# 22

# 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 ...