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PremierSolutions team-Stanley case-final memo
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Transcript of PremierSolutions team-Stanley case-final memo
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.