Reference no: EM132388950

Question 1

Consider a non-dividend paying stock subject to risk-free force of interest, r = 0.05 as well as volatility, σ = 0.2 and time horizon three months, i.e., T = 1/4 for a stock with current price S₀ = $45. Recall that

dS_t = S_t(rdt + σdB_t)              (1)

where {B_t}_t≥0 is standard Brownian motion. Furthermore, recall that Brownian motion is Gaussian with mean 0 and variance t. That is,

B_t ~ N(0, t)           (2)

for each t fixed, 0 ≤ t ≤ T. It follows that we can exploit this normality, thus leading to the discretisation

S_ti - St_i-1 = St_i-1, (rΔt + σ∈√Δt) (3)

for Δt = t_i, - t_i-1 where we assume that ∈ is a random draw from standard Gaussian distribution, i.e.

∈ ~ N(0, 1)

Given our time horizon of 3 months, we would like to simulate prices for Δt = 0.01, we simulate iteratively. In other words, after the first time period, we get that

St₁ - S₀ = S₀(rΔt + σ∈√Δt) (5)

Where we repeat until the final time step, T = 0.25. Your task is to simulate 100 such sample paths.

Question 2

Given the simulated sample paths from the previous question,

1. Find prices of European call options for strike prices, X = $40, $45, $50 (Note that the payoff of each sample path is max{S_T(i) - X, 0); 1, 2,...,100 (6)
Then take the discounted average of these. )

2. Find the continuous time equivalents for these calls, i.e. Black Scholes.

Recall that the formula for this is

C = S_oN.(d₁)- Xe^-rt N(d₂) (7)

Where

d₁ = (ln (S₀/X) + (r + σ²/2)T)/σ√T ; d₂ = d₁ - σ√T
What do you notice!

3. Given the same simulated sample paths calculate the price of put options for strikes X = $40, $45, $50.

4. Next, use put-call parity, i.e.
P = C-S_o + Xe^-rT

derive the prices, F, given the simulated call prices from item 1.

Question 3

Given your simulated stock price sample paths, calculate the following:

1. Straddle options for strike prices, X = $40, $45, $50
2. Covered call options for the same strikes.