𝑡) 1 2
𝑑2−𝑒𝑑1=𝑑𝑁; 𝑠 2 𝐾 =𝑑2; 𝑠 2 𝐾 =𝑑1 ∞− 𝜋2 𝑡−𝑇 𝜎 𝑡−𝑇 𝜎
and 𝜎 denotes the (lognormal) volatility of the forward LIBOR rate F 𝑡; 𝑇 , 𝑇 and 𝑍(𝑡; 𝑇)
the discount factor observed at time 𝑡 corresponding to expiry date 𝑇.
2. Swaption Valuation
• If the forward swap rate is assumed to be normally distributed, then the valuation of the payer and receiver swaption is given by:
𝑉 𝑡 =𝑁𝐵(𝑡;𝑇,𝑇){𝐹𝑡;𝑇,𝑇 −𝐾𝑁𝑑 +𝜎 𝑇−𝑡𝐺𝑑} 𝑃𝑠𝑒𝑠𝑒𝑠
𝑉𝑡=𝑁𝐵(𝑡;𝑇,𝑇){𝐾−𝐹𝑡;𝑇,𝑇 𝑁−𝑑+𝜎𝑇−𝑡𝐺𝑑} 𝑅𝑠𝑒𝑠𝑒𝑠
for a payer and receiver swaption, respectively, and where
𝐹𝑡;𝑇,𝑇 −𝐾 𝑒−𝑥2 𝑑=𝑠𝑒;𝐺𝑥=2
𝜎 𝑇−𝑡 2𝜋 𝑠
and 𝜎 denotes the (normal) volatility of the forward swap rate F 𝑡; 𝑇 , 𝑇 and 𝑠𝑒
𝐵(𝑡; 𝑇 , 𝑇 ) the annuity observed at time 𝑡 corresponding to dates 𝑇 and 𝑇 . 𝑠𝑒 𝑠𝑒
3. Swaption Volatility
• Recall from the previous lecture that for Caplets and Floorlets we made the assumption that the LIBOR forward rate was lognormal (or normal).
• Now we are making the assumption that the distribution of the forward swap rate is lognormal (or normal).
• Although this is market practice it is not theoretically consistent, since swap rates can be thought of as a weighted average of LIBOR rates, and a weighted average of a random variable which is lognormal is not lognormal.
• Recall also from the previous lecture that the Caplet and Floorlet valuation depended on the volatility of the LIBOR forward rate.
• The situation for the interest rate swaption is the same, but now the volatility input is the volatility of the forward swap rate.
• The most common swaption volatility that is quoted is the volatility corresponding to a strike corresponding to the forward swap rate (at-the-money).
3. Swaption Volatility
ATM Swaption Volatilities
1Y 2Y 3Y 4Y 5Y 7Y 10Y 15Y 20Y 25Y 30Y
1M 3M 6M 1Y 2Y 3Y 4Y 5Y 7Y 10Y 15Y 20Y 25Y 30Y
65.50 62.10 59.20 55.00 58.40 52.10 49.90 44.00 40.20 37.50 35.70 34.20 32.20 31.30 29.40 29.50 27.00 26.90 22.80 22.40 21.80 22.20 22.70 22.80 20.80 20.60 18.40 18.40
55.50 52.75 51.75 50.20 50.70 49.70 49.80 47.90 48.30 47.80 48.10 46.00 43.40 43.20 43.70 42.00 37.00 37.00 37.60 36.70 33.60 33.40 34.00 33.60 31.20 31.40 31.50 31.10 29.70 29.80 30.00 29.30 27.10 26.90 26.60 26.10 22.30 22.40 22.50 22.70 22.60 22.90 23.20 23.30 22.90 22.90 22.90 22.40 20.50 20.40 20.40 20.00 18.40 18.50 18.60 18.50
52.20 51.00 52.00 48.80 47.60 49.60 46.20 44.60 46.30 41.40 40.40 41.60 36.00 35.80 36.70 32.90 32.70 33.00 30.40 30.40 30.30 28.70 29.00 28.80 26.00 26.40 25.80 23.00 22.90 22.00 23.10 21.90 21.40 21.50 20.30 19.90 19.60 18.80 18.80 18.40 18.00 18.10
54.60 56.70 51.70 53.30 48.40 50.10 43.10 44.30 37.30 37.60 33.40 33.50 30.70 30.90 28.90 28.80 25.80 25.70 21.90 21.80 21.10 20.90 19.80 19.70 18.80 18.70 18.00 17.90
3. Swaption Volatility
• The previous slide shows an example of an ATM volatility grid.
• The volatilities shown are lognormal volatilities.
• In practice, ATM swaption straddle prices are quoted in the market from which one can imply the volatility.
• The disadvantage of quoting volatilities is that these are directly tied to a model whereas prices are model independent.
• So, if one wants to value a 10y into 5y ATM swaption then one reads uses the grid on the previous slide by looking across horizontally for the underlying swap tenor (5y in this example) and then looking down for the corresponding expiry (10y in this example).
• The volatility that one would read off would be 22.50%, and this is the number that I would insert into my Black swaption valuation model.
4. Example –
General Inputs
Quote Date
Set Evaluatio
n DatTeRUE
Days To Spo Spot Date Fixed Freq
Semiannual
Fixed Basis
Fixed Bus Da
Fixed Pmt Ca
Float Freq
30/360 (Bond B
lUenidteadr States::S
Float Basis
Float Ref Rat
Float Bus Da Float Pmt Ca Float Reset
Actual/360
lUenditaerdStates::S
UalneinteddaKr ingdom
ettlement ::Settlemen
Option Expiry
Swap Tenor
Swaption Details
Swaption #
Option Type
Volatility
Market Inputs
USD Yield Curv
USDSwaptionV
5. SABR Model
• As can be seen from the previous slide which shows an example of an ATM volatility grid covering both LIBOR and Swap underlyings, the number of possible expiry/tenor combinations is significant.
• The use of the Black model for valuation (and hedging) for each of these combinations obviously leads to inconsistencies which can only really be addressed with a single consistent term structure model.
• It is like having a different model for each underlying swap tenor and expiry.
• Furthermore, the Black model is not rich enough to generate the observed market smile, whereby options of underlying and expiry display different implied volatilities for different strikes.
• In the absence of having a term structure model which is rich enough and (numerically) quick enough for trading purposes, extensions of the Black model have been developed, the most popular being the SABR model.
5. SABR Model
• The SABR model is an example of a stochastic volatility model and defined by the following (correlated) stochastic differential equations (SDEs):
𝑑𝐹 𝑡 = 𝛼 𝑡 𝐹𝛽 𝑑𝑊 𝑡 1
𝑑𝛼 𝑡 = 𝜈 𝑡 𝛼 𝑑𝑊 𝑡 2
• The two Brownian motions are correlated via 𝐸 𝑑𝑊 (𝑡)𝑑𝑊 (𝑡) = 𝜌𝑑𝑡. 12
• SABR stands for Stochastic(S) Alpha(A), Beta(B), Rho(R).
• In the above, 𝐹 is the relevant forward rate, 𝛼 the volatility of the forward, 𝛽 a coefficient allowing one to ‘tune’ the model between purely normal (𝛽=0) and purely lognormal (𝛽=1), and 𝜈is the volatility of the volatility (Vol of Vol).
• The solution of the above SDE is carried out using singular perturbation techniques and is outlined in detail in the paper by Hagan et al.
5. SABR Model
• Under the SABR model, the prices of European options can be still be priced using the Black formula, but with the following Black volatility 𝜎 given by:
𝜎𝐹,𝐾= 𝛼 𝐵𝑍 𝐹𝐾 1−𝛽Τ2 1+1−𝛽2𝑙𝑜𝑔2 𝐹 +1−𝛽4𝑙𝑜𝑔4 𝐹 𝜒𝑍
1+ 1−𝛽 2 24
2 24 𝐾19202𝐾
𝛼 +1 𝜌𝛽𝜈𝛼 +2−3𝜌 𝜈2 𝑇+⋯
𝐹𝐾 1−𝛽 4 𝐹𝐾 1−𝛽 Τ2
Z = 𝜈 𝐹𝐾 1−𝛽 Τ2𝑙𝑜𝑔 𝐹 𝛼𝐾
1−2𝜌𝑍+𝑍2 +𝑍−𝜌 1−𝜌
• See reference “Managing Smile Risk” by Hagan et al for the derivation.
5. SABR Model
Swaption SABR Alpha (ATM Vol)
1Y 2Y 3Y 4Y 5Y 7Y 10Y 15Y 20Y 25Y 30Y
1M 3M 6M 1Y 2Y 3Y 4Y 5Y 7Y 10Y 15Y 20Y 25Y 30Y
16.00 13.50 6.50 7.00 6.60 6.40 6.70 6.30 6.20 6.50 6.30 6.20 6.25 6.10 5.90 6.20 5.90 5.70 6.00 5.80 6.70 5.75 5.70 6.50 5.50 5.40 6.10 5.25 5.00 5.80 4.50 4.30 4.90 4.00 3.80 4.30 3.75 3.60 4.00 3.50 3.30 3.70
5.20 5.10 5.10 5.00 6.10 5.90 6.00 5.80 5.70 5.60 5.50 5.30 6.40 6.20 6.30 6.10 5.90 5.80 5.60 5.40 4.90 4.90 4.40 4.40 4.20 4.20 3.80 3.70
4.50 6.50 6.30 8.60 4.50 6.60 6.40 5.10 6.60 6.70 6.50 5.20 5.00 5.00 5.60 5.40 5.10 4.90 5.40 5.20 5.50 4.80 5.40 5.20 5.80 5.50 5.30 5.10 5.70 5.40 5.20 5.10 5.50 5.20 5.10 4.90 5.20 5.00 4.80 4.80 4.60 4.50 4.50 4.40 4.10 4.10 4.10 4.00 3.90 3.80 3.80 3.70 3.50 3.50 3.50 3.40
8.20 7.90 4.90 4.70 5.00 4.90 5.20 5.00 5.10 4.90 5.10 4.90 5.00 4.90 5.00 4.80 4.80 4.70 4.60 4.50 4.30 4.20 3.90 3.80 3.70 3.60 3.30 3.20
5. SABR Model
Swaption SABR Beta
1Y 2Y 3Y 4Y 5Y 7Y 10Y 15Y 20Y 25Y 30Y
1M 3M 6M 1Y 2Y 3Y 4Y 5Y 7Y 10Y 15Y 20Y 25Y 30Y
0.70 0.70 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.60 0.55 0.55 0.60 0.55 0.55 0.60 0.55 0.55 0.60 0.55 0.55 0.60 0.55 0.55 0.60 0.55 0.55 0.60 0.55 0.55 0.60
0.50 0.50 0.50 0.50 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60
0.50 0.65 0.65 0.75 0.50 0.65 0.65 0.60 0.60 0.65 0.65 0.60 0.53 0.55 0.60 0.60 0.55 0.55 0.60 0.60 0.58 0.55 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60
0.75 0.75 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60
5. SABR Model
Swaption SABR Rho
1Y 2Y 3Y 4Y 5Y 7Y 10Y 15Y 20Y 25Y 30Y
1M 3M 6M 1Y 2Y 3Y 4Y 5Y 7Y 10Y 15Y 20Y 25Y 30Y
0.65 0.75 0.50 0.65 0.55 0.50 0.65 0.55 0.50 0.65 0.55 0.55 0.65 0.45 0.55 0.65 0.43 0.50 0.60 0.42 0.50 0.55 0.40 0.50 0.50 0.40 0.30 0.50 0.40 0.30 0.50 0.40 0.30 0.50 0.40 0.30 0.50 0.40 0.30 0.50 0.40 0.30
0.60 0.80 0.60 0.80 0.60 0.60 0.55 0.50 0.50 0.45 0.40 0.40 0.20 0.35 0.25 0.30 0.23 0.23 0.20 0.20 0.18 0.18 0.17 0.17 0.17 0.17 0.17 0.17
0.70 0.60 0.60 0.60 0.60 0.50 0.55 0.55 0.50 0.50 0.50 0.50 0.45 0.45 0.35 0.30 0.43 0.40 0.30 0.21 0.40 0.35 0.25 0.20 0.35 0.30 0.23 0.19 0.25 0.25 0.20 0.18 0.23 0.23 0.20 0.17 0.20 0.20 0.20 0.17 0.18 0.18 0.18 0.16 0.17 0.17 0.17 0.15 0.17 0.17 0.17 0.15 0.17 0.17 0.17 0.15
0.60 0.60 0.55 0.55 0.50 0.50 0.30 0.30 0.21 0.21 0.20 0.20 0.19 0.19 0.18 0.18 0.17 0.17 0.17 0.17 0.16 0.16 0.15 0.15 0.15 0.15 0.15 0.15
5. SABR Model
Swaption SABR Nu (Vol Of Vol)
1Y 2Y 3Y 4Y 5Y 7Y 10Y 15Y 20Y 25Y 30Y
1M 3M 6M 1Y 2Y 3Y 4Y 5Y 7Y 10Y 15Y 20Y 25Y 30Y
70.00 70.00 60.00 65.00 65.00 55.00 60.50 60.00 54.00 55.00 50.00 46.00 53.00 40.00 36.00 35.50 33.00 28.00 31.50 30.00 25.25 25.50 22.00 22.25 22.25 21.25 17.75 17.50 16.75 15.75 16.25 15.00 14.00 14.50 13.75 13.00 14.25 13.50 12.75 14.00 13.25 12.50
60.00 70.00 70.00 55.00 60.00 60.00 54.00 52.25 45.00 43.00 32.00 40.00 36.75 26.00 35.00 31.25 24.25 25.00 23.00 23.00 22.00 21.25 19.00 19.00 18.75 15.50 17.00 15.25 14.25 14.00 13.75 12.75 13.00 12.50 12.00 12.00 12.25 12.00 12.00 12.00 12.00 12.00
65.00 70.00 58.50 50.00 50.00 52.50 45.00 52.75 52.50 30.00 36.00 35.00 27.50 35.00 35.00 24.00 23.50 24.00 21.50 20.75 21.00 18.00 18.00 18.00 17.50 17.50 17.50 14.00 17.00 17.00 12.00 13.00 14.25 11.58 12.25 14.00 11.58 12.00 13.75 11.58 11.75 13.50
58.50 58.50 52.50 52.50 52.50 52.50 35.00 35.00 35.00 35.00 24.00 24.00 21.00 21.00 18.00 18.00 17.50 17.50 17.00 17.00 14.25 14.25 14.00 14.00 13.75 13.75 13.50 13.50
General Inputs
Swaption Details
Swaption #
Option Expiry
Swap Type
SABR Alpha
Volatility
Quote Date
Set Evaluatio
Days To Spo Spot Date
n DatTeRUE
Fixed Freq Fixed Basis
Fixed Bus Da
Fixed Pmt Ca
Semiannual
30/360 (Bond B
lUenidteadr States::S
Float Freq Float Basis
Float Ref Rat
Float Bus Da
Actual/360
Float Pmt Ca Float Reset
lUenditaerdStates::S
UalneinteddaKr ingdom
::Settleme
Market Inputs
VcID SABRAlpha SABRBeta SABRRho SABRNu
USD Yield Curv
USDSwaptionVcID#0001
USDSABRSwaptionAlpha#0001
USDSABRSwaptionBeta#0001
USDSABRSwaptionRho#0001
USDSABRSwaptionNu#0001
6. Example – SABR Swaption
7. Swaption Risk Sensitivities
• In the previous lecture we saw that interest rate caps and floors have both interest rate delta risk (i.e. sensitivity to the inputs of the yield curve used for projecting LIBOR rates and discounting cash flows) as well as sensitivity to the volatilities of the underlying forward LIBOR rates.
• The situation is the same for interest rate swaptions and a risk management vega report for a portfolio for swaptions would be a table like the one shown on slide 9, detailing the sensitivities of the portfolio to a 1% change in each lognormal volatility or a 0.01% change in the normal volatility.
• Under the SABR model, it is necessary to manage 4 separate matrices of inputs compared to a single matrix in the case of the Black model.
• The corresponding volatility risk report for the SABR model will therefore show sensitivities to each element of the 4 SABR matrices.
7. CMS Swap
• In lecture 5 under variations of plain vanilla swaps we mentioned the CMS swap.
• Recall that for in a standard plain vanilla interest rate swap there is a fixed leg
and a floating leg and that the floating leg index is LIBOR (e.g. LIBOR 3m, etc.).
• The terminology “floating” simply means that the future index fixings are not known today, in contrast to the fixed rate used for the fixed leg of the swap.
• An alternative floating index is a swap rate of a given tenor (e.g. 10y).
• So, instead of looking at the Bank of England LIBOR fixing page at 11am GMT on every fixing date, I must look on a (pre-agreed) Reuters, Bloomberg, etc. for the mid-market swap rate of the specified tenor, and this is rate that I insert into my floating rate schedule to determine the floating rate cash flow.
• Apart from the fact that the projection of the CMS rate is different, the mechanics of how that rate is used to determine the expected flows is the same.
• The spreadsheet CMS Swap.xlsx included provides the details.
7. CMS Swap
• In the previous lecture we studied the LIBOR in-arrears swap where the LIBOR rate was paid at the beginning of the accrual period (i.e. at the time of fixing), rather than the end of the accrual period.
• This led to the use of a so-called ‘unnatural’ pricing measure leading to the necessity to incorporate a convexity adjustment to the standard projected LIBOR rate, and this convexity adjustment was a function of the LIBOR volatility.
• In the case of the CMS swap we encounter a similar situation, namely that the use of an ‘unnatural’ pricing measure to calculate the expectation of the forward swap rate, leads to a convexity adjustment.
• The derivation of the convexity adjustment is similar to what was presented for the LIBOR in-arrears case in the previous lecture .
• See chapter 30 in Hull’s book on convexity adjustments and also the paper “Convexity Conundrums” by for the result.
8. Example – CMS Swap
General Inputs
Fixed Leg Pmt Details
CMS Leg Pmt Details
Quote Date Result Ccy Spot Date Start Date Maturity End Date Notional
Start Date Maturity
End Date Pay/Rec
Fwd Swap Coupon Coupon Freq Accrual Basis Bus Day Calendar
Start Date Maturity
End Date Pay/Rec Margin
Accrual Basis Bus Day Calendar
Reset Calendar
100,000,000
100,000,000
100,000,000
Market Inputs
Semiannual
Actual/360
Yc ID VcID
USD Yield Curve#0000
30/360 (Bond Basis)
Modified Follow
USDSwaptionVcID#0000
Modified Following
UnitedStates::S
UnitedStates::Settlement
UnitedKingdom
CMS Index Details
Swap Tenor Fixed Freq Fixed Basis Fl
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