PremierSolutions LLP

Buffalo Options

Jiang Tian

Liu Haobing

Luong Huy Quang

Ray Raktim

Wang Xiao

Buffalo Options

Table of Contents

1. Executive Summary ........................................................................................... 3

2. Description of the Deal ...................................................................................... 4

2.1 Bull Call Spread Option .................................................................................... 4

2.2 Yield Curve Spread Option............................................................................... 5

3. Valuation and Methodology .............................................................................. 9

3.1 Valuation Model ................................................................................................. 9

3.2 Calibration for the interest rate parameters ................................................... 9

3.3 Simulation Result ............................................................................................. 10

4. Risk Assessment ............................................................................................... 12

Buffalo Options

1. Executive Summary

PremierSolutions LLP, the financial solutions provider, has been approached by

Stanley Investments LLC to incorporate their views about the nature of the Federal Reserves’

tapering program into a tradable product. Stanley Investments is a hedge fund, headquartered

in South San Francisco, primarily trading on macroeconomic forecasts with AUM of USD

2.5 billion.

This memo serves to explain the deal to be brokered with the aforementioned client

and request for the approval of the senior management.

Ms. Kim, a senior portfolio manager at Stanley Investments, holds a strong view

about the nature and timing of the Fed’s tapering strategy. It is her belief that the short term

interest rates in a years’ time will be within a band somewhat higher than current spot rates.

Additionally, she believes that the slope of the term structure will flatten in one year.

PremierSolutions has been entrusted to structure and design exotic options to trade on the

views held by the client.

The interest rate structuring group under Brad Kaiser at the firm has the following

exotic options as a solution to the clients’ needs:

I) Bull Call Spread on Interest Rate

II) Yield Curve Spread Option

PremierSolutions strongly believes that trading these options are commensurate with

the beliefs held by Stanley Investments. This deal provides the opportunity for the firm to

establish strong business ties with Stanley Investments and also gain a foothold in the

lucrative West-Coast Asset Management business. A detailed report on costs and option

choices are provided in the report. The management is requested to analyse the option

suggestion and provide approval for taking the deal forward.

Buffalo Options

2. Description of the Deal

2.1 Bull Call Spread Option

Ms. Kim expects short-term interest rates in a year to be within a band somewhat

higher than current spot rates. Given this view, we recommend client to purchase a bull call

spread option that consists of buying a call option in the 3-month spot rate and

simultaneously selling another call on the same underlying and maturity but at higher strike.

With this option, client will make profit if the interest rate increase in the future but not

exceeding certain level. The price of this product also lower than purchasing a call alone.

Bull call spread on 3-month spot rate

Description Buy call at strike K1 and short call at strike K2

Strike K1 0.032

Strike K2 0.043

Maturity 1 year

Notional $100M

Market Price $524,400

Price to Stanley $577,000

Profit for PremierSolutions $52,600

Payoff graph:

The maximum payoff is at $1,100,000 and the minimum payoff is $0

-

2,00,000

4,00,000

6,00,000

8,00,000

10,00,000

12,00,000

2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8

Pay

off

($

)

Spot rate (%)

Option Payoff

Buffalo Options

To price this product, we use simulation based on two-factor HJM interest rate model

to estimate the spot rate in one year as well as discount factor for one year. We then vary the

value of strike K1 and K2 to derive the price table for Bull Call spread option.

P = D(0,1) * N * (Max(RT - K1, 0) – Max(RT – K2, 0))

Where:

- D(0,1) : 1-year discount factor

- N : Notional Amount

- RT : Estimated spot rate 1 year from now

- K1: Lower strike

- K2 : Higher strike

Bull Call Spread Price for different Strikes

Buy Strike\ Sell Strike

0.042 0.0425 0.043 0.0435 0.044 0.0445

0.031 546737.29 565546.28 584355.26 603164.25 620802.89 638436.32

0.0315 516172.69 534981.67 553790.66 572599.65 590238.29 607871.71

0.032 486781.52 505590.50 524399.49 543208.48 560847.12 578480.54

0.0325 457392.48 476201.46 495010.45 513819.44 531458.08 549091.50

0.033 428003.43 446812.42 465621.41 484430.40 502069.04 519702.46

0.0335 546737.29 565546.28 584355.26 603164.25 620802.90 638436.32

2.2 Yield Curve Spread Option

Ms. Kim’s second view is that one year from now, the current 1-year minus 3-month

zero-yield spread (44 bps) to shrink by half (22 bps). To trade on this view, client can

purchase a put option on the yield curve spread between 1-year and 3-month rate or she can

also sell a call on the same underlying. The later turn out to be better option since Ms. Kim

can obtain option premium upfront and use that to offset the cost associate with the bull

spread mention above.

Buffalo Options

Yield Curve Spread Option

Description Sell a call option on underlying

Underlying Spread between 1-year and 3-month zero yields

Maturity 1 year

Strike 22bps

Notional $100M

Market Premium $171,840

Premium passed to Stanley $155,000

Profit for PremierSolutions $16,840

Payoff graph:

However selling or writing a call option bear unlimited downside risk in case the

underlying value increase instead of decrease as expected by client. To cater for this, we also

provide alternative which is Yield curve spread option with downside protection by

purchasing a call on the same underlying with higher strike. The premium obtained for client

in this case will be lower but the downside loss will be contained.

(4,00,000)

(3,50,000)

(3,00,000)

(2,50,000)

(2,00,000)

(1,50,000)

(1,00,000)

(50,000)

-

0 5 10 15 20 25 30 35 40 45 50 55 60

Pay

off

($

)

1-yr rate minus 3-mth rate (bps)

Option Payoff

Buffalo Options

Yield Curve Spread Option with downside protection

Description Sell a call at strike K1 and Buy a call at higher strike K2

Underlying Spread between 1-year and 3-month zero yields

Maturity 1 year

Strike K1 22 bps

Strike K2 44 bps

Notional $100M

Market Premium $155,990

Premium passed to Stanley $140,000

Profit for PremierSolutions $15,990

Payoff graph:

The maximum payoff is at $0 and the minimum payoff is -$220,000

To price this product, we use simulation based on two-factor HJM interest rate model

to estimate the 1 year minus 3 month zero-yield spread. By choose different strike K1 and

K2, we derive the price table for this yield spread option.

P = D(0,1) * N * (Max(ST – K2, 0) – Max(ST – K1, 0))

Where:

- D(0,1) : 1-year discount factor

- N : Notional Amount

- ST : 1 Year minus 3 month zero yield spread

- K1: Lower strike

- K2 : Higher strike

(2,50,000)

(2,00,000)

(1,50,000)

(1,00,000)

(50,000)

-

0 5 10 15 20 25 30 35 40 45 50 55 60

Pay

off

($

)

1-yr rate minus 3-mth rate (bps)

Option Payoff

Buffalo Options

To price this product, we use simulation based on two-factor HJM interest rate model to

estimate the 1 year minus 3 month zero-yield spread. By choose different strike K1 and K2,

we derive the price table for this yield spread option.

P = D(0,1) * N * (Max(ST – K2, 0) – Max(ST – K1, 0))

Where:

- D(0,1) : 1-year discount factor

- N : Notional Amount

- ST : 1 Year minus 3 month zero yield spread

- K1: Lower strike

- K2 : Higher strike

Yield Spread Price for different Strikes

Buy Strike \ Sell Strike

0.0041 0.0042 0.0043 0.0044 0.0045 0.0046

0.0019 -174754.23 -178727.16 -182127.10 -184801.46 -187145.37 -189290.76

0.0020 -165114.62 -169087.55 -172487.50 -175161.85 -177505.77 -179651.15

0.0021 -155486.51 -159459.45 -162859.39 -165533.75 -167877.66 -170023.05

0.0022 -145943.31 -149916.24 -153316.18 -155990.54 -158334.45 -160479.83

0.0023 -136400.10 -140373.03 -143772.97 -146447.33 -148791.24 -150936.62

0.0024 -126934.85 -130907.78 -134307.72 -136982.08 -139325.99 -141471.38

The combined package will have the profit given below:

Derivative Structure for Stanley Investment LLC

Profit from Bull Call Spread Option $

52,600

Profit from Yield Curve Spread Option $

15,990

Total Profit $

68,590

Buffalo Options

3. Valuation and Methodology

3.1 Valuation Model

The interest rate curve is modelled by the two-factor HJM interest rate model for US

treasure rate term structure.

𝑑𝐹𝑡𝑇 = m(t, T)dt + 𝑆1(𝑡, 𝑇)𝑑𝑧1𝑡 + 𝑆2(𝑡, 𝑇)𝑑𝑧2𝑡

m(t, T) = ∑ 𝑆𝑖（𝑡, 𝑇）∫ 𝑆𝑖(𝑡, 𝑢)𝑑𝑢𝑇

𝑡

2

𝑖=1

𝑆1(𝑡, 𝑇) = 𝑎(𝑡) ∗ ∅

𝑆2(𝑡, 𝑇) = 𝑎(𝑡) ∗ (exp[(−2) ∗ (T − t − 0.25)] − 0.5)

3.2 Calibration for the interest rate parameters

a) Compute the zero coupon bond price. ∆t=0.25 year.

P(0, T + ∆t) =𝑃(0, 𝑇)

1 + 𝑓(0, 𝑇, 𝑇 + ∆t)

b) Compute the caplets market price using the Black ISDs

Market Price = P (0, T + ∆t) ∗ [∆t ∗ f(0, T, T + ∆t) ∗ N(𝑑1) − ∆t ∗ f(0, T, T + ∆t) ∗

N(𝑑2)]

𝑑1 =ln (

∆t ∗ f (0, T, T + ∆t)∆t ∗ f(0, T, T + ∆t)

) + 𝜎𝑖𝑚𝑝2 ∗ 𝑇/2

𝜎𝑖𝑚𝑝 ∗ √𝑇

𝑑2 = 𝑑1 − 𝜎𝑖𝑚𝑝 ∗ √𝑇

c) Calculate the current instantaneous forward rate

f(0, T) =lnP(0, T) − lnP(0, T + ∆t)

∆t

Buffalo Options

d) Simulate the price of the Caplet

f(∆t, T) = f(0, T) + m(0, T) ∗ ∆t + 𝑆1(0, 𝑇) ∗ 𝑍1𝑡+𝑆2(0, 𝑇) ∗ 𝑍2𝑡

Use the above function, we can simulate the instantaneous forward rate starting at

different time period and then find the instantaneous forward rate of f (T, T) at different

T. Here we will simulate 100 path. Next, we can price a series of Caplet.

P = P(0, T) ∗ max (f (T, T) − K, 0)

e) Get the interest rate parameters

Use the solver in Excel to let the simulated caplet price equals to the market

caplet price with the restriction of making the restriction of make the correlation between

1-year and 3-month US Treasury rates 0.81, we can get the interest rate parameters.

Parameters estimation

φ 𝒂𝟎 𝒂𝟏

7.1310 0.0051 0.0104

𝑎0 apply when T=0.25,0.5,0.75,1

𝑎1 apply when T=1.25,1.5,1.75

3.3 Simulation Result

After calibrating a(t) andφ, we can simulate the instantaneous forward rate.

The future 1-year rates are f (t,t+1) and 3-month rates are f(t,t+0.25).The 1-year/3-month

spread is calculated by take subtracting of corresponding two rates. The simulation result

below shows that both rates will increase but the 3-months rates will increase slightly

more and thus the spread will drop as t increases.

Buffalo Options

For the term structure of volatility for the 1-year and 3-month US zero rates and

spread. We consider the yield rates which are corresponding to f(t,t+1) and f(t,t+0.25)

respectively. Also, the 1-year/3-month yield spread is simply their difference. We

calculate the standard deviation of each target yield using the 100 path sample and draw

the term structure graph below. The result shows that the 1- year rate volatility increases a

lot while 3-month rate increase slightly. As a result, the volatility of the 1-year/3-month

yield spread will go up as t increases.

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.25 0.5 0.75 1

Mean of the forward rates and Spread

1-year rate 3-month rate 1-year/3-month spread

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.25 0.5 0.75 1

Term Structure Volatility for the US zero rate the Spread

3-month 1-year 1-year/3-month spread volatility

Buffalo Options

4. Risk Assessment

As Ms. Kim’s views regarding on interest rate multi-dimensional, the risk borne

within selling to client is consisted of two portions, interest rate risk and yield spread risk.

The following tables highlight the potential risk borne within selling the structure to client

‘Stanley Investment LLC’ and our approaches to hedging the corresponding risk.

Interest Rate Risk

No. Source of Risk

Description of Risk Risk Management Strategy Risk Assessment

1 Interest risk

Interest rate is changing bi-directionally. From client’s view short-term interest rate is moving up. If interest rate moves up, Premier Solution as counterparty will lose.

To reduce the exposure by entering a similar option with another counterparty in the market.

Low

2 Yield spread risk

The spread between 1 year and 3 month zero-yield spread may spread may shrink by half, then PremierSolution will loss on the price paid to Stanley on yield spread option.

To reduce the exposure by entering similar options with another counterparty in the market.

Low

Buffalo Options

Credit Risks:

No. Source of Risk

Description of Risk Risk Management Strategy Risk Assessment

1 Stanley Investment LLC

Possibility of Stanley defaulting on their liabilities to PremierSolution.

Stanley Investment has asset under management of USD 2.5 billion. Its fund’s strategy revolves around trading on macroeconomic forecasts expressed through taking positions in stocks, bonds, and exchange-trade and exotic derivatives on market variable. Some collateral will be post to reduce the counter party risk.

Low

With regard to the two alternatives involving the Yield Spread Option, the first one

with the sale of a single call option on a spread strike involves considerable downside risk if

the clients’ outlook about the future does not materialise. There is a possibility of the interest

rates increasing without much change in the shape of the term structure. An even riskier

outcome from the clients’ perspective is if the yield spread increases. The second alternative

is proposed to limit the possible unlimited downside loss. Although this would involve

foregoing upfront premium to buy the higher strike, it will limit the downside risk should

expectations fail to materialise.