The risk-neutral probability calculation itself is fine, just plug into the formula.
First, compute the risk-neutral probability using the formula p = (e^{rΔt} - d) / (u - d), ensuring consistency with the given per-period risk-free rate. Then, construct the binomial tree for the stock price, calculate the option payoff at maturity, and work backward to discount expected payoffs at each node to find the initial call price.
Pro tip: Always verify that the risk-neutral probability lies between 0 and 1; if not, check for arbitrage or input errors. Also, mention that the same risk-neutral probability applies to all nodes, simplifying calculations.
Use the formula p = (e^{rΔt} - d) / (u - d) with r = 0.05, Δt = 1, u = 1.25, d = 0.8. Calculate p and verify 0 < p < 1.
Calculate stock prices at each node: S0=100, up to 125 or down to 80 after one period, and then to 156.25, 100, or 64 after two periods.
At each final node, compute call payoff as max(S_T - K, 0) with K=100. Payoffs: max(156.25-100,0)=56.25, max(100-100,0)=0, max(64-100,0)=0.
At each earlier node, compute option value as e^{-rΔt} [p * V_up + (1-p) * V_down]. Start from period 1 nodes and then period 0.
After backward induction, the initial call price is the value at the root node. Present the numerical result clearly.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
You have to compare the hold value against the intrinsic value at every node, not just the terminal ones.
First, clarify the binomial tree parameters (u, d, p, r, Δt) and then compute the American put value via backward induction, comparing intrinsic value and continuation value at each node. Identify nodes where intrinsic value exceeds continuation value, as those are optimal early exercise points.
Pro tip: Emphasize that early exercise for an American put is only optimal when the stock price is sufficiently low, and that the early exercise boundary can be characterized by a critical stock price at each time step. This shows deeper understanding beyond just computing values.
Confirm the binomial tree parameters: up factor u, down factor d, risk-neutral probability p, risk-free rate r, and time step Δt. If not given, state typical assumptions (e.g., u=1/d, p=(e^{rΔt}-d)/(u-d)).
At maturity, compute the put payoff max(100 - S_T, 0) for each terminal node.
At each node, compute the continuation value as the discounted expected value of the two child nodes. Compare with intrinsic value max(100 - S, 0). The American put value is the maximum of the two.
Mark nodes where intrinsic value > continuation value. These are the nodes where early exercise is optimal. Optionally, describe the boundary of these nodes.
State the final American put price and list the early exercise nodes. Explain that early exercise occurs when the stock price is low enough that the benefit of immediate exercise outweighs waiting.
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I knew the answer is no, and I knew it had something to do with time value, but my explanation came out muddled.
Set up a one-period binomial model for a non-dividend-paying stock and compare the value of immediate exercise with the value of holding the option. Show that the no-arbitrage price of the American call equals the European call price, so early exercise is never optimal. Explain the intuition using the time value of money and the insurance value of waiting.
Pro tip: Emphasize that the result holds because the stock pays no dividends; if dividends were present, early exercise might be optimal to capture the dividend. Also, mention that this is a model-independent result under no-arbitrage, not just a binomial artifact.
Define a one-period binomial tree with stock price S, up factor u, down factor d, risk-free rate r, and strike price K. Assume no dividends.
At any node, compare the intrinsic value (S - K) with the continuation value (risk-neutral expected discounted payoff). Show that continuation value is always at least as large as intrinsic value.
Use the risk-neutral pricing formula to show that C >= S - K for all states, hence early exercise is never strictly optimal. This can be done by constructing a dominating portfolio or using Jensen's inequality.
Argue by backward induction that if early exercise is not optimal at any final step, it is not optimal at any earlier step, so the American call price equals the European call price.
Discuss that waiting preserves the option's time value and insurance against downside risk, while exercising forfeits these benefits and requires paying K earlier, incurring interest cost.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Use backward induction on the short-rate tree: first compute the bond price at t=1 in both up and down states by discounting the face value at the respective short rates, then compute the t=0 price by discounting the risk-neutral expected value of the t=1 prices at the initial short rate. Clearly state the given rates and probability, and show the calculations step by step.
Pro tip: Emphasize that this is a risk-neutral valuation, so the expected value uses the risk-neutral probability, not real-world probabilities. Also, mention that the same tree can be used to price derivatives like options, showing broader applicability.
List the up and down short rates for each period (r0, r1_up, r1_down) and the risk-neutral probability p=0.5. Note the face value F=100 and maturity T=2.
In the up state, the bond price is F/(1+r1_up). In the down state, it is F/(1+r1_down). These are the values at t=1 if the bond matures at t=2.
Calculate the risk-neutral expected value at t=0: E[P1] = p * P1_up + (1-p) * P1_down.
Discount the expected value by the initial short rate r0: P0 = E[P1] / (1+r0). This is the final price of the zero-coupon bond.
Check that the price is less than the face value (since rates are positive) and discuss any assumptions (e.g., no arbitrage, constant risk-neutral probability).
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
First, clarify the short-rate tree and any assumptions (e.g., risk-neutral probabilities, recovery). Then compute the bond price by backward induction using the tree, and solve for the yield to maturity that equates the price to the present value of cash flows. Finally, explain that the price is arbitrage-free because it equals the expected discounted payoff under the risk-neutral measure.
Pro tip: Emphasize that the yield to maturity is a single discount rate that equates the bond's price to its cash flows, but it is not the same as the short-rate tree; it's a derived summary measure. Also, mention that if the bond is callable or has credit risk, the approach would need adjustment.
State the given short-rate tree (e.g., annual rates) and confirm assumptions: risk-neutral probabilities, no arbitrage, and that coupons are paid at t=1 and t=2. If not provided, ask for the tree or assume a simple binomial tree.
At t=2, the bond pays 105 (coupon + face). Discount back to t=1 using the short rate at each node, add the coupon of 5, then discount back to t=0 using the initial short rate. Average across branches using risk-neutral probabilities.
Solve for y such that the bond price equals 5/(1+y) + 105/(1+y)^2. Use a numerical method (e.g., Newton-Raphson) if necessary, and report y as an annualized rate.
Articulate that the bond price is the expected present value of its cash flows under the risk-neutral measure, ensuring no arbitrage opportunities. Mention that any deviation would allow riskless profits via replication or hedging.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.