Options, Futures and Risk Management - Standard Gaussian Distribution - Management Assessment Answer

Finance Management Assessment Task

Question 1
Consider a non-dividend paying stock subject to risk-free force of interest, r = 0.045 as well as volatility, σ = 0.24 and time horizon three months, i.e., T = 1/4 for a stock with current price S0 = $45. Recall that dSt = St(rdt + σdBt) (1) where {Bt}t≥0 is standard Brownian motion. Furthermore, recall that Brownian motion is Gaussian with mean 0 and variance t. That is, Bt ∼ N (0, t) (2) for each t fixed, 0 ≤ t ≤ T. It follows that we can exploit this normality, thus leading to the discretisation Sti − Sti−1 = Sti−1 (r?t + σ√ ?t) (3) for ?t = ti − ti−1 where we assume that  is a random draw from standard Gaussian distribution, i.e.∼ N (0, 1) (4)
Given our time horizon of 3 months, we would like to simulate prices for ?t = 0.005, we simulate iteratively. In other words, after the first time period, we get that St1 − S0 = S0(r?t + σ√ ?t) (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 (i) T − X, 0} ; i = 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 = S0N (d1) − Xe−rtN (d2) (7)

where d1 = ln(S0/X) + (r + σ 2/2)T σ √ T ; d2 = d1 − σ √ T (8)

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 − S0 + Xe−rT (9) derive the prices, P, given the simulated call prices from item 1.
Question 3
We now wish to connect the Binomial model to the continuous time equivalent provided by Black Scholes Merton. To that end, we need to introduce a measure
of volatility as measured by the standard deviation, σ. To that end, consider the inputs r = 0.045, σ = 0.24, T = 1/4 and S0 = $45
and strikes X = $40, $45, $50. For n = 1, 2, . . . , 100 set
1. u = exp(σ p T /n) and d = 1/u.

2. R = (1 + r/n)T
3. q = (R − d)/(u − d)
4. q
0 = uq/R
5. a which is the smallest positive integer greater than

(ln(S0/X) + n ln(d))/ ln(d/u) (10) 2

6. P[J ≥ a] and P[J 0 ≥ a]
7. Call option price C0 = S0P[J 0 ≥ a] − XR−nP[J ≥ a] (11) Next, compare these to Black Scholes Merton prices.
Question 4
Provide an executive summary that discusses the results obtained in the three methods used in questions 2 and 3. What do you notice? How do these methods compare?

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