A one year European-style call option has a strike price of $4.
The option's underlying stock currently trades at $5, pays no dividends and its standard deviation of continuously compounded returns is 47% pa.
The risk-free interest rate is 10% pa continuously compounded.
Use the Black-Scholes-Merton formula to calculate the option price. The call option price now is:
Using the Black-Scholes-Merton formula, first find ##d_1## and ##d_2##:
###\begin{aligned} d_1 &= \dfrac{\ln[S_0/K]+(r+\sigma^2/2).T}{\sigma.\sqrt{T}} \\ &= \dfrac{\ln[5/4]+(0.1+0.47^2/2) \times 1}{0.47 \times \sqrt{1}} \\ &= \dfrac{0.223143551 + 0.21045}{0.47} \\ &= 0.922539471 \\ \end{aligned}### ###\begin{aligned} d_2 &= d_1-\sigma.\sqrt{T}=\dfrac{\ln[S_0/K]+(r-\sigma^2/2).T}{\sigma.\sqrt{T}} \\ &= 0.922539471 - 0.47 \times \sqrt{1} \\ &= 0.452539471 \\ \end{aligned}###Now we can find the European style call option price now ##(c_0)## using the Black-Scholes-Merton formula:
###\begin{aligned} c_0 &= S_0.N[d_1] - K_T.e^{-r.T}.N[d_2] \\ &= 5 \times N[0.922539471] - 4 \times e^{-0.1 \times 1} \times N[0.452539471] \\ &= 5 \times 0.821876374 - 4 \times e^{-0.1 \times 1} \times 0.674559804 \\ &= 4.10938187 - 2.441467805 \\ &= 1.66791407 \\ \end{aligned}### CommentsThe Delta of the call option is ##N[d_1]##. This is the option's gradient at ##S_0## on the call option price (##c_0## on the y-axis) versus underlying asset price (##S_0## on the x-axis) graph.
###\text{CallOptionDelta} = N[d_1] = N[0.922539471] = 0.821876374###So a one cent ($0.01) increase in the underlying stock price from $5 to $5.01 will result in a 0.8219 cent ($0.08219) increase in the option price.
The risk-neutral probability of the call option maturing in-the-money ##(S_T > K_T)## is ##N[d_2]##.
###\text{RiskNeutralProbabilityOfCallOptionMaturingInTheMoney} \\ = N[d_2] = N[0.452539471] = 0.674559804 ###A one year European-style put option has a strike price of $4.
The option's underlying stock currently trades at $5, pays no dividends and its standard deviation of continuously compounded returns is 47% pa.
The risk-free interest rate is 10% pa continuously compounded.
Use the Black-Scholes-Merton formula to calculate the option price. The put option price now is:
Using the Black-Scholes-Merton formula, first find ##d_1## and ##d_2##:
###\begin{aligned} d_1 &= \dfrac{\ln[S_0/K]+(r+\sigma^2/2).T}{\sigma.\sqrt{T}} \\ &= \dfrac{\ln[5/4]+(0.1+0.47^2/2) \times 1}{0.47 \times \sqrt{1}} \\ &= \dfrac{0.223143551 + 0.21045}{0.47} \\ &= 0.922539471 \\ \end{aligned}### ###\begin{aligned} d_2 &= d_1-\sigma.\sqrt{T}=\dfrac{\ln[S_0/K]+(r-\sigma^2/2).T}{\sigma.\sqrt{T}} \\ &= 0.922539471 - 0.47 \times \sqrt{1} \\ &= 0.452539471 \\ \end{aligned}###Now we can find the European style put option price now ##(p_0)## using the Black-Scholes-Merton formula:
###\begin{aligned} p_0 &= -S_0.N[-d_1] + K_T.e^{-r.T}.N[-d_2] \\ &= -5 \times N[-0.922539471] + 4 \times e^{-0.1 \times 1} \times N[-0.452539471] \\ &= -5 \times 0.178123626 + 4 \times e^{-0.1 \times 1} \times 0.325440196 \\ &= -0.89061813 + 1.177881868 \\ &= 0.287263739 \\ \end{aligned}### CommentsThe Delta of the put option is ##-N[-d_1]##. This is the option's gradient at ##S_0## on the put option price (##p_0## on the y-axis) versus underlying asset price (##S_0## on the x-axis) graph.
###\text{PutOptionDelta} = -N[-d_1] = -N[-0.922539471] = -0.178123626###So a one cent ($0.01) increase in the underlying stock price from $5 to $5.01 will result in a 0.178 cent ($0.00178) decrease in the option price.
The risk-neutral probability of the put option maturing in-the-money ##(S_T < K_T)## is ##N[-d_2]##.
###\text{RiskNeutralProbabilityOfPutOptionMaturingInTheMoney} \\ = N[-d_2] = N[-0.452539471] = 0.325440196 ###Question 903 option, Black-Scholes-Merton option pricing, option on stock index
A six month European-style call option on the S&P500 stock index has a strike price of 2800 points.
The underlying S&P500 stock index currently trades at 2700 points, has a continuously compounded dividend yield of 2% pa and a standard deviation of continuously compounded returns of 25% pa.
The risk-free interest rate is 5% pa continuously compounded.
Use the Black-Scholes-Merton formula to calculate the option price. The call option price now is:
Using the Black-Scholes-Merton formula, first find ##d_1## and ##d_2##. Remember to replace all instances of ##S_0## with ##S_0.e^{-q.T}## since the stock index pays a continuously compounded dividend yield ##q## pa.
###\begin{aligned} d_1 &= \dfrac{\ln[S_0.e^{-q.T}/K]+(r+\sigma^2/2).T}{\sigma.\sqrt{T}} \\ &= \dfrac{\ln[2700 \times e^{-0.02 \times 0.5}/2800]+(0.05+0.25^2/2) \times 0.5}{0.25 \times \sqrt{0.5}} \\ &= \dfrac{\ln[2673.134551/2800]+0.040625}{0.176776695} \\ &= \dfrac{-0.046367644+0.040625}{0.176776695} \\ &= -0.032485301 \\ \end{aligned}### ###\begin{aligned} d_2 &= d_1-\sigma.\sqrt{T}=\dfrac{\ln[S_0.e^{-q.T}/K]+(r-\sigma^2/2).T)}{\sigma.\sqrt{T}} \\ &= -0.032485301 - 0.25 \times \sqrt{0.5} \\ &= -0.209261996 \\ \end{aligned}###We can find the European style call option price now ##(c_0)## using the Black-Scholes-Merton formula, again replacing ##S_0## with ##S_0.e^{-q.T}## to take the continuously compounded dividend yield pa ##q## into account:
###\begin{aligned} c_0 &= S_0.e^{-q.T}.N[d_1] - K_T.e^{-r.T}.N[d_2] \\ &= 2700 \times e^{-0.02 \times 0.5} \times N[-0.032485301] - 2800 \times e^{-0.05 \times 0.5} \times N[-0.209261996] \\ &= 2700 \times e^{-0.02 \times 0.5} \times 0.487042519 - 2800 \times e^{-0.05 \times 0.5} \times 0.417121859 \\ &= 1301.930185 - 1139.104633 \\ &= 162.8255519 \\ \end{aligned}###Question 904 option, Black-Scholes-Merton option pricing, option on future on stock index
A six month European-style call option on six month S&P500 index futures has a strike price of 2800 points.
The six month futures price on the S&P500 index is currently at 2740.805274 points. The futures underlie the call option.
The S&P500 stock index currently trades at 2700 points. The stock index underlies the futures. The stock index's standard deviation of continuously compounded returns is 25% pa.
The risk-free interest rate is 5% pa continuously compounded.
Use the Black-Scholes-Merton formula to calculate the option price. The call option price now is:
Using the Black-Scholes-Merton formula, first find ##d_1## and ##d_2##. Remember to replace all instances of ##S_0## with ##F_{0,T}.e^{-r.T}##. This is because ##F_{0,T}##, the current ##(t=0)## futures price that matures at time ##T##, takes into account the market's estimate of the underlying stock index's continuously compounded dividend yield which is not given in the question (see the comment at the end of this question for an alternative method):
###\begin{aligned} d_1 &= \dfrac{\ln[F_{0,T}.e^{-r.T}/K]+(r+\sigma^2/2).T}{\sigma.\sqrt{T}} \\ &= \dfrac{\ln[2740.805274 \times e^{-0.05 \times 0.5}/2800]+(0.05+0.25^2/2) \times 0.5}{0.25 \times \sqrt{0.5}} \\ &= \dfrac{\ln[2673.134551/2800]+0.040625}{0.176776695} \\ &= \dfrac{-0.046367644+0.040625}{0.176776695} \\ &= -0.032485301 \\ \end{aligned}### ###\begin{aligned} d_2 &= d_1-\sigma.\sqrt{T}=\dfrac{\ln[F_{0,T}.e^{-r.T}/K]+(r-\sigma^2/2).T}{\sigma.\sqrt{T}} \\ &= -0.032485301 - 0.25 \times \sqrt{0.5} \\ &= -0.209261996 \\ \end{aligned}###Now we can find the European style call option price now ##(c_0)## using the Black-Scholes-Merton formula, again replacing ##S_0## with ##F_{0,T}.e^{-r.T}##:
###\begin{aligned} c_0 &= F_{0,T}.e^{-r.T}.N[d_1] - K_T.e^{-r.T}.N[d_2] \\ &= 2740.805274 \times e^{-0.05 \times 0.5} \times N[-0.032485301] - 2800 \times e^{-0.05 \times 0.5} \times N[-0.209261996] \\ &= 2740.805274 \times e^{-0.05 \times 0.5} \times 0.487042519 - 2800 \times e^{-0.05 \times 0.5} \times 0.417121859 \\ &= 1301.930185 - 1139.104633 \\ &= 162.8255519 \\ \end{aligned}###Commentary
Note that this question is almost identical to Question 903. This is because the futures price ##(F_{0,T})## of 2740.805274 points was chosen to match a 2% pa continuously compounded dividend yield ##(q)## in the underlying index:
###F_{0,T} = S_0.e^{(r-q).T}### ###2740.805274 = 2700 \times e^{(0.05-q) \times 0.5}### ###q = 0.05 - ln[2740.805274/2700]/0.5 = 0.02###So another way to solve this question is to work out ##q## (which equals ##r - ln[F_{0,T}/S_0]/T##) and then replace all instances of ##S_0## with ##S_0.e^{-q.T}## similarly to Question 903.
Question 794 option, Black-Scholes-Merton option pricing, option delta, no explanation
Which of the following quantities from the Black-Scholes-Merton option pricing formula gives the Delta of a European call option?
Where:
###d_1=\dfrac{\ln[S_0/K]+(r+\sigma^2/2).T)}{\sigma.\sqrt{T}}### ###d_2=d_1-\sigma.\sqrt{T}=\dfrac{\ln[S_0/K]+(r-\sigma^2/2).T)}{\sigma.\sqrt{T}}###No explanation provided.
Question 795 option, Black-Scholes-Merton option pricing, option delta, no explanation
Which of the following quantities from the Black-Scholes-Merton option pricing formula gives the Delta of a European put option?
No explanation provided.
Question 796 option, Black-Scholes-Merton option pricing, option delta, no explanation
Which of the following quantities from the Black-Scholes-Merton option pricing formula gives the risk-neutral probability that a European call option will be exercised?
No explanation provided.
Question 797 option, Black-Scholes-Merton option pricing, option delta, no explanation
Which of the following quantities from the Black-Scholes-Merton option pricing formula gives the risk-neutral probability that a European put option will be exercised?
No explanation provided.
Question 793 option, hedging, delta hedging, gamma hedging, gamma, Black-Scholes-Merton option pricing
A bank buys 1000 European put options on a $10 non-dividend paying stock at a strike of $12. The bank wishes to hedge this exposure. The bank can trade the underlying stocks and European call options with a strike price of 7 on the same stock with the same maturity. Details of the call and put options are given in the table below. Each call and put option is on a single stock.
| European Options on a Non-dividend Paying Stock | |||
| Description | Symbol | Put Values | Call Values |
| Spot price ($) | ##S_0## | 10 | 10 |
| Strike price ($) | ##K_T## | 12 | 7 |
| Risk free cont. comp. rate (pa) | ##r## | 0.05 | 0.05 |
| Standard deviation of the stock's cont. comp. returns (pa) | ##\sigma## | 0.4 | 0.4 |
| Option maturity (years) | ##T## | 1 | 1 |
| Option price ($) | ##p_0## or ##c_0## | 2.495350486 | 3.601466138 |
| ##N[d_1]## | ##\partial c/\partial S## | 0.888138405 | |
| ##N[d_2]## | ##N[d_2]## | 0.792946442 | |
| ##-N[-d_1]## | ##\partial p/\partial S## | -0.552034778 | |
| ##N[-d_2]## | ##N[-d_2]## | 0.207053558 | |
| Gamma | ##\Gamma = \partial^2 c/\partial S^2## or ##\partial^2 p/\partial S^2## | 0.098885989 | 0.047577422 |
| Theta | ##\Theta = \partial c/\partial T## or ##\partial p/\partial T## | 0.348152078 | 0.672379961 |
Which of the following statements is NOT correct?
To Delta and Gamma hedge the 1000 long put options, short 2078.4226 call options and long 2397.9617 stocks. This is the correct answer to option E about Gamma and Delta hedging.
To Gamma hedge the 1000 puts, set the portfolio Gamma equal to zero. Note that the Gamma of the shares is zero since their payoff is a straight line, there's no curvature:
###\Gamma_\text{portfolio} = n_\text{puts}.\Gamma_\text{put} + n_\text{calls}.\Gamma_\text{call} + n_\text{stocks}.\Gamma_\text{stock}### ###0 = 1000 \times 0.098885989 + n_\text{calls} \times 0.047577422 + n_\text{stocks} \times 0### ###\begin{aligned} n_\text{calls} &= \dfrac{-1000 \times 0.098885989}{0.047577422} \\ &= -2078.422592 \\ \end{aligned}###Since the number of calls to buy is negative, short 2078.422592 calls (sell the call options).
To Delta hedge the long 1000 puts and short 2078.422592 calls, set the portfolio Delta equal to zero. Note that the Delta of the shares is one since their payoff is a diagonal straight line with a gradient of one.
###\Delta_\text{portfolio} = n_\text{puts}.\Delta_\text{put} + n_\text{calls}.\Delta_\text{call} + n_\text{stocks}.\Delta_\text{stock}### ###0 = 1000 \times -0.552034778 + -2078.422592 \times 0.888138405 + n_\text{stocks} \times 1### ###\begin{aligned} n_\text{stocks} &= 1000 \times 0.552034778 + 2078.422592 \times 0.888138405 \\ &= 2397.961704 \\ \end{aligned}###Since the number of stocks to buy is positive, long 2397.961704 stocks.
To simply Delta hedge the long 1000 puts without Gamma hedging, set the portfolio Delta of puts and stocks equal to zero. This is the answer to option D. Again, the Delta of the shares is one since their payoff is a diagonal straight line with a gradient of one.
###\Delta_\text{portfolio} = n_\text{puts}.\Delta_\text{put} + n_\text{stocks}.\Delta_\text{stock}### ###0 = 1000 \times -0.552034778 + n_\text{stocks} \times 1### ###\begin{aligned} n_\text{stocks} &= 1000 \times 0.552034778 \\ &= 552.034778 \\ \end{aligned}###Since the number of stocks to buy is positive, long 552.034778 stocks.