The Mean Rate Of Return On A Stock Is Estimated At 20

The Mean Rate Of Return On A Stock Is Estimated At 20 While The V

The mean rate of return on a stock is estimated at 20% while the volatility is 40%. The risk-free interest rate is 5%. The assignment involves several analytical and financial computations related to stock price modeling and options valuation.

Paper For Above instruction

Understanding the dynamics of stock prices and their associated options is fundamental in financial mathematics. This paper addresses various facets of stock modeling and derivatives pricing, focusing on a stock with specified parameters, constructing probability measures, and evaluating options under different probability frameworks to identify potential arbitrage opportunities.

Part A: Mean Log Price Relative

The first part requires calculating the mean of the log price relative of the stock. Given the mean return (μ) is 20%, or 0.20, and the volatility (σ) is 40%, or 0.40, we model the stock price process as a geometric Brownian motion. The expected logarithmic return over a small period Δt is approximately (μ - 0.5σ^2)Δt, which aligns with the properties of Itô diffusion in continuous time models.

Assuming a single period of one year, the mean of the log price relative (ln(S_T/S_0)) is given by:

Mean log price relative = (μ - 0.5σ^2) × T

where T = 1 year. Substituting the values:

Mean log relative = (0.20 - 0.5×0.40^2) × 1 = (0.20 - 0.5×0.16) = (0.20 - 0.08) = 0.12

Thus, the mean of the log price relative is 0.12.

Part B: Constructing a 10-Period One-Year Tree of Stock Prices

Constructing a binomial tree involves discretizing the time horizon into 10 steps, each of duration 0.1 years. The parameters for the binomial model are derived from the continuous parameters: up and down factors (u and d) and the risk-neutral probability (p).

The up-factor (u) is calculated as:

u = e^(σ√Δt) = e^(0.40×√0.1) ≈ e^(0.40×0.3162) ≈ e^0.1265 ≈ 1.1347

The down-factor (d) is:

d = e^(-σ√Δt) ≈ e^(-0.1265) ≈ 0.8814

The stock price at each node is obtained by multiplying the previous price by either u or d depending on the path taken. Starting from an initial price S_0, say 100, the possible prices after ten steps are computed as:

- For n up-moves and (10 - n) down-moves, the price is:

S_n = S₀ × uⁿ × d^{10 - n}

Part C: Statistical Probabilities of Stock Prices

Under the real-world measure, the probabilities of paths depend on the statistical likelihood of each sequence of up/down moves. The probability of exactly n up-moves out of 10 is given by the binomial distribution:

P(n) = C(10, n) × pⁿ × (1 - p)^{10 - n}

where p is the statistical probability derived from the actual mean and variance of the stock return, calculated through historical data or assumed distribution.

Part D: Risk-Neutral Probabilities

The risk-neutral probability (q) adjusts the statistical probabilities so that the expected growth of the stock matches the risk-free rate. It is computed as:

q = (e^rΔt - d) / (u - d)

where r = 0.05 (5%). Plugging in the numbers:

q = (e^{0.05×0.1} - 0.8814) / (1.1347 - 0.8814) ≈ (e^{0.005} - 0.8814) / 0.2533 ≈ (1.005 - 0.8814) / 0.2533 ≈ 0.1236 / 0.2533 ≈ 0.4885

Part E: Graphical Comparison of Probabilities and Stock Prices

Graphing both the statistical and risk-neutral probabilities against the resulting stock prices involves plotting the binomial tree probabilities on the y-axis and stock prices on the x-axis, illustrating how each framework weights different outcomes. In practice, the statistical probabilities assign more weight to paths based on historical likelihoods, whereas the risk-neutral probabilities modulate these to ensure no arbitrage.

Part F: Final Cash Flows for Call and Put Options at Strikes 80 and 120

At maturity, the payoff of a call option with strike K is:

C = max(S_T - K, 0)

Similarly, the payoff for a put option is:

P = max(K - S_T, 0)

Using the possible stock prices at maturity from the binomial tree, we compute these payoffs for each terminal node at strike prices of 80 and 120.

Part G and H: Option Pricing Using Statistical and Risk-Neutral Probabilities

The present value of the options under each probability measure is calculated by discounting the expected payoffs using the relevant probabilities:

Price = e^-rT × Σ (Probability) × (Payoff)

For the statistical probabilities, this reflects the real-world likelihoods, whereas for risk-neutral probabilities, it corresponds to theoretical no-arbitrage pricing.

Part I and J: Arbitrage Opportunities and Failures

An arbitrage opportunity exists when a price discrepancy allows for riskless profit. When prices are derived under statistical probabilities, differences tend to be more reflective of real-world expectations, and arbitrage might occur if, for instance, the market misprices options relative to historical likelihoods. Conversely, using risk-neutral probabilities ensures no arbitrage because these are constructed to align with the fundamental theorem of asset pricing. The failure of arbitrage under risk-neutral measures confirms the theoretical robustness of this framework in preventing arbitrage opportunities.

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