Exam-Style Questions on Market-Making and Delta-Hedging

This is Section 51 of the Study Guide. See Section 1 here. See Section 2 here. See Section 3 here. See Section 4 here. See Section 5 here. See Section 6 here. See Section 7 here. See Section 8 here. See Section 9 here. See Section 10 here. See Section 11 here. See Section 12 here. See Section 13 here. See Section 14 here. See Section 15 here. See Section 16 here. See Section 17 here. See Section 18 here. See Section 19 here. See Section 20 here. See Section 21 here. See Section 22 here. See Section 23 here. See Section 24 here. See Section 25 here. See Section 26 here. See Section 27 here. See Section 28 here. See Section 29 here. See Section 30 here. See Section 31 here. See Section 32 here. See Section 33 here. See Section 34 here. See Section 35 here. See Section 36 here. See Section 37 here. See Section 38 here. See Section 39 here. See Section 40 here. See Section 41 here. See Section 42 here. See Section 43 here. See Section 44 here. See Section 45 here. See Section 46 here. See Section 47 here. See Section 48 here. See Section 49 here. See Section 50 here.

Note 51.1: Before, we solve the exam-style questions on market-making and delta-hedging, one additional formula is in order. Within the Black-Scholes framework, if a delta-hedged market-maker makes exactly zero profit during a specific time period, it can be assumed that a stock price moved by one standard deviation during that period. That is, the magnitude of the move will be σS_t√(h), where t is the time at which the original stock price existed, S_t is the stock price, h is the time period during which the stock price moves, and σ is the annual standard deviation of the stock price movement.

Note 51.2: Furthermore, it is possible to use MS Excel to find x when given N(x). Using the function "=NormSInv(N(x))", where you can substitute in the relevant value for N(x), will accomplish this aim. On the exam, you will need to use a chart of values to get the same results, but Excel is much more convenient to use when one is attempting to learn the problem-solving procedures as efficiently as possible.

The problems in this section were designed to be similar to problems from past versions of Exam 3F / Exam MFE. They use original exam questions as their inspiration - and the specific inspiration for each problem is cited so as to give students a chance to see the original. All of the original problems are publicly available, and students are encouraged to refer to them. But all of the values, names, conditions, and calculations in the problems here are the original work of Mr. Stolyarov.

Problem ESQMMDH1.

Similar to Question 33 from the Casualty Actuarial Society's Spring 2007 Exam 3:

Which of these statements about the role of option Greeks in market-making are correct?

(a) Hedging does not require the purchase of stock.
(b) If the gamma of a call is positive, then by writing the call, a market-maker will lose money in proportion to the square of the stock price change.
(c) If the gamma of a call is positive, then by purchasing the call, a market-maker will lose money in proportion to the square of the stock price change.
(d) If theta for a call is negative, the option buyer benefits from theta.
(e) If theta for an option is negative and gamma is positive, it is possible to benefit from theta and gamma simultaneously.

Solution ESQMMDH1. We examine these statements by keeping in mind the delta-gamma-theta approximation:
When a market-maker has purchased Δ shares and short-sold the call, his profit is

Profit = -(0.5є²Γ_t + θ_th + rh[Δ_tS_t - C(S_t)])

(a) is clearly untrue, because the market-maker needs to purchase Δ shares of stock for every call option he writes in order to offset the negative delta of the call option. We also note that the difference between Δ_tS_t and C(S_t) implies that there exists a net cost to the market-maker from purchasing the stock. This is the interest cost of the position.

(b) is correct. If the stock changes by є, the market-maker will lose -0.5є²Γ_t due to the effects of gamma.

For (c): For a market-maker who has purchased a call (and sold Δ shares), the sign of the profit equation will be reversed: Profit = (0.5є²Γ_t + θ_th + rh[Δ_tS_t - C(S_t)]). Thus, the market-maker will gain money in proportion to the square of the stock price change, and (c) is incorrect.
For (d): If theta for a call is negative, the option writer benefits from theta, and the buyer loses. We see that in the equation Profit = -(0.5є²Γ_t + θ_th + rh[Δ_tS_t - C(S_t)]) for the option writer, a negative theta implies that -θ_th is positive. So (d) is incorrect.

For (e): We consider the equation Profit = -(0.5є²Γ_t + θ_th + rh[Δ_tS_t - C(S_t)]). If theta is negative, then -θ_th is positive. If gamma is positive, then -0.5є²Γ_t is negative. Thus, if theta is negative and gamma is positive, the two Greeks will have the opposite effects on the market-maker profit, and it is impossible for the market-maker to benefit from gamma and theta simultaneously. So (e) is incorrect. Thus, only (b) is correct.

Problem ESQMMDH2.

Similar to Question 24 from the Casualty Actuarial Society's Fall 2007 Exam 3:

N50 is a market-marker who has 564 $200-strike put options on the stock of Invulnerable Co. in his portfolio. He also has 17 shares of the stock in his portfolio. He can write $177-strike put options on the stock. The delta and gamma values for the put options are as follows:
$177-strike put option: ∆ = -0.7, Γ = 0.33

$200-strike put option: ∆ = -0.2, Γ = 0.54

What would N50 need to do to delta- and gamma-neutralize his portfolio?

Solution ESQMMDH2.

First, we consider the current delta and gamma values of N50's portfolio:

From 17 shares: ∆ = 17, Γ = 0

From 564 $200-strike put options: ∆ = 564*-0.2 = -112.8, Γ = 0.54*564 = 304.56

Net ∆ = 17 - 112.8 = -95.8

We first try to gamma-neutralize the portfolio using the $177-strike puts; then we can delta-neutralize the remaining net delta with shares.

One $177-strike put option has Γ = 0.33, so a gamma of -304.56 is possessed by

-304.56/0.33 = -922.90909090909 $177-strike put options. So to gamma-neutralize the portfolio, N50 would need to write 922.90909090909 $177-strike put options.

That changes net delta by -(-0.7*922.90909090909) = 646.03636363636 - so that the new net delta is -95.8 + 646.03636363636 = 550.236363636. To delta-neutralize this portfolio, N50 will need to sell 550.236363636 shares of stock (each with ∆ = 1, as is always the case for shares of stock). Thus, to delta- and gamma-neutralize his portfolio, N50 will need to write 922.90909090909 $177-strike put options and sell 550.236363636 shares of stock.

Problem ESQMMDH3.

Similar to Question 10 from the Society of Actuaries' May 2007 Exam MFE:

There are two possible call options on Stock A: Call Ψ and Call Φ.

The calls have the following Greeks:
Φ: ∆ = 0.66, Γ = 0.04, vega = 0.05

Ψ: ∆ = 0.32, Γ = 0.10, vega = 0.006

Aethelred just sold 542 units of Call Φ. How many units of Stock A and Call Ψ would he need to buy or sell in order to both delta-hedge and gamma-hedge his position?

Solution ESQMMDH3.

We first note that vega values are completely irrelevant to this problem, so we only need to consider delta and gamma values. Selling 542 units of Call Φ means that Aethelred has a position delta of -0.66*542 = -357.2 and a position gamma of -0.04*542 = -21.68.

We first try to gamma-neutralize this portfolio. Each Call Ψ has a gamma of 0.10, so we need 21.68/0.1 = 216.8 Call Ψ options to obtain a gamma of 21.68. By buying 216.8 Call Ψ options, Aethelred will thus gamma-neutralize the portfolio. Doing so will increase the portfolio's net delta to -357.2 + 216.8*0.32 = -287.824, which leaves 287.824 in delta to be accounted for. This increase in delta can be obtained by purchasing 287.824 shares of Stock A.

Thus, Aethelred would need to buy 216.8 Call Ψ options and buy 287.824 shares of Stock A.

Problem ESQMMDH4.

Similar to Question 19 from the Society of Actuaries' May 2007 Exam MFE:

Assume that the Black-Scholes framework holds. The price of Blackscholesian Co. stock, which pays no dividends, is $566. A certain put option on this stock trades for $25. The option has a delta of -0.66 and a gamma of 0.04. Suddenly, the price of Blackscholesian Co. stock increases to $588 per share. Find the new put option price using the delta-gamma approximation.

Solution ESQMMDH4. We use the delta-gamma approximation, noting that C here stands for the put option price:

C(S_t+h) = C(S_t) + є∆(S_t) + (1/2)є²Γ(S_t), where C(S_t) = 25, є = 22, ∆(S_t) = -0.66, and Γ(S_t) = 0.04

Thus, C(S_t+h) = 25 + 22*-0.66 + (1/2)22²0.04 = C(S_t+h) = $20.16

Problem ESQMMDH5.

Similar to Question 10 from the Society of Actuaries' Sample MFE Questions and Solutions:

Assume the Black-Scholes framework. Xenophon, a market-maker who delta-hedges his position, has sold a four-year at-the-money European call option on Stock Q, which pays no dividends. Stock Q currently trades for $99 per share, and the annual continuously-compounded risk-free interest rate is 0.02. The delta of the call option is 0.725746935. There are 365 days in a year. After seven particular days, Xenophon has no profits or losses on his position. It is known that the annual standard deviation of the stock price movement is greater than 0.33. Determine the stock price movement during the course of these seven days.

Solution ESQMMDH5. As stated in Note 51.1, we can assume that the stock period moved by one one-week standard deviation during this period. So the stock price movement is σS_t√(h), where h = 7/365 and S_t = 99. All that remains to do is to find the value of σ.

Since the stock pays no dividends, ∆ = N(d₁) = 0.725746935.

As per Note 51.2, we use the input "=NormSInv(0.725746935)" in MS Excel to get d₁ = 0.6.

Thus, 0.6 = [(r + 0.5σ²)T]/[σ√(T)], since the stock pays no dividends and S = K, so ln(S/K) = 0.

Thus, 0.6 = [(0.02 + 0.5σ²)4]/[σ√(4)] and 0.6 = 0.04/σ + σ, and 0.6σ = 0.04 + σ², so

σ² - 0.6σ + 0.04 = 0 and, by the quadratic formula, σ = 0.0763932023 or σ = 0.5236067978. But we are given that σ > 0.33, so σ = 0.5236067978 and the stock price movement is σS_t√(h) = 0.5236067978*99√(7/365) = $7.178654513

See other sections of The Actuary's Free Study Guide for Exam 3F / Exam MFE.