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# Delta Equivalent Computation

Mauvais ouvrier ne trouve jamais bon outil.

## 22.1 PRINCIPLES

The terms delta equivalent or delta hedging refers to the representation techniques of options in gaps and refers to the hedging techniques used by A/L managers.

The term “delta hedging” means “hedging with derivatives computation” or “first-rate hedging”.

### 22.1.1 Delta equivalent computation in trading activities

These delta-hedging techniques refer to delta hedging practices used by front-office traders in trading books for products accounted as marked-to-market.

Let us show an example to explain how delta-hedging techniques operate in trading activities. We consider a trader selling 10 call options on a company X with these characteristics:

• stock price of company X: 5;
• strike call:5;
• 1-day exercise.

The counterparty who bought the call will receive tomorrow 10 times the maximum between 0 and (S–5) where S is the stock price of company X tomorrow.

We suppose that in one day, there are two possible evolutions for the company stock price:

• State 1: the company stock price is 6 (probability 50%).
• State 2: the company stock price is 4 (probability 50%).

The value of the call in the trading book is the expected pay-off of this call, i.e. 5 since:

• in state 1, the pay-off is 10 (probability 50%);
• in state 2, the pay-off is 0 (probability 50%).

How will the trader hedge this position?

Before hedging, the trader profit and loss (P&L) will be the following:

• State 1:5−10 = −5 (probability ...

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