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- Volatility Derivatives 1
- The world of Structured Products 4
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- Table of Contents
- Vanilla Options
- Volatility, Skew and Term Stru
- Option Sensitivies: Greeks
- Option Strategies
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- Option Sensitivies: Greeks

# Option Sensitivies: Greeks

__5. Option Sensitivities : Greeks__

The buying or selling of a derivative creates a position with various sources of risk, some of which may be unwanted risk. When a bank sells a derivative to a client, it should understand all the risks associated with the product and hedge its position accordingly. Once a sale is done, the product is added to an existing book of options, and it is the book that must be risk managed. In order to see where the risks lie, the trader hedging a derivative will need to know the sensitivity of the derivative’s price to the various parameters that impact its value. The sensitivities of an option’s price are commonly referred to as **the Greeks**.

To obtain the price of an exotic derivative, one is more often than not forced to use some model. It is imperative that one uses the correct model to price, but since the sensitivities of the price are also computed using a model, we discuss the implications of various models on the option price’s sensitivity. One must understand the implications of any model, and the impact of the model’s assumptions on the hedge ratios it generates; **if the model is wrong, then the hedge ratios computed using this model will typically also be wrong**.

In some cases, although uncommon, an exotic product may have all its cashflows aligned with those of derivatives that can be traded liquidly in the market and can thus be hedged by taking the opposite position in such derivatives at the onset of the contract. This is known as a **static hedge** because once this hedge is put into place, as an initial hedge, there is no need for further hedging irrespective of how the market moves. In this case, the hedge is **model independent**: **the cost/price of the derivative is the cost of its hedge** and we do not need any model, just market prices. The existence of a static hedge can thus provide us with both a price and a hedge.

In the general case of exotic structures, such a hedge is not possible and a **dynamic hedge** must be put in use. Once such a hedge is in place, it is sensitive to movements in the market and must be modified to still be a hedge. How often hedges should be adjusted is dependent on the nature of the sensitivity and its impact on the price. The day 1 hedge may consist of a static part that needs no further adjustment, and a dynamic part that will need adjustment through the life of the product.

**5.1. Delta**

**Delta** is the sensitivity of an option to the price of its underlying asset. To understand the concept of sensitivity we must first mention the **Taylor series**. This simply gives us the various orders of sensitivities. If we consider the price of an option as a function of the underlying’s price S and ask how much is the rate of change of this price if the underlying moves by an amount x, the answer is given by the 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 the underlying moves by an amount, say 1%, then the value of the derivative will move by 0.5635 × 1%.

In a book consisting of many options, some of the Deltas of the various options may cancel each other. The** linearity of addition of the Deltas **within a portfolio follows from the fact that each Delta is essentially **a mathematical derivative and the derivative is linear**. Consider a portfolio P consisting of n options: O_{1} , O_{2} , ..., O_{n} all written on the same underlying asset whose price we denote as S, then

Other than trading the underlying itself to Delta hedge, it is also possible to use **forwards or futures**. Recall that the value of the futures/forward contract at time t (assuming no dividends) with maturity T is given by F(t) = S(t) e^{r(T−t)}. So if the price of the underlying changes by x, the futures price changes by x*e^{r(T−t)}, that is, the **Delta of the futures contract is given by e ^{r(T−t)}**.

One can further exploit correlations between assets to Delta hedge. Specifically, if an option is written on an asset with price S_{1}, then it is possible to **use a second asset S _{2} to Delta hedge**. Because of the

**:**

__chain rule__If we use BS assumptions, specifically the log-normality of the asset prices, and denote ρ_{1,2 }to be the correlation between S_{1} and S_{2} then:

and accordingly we know the value of 2 in the previous equation.

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

The Delta of a European option is sensitive to the **time to expiry**, the **volatility** of the underlying asset, and the difference between the strike and spot prices (**moneyness**).

__To discuss__

Owing to the uncertainty involved in Delta hedging, and the **costs involved in buying/selling** the underlying asset, one would want 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**, because even if the underlying does not move, time will have elapsed and this has an impact on Delta, especially in the cases where there is little time left to expiry and the underlying is still close to the strike. The effect of time on Delta is known as ** Charm**.

Liquidity is also a concern. If the stock is illiquid and hard to trade, one must make adjustments. **In some cases it is difficult to short stocks, which may be necessary to Delta hedge, and borrow costs (repos) will need to be factored into the price.**

Other parameters impacting Delta hedging are dividends and interest rates. A desk selling exotic products will typically be structurally long the underlying assets from having to buy Delta in these assets. When **long Delta** in an underlying, the trader will be **long the 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, or priced at current levels and hedged using a dividend swap**. On a book level, large exposures to dividend fluctuations will need to be hedged.

Regarding interest rates, we note that when a trader needs to **buy Delta** of stock, the trader will have to **borrow money** in order to buy whatever units of stock are needed. **If rates go up, then it costs more to borrow money thus making the hedging process 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

__Exercise __

Imagine you are in charge of Delta hedging a portfolio of options. Assume short skew and volatility goes down. Would you end up buying or selling underlying shares?

The skew increases the price of OTM puts and ITM calls; and decreases the price of OTM calls and ITM puts. Being short the skew can mean being short OTM puts, short ITM calls, long OTM calls or long ITM puts. Let’s consider the case where you are long OTM call options. If volatility goes down, the Delta of OTM calls goes down (the probability of expiring ITM decreases). Since you are long the options, the portfolio overall Delta is then negative. Therefore, you have to buy shares to maintain zero sensitivity to the spot price of underlying shares.

**5.2. Gamma**

Gamma represents the **second-order** sensitivity of the option to a movement in the underlying asset’s price.

The price of an option as a function of the underlying price is **non-linear**. Gamma allows for a second-order correction to Delta to account for this **convexity**. This convexity in the underlying price is **what gives the option value**, and in order **to see the second-order effect in pricing we will always use models that assume** **some form of randomness in the asset’s price**.

The BS Gamma **for both calls and puts** is given by

The dollar or cash Gamma is given by Gamma times S^{2}.

The ** effect of volatility on Gamma**:

**a higher volatility lowers the Gamma of the call option when the underlying is near the strike, but raises it when the underlying moves away from the strike**. We can think of this effect in terms of the time value of European options. For low levels of volatility, the Gamma is low for deep ITM and OTM options because, for low levels of volatility, these options have little time value and can only gain time value if the underlying moves closer to the strike. On the other hand, a high volatility means that both ITM and OTM options have time value and so the Gamma sensitivity near the strike should not be too different from the Gamma away from the strike.

Gamma tells us how much Delta will move if the underlying moves. The Gamma of a European option is **high when the underlying trades near the strike and there is little time left to maturity**. Near these points, there will be the need for more frequent Delta hedging and thus inflict more hedging costs upon the trader.

The concept of a Delta-hedged portfolio of options means that the portfolio has been hedged by trading in the underlying assets against small movements in these assets. As a second-order effect, Gamma becomes increasingly significant when a large move in the underlying’s price occurs and the Delta moves with according significance. **To hedge this Gamma one will need to trade other European options** in a manner that the Gammas cancel out and yield a lower overall Gamma. Gamma represents the convexity (non-linear) of the option price, and **to remove (some of) this convexity one must use another convex instrument**. By lowering Gamma, we lower the need for the large and frequent rebalancing of Delta.

Like Delta, the **Gamma of a portfolio is the sum of the individual Gammas of the options in the portfolio**. As we move into exotic structures, we find that these may have quite different Gamma profiles to the European options seen here. Their gamma can change sign.

Gamma is important as it enables the traders to derive the profit on an option for any given stock move.

**Positive gamma** means that one needs to sell stocks if S goes up and buy stocks if S goes down to be delta hedged.

**Negative gamma** means that one needs to sell stocks if S goes down and buy stocks if S goes up to be delta hedged.

However, **gamma does not come for free**. The way an option holder pays for the right of buying low and selling high is **by means of the theta**, the time decay of an option.

In other words, the holder of an option needs to earn back the daily loss in value of the option by taking advantage of the moves of the underlying.

Obviously, the reverse applies to the seller of an option who makes money on the theta and loses it by rebalancing the delta by buying high and selling low.

**5.3. Vega**

Vega is the sensitivity of the option price to a movement in the volatility of the underlying asset.

Under Black–Scholes the Vega **of both calls and puts** is given by

where

Vega is greatest when ATM and decays exponentially on both sides thus giving the ** bell-shaped curve**. This makes sense intuitively because if we are ATM then a change in the volatility of the underlying asset

**can send the option either ITM or OTM thus the large effect on the price**.

For European options the Vega position is simple. Both calls and puts have positive Vega. **For more exotic structures, the Vega profile, like the Gamma profile, can change sign**, and whether we are short or long volatility depends on the underlying’s price.

A book of exotics, or even a single exotic product, can have different sensitivities to the various implied volatilities along the term structure, and these are referred to as ** Vega buckets**, each corresponding to the volatility sensitivity of a particular maturity on the term structure of implied volatilities.

The **ATM option is almost linear in volatility**. The prices of the **ITM and OTM options are convex in volatility up to a certain level** then become linear for large volatilities.

The ATM and ITM options are not worth zero when volatility is zero. In the case of the ATM, even if volatility were zero, the forward is not null and the call option still has a value of approximately r × T.

**5.4. Theta**

The Theta of an option is the rate at which the option price varies over time. Time is always moving forward and so even if all else remained the same, the option’s value will change as time goes by. An option that loses 0.05% per day is said to have a Theta of −0.05%. If we **buy a call option (or a put) we will have a negative Theta**, and vice versa.

Under BS, the Theta of a call option is given by

and for put options, Theta is

The Theta of the call option is always non-positive. As we can see, the options that are **close to the money near maturity will exhibit the most time decay**.

**5.5. Rho**

Rho is the Greek letter used to represent the sensitivity of an option’s price to a movement in interest rates. In the BS model, the Rho of a call option is given by

and the Rho of a put option is the negative of this. The **prices of call and put options are almost perfectly linear in interest rates**; the reason for this is that a change in rates

**only has a first-order effect**on the price of the option. This effect comes from the impact of an increase in rates on

**the**and also from

__cost of Delta hedging__**the option price. This last effect is generally smaller than the effect of rates on our Delta hedge. In the case of the call, the two effects work in opposite direction. In the case of the put, they both work in the same direction.**

__discounting__**A call option is delta hedged by selling delta shares. By selling short those shares, the trader receives money, which he can put in a saving account and earn interest on it.** Therefore the higher the IR, the higher the call price. The discounting effect slightly offsets this delta hedge impact though.

Similar arguments can be made in regards to the **price sensitivity to the dividend yield** of the underlying asset. If we sell a call option, we need to buy Delta of the asset. If we hold the asset we are long the dividends paid by this asset. If dividends are higher, it means that we make more money on our Delta hedge and thus **the cost of hedging is less and the option premium will be less**.

--> **If you make more money on your delta hedge, you are going to pay more (receive less) for it.**

**5.6. Relationships between the greeks**

Consider the equation, known as the **Black–Scholes PDE**, derived using the assumptions of the theory,

This relationship shows the trade-off between movements in the underlying asset (Delta and Gamma) and the time decay (Theta) of a European option. Here V represents the value of the call or put.

**5.7. Volga**

Vega–Gamma, or Volga, is the second-order sensitivity of the option price to a movement in the implied volatility of the underlying asset. When an option has such a second-order sensitivity we say it is **convex in volatility**, or has **Vega convexity**. ITM and OTM European options do exhibit Vega convexity but these can be captured in the skew.

Other structures exhibit a lot of Vega convexity and will result in losses if we do not use a model that prices this correctly. The reason is that as volatility moves, a **Vega convex payoff will have a Vega that now moves with the volatility** and this must be firstly priced correctly and then hedged accordingly.

**5.8. Vanna**

Vanna measures the sensitivity of the option price to a movement in both the underlying asset’s price and its volatility. We can thus think about Vanna as the sensitivity of the option’s Vega to a movement in the underlying’s price, also as the sensitivity of an option’s Delta to a movement in the volatility of the underlying. As such, Vanna gives important information regarding a Delta hedge by telling us by **how much this delta hedge will move if volatility changes**. It also tells us **how much Vega will change if the underlying moves** and can thus be important for a trader who is Delta or Vega hedging. If Vanna is large, then the Delta hedge is very sensitive to a movement in volatility.

**5.9. Multi-Asset Sensitivities**

**5.9.1. Cross Gamma **

The cross Gamma is the sensitivity of a multi-asset option to a movement in two of the underlying assets. The cross Gamma involving S_{i} and S_{j }is given by

It is the **effect of a movement in S _{i} on the Delta sensitivity of the option to S_{j}**. In multi-asset options, it is possible that the Delta w.r.t. one asset can be affected by a movement in another underlying asset even if the first asset has not moved.

**5.9.2. Correlation Delta = Cega**

The correlation Delta is the **first-order sensitivity of the price of a multi-asset option to a move in the correlations between the underlyings**. Correlations vary over time. A multi-asset product’s sensitivity to the correlation between a pair of underlying assets can vary as the other parameters change.

While correlation is not easily tradeable, there are some methods of trading correlation. Many correlation risks are not completely hedgeable, if at all, and in many cases traders must resort to maintaining dynamic margins for the unhedged correlation risk. Knowing the sign and magnitude of correlation sensitivity is again necessary in this case.

Some multi-asset derivatives are **convex in correlation**, meaning that the second-order effect on the price from a movement in the correlation is non-zero and needs to be taken into account.

**5.10. Approximations to BS and Greeks**

**….**