### Advanced Level

### Agenda

- Option values and risk management
- The Greeks (Delta, Gamma, Theta, Vega)
- Risk Management
- What Is Hedging?
- How Do Investors Hedge?
- Advanced techniques

**Disclaimer: Hypothetical performance results have many inherent limitations. No representation is being made that any account will or is likely to achieve profit or loss.**

### Option values and risk management

To value and price FX options, dealers and traders use FX option-pricing models.

To value FX calls and puts, the standard FX option-pricing model takes into account these factors:

- Price of the underlying currency
- Risk-free rate of interest prevailing in the home country of the base currency
- Risk-free rate of interest prevailing in the home country of the quote currency
- Strike price of the option
- Option’s time to expiration
- Expected volatility of the price of the base currency with respect to the quote currency

As the value of any of these factors changes over time, the model value of the call or put changes.

For FX calls, the relationships among the factors and the model value of the option are these:

- If the price of the underlying currency goes up, then the value of a call written on it goes up.
- If the risk-free rate of interest prevailing in the home country of the quote currency goes up, then the value of the call goes up.
- If the risk-free rate of interest prevailing in the home country of the base currency goes up, then the value of the call goes down.
- The lower the strike price of a call; the greater its value.
- As a call’s time to expiration grows shorter, its value decreases.
- The greater the expected volatility of the price of the base currency with respect to the quote currency, the greater the value of a call.

For FX puts, some of these relationships are the same. Others are opposite:

- If the price of the underlying currency goes up, then the value of a put written on it goes down. (Opposite relationship from call.)
- If the risk-free rate of interest prevailing in the home country of the quote currency goes up, then the value of the put goes down. (Opposite relationship from call.)
- If the risk-free rate of interest prevailing in the home country of the base currency goes up, then the value of the put goes up. (Opposite relationship from call.)
- The higher the strike price of a put; the greater its value. (Opposite relationship from call.)
- As a put’s time to expiration grows shorter, its value decreases. (Same relationship as for a call.)
- The greater the expected volatility of the price of the base currency with respect to the quote currency, the greater the value of a put. (Same relationship as for a call.)

From a trading perspective, the more important factors that affect an option’s value are:

- Strike price
- Price of the underlying currency
- Time to expiration
- Expected volatility of the price of the base currency with respect to the quote currency

The less important factors (barring dramatic changes in their value) are:

- Base-currency risk-free rate
- Quote-currency risk-free rate

Over a given option’s time to expiration, some of the more important factors that affect an option’s value will change:

- The strike price of the option will not change.
- Much more likely than not, the price of the underlying currency will change.
- The option’s time to expiration continuously decreases until it reaches zero at expiration.
- Marketplace expectations about the future volatility of the price of the base currency with respect to the quote currency may change.

When the value of any of these factors changes, the value of the option changes. To maximize profit opportunities and to manage risks, traders keep an eye on how the values of their positions will change if and when the values of the factors change. That is, they keep an eye on the Greeks of their existing positions and examine the Greeks of the positions they contemplate taking.

To manage risks, traders often add new positions to their portfolios for the sole purpose of bringing the Greeks back within acceptable levels.

### The Greeks

Relationships among an option’s value and the factors that influence that value are mathematically complex.

By convention, a symbol (usually a Greek letter) represents how much the value of an option will change as a given factor changes.

One symbol, gamma, represents how much another value, delta, will change as the price of the underlying currency changes.

The more important Greeks are:

- Delta
- Gamma
- Theta and
- Vega

### Delta

Rate of change of the value of an option or option portfolio to changes in the market price of the underlying currency when all else remains the same.

For example, if a call has a delta of 0.5 and the price of the underlying goes up by $0.0002, the value of the call could be expected to go up by approximately $0.0001.

A given delta calculation holds over only a very small change in the price of the underlying. That is, as the price of the underlying changes, delta changes.

For a deep in-the-money call, delta approaches 1.0. Every one unit change in the value of the underlying will produce an almost equal change in the value of the option.

The deeper a call is out-of-the-money, the closer its delta will be to 0.0. A small change in the market price of the underlying will produce very little change in the value of the call

Calls in between deep out of and deep in the money will have deltas between 0.0 and 1.0.

If a put is deep in-the-money, its delta will approach 1.0. A small drop in the market price of the underlying will produce an increase of almost equivalent magnitude in the value of the put.

The deeper a put is out of the money, the closer its delta will be to 0.0. A small change in the market price of the underlying will produce very little change in the value of the put.

Puts in between deep out of and deep in the money will have deltas between 0.0 and -1.0. If a put has a delta of 0.5 and the price of the underlying goes down by $0.0002, the value of the put could be expected to go up by approximately $0.0001.

To modify his or her option portfolio’s delta, a trader can buy or sell the underlying currency and/or options. For example, if a trader were long 100 call options and the options had a delta of .6, then he or she could sell short 60 of the underlying currency. The combined option and currency position then would have a delta of 0.0. As the price of the underlying changed (or as time passed), the option’s delta would change. To maintain delta neutrality (a delta of 0), the trader would need to change the amount of currency he or she is short.

**Key Terms: DELTA**

### Gamma

Rate of change of an option’s or option portfolio’s delta to changes in the market price of the underlying currency.

If an option’s or portfolio’s gamma is large, then its delta will change rapidly as the price of the underlying currency or currencies changes.

For an at-the-money option, gamma increases as the option’s time to expiration decreases. When an at-the-money option has very little time left until expiration, its gamma is very high. The very high gamma means the value of the option is very sensitive to changes in the market price of the underlying.

With a large gamma, to keep delta within acceptable limits, the trader will need to make adjustment to the portfolio more frequently than he or she would need to if gamma were low.

An underlying currency has a gamma of zero. Hence, to modify his or her option portfolio’s gamma, a trader would need to buy or sell options. If the trader wishes to manage his or her portfolio’s delta and gamma, then, after buying or selling options to adjust gamma, he or she will need to buy or sell the underlying to adjust delta.

**Key Terms: GAMMA**

### Theta

Rate of change of the value of an option or option portfolio to the passage of time (to decreases in the option’s or portfolio’s time to expiration) when all else remains the same.

When option prices are quoted, time is measured in years. When Theta is quoted, however, it is usually for the passage of one day.

Theta is almost always negative. If nothing else changes, options lose value as time passes.

Theta has the largest absolute magnitude (is most negative) for at-the-money options.

Because the passage of time is constant and completely predictable, traders do not ordinarily attempt to hedge theta.

**Key Terms: THETA**

### Vega

Rate of change of the value of an option or option portfolio to a change in the expected volatility of the underlying currency.

The standard model for valuing FX options assumes that volatility or expected volatility is constant over an option’s time to expiration. In reality, volatilities and expected volatilities change.

If vega is high, then the value of the option or portfolio is very sensitive to changes in expected volatility. If vega is low, then the value of the option or portfolio will change little with changes in expected volatility.

An underlying currency has a vega of zero. Hence, to modify his or her option portfolio’s vega, a trader would need to buy or sell options.

**Key Terms: VEGA**

### Risk management

Prudent traders manage their option portfolios so that they keep the Greeks within acceptable ranges.

If a trader buys or sell at- or near-the-money options and, over their times to expiration, those options go either deep into the money or deep out of the money, their gammas and vegas will become small and of little consequence. The more difficult risk-management task is to manage the gammas and vegas of options that are at the money right up until they expire.

### What Is Hedging?

##### The best way to understand hedging is to think of it as insurance.

When people decide to hedge, they are insuring themselves against a negative event. This doesn’t prevent a negative event from happening, but if it does happen and you’re properly hedged, the impact of the event is reduced. So, hedging occurs almost everywhere, and we see it everyday.

For example, if you buy house insurance, you are hedging yourself against fires, break-ins or other unforeseen disasters.

Portfolio managers, individual investors and corporations use hedging techniques to reduce their exposure to various risks. In financial markets, however, hedging becomes more complicated than simply paying an insurance company a fee every year. Hedging against investment risk means strategically using instruments in the market to offset the risk of any adverse price movements. In other words, investors hedge one investment by making another.

Technically, to hedge you would invest in two positions with negative correlations. Of course, nothing in this world is free, so you still have to pay for this type of insurance in one form or another.

Although some of us may fantasize about a world where profit potentials are limitless but also risk free, hedging can’t help us escape the hard reality of the risk-return tradeoff. A reduction in risk will always mean a reduction in potential profits. So, hedging, for the most part, is a technique not by which you will make money but by which you can reduce potential loss. If the investment you are hedging against makes money, you will have typically reduced the profit that you could have made, and if the investment loses money, your hedge, if successful, will reduce that loss.

### How Do Investors Hedge?

Hedging techniques in FX generally involve the use of options.

##### Here is an example:

Say you are long 100,000 EUR/USD. You believe the rate will rise over time; however, with volatility in the marketplace, you do not wish to have a margin call should the market turn down in the interim. To protect yourself from a fall in EUR/USD, you can buy a put option which gives you the right to sell EUR/USD at a specific price (strike price).

This strategy is known as a married put or the creation of a synthetic call. If the EUR/USD price tumbles below the strike price, these losses will be offset by gains in the put option.

### Advanced techniques

Consider the same scenario above; however, you feel the volatility will make the price of the EUR/USD go down no more than 5 basis points – for this example, consider the price now at 1.35 and you feel the lower limit is 1.30. You can go long the spot at 1.35, you can buy a 1.35 put and you can sell at 1.30 put. By creating a bear-put-spread, you are reducing the cost of the 1.35 put by taking in the premium of the 1.30.

Because the 1.30 is further out of the money, you are not completely offsetting the premium of the 1.35 put, but merely reducing the price. NOTICE – if you are wrong, and the market goes below 1.30, you no longer have any protection and you would be subject to a margin call.

### Summary

- Option values and risk management
- The Greeks (Delta, Gamma, Theta, Vega)
- Risk Management
- What Is Hedging?
- How Do Investors Hedge?

**Next: Level Review**