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5

\begin{align} \left( L\times(S_T+\alpha)-K \right)^ + {} & = max \{ L\times(S_T+\alpha)-K,0\} \\ {}&= max \left \{ L \times\left( S_T+\alpha-\frac{K}{L}\right),0\right \} \\ {}&\stackrel{\dagger}{=}L \times max \left \{ S_T+\alpha-\frac{K}{L} ,0\right \} \\ {}& =L \times max \left \{ S_T- \left( \frac{K}{L} -\alpha\...

2

I adjusted your function slightly: import numpy as np from matplotlib import pyplot as plt def terminal_value(S0, sigma, M): S = np.zeros(M) S[0] = S0 for i in range(1, M): S[i] = S[i-1] + sigma * np.random.standard_normal() * np.sqrt(1/M) return S for i in range(10): series = terminal_value(100, 10, 100) plt.plot(series) ...

1

It sounds to me like you understand everything apart from how Girsanov's theorem defines the EMM. Girsanov's theorem tells us that if $B_t$ is standard Brownian motion under $P$, then for any adapted process $\gamma_t$ (satisfying certain conditions) the process $\hat{B}_t$ defined by: $$d\hat{B}_t = \gamma_t dt +dB_t$$ is ...

1

You can compute the implied volatility in terms of the Bachelier model. That is, compute the volatility parameter in the Bachelier that corresponds to the BS price.

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