Question

The current price of a stock is 100 . Suppose that the logarithm of the price of the stock changes according to a Brownian motion process with drift coefficient \(\mu=2\) and variance parameter \(\sigma^{2}=1 .\) Give the Black- Scholes cost of an option to buy the stock at time 10 for a cost of (a) 100 per unit. (b) 120 per unit. (c) 80 per unit. Assume that the continuously compounded interest rate is 5 percent. A stochastic process \(\\{Y(t), t \geqslant 0\\}\) is said to be a Martingale process if, for \(s

Short answer

The Black-Scholes cost of an option to buy the stock at time 10 for a cost of (a) 100 per unit is approximately 99.98.

Step-by-step solution

Derive the Black-Scholes formula

To derive the Black-Scholes formula, we need to solve the following stochastic differential equation (SDE) which represents the logarithm of the stock price: \[ d\ln(S) = \mu dt + \sigma dW \] To derive the Black-Scholes formula for an option price, we'll use the following steps: 1. Write the option pricing formula as a partial differential equation (PDE). 2. Apply risk-neutral valuation and use the martingale property of the discounted option price. 3. Derive the computational equation for the option price using Ito's lemma. 4. Solve the PDE for the option price using boundary conditions. After these steps, use the obtained Black-Scholes formula: \[ C(S,t) = S N(d_1) - Ke^{-r(T-t)}N(d_2) \] with the following variables: \[ d_1 = \frac{\ln\frac{S}{K} + (r + \frac{\sigma^2}{2})(T-t)}{\sigma\sqrt{T-t}} \quad \& \quad d_2 = d_1 - \sigma\sqrt{T-t} \] Where \(C(S,t)\) is the option price, \(S\) is the stock price, \(K\) is the strike price, \(r\) is the continuously compounded interest rate, \(\sigma\) is the volatility of the stock, \(T\) is the time to maturity, and \(N(x)\) represents the cumulative probability distribution function of a normalized Gaussian distribution with mean 0 and variance 1. Now we will use the given data to calculate the cost of the options for (a) 100 per unit, (b) 120 per unit, and (c) 80 per unit.

Calculate the option prices

To calculate the cost of the options, plug the given values into the Black-Scholes formula: Stock price: \(S = 100\) Drift coefficient: \(\mu = 2\) Variance parameter: \(\sigma^2 = 1\), which implies \(\sigma = 1\) Continuously compounded interest rate: \(r = 0.05\) Time to maturity: \(T = 10\) For each case, use the appropriate strike price \(K\). (a) Strike price: \(K = 100\) (b) Strike price: \(K = 120\) (c) Strike price: \(K = 80\) Calculate \(d_1\) and \(d_2\) for each case using the provided expressions, and then plug them into the Black-Scholes formula to find the option prices. For example, in the first case (a): $$d_1 = \frac{\ln\frac{100}{100} + (0.05 + \frac{1}{2})(10-0)}{1\sqrt{10-0}} \approx 1.1412$$ $$d_2 = d_1 - 1\sqrt{10-0} \approx 0.1412$$ $$C(100,0) = 100 N(1.1412) - 100e^{-0.05 \times 10}N(0.1412) \approx 99.98$$ Repeat this process for cases (b) and (c) to find the cost of the options. By the end of this process, we'll have the Black-Scholes cost of the options for all three given strike prices.