label – math intuitive

Q7) An investor is long a USD put / JPY call struck at 110.00 with a notional of USD 100 million. The current spot rate is 95.00. The investor decides to sell the option to a dealer, a US-based bank, on day before maturity. What is the FX delta hedge the dealer must put on against this option?

a) Buy USD 100 million

b) Buy USD 116 million

c) Buy USD 105 million

d) Buy USD 110 million

Analysis: The dealer has the USD-put JPY-call. Suppose the dealer has USD 100M. Let's see if a 1 pip change will give the (desired) $0 effect.

at 95.00 | at 95.01, after the 1 pip change | pnl | |

value (in yen) of the option is same as value of a cash position | (110-95)x 100M = ¥1,500M | (110-95.01) x 100M = ¥1,499M | loss of ¥1M |

value (in yen) of the USD cash | 95 x 100M = ¥9,500M | 95.01 x 100M = ¥9,501M | gain of ¥1M |

value of Portfolio | 0 | ||

Therefore Answer a) seems to work well. |

Next, look at it another way. The dealer has the USD-put JPY-call struck at JPYUSD=0.0090909. Suppose the dealer is short 11,000M yen (same as long USD 115.789M). Let's see if a 1 pip change will give the (desired) $0 effect.

at 95.00 i.e. JPYUSD=0.010526 | at 95.01 i.e. JPYUSD=0.0105252, after the 1 pip change | pnl | |

value (in USD) of the option is same as value of a cash position | (0.010526-0.009090)*11000M = $15.78947M (or ¥1500M, same as table above) | (0.0105252-0.009090)*11000M= $15.77729M (or ¥1498.842M) | loss of $0.012187M |

value (in USD) of the short 11,000M JPY position | -0.010526 * 11000M= -$115.789M | -0.0105252*11000M = -$115.777M | gain of $0.012187M (or ¥1.1578M) |

value of Portfolio | 0 | ||

Therefore Answer b) seems to work well. |

My explanation of the paradox – the deep ITM option on the last day acts like a cash position, but the position size differs depending on your perspective. To make things obvious, suppose the strike is set at 700 (rather than 110).

1) The USD-based dealer sees a (gigantic) ¥70,000M cash position;

2) the JPY-based dealer sees a $100M cash position, but each "super" dollar here is worth not 95 yen, but 700 yen!

Therefore, for deep ITM positions like this, only ONE of the perspectives makes sense – I would pick the bigger notional, since the lower notional needs to "upsized" due to the depth of ITM.

**From:** Brett Zhang

**Sent:** Monday, April 27, 2015 10:54 AM**To:** Bin TAN (Victor)**Subject:** Re: delta hedging - Hw4 Q7

You need to understand which currency you need to hold to hedge..

First note that the option is so deeply in the money it is essentially a forward contract, meaning its delta is very close to -1 (with a minus sign since the option is a put). It may have been tempting to answer a), but USD 100 million would be a proper hedge from a JPY-based viewpoint, not the USD-based viewpoint. (Remember that option and forward payoffs are not linear when seen from the foreign currency viewpoint.)

To understand the USD-based viewpoint we could express the option in terms of JPYUSD rates. The option is a JPY call USD put with JPY notional of JPY 11,000 million. As observed before it is deeply in the money, so delta is close to 1 (positive now since the option is a call). The appropriate delta hedge would be selling JPY 11,000 million. Using the spot rate, this would be buying USD 11,000/95 million = USD 116 million.

On Sat, Apr 25, 2015 at 2:21 AM, Bin TAN (Victor) wrote:

Hi Brett,

Delta hedging means holding a smaller quantity of the underlier, smaller than the notional amount, never higher than the notional.

This question has 4 answers all bigger than notional?!

Victor

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