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## Tuesday, April 28, 2015

### paradox - FX homework (thanks to Brett Zhang

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?

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