Part-B
Answer to question -1
Daily treasure yield curve rates on 12/04/2017 using Daily Treasury Yield Curve Rates
| Daily Treasure Yield Curve Rates On 12/04/2017 | |
| Date | interest |
| 1 Mo | 1.16 |
| 3 Mo | 1.29 |
| 6 Mo | 1.45 |
| 1 Yr | 1.66 |
| 2 Yr | 1.80 |
| 3 Yr | 1.93 |
| 5 Yr | 2.15 |
| 7 Yr | 2.29 |
| 10 Yr | 2.37 |
| 20 Yr | 2.58 |
| 30 Yr | 2.77 |
Part-B
Answer to question -1
Daily treasure yield curve rates on 12/04/2017 using Daily Treasury Yield Curve Rates
| Daily Treasure Yield Curve Rates On 12/04/2017 | |
| Date | interest |
| 1 Mo | 1.16 |
| 3 Mo | 1.29 |
| 6 Mo | 1.45 |
| 1 Yr | 1.66 |
| 2 Yr | 1.80 |
| 3 Yr | 1.93 |
| 5 Yr | 2.15 |
| 7 Yr | 2.29 |
| 10 Yr | 2.37 |
| 20 Yr | 2.58 |
| 30 Yr | 2.77 |
Part4: Black-Scholes and Binomial Option Pricing
Where:
- T : Time to maturity
- S0: Asset price X : Exercise price
- r: Continuously compounded risk free rate
- σ : Volatility of continuously compounded return on the stock
- N(∗) : Cumulative Normal Probability
| Black-Scholes Option Pricing Model with Dividends | |
| Current Stock Price | $ 200.00 |
| Exercise Price | $ 195.00 |
| Risk-Free Interest Rate | 4.00% |
| Expected Life of Option | 5 |
| Volatility | 20.0% |
| Dividend Yield | 0% |
| Call Option Value | $ 55.86 |
In the given Black Scholes, there is following options have been taken into consideration.
, which is given by • call option : c = S0N(d1) − Xe−rT N(d2) • put option : p = Xe−rT N(−d2) − S −rT 0 N(−d1) d1 = ln(S0/X) + (r + σ 2/2)T σ √ T d2 = ln(S0/X) + (r − σ 2/2)T σ √ T That is, d2 = d1 − σ √ T
Binomial tree model
| Binomial Tree Model | ||||||||||||||
| Tree model | ||||||||||||||
| Stocks | Currency | Futures | ||||||||||||
| S | 100 | 100 | S | 0.75 | 0.7501 | F | 525 | 525.0001 | ||||||
| K | 200 | 200 | X | 0.75 | 0.75 | X | 525 | 525 | ||||||
| r | 0.07 | 0.07 | r(domestic) | 0.07 | 0.07 | r | 0.06 | 0.06 | ||||||
| T | 0.5 | 0.5 | T | 0.75 | 0.75 | T | 0.416667 | 0.416667 | ||||||
| sigma | 0.3 | 0.3 | sigma | 0.04 | 0.04 | sigma | 0.151739 | 0.151739 | ||||||
| delta | 0.00001 | r(f) (foreign) | 0.09 | 0.09 | delta | 0.0001 | ||||||||
| dividend rate q | 0 | delta | 0.0001 | |||||||||||
| d(1) | -2.99647 | -2.99647 | d(1) | -0.41569 | -0.41184 | d(1) | 0.048974 | 0.048975 | ||||||
| d(2) | -3.2086 | -3.2086 | d(2) | -0.45033 | -0.44648 | d(2) | -0.04897 | -0.04897 | ||||||
| N(d1) | 0.001366 | 0.001366 | N(d1) | 0.338818 | 0.340227 | N(d1) | 0.51953 | 0.519531 | ||||||
| N(d2) | 0.000667 | 0.000667 | N(d2) | 0.326235 | 0.327624 | N(d2) | 0.48047 | 0.480471 | ||||||
| C | 0.007768 | 0.007768 | C | 0.005364 | 0.005396 | C | 20 | 20.00005 | ||||||
| P | 93.12885 | 93.12884 | P | 0.015959 | 0.015898 | P | 20 | 19.99995 | ||||||
| Call | Put | Call | Put | Call | Put | |||||||||
| DELTA = | 0.001366 | -0.99863 | 0.317361 | -0.61737 | 0.506703 | -0.46861 |
Monte Carlo
| Percentage Made | 90% |
| Percentage of First Ten Made | 90% |
| Attempt | Result |
| 1 | 1 |
| 2 | 1 |
| 3 | 1 |
| 4 | 1 |
| 5 | 1 |
| 6 | 1 |
| 7 | 0 |
| 8 | 1 |
| 9 | 1 |
| 10 | 1 |
| 11 | 1 |
| 12 | 1 |
| 13 | 1 |
| 14 | 1 |
| 15 | 1 |
| 16 | 1 |
| 17 | 0 |
| 18 | 1 |
| 19 | 1 |
| 20 | 1 |
| 21 | 1 |
| 22 | 1 |
| 23 | 1 |
| 24 | 1 |
| 25 | 1 |
| 26 | 1 |
| 27 | 1 |
| 28 | 1 |
| 29 | 1 |
| 30 | 1 |
| 31 | 1 |
| 32 | 1 |
| 33 | 1 |
| 34 | 0 |
| 35 | 0 |
| 36 | 1 |
| 37 | 1 |
| 38 | 0 |
| 39 | 1 |
| 40 | 1 |
| 41 | 1 |
| 42 | 1 |
| 43 | 1 |
| 44 | 1 |
| 45 | 1 |
| 46 | 1 |
| 47 | 1 |
| 48 | 1 |
| 49 | 1 |
| 50 | 1 |
| 51 | 1 |
| 52 | 1 |
| 53 | 1 |
| 54 | 1 |
| 55 | 1 |
| 56 | 1 |
| 57 | 1 |
| 58 | 1 |
| 59 | 1 |
| 60 | 1 |
| 61 | 1 |
| 62 | 1 |
| 63 | 1 |
| 64 | 1 |
| 65 | 0 |
| 66 | 1 |
| 67 | 1 |
| 68 | 1 |
| 69 | 1 |
| 70 | 1 |
| 71 | 1 |
| 72 | 1 |
| 73 | 1 |
| 74 | 1 |
| 75 | 1 |
| 76 | 1 |
| 77 | 1 |
| 78 | 0 |
| 79 | 1 |
| 80 | 1 |
| 81 | 0 |
| 82 | 1 |
| 83 | 0 |
| 84 | 1 |
| 85 | 1 |
| 86 | 1 |
| 87 | 1 |
| 88 | 1 |
| 89 | 1 |
| 90 | 1 |
| 91 | 1 |
| 92 | 1 |
| 93 | 1 |
| 94 | 1 |
| 95 | 1 |
| 96 | 1 |
| 97 | 1 |
| 98 | 1 |
| 99 | 0 |
| 100 | 1 |
Compute the Greeks that include delta, gamma, Vega, rho, and theta for the put option
K > ST : the given exercise would be assessed
Delta: ∆ = ∂V ∂S = −N(−d1
- . (ii) K < ST : the holder makes a profit of ST − K by exercising the option.
- Thus the price or value of the option at maturity is CT = (ST − K) + = max(ST − K, 0).
Ct − Pt = St − Ke−r(T −t) . the undertaken holder values would be assessed through the process.
Ct − Pt > St − Ke−r(T −t)
(Codagnone, Abadie, & Biagi,2016).
Where: V : Value of a put option, which is the same as p
N0 (d1) = e − d 2 1 2 · √ 1 2π
Using a four-step binomial model (∆t = 0.5).
| Binomial model | |
| Current Stock Price | $ 200.00 |
| Exercise Price | $ 195.00 |
| Risk-Free Interest Rate | 4.00% |
| Expected Life of Option | 5 |
| Volatility | 20.0% |
| Dividend Yield | 0% |
| Call Option Value | $ 54.38 |
American put option price
| Binomial model | |
| Current Stock Price | $ 200.00 |
| Exercise Price | $ 198.00 |
| Risk-Free Interest Rate | 4.00% |
| Expected Life of Option | 5 |
| Volatility | 20.0% |
| Dividend Yield | 0% |
| Call Option Value | $ 52.38 |
Two option prices and provide your finding from the results
Both two options are based on the interest rate of return, dividend yield and exercise price. It is analyzed that the American put option price is showing the lower amount of value which is 2 points lower as compared to the four year binomial model.
References
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