Learning Module 10 Valuing a Derivative Using a One-Period Binomial Model

In a one-period binomial model the underlying price S0 = 50. After the period it will be either S1_u = 65 or S1_d = 40. A European call with strike X = 55 expires at period end. If the risk-free rate for the period is 2 percent, what is the hedge ratio h for a short call position replicated by buying h units of the underlying?

A stock S0 = 80 will be either 110 (up) or 60 (down) next period. A call with X = 100 has payoffs 10 and 0 in the states. If r = 5 percent, what is the risk-neutral probability pi of an up move?

True or false: In the one-period binomial model, the actual real-world probability q of an up move affects the no-arbitrage option price.

Stock S0 = 100 is expected to go to either 130 or 70. Consider a put with X = 90 and r = 0. What is the put payoff vector (p1_u, p1_d)?

A put's payoffs are (0, 20) as in Question 5. With pi = 0.58333 and r = 5 percent, what is the put price p0?

Which of the following statements best describes why put and call prices derived from the one-period binomial model do not require investors' risk preferences?

Stock S0 = 16 moves to either 20 or 12. If r = 5 percent, what is the risk-neutral probability pi?

In a one-period model, increasing the spread between Ru and Rd while keeping S0 and r fixed will generally:

You sell a call and buy h units of the underlying to form a risk-free portfolio. If V1 is the common payoff in both states and r is the period rate, which equation gives c0 (call price)?

If a call option is overpriced relative to the binomial no-arbitrage price, an arbitrageur should:

Which of the following is a correct expression of the hedge ratio for any derivative whose value at expiry depends only on the underlying price in the two states?

Given S0 = 120, S1_u = 150, S1_d = 90, X = 110, r = 4 percent. A call's payoffs are (40, 0). Compute c0 using risk-neutral pricing.

If the risk-free rate increases while Ru and Rd remain unchanged, which of the following is typically true for European calls and puts (other things equal)?

A derivative has payoffs c1_u = 30 and c1_d = 5. Under S1_u = 150 and S1_d = 100 and S0 = 120, what is the hedge ratio h?

Which of the following best explains why replication pricing yields the same option price as discounted risk-neutral expected payoff?

If a one-period binomial model gives h = 0.2 and S0 = 80 and the common payoff V1 = 12 with r = 5 percent, what is the call price c0?

A European call in a one-period model has c1_u = 12 and c1_d = 0. If S1_u = 132, S1_d = 88 and S0 = 110, what is h?

Which factors determine the risk-neutral probability pi in a one-period binomial model?

Consider S0 = 300, S1_u = 360, S1_d = 240, X = 320, r = 0. What is the call's up and down payoff vector (c1_u, c1_d)?

If the hedged portfolio yields V1 = 50 in both states and r = 10 percent, what is present value V0 used in replication pricing?

A trader constructs a synthetic long underlying by combining a long call, short put (same X and T) and lending PV(X). Which parity principle justifies this?

In a one-period binomial model, if S0 = 50, Ru = 1.4, Rd = 0.8, and r = 5 percent, what is Ru - Rd?

Which of the following is NOT required to compute the no-arbitrage price of a European option in a one-period binomial model?

You observe a call priced below its binomial no-arbitrage value. Arbitrage strategy to exploit this mispricing is to:

A one-period call has c1_u = 8, c1_d = 1. Under S1_u = 120, S1_d = 80, S0 = 100, r = 2 percent. What is the hedge ratio h?

Which statement about multi-period extension of the one-period binomial model is correct?

When forming a hedged portfolio by selling one call and buying h shares, the portfolio is risk-free because:

If a put is overpriced in the market relative to its binomial no-arbitrage price, an arbitrageur should:

Which of the following increases the risk-neutral probability pi for an up move, holding Ru and Rd constant?

Why does increasing Ru and decreasing Rd (wider up/down moves) raise both call and put prices?

A protective put portfolio equals which of the following at inception according to put-call parity?

If S0 + p0 > c0 + PV(X) in the market, an arbitrageur should:

A stock S0 = 295, forward price F0(T) = 300.84, r and T such that PV(X)=295. Which statement about a call and put with X = F0(T) is correct today?

You calculate a put price via binomial model as 6.20. Market put trades at 7.00. Which elementary arbitrage is suggested by the chapter?

Which statement about the hedge ratio h is correct as option goes deeper in-the-money (all else equal)?

Given S0 = 50, S1_u = 75, S1_d = 25, a call with X = 60 has payoffs (15, 0). If r = 0 and you short the call and buy h underlying to hedge, what is V1 (common payoff) you will obtain?

If option c0 equals 4.57 as earlier example and S0 = 80, h = 0.2, what is current value of hedged portfolio V0?

A one-period model yields pi = 0.48. Option payoffs are c1_u = 10 and c1_d = 0. With r = 5 percent, what is c0?

A stock pays no dividends. If you observe a call and put with same X and T, which parity holds that links calls, puts, underlying, and bond?

A trader wants to replicate a sold call when exercise is certain at maturity. Which replication matches the sold-call payoff according to the chapter?

You price a call using replication and obtain c0 = 5.00. Using put-call parity with S0 = 50 and PV(X) = 45, what should the put p0 equal?

If an option's time-to-expiration shortens (T decreases) while other inputs unchanged, what is the typical effect on option time value?

A bank observes a one-period call price computed as 6 by the binomial model but market call is 8. If frictionless trading is possible, net initial cash flow from arbitrage by following correct strategy?

Which of the following best describes why both put and call premiums increase when volatility rises (holding other inputs constant)?

In the one-period binomial model, which of the following reasons explains why the underlying's expected return mu does not affect option price?