Quant and Trading Interview Questions about Option Greeks and Risk Management

It is crucial to have a strong understanding of the main option Greeks for quant or trading positions.

You will find below several questions you could be asked in an interview for a quant or a trading role, with possible answers.

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1. What are the main option Greeks and what are they used for?

Option Greeks are used to assess and manage the risk associated with options positions. They are key for pricing options, managing risk, and building options strategies effectively in quantitative finance and trading.

The Black-Scholes price of an option P(t, S, σ, r) is a function of time (t), the stock price (S), the implied volatility (σ) and interest rate (r). Theta (θ), Delta (Δ), Vega (ν), Rho (ρ) are the first order Greeks:

• The Theta (θ) or time decay measures the rate at which the value of the option declines due to the passage of time.
• The Delta (Δ) of an option measures the change in the option’s price resulting from a change in the underlying asset. For option traders, minus delta indicates how many underlying assets are needed to be purchased or sold to hedge the directional exposure of the option position. For call options, delta ranges from 0 to 1, while for put options, it ranges from -1 to 0.
• Vega (ν) measures the change in the option’s price resulting from a change in the implied volatility. Vega is essential because it helps traders assess how sensitive their options positions are to shifts in market volatility.
• Rho (ρ) measures the option’s sensitivity to changes in the risk-free rate of interest. It is most relevant for long-term options and is less critical for short-term ones.

The most important second order Greeks are:

• Gamma (Γ): it measures the rate of change of the delta of an option relative to the underlying asset price. It is the second order partial derivative of the value of the option with respect to the price of the underlying asset.
• Vanna: it measures the option’s sensitivity to small changes in the underlying asset price and volatility.  It is the sensitivity of the delta to changes in the volatility or the sensitivity of the vega to changes in the underlying asset price.
• Volga or Vomma: it measures the sensitivity of the vega to a change in the implied volatility.  It is the second order partial derivative of the price of the option with respect to the implied volatility.
• Charm: it is the option’s sensitivity to small changes in the underlying asset price and passage of time. It is also the sensitivity of the delta to the passage of time or the sensitivity of the theta to changes in the underlying asset price.

2. Can you please write the P&L decomposition of an option portfolio with these Greeks?

With a second order Taylor expansion of the option value over a short time interval delta t we can decompose the variation of the option price as following.

The residual regroups other second order and higher order sensitivities.

3. What is approximately the delta of an at-the-money call option when it is close to expire? Why? Avoid the maths as much as possible in your explanations.

The call option gives the right to buy the underlying asset at a given price K. If the asset price is very close to K, the option is at-the-money (ATM), then it has approximately:

• 50% chance of being exercised. If it is the case, then the delta would be 1.
• 50% chance of not being exercised, then the delta would be equal to 0.

So the delta of an ATM call option close to expire is close to 0.5.

4. Is the delta of a call option higher or lower than the probability of being exercised? Why?

The delta of a call option is higher than the probability that the option will be exercised.

In addition to the probability of being exercised, the delta incorporates by how far the option can be in-the-money.

5. How do the Delta, Theta, Vega and Gamma change for call options versus put options?

For call options, the Delta is positive. The price of the option increases with the underlying asset price as it becomes more likely to be executed. For put options it is the opposite, the Delta is negative, the price of the put option decreases when the underlying asset price increases as it becomes more out-of-the-money with a lower probability to be exercised.

Theta, Vega, and Gamma behave similarly for both call and put options but may differ in magnitude depending on the specific option contract.

Theta is typically negative for both call and put options. As time passes and the option gets closer to expiry, its premium decreases, the value of the option erodes due to time decay.

Vega is positive for both call and put options. It measures the sensitivity of the option price to changes in implied volatility. As the implied volatility increases, the price of both calls and puts tends to increase.

Gamma, which measures the rate of change of the delta with respect to the underlying asset price is positive for both call and put options. It is a measure of the convexity of the option, it is larger for at-the-money options and it decreases as options are more out-of-the-money.

6. What are the main risks of a long straddle strategy?

A long straddle is an option strategy where an investor simultaneously purchases both a call option and a put option with the same strike price and expiration date. The main goal of a long straddle is to benefit from significant price movement in the underlying asset, regardless of whether it moves up or down.

In terms of Greeks, a long straddle has:

• A low delta, has the positive delta of the long call option is offset by the negative delta of the long put option.
• A negative Theta (time decay), both the call and the put options lose value as time passes.
• A positive Gamma as both the long call and the long put have a positive gamma exposure. It means that the strategy would gain if there is an important movement of the underlying asset on both sides, with a gamma P&L higher than the negative time decay.
• A positive Vega, as both the long call and the long put positions have a positive Vega exposure. It means that the strategy would benefit if there is an increase of the implied volatility while it woud lead to losses if the implied volatility decreases.

7. Discuss the concept of dynamic Delta-hedging vs static Delta-hedging. When and why would a trader choose one approach over the other?

Dynamic Delta-hedging involves continuously adjusting the hedge ratio based on changing market conditions, while static Delta-hedging maintains a constant hedge ratio.

Traders may choose dynamic Delta hedging when they anticipate frequent and significant changes in market conditions, while static Delta hedging can be simpler for more stable markets.

Dynamic delta hedging comes with higher transaction costs, so the trader has to be careful in the frequency of his rebalancing.

8. Plot the P&L of: 1/ a call option, 2/ its delta position, 3/ the delta-hedged option position with respect to the price of the underlying asset

P&L of a delta-hedged call option

9. Write the decomposition of the P&L of a long delta-hedged option position between t and t+ẟt from its Gamma and its Theta P&Ls. If the implied volatility is higher than the realised volatility, is the P&L of the position positive or negative? We assume that the risk-free interest rate is equal to zero, there is no dividend payment, and there is no change of the implied volatility.

Theta, the time decay of the option, is the cost of gamma. It is the cost for the convexity of the option measured by the gamma.

We have, with σ_r the realised volatility:

And if we assume that r = q = 0 we have:

So the P&L of the delta-hedged option between t and t + δt is close to:

when neglecting higher order Greeks.

If the implied volatility is higher than the realised volatility, the P&L of the strategy is negative.

10. Let’s assume stock price = 100, option gamma = 0.2, option delta = 0.5,  and that the stock price goes up + 5% very quickly. What is approximately the P&L of a long position on the option (quantity = 1). Tips: decompose the P&L with delta and gamma neglecting other Greeks.

The P&L of the long position on the option can be approximated as following, keeping the Delta and Gamma P&L and neglecting other Greeks:

Delta P&L = 0.5 x 5% x 100 = 2.5

Gamma P&L = 1/2 x 0.2 x (5% x 100) ^2 = 2.5

P&L ≈ Delta P&L + Gamma P&L

P&L ≈ 2.5 + 2.5 = 5

Below is a series of questions related to a risk-reversal strategy:

11. What is a risk-reversal strategy?

A risk reversal strategy involves simultaneously buying a call option and selling a put option (long risk-reversal), or vice versa (short risk-reversal), typically with the same expiration date but different strike prices. This strategy is used to speculate on the direction of the underlying asset’s price movement or to hedge exposure to that asset.

12. Can you draw the payoff of such strategy?

13. In order to delta-hedge a long risk-reversal position, will you have to buy or to sell the underlying asset?

The delta of a long risk-reversal strategy is positive. It is the sum of the delta of a long call position, positive, and a short put position, positive as well.

So we will have to sell the underlying asset in order to offset the delta of the risk-reversal position.

14. What about the gamma of a long risk-reversal?

It depends. The gamma of the long call position is positive while the gamma of the short put is negative. So if we structure the strategy such that the two gamma balance each other, the overall gamma will be near zero.

15. Let’s assume that the gamma of a long risk-reversal strategy was initially close to zero, that both the long call and the short put are out-of-the-money, and that the underlying asset price increases and gets closer to the strike price of the call. How does the gamma of the position evolve?

The gamma of the long call will increase as the option will get closer to the money while the gamma of the short put will be less negative, the option being more out-of-the-money. So all-in-all the gamma of the risk-reversal strategy will increase.

16. Draw the Gamma of a long risk-reversal strategy as a function of the underlying asset price

The initial gamma of the strategy is close to zero.

When the asset price increases (resp. decreases) and gets closer to the strike of the long call (resp. the short put), the gamma increases (resp. decreases).

The value of the gamma is maximal (resp. minimal) when the asset price is close to the strike price of the call option (resp. the put option).

When the asset price is higher (resp. lower) than the strike price of the call option (resp. put option), the gamma starts to decrease (resp. increase) and converges to zero on both sides.

We used the following parameters to plot the chart:

17. And what about the vega of the strategy?

This is very similar to what we did before for the gamma.

The vega of the long call position is positive while the vega of the short put is negative. So if we structure the strategy such that the two vega balance each other, the overall vega will be near zero initially.

When the asset price increases (resp. decreases) and gets closer to the strike of the long call (resp. the short put), the vega increases (resp. decreases).

The value of the vega is maximal (resp. minimal) when the asset price is close to the strike price of the call option (resp. the put option).

When the asset price is higher (resp. lower) than the strike price of the call option (resp. put option), the vega starts to decrease (resp. increase) and converges to zero on both sides.

We used the following parameters to plot the chart:

18. What is the vanna of an option?

The vanna measures the option’s sensitivity to small changes in the underlying asset price and volatility.  It is the sensitivity of the delta to changes in the volatility or the sensitivity of the vega to changes in the underlying asset price.

19. Is the vanna of an out-of-the-money call option positive or negative? What about an out-of-the-money put?

While both long call and put options have a positive vega, a higher volatility means a higher price, the Vanna is positive in general for OTM call option and negative for OTM put option.

When the option is very OTM its delta decreases in absolute value and tends to zero particularly when the option becomes close to expiry. As the implied volatility increases, the probability that the option moves in the money and that the delta becomes more positive for call options and more negative for put options increases.

So a higher volatility means a higher delta for an OTM call and a lower delta for an OTM put.

20. So what about the vanna of our long risk-reversal strategy?

A long risk-reversal strategy is the sum of a long call and a short put positions. The vanna of each leg is positive initially, when both options are out-of-the-money, so the vanna of the risk-reversal strategy is positive.

We can see it on the previous chart which shows the vega of the risk-reversal as a function of the asset price. The vanna is the slope of the curve. So we see that it is positive initially, while it will become negative if the call or the put option are in the money.

We used the following parameters to plot the chart:

Below is a series of questions related to a risk-reversal strategy:

Below is a series of questions related to the Black-Scholes delta:

21. Write the Black-Scholes formula for a call option and identify the Delta in the Formula. What is the second term? We assume that there is no dividend payment in this question and the following ones.

22. Do the same for a European Put Option.

23. Verify that we have the call / put parity. What is the relationship between the delta of a call and a put option on the same underlying asset with same expiry and strike?

24. Demonstrate that the delta of a call option is equal to N(d1) by calculating the partial derivatives of the Black-Scholes call price with respect to the underlying asset price.

25. Draw the delta profiles of a call option far from expiry and close to expiry as a function of the underlying asset price.

26. Do the same for a digital call option.

For a digital call option, the delta will tend to zero on the right and on the left of the strike price, while it will tend to infinity at the strike price when the time to expiry tends to zero.