# Introduction

Delta:

- is the first-order sensitivity of the price to a movement in S.

- gives the equity sensitivity of the option.

- is normally quoted in percent --> it says how much of % of the actual stock is required to hedge the option.

- is closely related to the probability of finishing ITM (small difference though!)

For calls, it lies between:

- 0% --> no equity sensitivity --> far OTM

- 100% --> trades like a stock --> far ITM

For puts, it lies between:

- 0% --> no equity sensitivity --> far OTM

- -100% --> trades like short stock --> far ITM

If a call option has a delta of 50% and the underlying rises €1, the call option increases in value €0.50 (= €1 * 50%).

Let's go back to our Taylor Series.

....

The first derivative w.r.t. S on the right-hand side is the first-order sensitivity of the price to a movement in S, the Delta.

If x is small, meaning there is only a small movement in S, then the price of the derivative will move by Delta times x.

A Delta of 0.5635 means that if underlying moves by 1%, then the value of the derivative will move by 0.5635 × 1%.

Linearity of addition of the deltas --> delta of a book of options = sum of individual deltas.

--> Each Delta is a mathematical derivative and the derivative is linear.

To delta hedge, one can use:

- the underlying itself

- forwards or futures

- another correlated asset

Delta Hedging using Forwards/Futures:

Other than trading the underlying itself to Delta hedge, it is also possible to use forwards or futures.

The value of the futures/forward contract at time t (assuming no dividends) with maturity T is given by F(t) = S(t) er(T−t).

So the Delta of the futures contract is given by er(T−t).

Delta Hedging using another correlated asset:

One can further exploit correlations between assets to Delta hedge.

Specifically, if an option is written on an asset with price S1, then it is possible to use a second asset S2 to Delta hedge.

Because of the chain rule: ....

Under BS assumptions, the Deltas for call and put options are given by:

Call delta: N(d1)
Put delta: −N(−d1) = N(d1) − 1

The Delta of a European option is therefore sensitive to:

- time to expiry

- volatility of underlying

- moneyness

Impact of Time to Maturity

At maturity, delta has a digital shape around the strike.

Once we move ourselves away from maturity, the delta becomes much smoother shaped.

The further we are from maturity, the flatter the curve looks.

The delta converges faster and faster to its intrinsic value as time to maturity goes to zero.

The option picks its direction. That is true, except for ATM option, where the delta remains flatter.

Main variation from the delta is always located in the ATM range.

The closer we are to maturity, the tighter the range is where the delta changes from small value to full value.

Impact of time on Delta

Owing to uncertainty involved and costs in buying/selling underlying --> willingness to keep Delta hedging to a minimum.

Traditionally Delta hedges are rebalanced on a daily basis.

One should adjust the effect of time on Delta for holidays and weekends.

Even if the underlying does not move, time will have elapsed --> impact on Delta --> especially if little time left to expiry.

Charm = effect of time on Delta.

Impact of Volatility on Delta: Delta Attacked by Volatility

The trader wants to know and understand how the delta will change when volatility changes.

This will allow the trader to anticipate the hedge adjustments and take a position accordingly.

Higher volatility:

- higher delta for OTM options

- lower delta for ITM options

So for more volatile stocks, the delta is less pronounced.

The more volatility, the less ITM this ITM option really is.

The more volatility, the less OTM this OTM option really is.

Liquidity:

Liquidity is also a concern. If the stock is illiquid and hard to trade, one must make adjustments.

In some cases, it might be difficult to short stocks and borrow costs (repos) will need to be factored into the price.

Dividends:

A desk selling exotic products will be structurally long underlyings from having to buy Delta in these assets.

Long underlying --> long dividends.

Dividends are a necessary input to obtain a correct price and hedge, but they are uncertain.

Expectations regarding dividends:

- can be factored into the price in the form of a term structure of dividend yields.

- can be priced at current levels and hedged using a dividend swap.

On a book level, large exposures to dividend fluctuations will need to be hedged.

Interest Rates:

If IR goes up --> it costs more to borrow money --> hedging process is more expensive.

There are different ways of thinking about delta:

• The rate of change of an option value relative to a change in the underlying
• The derivative of the graph of an option value in relation to the stock price
• The equivalent of the underlying shares represented by an option position (hedge ratio)
• The estimate of the likelihood of an option expiring ITM

The delta for the call and put have the same shape. This is a consequence of the call-put parity.

Both deltas are increasing with the stock value.