**Module 6，13章，**

**The valuation of European call based on the Black-Scholes-Merton model.**

**The valuation of European put based on the Black-Scholes-Merton model.**

**See the example on page 305.**

**13.13. What is the price of a European call option on a non-dividend-paying stock when the**

**stock price is $52, the strike price is $50, the risk-free interest rate is 12% per annum, the**

**volatility is 30% per annum, and the time to maturity is three months?**

, , , , and .

The price of the European call is

** **

**13.14. What is the price of a European put option on a non-dividend-paying stock when the**

**stock price is $69, the strike price is $70, the risk-free interest rate is 5% per annum, the**

**volatility is 35% per annum, and the time to maturity is six months?**

, , , , and .

The price of the European put is

**Module 8**

**Value-at-risk (VaR).**

**Two-asset portfolio.**

**Diversification benefit.**

**See the example on page 436-438.**

The volatility of SBN company is 1.5% per day and the size of the position is $6 million. Assuming that the change is normally distributed, find a one-day 97% VaR and 10-day 90% VaR.

VaR for 97% Confidence level

10-day VaR for SBN = $6000000 x 1.5% x 1.880793608 x 10^0.5

= **$535,283.24**

VaR for 90% Confidence level

10-day VaR for SBN = $6000000 x 1.5% x 1.281551566 x 10^0.5

= **$364,735.97**

Consider a portfolio consisting of $4 million invested in Stock Fund and $6 million in Bond Fund. The daily volatility of Stock Fund is 1% and the daily volatility of Bond Fund is 2%. The correlation coefficient between two funds is -0.4 and they are normally distributed.

- a) Find the 10-day 99% VaR.

Standard Deviation of portfolio when Corr = – 40%

Standard deviation of stock = SdA = 1% ; Standard deviation of bond = SdB = 2%;

Weight of stock = wA = 40% ; Weight of bond = wB = 60%

Standard Deviation of portfolio = (wA^2*sdA^2+wB^2*sdB^2+2*Corr*wA*sdA*wB*sdB)^0.5

= (40%^2*1%^2+60%^2*2%^2+2*-40%*40%*1%*60%*2%)^0.5

= 1.102724%

10-day VaR of portfolio at 99% confidence level

= Portfolio value x Standard deviation daily x Z-score for 99% confidence level x Days^0.5

= 10,000,000 x 1.102724% x 2.326347874 x 10^0.5

**= $811,225.30**

.

- b) Find the diversification benefit.

Diversification benefit = VaR of stock + VaR of Bond – VaR of Portfolio

VaR of stock

= Stock value x Standard deviation daily x Z-score for 99% confidence level x Days^0.5

= 4000000 x 1% x 2.326347874 x 10^0.5

= $294,262.32

VaR of Bond

= Bond value x Standard deviation daily x Z-score for 99% confidence level x Days^0.5

= 6000000 x 2% x 2.326347874 x 10^0.5

= $882,786.95

Diversification benefit = $294,262.32 + $882,786.95 – $811,225.30

Diversification benefit = **$365,823.97**

**Module 8**

**Weather derivatives.**

**HDD and CDD.**

**See the example on page 515-516.**

Suppose that you buy a weather call option with strike price = 200 based on HDD because you are concerned about unexpectedly cool weather in summer. The payment rate on the option contract is $1,000 and the payment cap is $200,000.

- a) If the cumulative HDD = 320, what is your payoff?

payoff on the option = min(200,000, 320 – 200 * 1000)

payoff on the option = 120,000

- b) If the cumulative HDD = 450, what is your payoff?

payoff on the option = min (200,000, 450 – 200* 1000)

payoff on the option = min (200,000 , 250,000) = 200,000

**Module 6 **

**Risk-neutral valuation.**

**Risk-neutral valuation vs Real World valuation.**

**See the example on page 272-274.**

**12.10. A stock price is currently $80. It is known that at the end of four months it will be either**

**$75 or $85. The risk-free interest rate is 5% per annum with continuous compounding.**

**What is the value of a four-month European put option with a strike price of $80? Use**

**no-arbitrage arguments.**

The value of the option is therefore $1.80.

,

**12.11. A stock price is currently $40. It is known that at the end of three months it will be either**

**$45 or $35. The risk-free rate of interest with quarterly compounding is 8% per annum.**

**Calculate the value of a three-month European put option on the stock with an exercise**

**price of $40. Verify that no-arbitrage arguments and risk-neutral valuation arguments**

**give the same answers.**

the value of the option is $2.06.

The expected value of the option in a risk-neutral world is

This has a present value of

This is consistent with the no-arbitrage answer.

**12.18. The futures price of a commodity is $90. Use a three-step tree to value (a) a nine-month**

**American call option with strike price $93 and (b) a nine-month American put option**

**with strike price $93. The volatility is 28% and the risk-free rate (all maturities) is 3%**

**with continuous compounding.**

The tree for valuing the call is in Figure S12.5a and that for valuing the put is in Figure S12.5b. The values are 7.94 and 10.88, respectively.

**CALL PUT**

**Module 7**

**Dynamic delta hedging.**

**See the example on page 365-367.**

** **

**A trader sells 10 call option contracts on a certain stock. The option price is $6, the stock price is $40, and the option’s delta is 0.4. The trader’s portfolio consists of short call and long stock.**

**a) Find the number of stocks purchased to make your portfolio delta-neutral.**

Delta for 1 short call option -Δ= -0.4

The total delta of options 10 x (- 0.4) = – 4.

The delta of 1 stock 1

Therefore, the number of stocks purchased to make the portfolio delta neutral = 4/1 = 4

**b) When the stock price increases by $2, prove that your portfolio is delta-neutral by showing that gain and loss are offset.**

When the stock price increase, the value of the stock = 42 x 4 = 168

Gain on stocks = 168 – (4 x 40) = 8

Value of the option position = 10 x (6- (2 x 0.4)) = 52

Loss on options = (10 x 6) – 52 = 8

Therefore, gain and loss are offset.

** **