L-questions are questions about the lecture and extra materials, H(ull)-questions are question from the book.

H8. 1 The delta of a derivatives portfolio dependent on an index is −2,100. The index is currently 1,000. Estimate what happens to the value of the portfolio when the index increases to 1,005.

8.1 The value of the portfolio decreases by $10,500.

H8.2 The vega of a derivatives portfolio dependent on the dollar–sterling exchange rate is 200 (per %). Estimate the effect on the portfolio of an increase in the volatility of the exchange rate from 12% to 14%.

8.2 The value of the portfolio increases by $400.

H8.3 The gamma of a delta-neutral portfolio is 30. Estimate what happens to the value of the  portfolio when the price of the underlying asset (a) suddenly increases by $2 and (b) suddenly decreases by $2.

8.3 In both cases, it increases by 0.5 × 30 × 22 or $60.

H8.4 What does it mean to assert that the delta of a call option is 0.7? How can a short position in 1,000 options be made delta neutral when the delta of a long position in each option is 0.7?

8.4 A delta of 0.7 means that, when the price of the stock increases by a small amount, the price of the option increases by 70% of this amount. Similarly, when the price of the stock decreases   by a small amount, the price of the option decreases by 70% of this amount. A short position in  1,000 options has a delta of −700 and can be made delta neutral with the purchase of 700 shares.

H8.6 What is meant by the gamma of an option position? What are the risks in the situation where the gamma of a position is large and negative and the delta is zero?

8.6 The gamma of an option position is the rate of change of the delta of the position with respect to the asset price. For example, a gamma of 0.1 would indicate that, when the asset price increases by a certain small amount, delta increases by 0.1 of this amount. When the gamma of an option writer’s position is large and negative and the delta is zero, the option writer will lose significant amounts of money if there is a large movement (either an increase or a decrease) in the asset price.

L8.15.

The gamma and vega of a delta-neutral portfolio are 50 per $ per $ and 25 per %, respectively. Estimate what happens to the value of the portfolio when there is a shock to the market causing the underlying asset price to decrease by $3 and its volatility to increase by 4%.

With the notation of the text, the increase in the value of the portfolio is

0.5´Gamma´ (DS)  +Vega´ Ds2

This is

0.5 × 50 × 32 + 25 × 4 = 325

The result should be an increase in the value of the portfolio of $325.

L8.17.

Afinancial institution has thefollowing portfolio of over-the-counter options on sterling:

A traded option is available with a delta of 0.6, a gamma of 1.5, and a vega of 0.8. (a) What position in the traded option and in sterling would make the portfolio both gamma neutral and delta neutral?

(b) What position in the traded option and in sterling would make the portfolio both vega neutral and delta neutral?

The delta of the portfolio is

−1,000 × 0.50 − 500 × 0.80 − 2,000 × (−0.40) − 500 × 0.70 = −450 The gamma of the portfolio is

−1,000 × 2.2 − 500 × 0.6 − 2,000 × 1.3 − 500 × 1.8 = −6,000

The vega of the portfolio is

−1,000 × 1.8 − 500 × 0.2 − 2,000 × 0.7 − 500 × 1.4 = −4,000

(a) A long position in 4,000 traded options will give a gamma-neutral portfolio since the long position has a gamma of 4,000 × 1.5 = +6,000. The delta of the whole portfolio (including traded options) is then:

4,000 × 0.6 − 450 = 1,950

Hence, in addition to the 4,000 traded options, a short position in £1,950 is necessary so that the portfolio is both gamma and delta neutral.

(b) A long position in 5,000 traded options will give a vega-neutral portfolio since the long position has a vega of 5,000 × 0.8 = +4,000. The delta of the whole portfolio (including traded options) is then

5,000 × 0.6 − 450 = 2,550

Hence, in addition to the 5,000 traded options, a short position in £2,550 is necessary so that the portfolio is both vega and delta neutral.

L8.18.

Consider again the situation in Problem 8.17. Suppose that a second traded option with a delta of 0.1, a gamma of 0.5, and a vega of 0.6 is available. How could the portfolio be made delta, gamma, and vega neutral?

Let w1 be the position in the first traded option and w2 be the position in the second traded option. We require:

6,000 = 1.5w1 + 0.5w2

4,000 = 0.8w1 + 0.6w2

The solution to these equations can easily be seen to be w1 = 3,200, w2 = 2,400. The whole portfolio then has a delta of

−450 + 3,200 × 0.6 + 2,400 × 0.1 = 1,710

Therefore, the portfolio can be made delta, gamma, and vega neutral by taking a long position in 3,200 of the first traded option, a long position in 2,400 of the second traded option, and a short position in £1,710.

L8.2 Consider a European option on a non-dividend paying stock with a strike price of 70 and a maturity of 2 years. The current stock price is 50 and the stock volatility is 30%. The risk-free rate is 5 percent.

Use the Python code of the lecture and plot the Black-Scholes option price against the current stock price.

What is the delta of the option approximately if the current stock price equals 10? And what is the current stock price is close to 120? Explain why.

Answer

import numpy as np

import matplotlib.pyplot as plt

from scipy.stats import norm

def black_scholes_call(S, K, T, rf, s):

d1 = (np.log(S/K) + (rf + (s**2)/2) * T) / (s * np.sqrt(T))

d2 = d1 - s * np.sqrt(T)

call_price = S * norm.cdf(d1) - K * np.exp(-rf * T) * norm.cdf(d2)

return call_price

S = np.linspace(0, 130, 100)

K = 70

s = 0.3

T = 2

rf = 0.05

option_prices_S = black_scholes_call(S, K, T, rf, s)

plt.figure()

plt.plot(S, option_prices_S)

plt.xlabel('Stock Price (S)')

plt.ylabel('Option Price')

plt.title('Option Price of European Call Option for Varying Stock Price')

plt.show()

What is the delta of the option approximately if the current stock price equals 10? And what is the current stock price is close to 120?

S0 = 10: The delta of the option is close to zero

S0 = 120: The delta of the option is close to one

The delta of an option represents the sensitivity of the option price to changes in the underlying asset's price. When an option is deep out of the money, it means the strike   price of the option is significantly higher (for a call option) or lower (for a put option)   than the current market price of the underlying asset. In this scenario, the probability of the option ending up in the money by expiration is very low.

Intuitively, when the option is deep out of the money, its value is primarily determined by the probability of it moving into the money before expiration. Since the probability is low, any small change in the underlying asset's price would have a minimal impact on    the option's value. Therefore, the delta of the option approaches zero, indicating a low    sensitivity to changes in the underlying asset's price.

On the other hand, when an option is deep in the money, it means the strike price of the option is significantly lower (for a call option) or higher (for a put option) than the current market price of the underlying asset. In this case, the probability of the option ending up in the money by expiration is very high.