# Fight Finance

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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? Which of the following quantities from the Black-Scholes-Merton option pricing formula gives the Delta of a European call option? Where: $$d_1=\dfrac{\ln⁡[S_0/K]+(r+\sigma^2/2).T)}{\sigma.\sqrt{T}}$$ $$d_2=d_1-\sigma.\sqrt{T}=\dfrac{\ln⁡[S_0/K]+(r-\sigma^2/2).T)}{\sigma.\sqrt{T}}$$ Which of the following quantities from the Black-Scholes-Merton option pricing formula gives the Delta of a European put option? 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? 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? A 12 month European-style call option with a strike price of$11 is written on a dividend paying stock currently trading at $10. The dividend is paid annually and the next dividend is expected to be$0.40, paid in 9 months. The risk-free interest rate is 5% pa continuously compounded and the standard deviation of the stock’s continuously compounded returns is 30 percentage points pa. The stock's continuously compounded returns are normally distributed. Using the Black-Scholes-Merton option valuation model, determine which of the following statements is NOT correct.

Which of the following statements about an option (either a call or put) and its underlying stock is NOT correct?

 European Call Option on a non-dividend paying stock Description Symbol Quantity Spot price ($) $S_0$ 20 Strike price ($) $K_T$ 18 Risk free cont. comp. rate (pa) $r$ 0.05 Standard deviation of the stock's cont. comp. returns (pa) $\sigma$ 0.3 Option maturity (years) $T$ 1 Call option price ($) $c_0$ 3.939488 Delta $\Delta = N[d_1]$ 0.747891 $N[d_2]$ $N[d_2]$ 0.643514 Gamma $\Gamma$ 0.053199 Theta ($/year) $\Theta = \partial c / \partial T$ 1.566433

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:

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:

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:

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:

A bank sells a European call option on a non-dividend paying stock and delta hedges on a daily basis. Below is the result of their hedging, with columns representing consecutive days. Assume that there are 365 days per year and interest is paid daily in arrears.

 Delta Hedging a Short Call using Stocks and Debt Description Symbol Days to maturity (T in days) 60 59 58 57 56 55 Spot price ($) S 10000 10125 9800 9675 10000 10000 Strike price ($) K 10000 10000 10000 10000 10000 10000 Risk free cont. comp. rate (pa) r 0.05 0.05 0.05 0.05 0.05 0.05 Standard deviation of the stock's cont. comp. returns (pa) σ 0.4 0.4 0.4 0.4 0.4 0.4 Option maturity (years) T 0.164384 0.161644 0.158904 0.156164 0.153425 0.150685 Delta N[d1] = dc/dS 0.552416 0.582351 0.501138 0.467885 0.550649 0.550197 Probability that S > K at maturity in risk neutral world N[d2] 0.487871 0.51878 0.437781 0.405685 0.488282 0.488387 Call option price ($) c 685.391158 750.26411 567.990995 501.487157 660.982878 ? Stock investment value ($) N[d1]*S 5524.164129 5896.301781 4911.152036 4526.788065 5506.488143 ? Borrowing which partly funds stock investment ($) N[d2]*K/e^(r*T) 4838.772971 5146.037671 4343.161041 4025.300909 4845.505265 ? Interest expense from borrowing paid in arrears ($) r*N[d2]*K/e^(r*T) 0.662891 0.704985 0.594994 0.551449 ? Gain on stock ($) N[d1]*(SNew - SOld) 69.052052 -189.264008 -62.642245 152.062648 ? Gain on short call option ($) -1*(cNew - cOld) -64.872952 182.273114 66.503839 -159.495721 ? Net gain ($) Gains - InterestExpense 3.516209 -7.695878 3.266599 -7.984522 ? Gamma Γ = d^2c/dS^2 0.000244 0.00024 0.000255 0.00026 0.000253 0.000255 Theta θ = dc/dT 2196.873429 2227.881353 2182.174706 2151.539751 2266.589184 2285.1895 In the last column when there are 55 days left to maturity there are missing values. Which of the following statements about those missing values is NOT correct? A hedge fund issued zero coupon bonds with a combined$1 billion face value due to be paid in 3 years. The promised yield to maturity is currently 6% pa given as a continuously compounded return (or log gross discrete return, $LGDR=\ln[P_T/P_0] \div T$).

The hedge fund owns stock assets worth \$1.1 billion now which are expected to have a 10% pa arithmetic average log gross discrete return $(\text{AALGDR} = \sum\limits_{t=1}^T{\left( \ln[P_t/P_{t-1}] \right)} \div T)$ and 30pp pa standard deviation (SDLGDR) in the future.

Analyse the hedge fund using the Merton model of corporate equity as an option on the firm's assets.

The risk free government bond yield to maturity is currently 5% pa given as a continuously compounded return or LGDR.

Which of the below statements is NOT correct? All figures are rounded to the sixth decimal place.