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
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
VaR for 90% Confidence level
10-day VaR for SBN = $6000000 x 1.5% x 1.281551566 x 10^0.5
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
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
- 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
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
Diversification benefit = $294,262.32 + $882,786.95 – $811,225.30
Diversification benefit = $365,823.97
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
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
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.
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.