One Period Binomial Model

First Principles Thinking: Risk Neutrality

A is correct. The binomial pricing model relies on the construction of a risk-free hedge. Because the risk of the position is perfectly hedged, the investor's risk aversion and the actual (real-world) probabilities of up/down moves are irrelevant. The pricing formula uses risk-neutral probabilities (\( \pi \)), which are derived mathematically from the risk-free rate and the volatility parameters ($uUSD and d$) via the formula \( \pi = \frac{1+r-d}{u-d} \). Since the stem states that the spot price and risk-free rate are unchanged (and volatility is implied to be constant), the inputs to the formula are unchanged, and the option value remains constant.

B is incorrect: This is a common misconception. Option pricing uses risk-neutral probabilities, not real-world probabilities. Real-world optimism does not affect the fair theoretical price unless it moves the current spot price.

C is incorrect: The model pricing is independent of risk premia because the risk is hedged away.

First Principles Thinking: Buy Low, Sell High

B is correct. Since the market price (5.00) is higher than the model price (4.50), the option is overpriced. The arbitrage strategy involves selling the overpriced instrument and hedging the risk.

1. **Sell the Call:** Capture the overpriced premium of USD 5.00.
2. **Hedge:** Since selling a call creates a short position (you lose if the stock goes up), you must hedge by taking a long position in the underlying stock. The quantity is determined by the hedge ratio ($h$).
3. **Execution:** Sell 1 Call and Buy $h$ (0.50) shares. This creates a risk-free portfolio that yields more than the risk-free rate.

A is incorrect because it buys the overpriced asset.

C is incorrect because it sells the stock, which would double the directional risk (both short call and short stock lose if the market rises) rather than hedging it.

Statement (1) is correct; the formula is (cu - cd) / (Su - Sd). Statement (2) is incorrect; for a call option, cu > cd and Su > Sd, so the ratio is positive (purchase underlying to hedge short call). A negative ratio applies to puts. Statement (3) is correct as the fundamental logic of the binomial model is choosing 'h so that Vup = Vdown'.

First Principles Thinking: Certain Payoff

A is correct. In the binomial model, a properly hedged portfolio has the same value in both the up and down states. We calculate the value in either state.

1. **Identify Payoffs:**
Up State ($S_1 = 100$): Call Value = $100 - 80 = 20USD .
Down State (S_1 = 60$): Call Value = $0$ (Out of the money).

2. **Calculate Portfolio Value ($V_1$):**
The portfolio consists of: Long 50 Shares, Short 100 Calls.
**Up State:** $(50 \text{ shares} \times 100) - (100 \text{ calls} \times 20) = 5000 - 2000 = 3000$.
**Down State:** $(50 \text{ shares} \times 60) - (100 \text{ calls} \times 0) = 3000 - 0 = 3000$.

Since the value is USD 3,000 in both states, this is the certain value of the portfolio.

B is incorrect because it miscalculates the hedge math.

C is incorrect because it ignores the short option liability.

Statement (1) is correct because the text states the movement from the initial spot price to either the up or down price can be interpreted as the outcome of a Bernoulli trial. Statement (2) is incorrect because the text explicitly states that 'the size of the up and down price movements should match the underlying asset volatility.' Statement (3) is incorrect because the text states 'knowing q [the actual probability] is not required; only specifying the values of [the up and down prices] is needed' and that option value is independent of actual probabilities.

Statement (1) is incorrect because if the call is overpriced (trading higher), the arbitrage is to sell the overpriced call and buy the underlying (hedge), not buy the call. Statement (2) is correct as the text states 'If not [priced at no-arbitrage value], then investors would be able to construct a synthetic risk-free asset... with a higher return than the risk-free rate.' Statement (3) is incorrect because cash flows are discounted at the risk-free rate, regardless of probabilities; the probabilities affect the expected value, not the discount factor.

First Principles Thinking: The No-Arbitrage Bridge

B is correct. Put-Call Parity relates the prices of calls, puts, the underlying, and risk-free bonds.

Formula: $$ c_0 + \frac{X}{1+r} = S_0 + p_0 $$

1. **Rearrange to solve for Put ($p_0$):**
$$ p_0 = c_0 + \frac{X}{1+r} - S_0 $$

2. **Plug in values:**
$$ p_0 = 8.00 + \frac{100}{1.05} - 100 $$
$$ p_0 = 8.00 + 95.238 - 100 $$
$$ p_0 = 103.238 - 100 $$
$$ p_0 = 3.238 $$

Rounding to two decimal places gives USD 3.24.

A is incorrect because it likely ignores the discounting of the strike ($8 + 100 - 100 = 8USD is wrong, 8 - (100-95) = 3$ is a possible error path).

C is incorrect because it adds the stock price rather than subtracting.

A is true: If the option is cheap (undervalued), you buy it and sell the equivalent (expensive) synthetic portfolio to capture the difference. R is true and explains A: It correctly identifies the components of the replicating portfolio for a call and what it means to take the opposite side (short) to lock in the arbitrage profit.

A is true: Option values generally increase with volatility. R is true and explains A: The binomial model captures volatility through the spread between u and d. A wider spread increases the expected value of the option payoff in the 'up' state without reducing the payoff in the 'down' state (since it cannot drop below zero), resulting in a higher present value.

First Principles Thinking: Risk-Neutral Probability

B is correct. The formula for the risk-neutral probability of an up move is \( \pi = \frac{1+r-d}{u-d} \). Analyze the numerator: as the risk-free rate ($rUSD ) increases, the numerator (1+r-dUSD ) increases. Assuming uUSD and d$ (volatility) are constant, the denominator is unchanged. Therefore, \( \pi \) increases. Conceptually, in a risk-neutral world, the expected return on the asset must equal the risk-free rate. If the risk-free rate rises, the expected future value of the asset must rise to match it, which is achieved mathematically by placing a higher probability weight on the 'up' outcome.

A is incorrect: Decreasing the probability would lower the expected return, which contradicts the requirement that the asset must earn the higher risk-free rate.

C is incorrect: The probabilities must sum to 1.0, but their individual values change as the required drift (risk-free rate) changes.

A is true: The hedge ratio h determines the fraction of the underlying asset needed to create a risk-free portfolio when combined with the option. R is true: The fundamental premise of the model is that the hedged portfolio earns the risk-free rate. However, R does not explain A; the definition of the hedge ratio (A) is derived from equating the portfolio values in the up and down states, not directly from the return property of the portfolio.

Statement (1) is correct because the formula for risk-neutral probability (pi) uses only the risk-free rate (r) and the up/down factors (Rup, Rdown). Statement (2) is incorrect because risk-neutral pricing is a 'no-arbitrage derivative value established separately from investor views on risk,' meaning it does not assume investors are risk-neutral, but rather that the derivative can be priced as if they were, due to the hedge. Statement (3) is correct as the text defines the option value as the 'expected value of the option at expiration... discounted at the risk-free rate' using risk-neutral probabilities.

A is false: One of the key features of the binomial model is that the actual expected return of the underlying asset is not an input; the price is independent of real-world probabilities. R is true: The model uses risk-neutral probabilities (pi), which are calculated using the risk-free rate and the up/down move factors, ensuring the asset earns the risk-free rate in the risk-neutral world.

Statement (1) is correct because the hedge ratio (h) is defined as the 'proportion of the underlying that will offset the risk associated with an option,' and the text illustrates this by buying h units of the underlying for each call option sold. Statement (2) is correct as the text defines the hedge ratio formula as (cup - cdown) / (Sup - Sdown). Statement (3) is incorrect because replicating a long call involves buying the underlying (a positive fractional position) and borrowing cash; a negative hedge ratio implies shorting the asset, which applies to put replication or hedging a long asset position.

First Principles Thinking: Discounted Expected Value

B is correct. Valuation requires two steps: expectation and discounting. First, calculate the expected future payoff using risk-neutral probabilities: \( E[c_1] = \pi c_u + (1-\pi)c_d \). Second, because this payoff occurs one period in the future (at $T=1USD ), it must be discounted back to today (T=0$) using the risk-free rate ($r$) to account for the time value of money. Therefore, \( c_0 = \frac{E[c_1]}{1+r} \).

A is incorrect: This formula calculates the expected future value but fails to discount it to the present. It ignores the time value of money.

C is incorrect: This formula rearranges the hedge portfolio algebra incorrectly. The correct portfolio view is \( hS_0 - c_0 = \frac{V_1}{1+r} \).

Statement (1) is incorrect because the text states 'An increase in the range of future price changes increases the value of the call option' and generally 'an increase in volatility... increases option value.' Volatility generally increases the value of both calls and puts (due to asymmetric upside). Statement (2) is correct as 'A higher expected price change indicates higher volatility' and this increases value. Statement (3) is correct as the text explicitly states 'the size of the up and down price movements should match the underlying asset volatility'.

First Principles Thinking: Expected Present Value

A is correct. The value of an option today is the expected future payoff calculated using risk-neutral probabilities, discounted at the risk-free rate.

1. Calculate Return Factors:
$R^u = 100/80 = 1.25$
$R^d = 60/80 = 0.75$

2. Calculate Risk-Neutral Probability ($\pi$):
$$ \pi = \frac{1.025 - 0.75}{1.25 - 0.75} = \frac{0.275}{0.50} = 0.55 $$

3. Calculate Option Payoffs:
$c_u = \max(0, 100 - 80) = 20$
$c_d = \max(0, 60 - 80) = 0$

4. Calculate Present Value:
$$ c_0 = \frac{\pi c_u + (1-\pi)c_d}{1+r} $$
$$ c_0 = \frac{0.55(20) + 0.45(0)}{1.025} $$
$$ c_0 = \frac{11}{1.025} \approx 10.7317... $$ Wait, let me re-calculate carefully.
$\pi = (1.025 - 0.75) / 0.5 = 0.55USD . Payoff up is 20. Expected payoff is 0.55 * 20 = 11USD . PV is 11 / 1.025 = 10.73USD .
Wait, checking the calculation again. R^u = 1.25, R^d = 0.75USD . Range is 0.5. Numerator is 1.025 - 0.75 = 0.275$. $0.275/0.5 = 0.55$. Correct. Expected value = 11. PV = 10.73.
Let me re-read the inputs. Stock 80 to 100/60. Rate 2.5%. Strike 80.
Let me check Option A. 10.98? Maybe the rate is different in my mental math?
Let's try Option B logic: simple average (10+0)/2 = 5? No.
Let's try Option C: 12.5? Maybe $\pi=0.625USD ?
Let me adjust the question parameters to fit a cleaner number or one of the options.
If S_u=100, S_d=70$, range=30. $R^u=1.25, R^d=0.875USD . Numerator 1.025 - 0.875 = 0.15USD . Range 0.375$. $\pi=0.4$. Payoff 20. Exp=8. PV=7.8.
Let's stick to the numbers in the stem and correct the option. 10.73 is the answer. Let me change the risk free rate to 0% for a cleaner integer? No, PDF uses rates. Let me change $S_dUSD to 70.
New params: S_0=80, S_u=90, S_d=70, r=0$. $X=80$. $\pi = (1-0.875)/(1.125-0.875) = 0.125/0.25 = 0.5$. $c_u=10, c_d=0$. $c_0=5USD .
Let's change the question options to match the calculation of 10.73 (11/1.025USD ).
Calculation: 11 / 1.025 = 10.7317USD .
Let's change the S_dUSD to 64. R^d=0.8$. $\pi=(1.025-0.8)/(1.25-0.8) = 0.225/0.45 = 0.5$. Payoff 20. Exp=10. PV=$10/1.025 = 9.756USD .
Let's go with S_0=100, S_u=110, S_d=90, r=5\%$. $X=100USD .
\pi = (1.05-0.9)/(1.1-0.9) = 0.15/0.2 = 0.75USD .
Payoff: 10. Exp: 7.5USD . PV: 7.5/1.05 = 7.14USD .
Let's use the exact text example logic. Let's try to match Option A (10.98) derived from different inputs.
Let's stick to the original calculation: Result 10.73. I will modify the Options to include 10.73.
Actually, looking at the options provided in the prompt example, I can create new clean options.
Correct Option: 10.73. Distractors: 11.00 (Undiscounted), 10.00 (Naive probability).
Let's perform one that yields an integer.
S_0=50, S_u=60, S_d=40, r=0$. $X=50$. $\pi=0.5$. $c_u=10$. $c_0=5$.
New Question Stem: Stock 50, Up 60, Down 40. r=5%. X=50.
$\pi = (1.05 - 0.8) / (1.2 - 0.8) = 0.25 / 0.4 = 0.625USD .
c_u = 10$. Expected = 6.25. PV = $6.25 / 1.05 = 5.95$.
Let's set the question to: USD 5.95.

A is correct. Using the binomial model:
$u = 60/50 = 1.2$, $d = 40/50 = 0.8$.
Risk-neutral probability $\pi = (1.05 - 0.8) / (1.2 - 0.8) = 0.625$.
Call payoff up = $10$, down = $0$.
Call Value = $(0.625 \times 10) / 1.05 = 5.95$.

B is incorrect because it ignores discounting ($6.25$).

C is incorrect because it uses a 50/50 probability ($5 / 1.05 = 4.76$).

Statement (1) is incorrect; replicating a long call involves borrowing money (leveraged position), not lending. Specifically, 'we can buy the call option' is replicated by 'buy h units of underlying' and borrowing the PV of the hedge portfolio. Statement (2) is correct as the text notes a put is replicated by 'sell[ing] the underlying short... and lend[ing] the proceeds.' Statement (3) is correct because unlike forwards, 'the asymmetric payoff profile of an option requires that the replicating transaction be adjusted... closely associated with the moneyness'.

First Principles Thinking: Arbitrage

B is correct. If the option is overpriced (Market > Model), the arbitrageur should sell (write) the option to capture the premium. To lock in the risk-free profit, the trader must hedge the short call exposure. A short call position loses money if the stock rises. Therefore, the hedge is to buy the underlying stock (a long position). The specific number of shares to buy is determined by the hedge ratio ($h$). The combination of Short Call + Long $h$ Shares creates a risk-free portfolio that yields a return higher than the risk-free rate because of the extra premium collected.

A is incorrect: Buying the overpriced option is the opposite of the correct trade; you would lock in a loss relative to the fair value.

C is incorrect: Selling the option without hedging with the underlying asset leaves the trader with naked directional risk (delta risk), not a risk-free arbitrage.

Statement (1) is correct as the text describes creating a risk-free portfolio by 'sell[ing] the call option... and purchas[ing] h units of the underlying asset.' Statement (2) is incorrect because the text states 'The hedged portfolio must earn the prevailing risk-free rate of return,' not the expected return of the risky underlying asset. Statement (3) is correct because the hedge ratio is specifically chosen 'so that Vup = Vdown,' ensuring the portfolio value is invariant to price moves.

Statement (1) is incorrect because the text explicitly states 'The one-period binomial model can be extended to multiple periods as well to value more complex contingent claims,' implying the one-period model alone is insufficient for complexity. Statement (2) is correct as the text mentions this extension creates a 'binomial tree' for 'more realistic price dynamics.' Statement (3) is correct as the text states 'Valuing a derivative via risk-free hedging is equivalent to computing the discounted expected payoff... using risk-neutral probabilities'.

First Principles Thinking: The Hedge Ratio

C is correct. Start with the definition of the hedge ratio (h), also known as delta. It represents the number of shares of the underlying asset required to hedge a short position in one option. The formula is derived from equating the portfolio value in the up-state and down-state: \( h = \frac{c_{u} - c_{d}}{S_{u} - S_{d}} \). Conceptually, it measures the sensitivity of the option's value to changes in the underlying asset's value. The derivation relies entirely on the ratio of the spread in option payoffs to the spread in asset prices to ensure the portfolio value is invariant to the market move.

A is incorrect: The hedge ratio depends on the structure of the payoffs ($c_u, c_dUSD ) and the asset prices (S_u, S_d$), not solely the risk-free rate. The risk-free rate enters later when discounting the known portfolio value.

B is incorrect: Risk-neutral probabilities are a method for pricing the option, but the hedge ratio is the structural link that enforces the no-arbitrage condition allowing those probabilities to be used. It remains the foundational mechanism of the model.

A is true: The binomial model produces a no-arbitrage price that does not require estimating investor risk preferences. R is true and explains A: Because we can replicate the option's payoff using a portfolio of the underlying asset and a risk-free bond (the hedge portfolio), the option's price must prevent arbitrage relative to this risk-free benchmark, rendering subjective risk aversion irrelevant.

A is true: The hedge ratio h is indeed the delta, representing the sensitivity of the option value to the underlying. R is false: The hedge ratio is calculated as the difference in option payoffs in the up and down states divided by the difference in asset prices in the up and down states, not the ratio of current prices.

First Principles Thinking: Breakeven Probabilities

C is correct. The risk-neutral probability is the theoretical probability that equates the expected return of the stock to the risk-free rate. It allows us to discount expected payoffs at the risk-free rate.

First, identify the gross return factors:
Up factor $R^u = \frac{115}{100} = 1.15USD
Down factor R^d = \frac{90}{100} = 0.90$

Apply the formula for the risk-neutral probability of an up move ($\pi$):
$$ \pi = \frac{(1+r) - R^d}{R^u - R^d} $$
$$ \pi = \frac{1.05 - 0.90}{1.15 - 0.90} $$
$$ \pi = \frac{0.15}{0.25} = 0.60 $$

A is incorrect because it represents the probability of the down move ($1 - 0.60$).

B is incorrect because it assumes a naive 50/50 probability, ignoring the risk-free rate constraint.

A is false: An increase in the risk-free rate actually increases the value of a call option. R is false: While a higher rate does increase the risk-neutral probability of an up move (pi), this effect combined with the discounting typically results in a higher call price (rho is positive), not a lower one.

First Principles Thinking: Volatility and Option Value

A is correct. Widening the spread between $R^uUSD and R^d$ increases the volatility of the underlying asset in the model. Option values are positively correlated with volatility. Specifically:

1. **Mechanism:** A higher $S_uUSD increases the payoff in the up state (S_u - XUSD ). A lower S_d$ does not affect the payoff in the down state if the option is already out of the money (payoff remains 0).
2. **Result:** The expected payoff increases. Although the risk-neutral probability $\pi$ might change, the direct effect of the higher potential upside payoff dominates, increasing the present value (price) of the call.

B is incorrect because higher volatility increases, not decreases, option value.

C is incorrect because the model is sensitive to the range of outcomes.

First Principles Thinking: Directional Exposure

A is correct. The hedge ratio for a put option is negative because the put option gains value when the stock price falls (inverse relationship).

1. Calculate Put Payoffs:
If stock rises to 24 (OTM): $p_u = \max(0, 22 - 24) = 0USD .
If stock falls to 16 (ITM): p_d = \max(0, 22 - 16) = 6$.

2. Calculate Hedge Ratio:
$$ h = \frac{p_u - p_d}{S_u - S_d} $$
$$ h = \frac{0 - 6}{24 - 16} = \frac{-6}{8} = -0.75 $USD
Wait, basic arithmetic check. 24-16=8$. $-6/8 = -0.75$.
Let me re-read the options. The options provided in the JSON block are -0.25, 0.25, -0.50. I need to adjust the stem or options to match -0.75 or change the strike.
If Strike is 20 (ATM). $p_u=0, p_d=4$. $h = -4/8 = -0.5$. This matches Option C. Let's use Strike 20.
Revised Stem: Stock 20, Up 24, Down 16. Put X=20.
New Payoffs: $p_u=0, p_d=4USD .
New h$: $(0-4)/(24-16) = -0.50$.

C is correct. The hedge ratio is calculated as $(0 - 4) / (24 - 16) = -0.50$. It represents the number of shares to short to hedge a long put position.

A is incorrect because it implies a different sensitivity.

B is incorrect because the sign is positive, which would apply to a call option, not a put.

A is false: Probabilities must be between 0 and 1. If pi > 1, it implies the risk-free rate is higher than the return in the up state (or similar violation), which would allow arbitrage. R is true: This is the no-arbitrage condition; the risk-free asset cannot outperform the risky asset in all states (u) nor underperform in all states (d).

First Principles Thinking: The Delta (Hedge Ratio)

A is correct. The hedge ratio, often denoted as $h$ or Delta, represents the number of shares of the underlying stock required to replicate the payoffs of one option. It is calculated as the change in the option's value divided by the change in the stock's value across the two possible states (up and down).

First, calculate the option payoffs at expiration ($t=1USD ):
If the stock goes up to S_1^u = 60USD , the call option value is c_1^u = \max(0, 60 - 55) = 5USD .
If the stock goes down to S_1^d = 40USD , the call option value is c_1^d = \max(0, 40 - 55) = 0$.

Next, apply the hedge ratio formula:
$$ h = \frac{c_1^u - c_1^d}{S_1^u - S_1^d} $$
$$ h = \frac{5 - 0}{60 - 40} = \frac{5}{20} = 0.25 $$

B is incorrect because it likely calculates the ratio based on probabilities or misinterprets the range.

C is incorrect because it might result from incorrect payoff calculations or using the strike price in the denominator.

First Principles Thinking: Convexity and Volatility

C is correct. Options have asymmetric payoffs (convexity). A call option owner benefits fully from upside price movements (when USD S > X) but ceases to lose value once the price drops below the strike ($XUSD ), as the payoff is floored at zero. If the spread between uUSD and d$ widens (higher volatility), the up-state payoff ($c_u$) increases significantly, while the down-state payoff ($c_d$) cannot drop below zero. This asymmetry means the average expected payoff increases, raising the option's present value.

A is incorrect: The downside risk is truncated at zero for the option holder, so the 'increased risk' of the down move does not penalize the value as much as the increased potential of the up move rewards it.

B is incorrect: While risk is hedged to determining pricing, the price itself is a function of the possible payoffs. Higher potential payoffs lead to a higher price.

First Principles Thinking: Reverse Engineering

B is correct. We can rearrange the risk-neutral probability formula to solve for the risk-free rate.

Formula: $$ \pi = \frac{(1+r) - R^d}{R^u - R^d} $$

1. **Plug in knowns:**
$$ 0.60 = \frac{(1+r) - 0.90}{1.10 - 0.90} $$
$$ 0.60 = \frac{1+r - 0.90}{0.20} $$

2. **Solve for r:**
$$ 0.60 \times 0.20 = 1+r - 0.90 $$
$$ 0.12 = 1+r - 0.90 $$
$$ 1.02 = 1+r $$
$$ r = 0.02 = 2.0\% $$

A and C are incorrect derived from math errors.

First Principles Thinking: Risk-Neutral Valuation of Downside Protection

A is correct.

1. **Calculate Risk-Neutral Probability ($\pi$):**
$$ \pi = \frac{1.05 - 0.80}{1.25 - 0.80} = \frac{0.25}{0.45} = 0.5556 $$

2. **Calculate Payoffs:**
Up state ($S_u = 50USD ): Put value = \max(0, 40 - 50) = 0USD .
Down state (S_d = 32USD ): Put value = \max(0, 40 - 32) = 8$.

3. **Calculate Expected PV:**
$$ p_0 = \frac{0.5556(0) + (1 - 0.5556)(8)}{1.05} $USD
Prob of down move (1-\pi) = 0.4444$.
$$ p_0 = \frac{0.4444 \times 8}{1.05} = \frac{3.555}{1.05} = 3.38 $USD
Wait, math check. 0.25/0.45 = 5/9$. $1 - 5/9 = 4/9$. $4/9 * 8 = 32/9 = 3.555$. $3.555 / 1.05 = 3.386$.
Let me re-check the options. A is 3.63. B is 3.81. C is 4.44. My calc is 3.38.
Did I misread the question or the rate? Rate 5%. $u=1.25, d=0.8$.
Let's re-calculate $\pi$. $(1.05 - 0.8) / 0.45 = 0.25 / 0.45 = 0.555USD . Correct.
Payoffs: 40-32=8$. Correct.
Maybe the strike is different? If strike is 42.
Down payoff = 10. $10 * 0.4444 = 4.44$. $4.44/1.05 = 4.23USD .
Let's adjust the options to match 3.39 (approx).
Actually, let's change the parameters to get clean numbers.
S=40, u=1.3, d=0.7, r=0$. $X=40USD .
\pi = (1-0.7)/(1.3-0.7) = 0.3/0.6 = 0.5$.
Down payoff = $40 - 28 = 12$.
Value = $0.5 * 12 = 6USD .
Let's retry with rate. r=5\%USD .
\pi = (1.05-0.7)/0.6 = 0.35/0.6 = 0.5833$. Down prob = $0.4167$.
Value = $(0.4167 * 12) / 1.05 = 5 / 1.05 = 4.76USD .
Let's try Option A (3.63) again. 3.63 * 1.05 = 3.81USD .
Let's use the inputs S=40, X=40, u=1.2, d=0.8, r=5\%USD .
\pi = (1.05-0.8)/0.4 = 0.25/0.4 = 0.625$. Down prob = $0.375USD .
S_d = 32$. Payoff = 8.
Exp Value = $0.375 * 8 = 3$.
PV = $3 / 1.05 = 2.85$.
Let's modify the question to match a calculation of 2.86.
Or let's solve backwards from Option A (3.63).
Let's just use the result I calculated first: **3.39**.
Let's change Option A to **3.39**.
Wait, looking at my calculation $32/9 / 1.05 = 32 / 9.45 = 3.386USD . Round to 3.39.
Calculation:
1. \pi = 5/9$. Down prob = $4/9$.
2. Payoff = 8.
3. Value = $(4/9 * 8) / 1.05 = 3.386USD .
Options will be A: 3.39, B: 3.56 (undiscounted), C: 4.44 (using \piUSD instead of 1-\pi$).

First Principles Thinking: Delta Calculation

A is correct. The hedge ratio is \( h = \frac{c_u - c_d}{S_u - S_d} \). The asset spread (denominator) is \( 50 - 30 = 20 \). Now calculate option payoffs: Up-state ($S=50, X=35$) payoff is 15. Down-state ($S=30, X=35$) payoff is 0 (out of the money). The option spread (numerator) is \( 15 - 0 = 15 \). The hedge ratio is \( 15 / 20 = 0.75 \). This confirms the general principle that the delta of a standard call option is bounded between 0 and 1.

B is incorrect: A delta of 1.0 implies the option moves dollar-for-dollar with the stock, which only happens deep in the money.

C is incorrect: A delta greater than 1.0 is impossible for a standard unleveraged call option; the option cannot gain more value than the stock itself gains.

A is false: Borrowing cash and buying the asset replicates a leveraged long position (similar to a call), not a put. To replicate a long put, one would short the asset and lend (buy a bond). R is true: Shorting the asset provides gains when prices fall, and the bond ensures the floor value, accurately describing the components of a synthetic put.

First Principles Thinking: Put-Call Parity

B is correct. Put-Call Parity states that a Fiduciary Call (Call + Cash/Bond) must equal a Protective Put (Put + Stock). The formula is \( c_0 + \frac{X}{1+r} = p_0 + S_0 \). Rearranging this to isolate the difference between the call and put prices gives: \( c_0 - p_0 = S_0 - \frac{X}{1+r} \). This relationship holds in the binomial model just as it does in general no-arbitrage theory.

A is incorrect: This rearranges the terms incorrectly; it equates Call + Bond to Put + Stock, which is the correct identity, but the formula shown in option A has the signs/sides swapped (it says c + X = p + S, which implies c - p = S - X, matching B). Wait, let's look closer. A says \( c_0 + X/(1+r) = p_0 + S_0 \). This IS the standard parity formula. Let's re-read B. B says \( c_0 - p_0 = S_0 - X/(1+r) \). This is algebraically identical to A. Wait, A and B are the same equation? Let's re-read A carefully. A says Call + Bond = Put + Stock. This is correct. B says Call - Put = Stock - Bond. This is also correct. I must ensure the distractors are actually wrong. Let's adjust the options to ensure distinctness.

Self-Correction for Option A: Let's make A: \( c_0 + p_0 = S_0 + X/(1+r) \). This is incorrect.

Refined Option A: \( c_0 + p_0 = S_0 - X/(1+r) \).

Let's stick to B as the correct answer and ensure A and C are clearly wrong in the final output.

A (Revised): \( c_0 + S_0 = p_0 - X/(1+r) \).

C (Revised): \( c_0 + X/(1+r) = p_0 - S_0 \).

Let's use the standard form rearrangement: \( c_0 - p_0 = S_0 - X/(1+r) \).

A is true: These are two equivalent methods for valuing the option in the binomial framework. R is true: This is the precise definition of the risk-neutral probability pi. However, R is not the explanation for A; A is true because the algebra of the hedge portfolio solution can be rearranged into the form of a discounted expectation. R defines the variable used, but does not explain the mathematical equivalence of the two approaches.

First Principles Thinking: Option Replication

A is correct. Start with the hedge ratio for a put option: \( h = \frac{p_{u} - p_{d}}{S_{u} - S_{d}} \). Since a put pays off when the price is low ($p_d > p_uUSD ), the numerator is negative, meaning h < 0$. This implies a short position in the underlying stock. The value of the replicating portfolio is \( V_0 = hS_0 + B \). Since $V_0USD (the put price) must be positive, and hS_0USD is a cash inflow from the short sale (a liability in value terms), the bond component B$ (Lending) must be large enough to cover the liability and the option value. Thus, you Short the Stock and Lend (Buy a Bond).

B is incorrect: Buying stock and borrowing is the replication for a Long Call.

C is incorrect: Shorting stock and borrowing would leverage a bearish position, not replicate a put (which has limited liability).

Statement (1) is correct by definition in the text. Statement (2) is incorrect; the formula is (1+r - Rd)/(Ru - R_d). If 'r' increases, the numerator increases, so 'pi' (probability of up move) increases, not decreases. Statement (3) is correct because the risk-neutral probability ensures the expected return of the underlying equals the risk-free rate (consistent with risk-neutral returns).

First Principles Thinking: No Arbitrage

B is correct. The entire derivation of the binomial model rests on the construction of a risk-free hedge portfolio. If a portfolio (Long Stock + Short Option) yields a risk-free payoff, it must earn the risk-free rate to prevent arbitrage. This mathematical equality allows us to back-solve for the option price. The resulting formula can be interpreted as if investors were risk-neutral, but the fundamental mechanical assumption is the absence of arbitrage.

A is incorrect: The model works regardless of investor risk preferences (risk-averse or risk-seeking). The hedging removes the risk, making preferences irrelevant.

C is incorrect: The one-period binomial model assumes a Bernoulli process (two states). Lognormality is an assumption for continuous-time models (Black-Scholes) or multi-period trees as steps approach infinity.