Risk Measurement

Question 1

The Delta

Answer 1

• A measure of an options risk with respect to the direction of movement in the underlying contract
  ○ Positive delta = desire for upward movement
  ○ Negative delta = desire for downward movement
  ○ The delta of a call at any given underlying price is the slope of the graph: the rate of change in an options value with respect to movement in the underlying contract
  ○ Delta of a call has an upper bound of 1 meaning it can never faster than the market
  ○ Puts are the same just go from 0 to-1
• Hedge Ratio
  ○ The proper number of underlying contracts to option contracts required for such a hedge
  ○ For a call option w a delta of 50, the proper hedge ratio is 2/1. For every two options purchased, we need to sell one underlying contract
  ○ A position that is delta neutral has no particular preference for either upward or downward movement in the price of the underlying contract
  § A trader who is trying to capture the theoretical value of an option must start with and maintain a delta neutral position over the life of the option- Probability
  ○ Ignoring the sign of a delta, the delta is approximately equal to the probability that the option will finish in the money
  § As the delta moves closer to 0, the option becomes less and less likely to finish in the money
  § This is why ATM options have 50 deltas, random walks suggest that this has a 50p chance of landing ITM
• The delta of an underlying contract is always 100

Question 2

The Gamma

Answer 2

• Sometimes referred to as an options curvature, is the rate of change in the delta as the underlying price changes
• The gamma is usually expressed in deltas gained or lost per one point change in the underlying
  ○ If an option has a gamma of 5, for each point rise in the price of the underlying, the option will gain 5 deltas
  ○ If the option initially has a delta of 25 and the underlying moves up one full point, the new delta of the option will be 30
• Gamma values are always positive and when a trader is long options, either calls or puts, he is said to be long gamma
• The rate of change in the delta will be determined by the size of the gamma position
• A negative gamma position is a good indication that a trader either wants the market to stand still or move only very slowly
  ○ Positive gamma would be the opposite
  Where delta can be thought of as directional risk, gamma can be thought of as magnitude risk

Question 3

The Theta

Answer 3

• the rate at which an option loses value as time passes, assuming that all other market conditions remain unchanged
  ○ It is usually expressed as value lost per one day’s passage of time
  ○ Common to express theta as a negative number as all options lose value when time passes
• If an option is exactly ATM as time passes, the theta of the option increases
  ○ The theta becomes increasingly large as expiration approaches
• Theta and interest rates
  ○ All else equal, because a call is discounted by the interest rates to the present value, the effect has negative time value therefor positive theta
  § Ex call is worth 40 but the PV is 39. As it approaches expiry the value moves to 40 hence capturing positive theta
  § Note this is only the case when the option is subject to stock type settlement, it must also be deeply in the money and have a European style exercise
• Gamma and theta are almost always of the opposite sign
  ○ From the negative gamma, a trader will monetize his position via time passing and holding a positive theta
  ○ Large gamma will mean large theta and vice versa, but a trader cannot ever have it both ways
  That is small gamma and large theta

Question 4

The Vega

Answer 4

• Impact of changes in volatility on the price of an option
• There is no one generally accepted term for an options theoretical value to a change in volatility, the most commonly used term in trading is Vega (this is not actually a Greek letter)
  ○ Vega is by no means universal, and is sometimes called Kappa
• Usually expressed as the change in theoretical value for each one percentage point change in volatility
  ○ Bc all options gain value for rising vol, the Vega for both P & C are positive
  ○ If an option has a Vega of .15, for each percentage point increase in vol, the option will gain .15 in theoretical value
• Note that gamma is a measure of if we want higher or lower realized volatility; while Vega is a measure of whether we want higher or lower implied volatility

Question 5

The Rho

Answer 5

• The sensitivity of an option’s theoretical value to a change in interest rates
• Usually expressed as a change in value for each one percentage point change in interest rates
  ○ Cannot generalize about rho bc its characteristics depend on the type of underlying instrument and the settlement procedure for the option
• Futures type settlement
  ○ Rho must be 0 because no cash flows result from either a trade in the underlying contract or a trade in options
• Stock type settlement
  ○ The rho associated w calls and puts is negative
  ○ An increase in int rates will decrease the value of such options because it raises the cost of carry of such options
  ○ Stock options: calls will have positive rho values, and puts will have negative rho values
• Rho is commonly viewed as the least important greek in valuing options