Learning Module 8 Pricing and Valuation of Options

A European call option on a non-dividend-paying stock has strike X = 50, time to maturity 3 months, risk-free rate 2% (annual, continuously compounded is not assumed; use discrete discount), current stock price S0 = 57.50 and current call premium c0 = 10. Calculate the option's exercise value and time value (use PV(X) = X / (1 + r)^(T) with T in years).

Which expression is the correct lower bound for a European call option (no dividends) with current spot S_t, strike X and time to expiration (T - t) and continuous discount replaced by discrete discount at risk-free rate r?

A European put on a non-dividend-paying stock has strike X = 100, time to maturity six months, S_t = 95, risk-free rate r = 1% annual. Using discrete discounting PV(X) = X*(1 + r)^{-(T - t)}, what is the put's exercise value at time t?

An option is described as deep-in-the-money. Which statement about sensitivity of its price to small changes in the underlying is most accurate?

Holding all else equal, how does an increase in volatility of the underlying affect European call and put prices?

If the underlying pays dividends (or other income) that a derivative holder does not receive, how does that affect the value of a European call and a European put (all else equal)?

Using put-call parity S0 + p0 = c0 + PV(X), solve for the put premium p0 in terms of call c0, spot S0 and PV(X).

An investor constructs a fiduciary call: long call c0 + long risk-free bond PV(X). Which single combination is equivalent to this fiduciary call at inception under put-call parity?

A covered call position (long underlying, short call) is equivalent to which alternative position using put-call parity (strike X, PV(X) defined)?

Which of the following is TRUE about put-call forward parity (forwards used instead of underlying)?

Consider a one-period binomial model: S0 = 80, S1_up = 110, S1_down = 60, strike X = 100, and risk-free rate r = 5% for the period. Payoff of call at maturity is c1_up = 10, c1_down = 0. Compute the hedge ratio h (units of underlying per option sold) that makes portfolio h*S - 1*call riskless at maturity.

With the same binomial data as Q11 (S0=80, S1_up=110, S1_down=60, c1_up=10, c1_down=0, r=5%), compute the certain portfolio payoff V1 and the present value V0 of that hedge portfolio if you sell the call and buy h underlying units with h=0.2. Then compute c0 from c0 = h*S0 - V0.

In the one-period binomial model, the risk-neutral probability pi is computed as pi = (1 + r - Rd) / (Ru - Rd). Using the data Ru = 1.375, Rd = 0.75, r = 5% (per period), compute pi.

Using the risk-neutral probability pi = 0.48 from Q13 and call payoffs c1_up = 10, c1_down = 0 and discount factor 1/(1 + r) = 1/1.05, compute c0 as risk-neutral discounted expected payoff and verify it matches the hedging result.

Why are real-world (actual) probabilities of up or down moves not needed in one-period binomial option pricing?

A European put option is currently priced at p0. According to put-call parity, which portfolio replicates a long put?

An option's time value tends to do what as time to expiration decreases (other factors constant)?

Consider a European call with strike X and maturity T. Which factor change will increase the call value but decrease the put value (holding other factors constant)?

A stock is currently S0 = 295 and a six-month forward price F0(T) = 300.84, risk-free rate 4% (annual). If both a call and a put have strike X = F0(T) and are European with six months to expiry, what can be said about the call's and put's exercise values at inception?

A trader observes a six-month call with strike 325 trading at 46.41. If in three months the stock is at 325 and PV(X) given three months remaining implies intrinsic value is 27.10, what is the call's time value at that time?

Which of the following correctly states the upper bound for a European put option price (no dividends) with strike X?

A dealer faces a client who wants to hedge a 25,000,000 liability by entering a pay-fixed, receive-floating FRA based on 3-month MRR. If the implied forward rate is 2.95% and the realized MRR is 3.25%, what is the net FRA cash settlement before discounting (use period = 0.25 years)?

If futures prices are positively correlated with interest rates, which contract is more attractive to a holder of a long position compared to an identical forward position and why?

An investor wants to synthetically create a long underlying position using options and bonds. Which position is equivalent to long underlying according to put-call parity?

Which of the following best describes why swap contracts are often preferred to a series of FRAs by issuers and investors?

A firm issues risky zero-coupon debt with face value D. Using option intuition from put-call parity, what option position does equity resemble on firm assets?

If put-call parity is violated and S0 + p0 > c0 + PV(X), which arbitrage strategy yields an immediate positive cash flow at t=0 and zero net cash flows at T?

A portfolio manager plans to increase portfolio duration to gain from falling rates. Which swap position would be appropriate compared with buying bonds?

A one-period binomial model has S0 = 16, S1_up = 20, S1_down = 12 and risk-free rate r = 5% for the period. Compute risk-neutral probability pi of up move.

In a one-period binomial with S0=50, X=55, Ru=1.2, Rd=0.8, r=5% per period, compute call price using risk-neutral probabilities.

Which of the following describes dynamic replication for an option?

A firm has asset value V0 = E0 + PV(D). Using put-call parity logic, express V0 in terms of call price on firm assets c0, PV(D), and put price p0.

An investor holds a long call on a stock and simultaneously borrows PV(X) and buys the stock. At maturity, what is the combined net payoff relative to exercising the call if stock price ST > X?

Which of the following changes will usually increase both call and put option values (all else equal)?

In the one-period binomial model, if the risk-free rate increases while Ru and Rd remain unchanged, how does the risk-neutral probability pi change (up to sign)?

A trader sees a European call priced above the no-arbitrage binomial value computed from Ru, Rd and r. Which arbitrage action is appropriate?

Given S0 + p0 = c0 + PV(X), what happens to c0 if p0 rises while S0 and PV(X) remain unchanged?

Which of the following best explains why put-call parity holds for European options (no dividends) in an arbitrage-free market?

A bank clears OTC forwards through a central counterparty and thus imposes margin-style requirements. How does central clearing affect forward vs futures price differences?

A one-period binomial put has payoffs p1_up = 0 and p1_down = 5.27, risk-free rate 0.37% for the period, risk-neutral probability of down move = 0.53. What is the put's no-arbitrage price p0?

If a six-month put is trading above its no-arbitrage price, which trade sequence would produce arbitrage profit given put-call parity (assume you can borrow/lend at risk-free rate and trade options and underlying)?

Which factor typically has the same directional effect on both call and put option prices?

A six-month call and put share strike X = 120, current spot S0 = 127.50, PV(X) over six months is 117.67. If call c0 = 22.60, compute put p0 by put-call parity p0 = c0 + PV(X) - S0.

Consider two European option strategies at t=0: Portfolio A is long one call and long PV(X) (fiduciary call); Portfolio B is long underlying and long put (protective put). At maturity, how do their payoffs compare?

A portfolio holds long fixed-rate debt and short a put option on firm value. Who benefits more from a decrease in volatility of firm assets?

In a one-period binomial, if the spread Ru - Rd widens while r remains constant, what is the typical effect on an (OTM) put option value, all else equal?

Which statement correctly describes the relationship between a short forward position and a sold put when X = F0(T) at maturity?

Which of the following most accurately summarizes why options are worth paying a premium at inception unlike forwards?

A six-month European call on a stock paying no dividends trades at 3. A 6-month forward on the stock has forward price 128.76. Strike X=130. Using put-call forward parity, compute the put premium p0 ≈ ? (risk-free rate annualized small used to get PV factors consistent with forward given).

Which of the following best explains the meaning of 'risk-neutral valuation' in option pricing?