A Tale of Two Futures:
$ versus ¥ Nikkei 225 Index
Futures
Christopher Ting
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Learning Objectives
• Define quanto
• Understand inter-market spread trading strategy
• Analyze the P&L of a short quanto position
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Quanto
• Quantos are derivatives where the payoff
is defined using variables measured in
one currency but paid in another
currency
• Example: futures contract providing a
payoff of NT – K dollars ($) to the
counterparty holding the long position.
– Here, NT is the Nikkei 225 index value at
maturity T and K is the futures price
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Nikkei 225 Futures in USD and JPY
• Contract Multiplier USD 5 for NKD; JPY 500
for NIY
• Minimum Price Change (Tick) 5 index points
• Final Settlement: Cash-settled to Special
Opening Quotation of the Nikkei 225 Index on
2nd Friday of the contract expiry month
• Last Trading Day 3:15 p.m. Central Time on
the day preceding final settlement – usually the
Thursday prior to 2nd Friday of the contract
expiry month
• Contract Months: Quarterlies for NKD;
Quarterlies and Serials for NIY
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Trading Hours (before 2011)
• At 3:30 p.m. Singapore Time, T+1
session for Nikkei index futures opens
– Simex: (multiplier 5, tick size 5 index
points)
– Osaka: Big (multiplier 5, tick size 10 index
points) and Mini (multiplier 1, tick size 5
index points)
• At 4 p.m. Singapore Time, NKD futures
market opens
• At 7 p.m. Singapore Time, NIY futures
market opens
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NKD: Dollar-Denominated Futures
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NIY: Yen-Denominated Futures
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Arbitrage Opportunity?
At 14:02, NKD @ 10,800
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Arbitrage Opportunity?
At 14:02, NIY @ 10,735
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Motivating Questions
• Why was the market price of NKD 65 points
higher than that of NIY on Jan 5?
• Risk-free arbitrage opportunity?
– Short NKD and long NIY?
• The exchange rate on Jan 5, 2010 at 14:00
Central Time
– Cash Market: ¥91.71 per $1
– Futures Market: front quarter JPY/USD futures (6J)
price was 109,040, which was equivalent to ¥91.71
per dollar.
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Follow-up Question
• What should the futures price of NKD
be relative to the futures price of NIY?
• What should be the spread between
these two futures prices?
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NKD – NIY Spread
• At time t=0, let N0 be the cash Nikkei
index value, and the futures prices F$
and F¥ are
F¥ = N0 (1+ r T )
F$ = N0 (1+ (r + ns)T)
= F¥ + N0  nsT
• So the fair-value spread is
F$ – F¥ = N0  ns T
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Behavior of the NKD – NIY Spread
• When cash market N goes up, dollar tends to
strengthen (S increases)
• In other words, when dollar strengthens (S
increases), cash market N tends to go up.
– Why?
– Dollar strengthening means Yen depreciating, which
will be helpful to export-oriented companies in Nikkei
225 index N, so N tends to go up.
• Thus the correlation between the (percentage)
change in N and the (percentage) change in S is
positive.
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Illustration
• Suppose the correlation is 30%, the volatility of
Nikkei 225 index return is 50%, and the
volatility of the yen-per-dollar exchange rate is
15%. The index level is at 10,680 and the time
to maturity is 3 months.
• The spread is about 60 index points:
10,680  0.3  0.5  0.15  3/12 = 60.1
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Money-Making Opportunity?
• At maturity, T = 0, the NKD – NIY spread is
zero. This is the time decay effect.
• Since the NKD – NIY spread is positive, one can
take a short position in this spread (i.e. sell NKD
and buy NIY), and hold this spread position
until maturity to benefit from the time decay.
• Is it a good money-making opportunity?
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Profit and Loss
• At time t=0, sell short one NKD contract at a
price of F$, and buy R number of NIY contracts
at a price of F¥.
• At maturity T, ST is the spot yen per dollar
exchange rate
• Let NT be the settlement price of the futures
contract, which is based on the SOQ of cash
Nikkei 225 index value. The position’s payoff at
maturity is, in dollars
–5  (NT – F$) + R  500  (NT – F¥) / ST
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Profit and Loss (cont)
• Suppose the ratio R is chosen to be
S0
R=
100
• Then the payoff is
–5  (NT – F$) + 5  S0  (NT – F¥) / ST
which is

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 S0

5  1 NT  F¥  5( F$  F¥)
 ST



 S0 
P & L  5  1NT  F ¥  5 N 0  n sT
 ST 
P&L Example: Normal
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• Same parameters as in the illustration, the spread
is 60 points. Thus, gain from time decay is
$5  60 = $300.
• Suppose S0 = 90 yens per dollar.
• So the ratio R is short NKD contracts and long 9
NIY contracts.
• Suppose ST is 87 yens per dollar, i.e., dollar
weakens, and the settlement is 800 points lower,
i.e., NT – F¥ = –800 at maturity.
• Then 5  (90 – 87)/87 = 15/87, and the P&L per
NKD contract is
–$15  800/87 + $300 = $116.09
P&L Example: Market Crashes
• Suppose the market crashes, and ST is 85
yens per dollar, i.e., dollar weakens
substantially, and the settlement is 2,000
points lower, i.e., NT – F¥ = –2,000.
• Then 5 (90 – 85)/85 = 25/85, and the P&L
at maturity is, for every NKD contract,
–$25  2,000/85 + $300 = –$288.24
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P&L Example: Market Rallies
• Suppose the market rallies, and ST is 95
yens per dollar, i.e., dollar strengthens,
and the settlement is 2,000 points higher,
i.e., NT – F¥ = 2,000.
• Then 5 (90 – 95)/95 = –25/95, and the
P&L at maturity per NKD contract is
–$25  2,000/95 + $300 = –$226.32
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Bottom Line
• When market is quiet, i,e., the markets
neither crash nor rally, short quanto
position will make money
• But it will lose money if extreme
conditions (either up or down) prevail.
– Don’t be the next Nick Leeson!
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