Problem 2: Recall the formula for computing the price Co of an option (derivative of the BLM stock prices) That yields a payoff at time T, denote by CT: 1 Co= E* (CT), (1+r)T where* refers to the fact that we must use the value p* instead of the original (real) P for the up/down probability of the BLM. (The real value of P is not needed for pricing.) Also recall that for Cr= (ST-K)+, the European call option, the expected value, E* (ST-K) can be computed explicitly yielding the famous Black-Scholes-Merton option pricing formula: Σ(1) Φ You are to use this formula to exactly obtain the price on the one hand, and then use Monte Carlo simulation on the other hand to compare and thus see how accurate the Monte Carlo method can be. Co= #write your code here 1 (1+r)T (p*)*(1-p*)-(ud- So - K)+. T-k Here are the parameters to use: T = 10, r = 0.05, u = 1.15, d = 1.01, So= 50, K = 70. Recall that p* = 1+r-d u-d For the Monte Carlo, use n = 100, n = 1000, n = 10, 000 iid copies of Cr (for averaging) to see how it gets more accurate as n increases.
Problem 2: Recall the formula for computing the price Co of an option (derivative of the BLM stock prices) That yields a payoff at time T, denote by CT: 1 Co= E* (CT), (1+r)T where* refers to the fact that we must use the value p* instead of the original (real) P for the up/down probability of the BLM. (The real value of P is not needed for pricing.) Also recall that for Cr= (ST-K)+, the European call option, the expected value, E* (ST-K) can be computed explicitly yielding the famous Black-Scholes-Merton option pricing formula: Σ(1) Φ You are to use this formula to exactly obtain the price on the one hand, and then use Monte Carlo simulation on the other hand to compare and thus see how accurate the Monte Carlo method can be. Co= #write your code here 1 (1+r)T (p*)*(1-p*)-(ud- So - K)+. T-k Here are the parameters to use: T = 10, r = 0.05, u = 1.15, d = 1.01, So= 50, K = 70. Recall that p* = 1+r-d u-d For the Monte Carlo, use n = 100, n = 1000, n = 10, 000 iid copies of Cr (for averaging) to see how it gets more accurate as n increases.
Computer Networking: A Top-Down Approach (7th Edition)
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ISBN:9780133594140
Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
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Using python
![# Problem 2: Pricing an Option Using the Black-Scholes-Merton Model
### Formula for Option Pricing:
To compute the price \( C_0 \) of an option, which is a derivative of BLM stock prices with a payoff at time \( T \), denoted by \( C_T \), we use the following formula:
\[ C_0 = \frac{1}{(1 + r)^T} E^* (C_T) \]
where:
- The asterisk (*) indicates the use of the value \( p^* \) instead of the original probability \( P \) for the up/down probability of the BLM model. The actual value of \( P \) is not needed for pricing.
- For \( C_T = (S_T - K)^+ \), representing a European call option, the expected value \( E^* ((S_T - K)^+) \) can be computed using the Black-Scholes-Merton option pricing formula:
\[ C_0 = \frac{1}{(1 + r)^T} \sum_{k=0}^{T} \binom{T}{k} (p^*)^k (1 - p^*)^{T-k} (u^k d^{T-k} S_0 - K)^+ \]
### Instructions:
- Utilize the above formula to precisely calculate the option price.
- Additionally, perform a Monte Carlo simulation to verify the accuracy of your result.
### Parameters:
- Time, \( T = 10 \)
- Interest rate, \( r = 0.05 \)
- Upward movement factor, \( u = 1.15 \)
- Downward movement factor, \( d = 1.01 \)
- Initial stock price, \( S_0 = 50 \)
- Strike price, \( K = 70 \)
Recall the probability adjustment formula:
\[ p^* = \frac{1 + r - d}{u - d} \]
### Monte Carlo Simulation:
- Conduct simulations with \( n = 100 \), \( n = 1000 \), and \( n = 10,000 \) independent and identically distributed copies of \( C_T \) for averaging, and observe the increase in accuracy as \( n \) increases.
```python
# Code placeholder
# You can write your code here
# Feel free to add more cells if needed
```](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffd86cda1-af2d-4585-b012-7ab18b7ef9b3%2F6b4855e5-b6eb-4c46-9ce0-39cbd0dab519%2Fqcsfywm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:# Problem 2: Pricing an Option Using the Black-Scholes-Merton Model
### Formula for Option Pricing:
To compute the price \( C_0 \) of an option, which is a derivative of BLM stock prices with a payoff at time \( T \), denoted by \( C_T \), we use the following formula:
\[ C_0 = \frac{1}{(1 + r)^T} E^* (C_T) \]
where:
- The asterisk (*) indicates the use of the value \( p^* \) instead of the original probability \( P \) for the up/down probability of the BLM model. The actual value of \( P \) is not needed for pricing.
- For \( C_T = (S_T - K)^+ \), representing a European call option, the expected value \( E^* ((S_T - K)^+) \) can be computed using the Black-Scholes-Merton option pricing formula:
\[ C_0 = \frac{1}{(1 + r)^T} \sum_{k=0}^{T} \binom{T}{k} (p^*)^k (1 - p^*)^{T-k} (u^k d^{T-k} S_0 - K)^+ \]
### Instructions:
- Utilize the above formula to precisely calculate the option price.
- Additionally, perform a Monte Carlo simulation to verify the accuracy of your result.
### Parameters:
- Time, \( T = 10 \)
- Interest rate, \( r = 0.05 \)
- Upward movement factor, \( u = 1.15 \)
- Downward movement factor, \( d = 1.01 \)
- Initial stock price, \( S_0 = 50 \)
- Strike price, \( K = 70 \)
Recall the probability adjustment formula:
\[ p^* = \frac{1 + r - d}{u - d} \]
### Monte Carlo Simulation:
- Conduct simulations with \( n = 100 \), \( n = 1000 \), and \( n = 10,000 \) independent and identically distributed copies of \( C_T \) for averaging, and observe the increase in accuracy as \( n \) increases.
```python
# Code placeholder
# You can write your code here
# Feel free to add more cells if needed
```
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