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Financial Engineering — exercise notes

Exercise 1 #

  1. Explain what financial engineering is and why it matters in modern finance.

    1. How do we price and hedge financial derivatives, price = cost of replication

    2. Assumes no-arbitrage, replication, risk-neutral valuation

  2. State and apply the no-arbitrage principle and the Law of One Price.

    1. $V_0(\pi) = 0$zero initial investment,$V_T(\pi) \geq 0$no risk of loss, $\mathbb{P}(V_T(\pi) > 0) > 0$positive probability of profit

    2. Assume: Assets are infinitely divisible; Unrestricted short selling; No spread, transaction costs, or taxes; Borrowing and lending at the same risk-free rate

    3. Law of one price: if portfolio have same payoff at maturity, then they must have same price earlier

  3. Compute present values and discount factors using continuous compounding.

    1. $PV=e^{-rT}$

  4. Derive the no-arbitrage forward price using cost-of-carry arguments.

    1. ${F_{0,T} = S_0 \, e^{rT}}$, $F_{t,T} = S_t \, e^{(r-q)(T-t)}$, forwards are more expensive by bond return - dividend

  5. Plot and interpret payoff diagrams for forwards, calls, and puts.

  6. Verify put-call parity both algebraically and numerically.

    1. $C_t - P_t = S_t - K e^{-r(T-t)}$

  7. Construct a simple static replicating portfolio.

  • Risk Neutral pricing

    • $V_t = e^{-r(T-t)} \, \mathbb{E}^{\mathbb{Q}}!\left[\, H \,\middle|\, \mathcal{F}_t \right]$where $H = h(S_T)$ is the derivative's payoff

    • Market is arbitrage free <=> there exists an equivalent martingale measure Q

    • Market is complete (all payoff can be replicated) <=> Q is unique

    • $\Delta \cdot S_u + B \cdot (1+r) = C_u \quad \text{and} \quad \Delta \cdot S_d + B \cdot (1+r) = C_d$

    • $\Delta = \frac{C_u - C_d}{S_u - S_d}, \qquad B = \frac{C_d - \Delta \cdot S_d}{1+r}$

    • $C_0 = \Delta \cdot S_0 + B$

    • $q = \frac{S0(*rate) - Sd}{S_u - S_d}$

    • $C0_{rn} = \frac{q * Cu + (1 - q) * Cd}{1 + r}$

  • P&L diagram

Exercise 2 ipynb #

  1. Construct a one-period binomial model and derive the no-arbitrage option price via replication.

  2. Derive the risk-neutral probability $q$ and understand its economic meaning.

    1. $\boxed{q = \frac{(1+r) - d}{u - d}}, \qquad 1 - q = \frac{u - (1+r)}{u - d}$expectation of payout is 1+r

    2. $\boxed{\Delta = \frac{C_1^u - C_1^d}{(u - d) S_0}}, \qquad \alpha = \frac{u \cdot C_1^d - d \cdot C_1^u}{(u - d)(1+r)}$

  3. Build a replicating portfolio (delta hedge) and verify it replicates a contingent claim.

  4. Extend to the multi-period (CRR) binomial tree using backward induction.

    1. Formulas:

      1. $H_k^{(i)} = \frac{1}{R}\left(q \cdot H_{k+1}^{(i+1)} + (1-q) \cdot H_{k+1}^{(i)}\right)$

      2. $H_0 = \frac{1}{R^N} \sum_{i=0}^{N} \binom{N}{i} q^i (1-q)^{N-i} \, H!\left(S_0 u^i d^{N-i}\right)$

      3. $C_0 = S_0 \, \Psi(A, N, q') - K \, R^{-N} \, \Psi(A, N, q)$where $\Psi(A, N, q) = \sum_{j=A}^{N} \binom{N}{j} q^j (1-q)^{N-j}$ is the complementary binomial CDF and $q' = qu/R$.

    2. option_tree[k] = (1 / R) * (q * option_tree[k + 1][1:] + (1 - q) * option_tree[k + 1][:-1])

    3. delta_tree[k] = (option_tree[k + 1][1:] - option_tree[k + 1][:-1]) / (stock_tree[k + 1][1:] - stock_tree[k + 1][:-1])

  5. Implement the Cox-Ross-Rubinstein parameterization $u = e^{\sigma\sqrt{\Delta t}}$, $d = e^{-\sigma\sqrt{\Delta t}}$.

    1. Volatility so we converge to black scholes, sqrt because we want variance to grow linearly in time

    2. $q = \frac{R - d}{u - d} = \frac{e^{r\Delta t} - e^{-\sigma\sqrt{\Delta t}}}{e^{\sigma\sqrt{\Delta t}} - e^{-\sigma\sqrt{\Delta t}}}$

  6. Demonstrate numerically that the binomial price converges to Black-Scholes as $n \to \infty$.

  7. Compute hedge ratios (delta) at each node of the tree.