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**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, 10,800

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**Arbitrage Opportunity?**

At 14:02, 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 So the fair-value spread is F¥ = N0 (1+ r T ) F$ = N0 (1+ (r + ns)T) = F¥ + N0 nsT F$ – F¥ = N0 ns 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**

Then the payoff is which is –5 (NT – F$) + 5 S0 (NT – F¥) / ST

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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 = $116.09

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**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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