All topics from Lectures 0 􀀀 12 are relevant for the midterm, including regular expressions
(but not including the pumping lemma). You should, in particular be comfortable with all
questions that have appeared in the rst two assignments, or have been solved during the
tutorials. Below, I am listing some additional practice questions.

Problem 1.

1 [ (14 marks) ] (Drifting Binomial Lattice)
In lectures we have considered the non-drift lattice. In this assignment we consider the Drifting Lattice described in Section $5.3$ of the course notes, i.e.,
\begin{aligned} u= & \exp \left[\sigma \sqrt{\Delta t}+\left(r-\sigma^2 / 2\right) \Delta t\right] \ d= & \exp \left[-\sigma \sqrt{\Delta t}+\left(r-\sigma^2 / 2\right) \Delta t\right] \ & S_{j+1}^{n+1}=u S_j^n ; S_j^{n+1}=d S_j^n \ q^*= & \frac{1}{2} \end{aligned}
Table 1: Some typical option parameters
\begin{tabular}{|c|c|}
\hline$\sigma$ & $20 \%$ \
$r$ & $3 \%$ \
Time to expiry T & 1 year \
Strike Price & $\$ 10$\ Initial asset price$S(0)$&$\$10$ \
\hline
\end{tabular}
Implement European option pricing using this binomial lattice. Your code should take only $O(N)$ storage NOT $O\left(N^2\right), N=T / \Delta t$. This can be done efficiently if you store the payoff in an array of size $O(N)$ and index it appropriately.
(a) (9 marks) Write a code to price an European put option with the Drifting Lattice described (see also in Section $5.3$ of the course notes). Using data in Table 1, assuming no dividend, write a

binomial pricing code. Test your code for standard European call and put by comparing your results with the exact solutions from the Matlab function blsprice as follows. Let $V_0^{\text {exact }}$ be the exact initial price $V(S(0), 0)$ (from the Black-Scholes formula), and $V^{\text {tree }}(\Delta t)$ be the value from a lattice pricer with the time step $\Delta t$. If convergence rate is linear, then it can be shown that
$$V_0^{\text {tree }}(\Delta t)=V_0^{\text {exact }}+\alpha \Delta t+o(\Delta t)$$
where $\alpha$ is a constant independent of $\Delta t$. Then the ratio below
$$\lim _{\Delta t \rightarrow 0} \frac{V_0^{\text {tree }}((\Delta t) / 2)-V_0^{\text {tree }}(\Delta t)}{V_0^{\text {tree }}((\Delta t) / 4)-V_0^{\text {tree }}((\Delta t) / 2)}$$
would be 2. If convergence rate is quadratic, then
$$V_0^{\text {tree }}(\Delta t)=V_0^{\text {exact }}+\alpha(\Delta t)^2+o\left((\Delta t)^2\right)$$
where $\alpha$ is some constant independent of $\Delta t$. What is the ratio when convergence is quadratic? Does your convergence table indicate a linear or quadratic convergence rate? Explain.
Show tables with the initial option value $V(S(0), 0)$ as a function of $\Delta t$. Start off with a timestep $(\Delta t)^0=.05$, and show the option value for $(\Delta t)^0 / 2,(\Delta t)^0 / 4, \ldots$. You should see your results converging to the blsprice value. Your tables should look like Table 2.
Table 2: Convergence Test
\begin{tabular}{|c|c|c|c|}
\hline$\Delta t$ & Value & Change & Ratio \
\hline \hline $.05$ & $V_1$ & & \
$.025$ & $V_2$ & $V_2-V_1$ & \
$.0125$ & $V_3$ & $V_3-V_2$ & $\frac{V_2-V_1}{V_3-V_2}$ \
$\ldots$ & $\ldots$ & $\ldots$ & $\ldots$ \
\hline
\end{tabular}
(b) (5 marks) For (1), the strike price may not equal to any binomial node $S_j^N, j=0, \ldots, N$. implement the smoothing payoff method at the $t=T$ (Section $5.4$ of the notes). Repeat the experiment of question 1. What do you see?

To implement the binomial pricing method for a European put option using the drifting lattice described in Section 5.3 of the course notes, we can follow these steps:

1. Define the option parameters:
pythonCopy codesigma = 0.2
r = 0.03
T = 1.0
K = 10.0
S0 = 10.0
N = 1000  # number of time steps

1. Calculate the lattice parameters:
pythonCopy codedt = T / N
u = np.exp(sigma * np.sqrt(dt) + (r - sigma ** 2 / 2) * dt)
d = np.exp(-sigma * np.sqrt(dt) + (r - sigma ** 2 / 2) * dt)
q = 0.5

1. Initialize the lattice values for the stock prices and option payoffs:
pythonCopy codeS = np.zeros((N+1, N+1))
V = np.zeros((N+1, N+1))
for j in range(N+1):
for i in range(j+1):
S[i, j] = S0 * (u ** i) * (d ** (j-i))
V[i, j] = max(K - S[i, j], 0)

1. Backward induction to calculate the option price at time 0:
pythonCopy codefor j in range(N-1, -1, -1):
for i in range(j+1):
V[i, j] = np.exp(-r*dt) * (q*V[i, j+1] + (1-q)*V[i+1, j+1])

1. Implement the smoothing payoff method at time T if the strike price is not equal to any binomial node:
pythonCopy codeif K not in S[:, N]:
j_star = np.argmax(S[:, N] > K) - 1
m_star = (V[j_star, N] - V[j_star+1, N]) / (S[j_star, N] - S[j_star+1, N])
b_star = V[j_star, N] - m_star * S[j_star, N]
for j in range(j_star+1):
V[j, N] = max(K - S[j, N], m_star*S[j, N]+b_star)

1. Calculate the initial option value at time 0:
pythonCopy codeput_price = V[0

To implement the smoothing payoff method at $t=T$, we first find the two nodes $S_{j}$ and $S_{j+1}$ such that $S_j \le K \le S_{j+1}$. Then we define the smoothed payoff as follows:

$\tilde{V}j=\frac{K-S_j}{S{j+1}-S_j} V_{j+1}+\frac{S_{j+1}-K}{S_{j+1}-S_j} V_j$

where $V_j$ and $V_{j+1}$ are the option prices at nodes $S_j$ and $S_{j+1}$, respectively.

We modify our previous code to implement the smoothing payoff method at $t=T$ when the strike price does not coincide with any binomial node. The modified code is shown below:

pythonCopy codeimport numpy as np
from scipy.stats import binom
from scipy.stats import norm

def european_put_binomial(S0, K, T, r, sigma, N):
delta_t = T / N
u = np.exp(sigma * np.sqrt(delta_t) + (r - sigma ** 2 / 2) * delta_t)
d = np.exp(-sigma * np.sqrt(delta_t) + (r - sigma ** 2 / 2) * delta_t)
p_star = 0.5
nodes = np.zeros(N + 1)
for j in range(N + 1):
nodes[j] = S0 * u ** (N - j) * d ** j
put_values = np.maximum(K - nodes, 0)
for i in range(N - 1, -1, -1):
put_values = np.exp(-r * delta_t) * (p_star * put_values[:-1] + (1 - p_star) * put_values[1:])
if K not in nodes:
j = np.argmax(nodes > K) - 1
put_values[j] = (K - nodes[j])/(nodes[j+1]-nodes[j])*put_values[j+1] + \
(nodes[j+1]-K)/(nodes[j+1]-nodes[j])*put_values[j]
return put_values[0]

# Example usage
S0 = 10
K = 11
T = 1
r = 0.03
sigma = 0.2
N = 100
put_price = european_put_binomial(S0, K, T, r, sigma, N)
print(f"The European put option price is: {put_price:.4f}")


We can now test the modified code for the case when the strike price $K=11$. The convergence table is shown below:

Problem 2.

$2[(12$ marks) $]$ (Drifting Lattice and Dividend) Assume that the underlying stock pays dividend at the dividend payment time $t_d$ where $T$ is the expiry. The amount of dividend payment $\mathcal{D}$ at each $t_d$ is specified below $$\mathcal{D}=\max \left(\rho \cdot S\left(t_d^{-}\right), \mathcal{D}_0\right)$$ where $S\left(t_d^{-}\right)$is the stock price immediately before dividend payment date $t_d, \rho>0$ is a constant dividend rate, and $\mathcal{D}_0>0$ is a constant, specifying a dividend floor. Note that, when the stock pays dividend at $t$, the stock price at $t^{+}, S\left(t^{+}\right)$, becomes $S\left(t^{-}\right)-\mathcal{D}$, where $t^{-}$and $t^{+}$denote immediately before and after the time of stock dividend payment respectively. Under no arbitrage, the option value immediately before dividend payment and immediately after dividend payment should remain the same. Your handling of dividend should not require change to the lattice (1). In $\$ 5.5 .2$of the course notes, the case when dividend is a fixed amount is discussed. Modify the binomial lattice call and put option pricing to handle the dividend payment case in this question using interpolation similar to the technique described in$\S 5.5 .2$. (a) (6 marks) Now, repeat Question 1 (a) (including checking for convergence), which implements smoothed payoff at T, but under above dividend payment specifications with$t_d=\frac{T}{2}, \mathcal{D}_0=0.3$and$\rho=0.3 \%$. To modify the binomial lattice call and put option pricing to handle the dividend payment case, we use the following steps: 1. Compute the ex-dividend stock price$S^{ex}$, which is the stock price immediately after the dividend payment, using the formula: $$S^{ex} = S^{-} – \mathcal{D}$$ 1. Compute the risk-neutral probabilities$p_u$and$p_d$using the formula: $$p_u = \frac{e^{(r-\delta) \Delta t} – d}{u-d}$$ $$p_d = 1 – p_u$$ where$r$is the risk-free interest rate,$\delta$is the dividend yield,$\Delta t$is the time step,$u$and$d$are the up and down factors respectively. 1. Compute the option prices at the final nodes using the formula: $$C_N = \max(S_N – K, 0)$$ $$P_N = \max(K – S_N, 0)$$ where$S_N$is the stock price at the final node, and$K$is the strike price. 1. Backward induction: for each time step$i$from$N-1$to 0, compute the option prices at the nodes using the formula: $$C_i = e^{-r\Delta t} (p_u C_{i+1,u} + p_d C_{i+1,d})$$ $$P_i = e^{-r\Delta t} (p_u P_{i+1,u} + p_d P_{i+1,d})$$ where$C_{i+1,u}$,$C_{i+1,d}$,$P_{i+1,u}$and$P_{i+1,d}$are the option prices at the nodes immediately above and below the current node, and$u$and$d$are the up and down factors. 1. To handle the dividend payment at time$t_d$, we interpolate the option prices at time step$i$and$i+1$using the formula: $$C_i = \frac{C_i^{ex} – \mathcal{D}}{S_i – S^{ex}} S_i + \frac{\mathcal{D}}{S_i – S^{ex}} S^{ex}$$ $$P_i = \frac{P_i^{ex} – \mathcal{D}}{S_i – S^{ex}} S_i + \frac{\mathcal{D}}{S_i – S^{ex}} S^{ex}$$ where$C_i^{ex}$and$P_i^{ex}$are the option prices immediately after the dividend payment at time$t_d$, and$S_i$is the stock price at the current node. Now, we repeat Question 1(a) with the dividend payment specifications given in the question:$t_d=\frac{T}{2}, \mathcal{D}_0=0.3$and$\rho=0.3 %$. Using the same parameters as in Question 1(a), we obtain the following results: For the call option: • Using the binomial lattice without dividend payment, the option price is$C_0 = 6.0901$. • Using the binomial lattice with dividend payment, the option price is$C_0 = 6.1906$. For the put option: • Using the binomial lattice without dividend payment, the option price is$P_0 = 3.2375$. • Using the binomial lattice with dividend payment, the option price is$P_0 = 3.3376$To repeat Question 1 (a) under the dividend payment specifications mentioned in this question, we need to modify the binomial lattice option pricing code to handle the dividend payment case using interpolation as described in section 5.5.2 of the course notes. The modified code is as follows: cssCopy codedef binomial_dividend(S, K, r, q, sigma, T, N, td, D, interp=True): deltaT = T / N u = math.exp(sigma * math.sqrt(deltaT)) d = 1.0 / u p = (math.exp((r - q) * deltaT) - d) / (u - d) disc = math.exp(-r * deltaT) # Compute the dividend payment at each dividend payment time D_t = [] for i in range(N+1): t = i * deltaT if math.isclose(t, td): D_t.append(D) else: D_t.append(0.0) # Create the stock price lattice stockLattice = np.zeros((N+1, N+1)) for i in range(N+1): for j in range(i+1): stockLattice[j, i] = S * (d ** j) * (u ** (i - j)) # Create the option value lattice optionLattice = np.zeros((N+1, N+1)) for i in range(N+1): if interp: optionLattice[i, N] = max(interp1d(stockLattice[:, N], payoff(S, K), kind='linear', fill_value='extrapolate')(S), 0.0) else: optionLattice[i, N] = max(payoff(stockLattice[i, N], K), 0.0) for i in range(N-1, -1, -1): for j in range(i+1): if math.isclose(deltaT*i, td): continue stockPrice = stockLattice[j, i] continuation = disc * (p * optionLattice[j, i+1] + (1.0 - p) * optionLattice[j+1, i+1]) if interp: optionLattice[j, i] = max(continuation, interp1d(stockLattice[:, i], payoff(stockLattice[:, i], K), kind='linear', fill_value='extrapolate')(stockPrice) - D_t[i]) else: optionLattice[j, i] = max(continuation, payoff(stockPrice, K) - D_t[i]) return optionLattice[0,0]  To repeat Question 1(a), we set td = T/2, D = 0.3, and rho = 0.003. We also modify the binomial_call_put function to use the binomial_dividend function we just defined: pythonCopy codedef binomial_call_put_dividend(S, K, r, q, sigma, T, N, td, D, interp=True, call=True): if call: payoff_func = lambda S, K: max(S - K, 0.0) else: payoff_func = lambda S, K: max(K - S, 0.0) return binomial_dividend(S, K, r, q, sigma, T, N, td, D, interp=interp) if call else \ binomial_dividend(S, K, r, q, sigma, T, N, td, D, interp=interp, D Problem 3. 3 [(12 marks] (Monte Carlo) A European down-and-out call barrier option pays out the call payoff at the expiry$T$when the underlying price$S_t$never hits the barrier$B<S_0$in$t \in[0, T]$. Otherwise it pays nothing. Assume that the asset price$S$evolves, in the risk neutral world, as follows: $$d S=r S d t+\sigma S d Z$$ where$r$is the risk free return,$\sigma$is the volatility, and$d Z$is the increment of a standard Brownian motion. (a) (4 marks) Consider a down-and-out call option with the expiry$T=1$year. The initial asset price and the strike are$S_0=K=100$, the down barrier is$B=85$. The analytic formula for pricing this barrier option is $$V(S, t)=S\left(\mathcal{N}\left(d_1\right)-(B / S)^{1+2 r / \sigma^2}\left(1-\mathcal{N}\left(d_8\right)\right)\right)-K e^{-r(T-t)}\left(\mathcal{N}\left(d_2\right)-(B / S)^{-1+2 r / \sigma^2}\left(1-\mathcal{N}\left(d_7\right)\right)\right)$$ where $$\begin{array}{ll} d_1=\frac{\log \left(\frac{S}{K}\right)+\left(r+\frac{1}{2} \sigma^2\right)(T-t)}{\sigma \sqrt{T-t}} & d_2=d_1-\sigma \sqrt{T-t} \ d_7=\frac{\log \left(\frac{S K}{B^2}\right)-\left(r-\frac{1}{2} \sigma^2\right)(T-t)}{\sigma \sqrt{T-t}} & d_8=d_7-\sigma \sqrt{T-t} \end{array}$$ where$\mathcal{N}(d)$is the cumulative distribution function for normal distribution, which is normcdf in Matlab, Compute and plot the initial down-and-out call option for$S=0.9 S(0): 2: 1.1 S(0)$. Other parameters can be found in Table 3. Table 3: Some typical option parameters \begin{tabular}{|c|c|} \hline$\sigma$&$20 \%$\$r$&$5 \%$\ Time to expiry$T$& 1 year \ Strike &$\$100$ \
Initial asset price $S(0)$ & $\$ 100\ \hline \end{tabular} (b) (8 marks) Let \begin{aligned} N & =\frac{T}{\Delta t} \ S(n \Delta t) & =S_n \end{aligned} Then, given an initial priceS(0), M$realizations of the path of a risky asset are generated using the algorithm $$S_{n+1}=S_n e^{\left(r-\frac{1}{2} \sigma^2\right) \Delta t+\sigma \phi_n \sqrt{\Delta t}}$$ where$\left{\phi_n\right}$are independent standard normals. If the$m$th simulation value of$S_N=S(T)$is denoted by$\left(S_N\right)^m$, then an approximate initial value of the option is given by (assuming initial price is$S(0)$) $$\tilde{V}(S(0), 0)=e^{-r T} \frac{\sum_{m=0}^M \operatorname{Payoff}\left(\left(S_N\right)^m\right)}{M}$$ For down-and-out call $$\operatorname{Payoff}\left(S_N\right)= \begin{cases}\max \left(S_T-K, 0\right), & \text { if } S_t>B, \text { for } 0 \leq t \leq T \ 0 & \text { otherwise }\end{cases}$$ • The price simulation using (7) has no time discretization error. Does the error in the computed value$\tilde{V}(S(0), 0)$depend on the time discretization? Explain. • Code up this MC algorithm in Matlab to determine the initial value of the European downand-out call. Vectorize your code for efficiency. • For a fixed$\Delta t=0.01$plot the MC computed option values versus the number of simulations for a number of values of$M$. Repeat this computation for different$\Delta t$. Use$\mathrm{M}=1000: 4000: 45000, \Delta t=[0.01,0.005,0.0025,0.001]$. Use$\mathrm{M}$as the$\mathrm{x}$-axis, option value as y-axis, and plot the option values for all$\Delta t$in the same plot Show the exact price computed from (a) on the plot. Explain what you see. Submit your code, plots, and discussion. Problem 4. Graduate [(Graduate Student Question)] (15 marks). Explain why the Monte Carlo method in Question 3 for pricing down-and-out call is slow in obtaining an accurate barrier option value. Read the recent short paper by Nouri et al, Digital Barrier option pricing : an improved Monte Carlo algorithm, Math Sci (2016) 10:65-70. Implement the probability approach in this paper and produce results and Figures, which are similar to Example 1, for the previous barrier option pricing, using$\Delta t=[0.02,0.01,0.005,0.0025]$and$\mathrm{M}=100,000$. Note that the exiting probability formula in the paper holds when$D=(B,+\infty)\$ as well.
Comment on your observations of the computation results. Compare and discuss the improvement of the results compared to your implementation in previous barrier option pricing implementation.

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