A Tale of Two Futures:$ versus ¥ Nikkei 225

Index Futures

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Christopher Ting

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 price3

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?

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At 14:02, NKD @ 10,800

Arbitrage Opportunity?

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At 14:02, NIY @ 10,735

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

• So the fair-value spread is12

F¥ = N0 (1+ r T )F$ = N0 (1+ (r + ns)T) = F¥ + N0 nsT

F$ – F¥ = N0 ns T

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

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–5 (NT – F$) + R 500 (NT – F¥) / ST

Profit and Loss (cont)• Suppose the ratio R is chosen to

be

• Then the payoff is

which is

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100= 0SR

–5 (NT – F$) + 5 S0 (NT – F¥) / ST

)(515 $0 FFFN

SS

TT

¥¥

TNFNSS

snTT

00 515L&P

¥

P&L Example: Normal• 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 = $116.0918

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 = –$288.2419

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 = –$226.3220

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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