PoF MIDTERM 2 GOOD

If the SML is flatter or inversed, higher-beta stocks might be “underpriced” relative to the CAPM prediction, while lower-beta stocks might be “overpriced.” A possible strategy is to go long high-beta stocks and short low-beta stocks, aiming to profit from the mispricing once prices adjust. The idea is that returns will converge toward the flatter slope rather than the steeper slope implied by the textbook CAPM, thereby generating alpha on the long–short position.

A linear stochastic discount factor (SDF) takes the form m(t+1) = a + b′f(t+1). Taking expectations of m(t+1)*R(t+1) = 1 yields E[R(t+1)] − Rf = β′λ, where β is the asset’s exposure to the factors f(t+1), and λ is the factor risk premium vector. Thus, a linear SDF directly implies the standard beta-pricing relationship in factor models.

You would typically run time-series regressions of asset excess returns on the proposed factors and examine whether the intercepts (alphas) are statistically zero. Then you would assess whether the factor loadings (betas) explain the cross-section of average returns. Additionally, you might check the model’s goodness-of-fit (R²) and analyze whether any pricing errors remain systematically large.

Sharpe ratios measure reward-to-volatility trade-offs. When testing a factor model, one examines the Sharpe ratio of any residual (alpha) strategies. If a model leaves large alpha unexplained and the residual variance is small, it could imply an unreasonably high Sharpe ratio (an apparent arbitrage). A valid factor model should not permit an alpha strategy with an abnormally large Sharpe ratio.

Under FF3, the expected excess return on an asset is given by E(Ri) − Rf = bi,Mkt * [E(RM) − Rf] + si * E(SMB) + hi * E(HML). In contrast, the CAPM predicts E(Ri) − Rf = βi * [E(RM) − Rf]. FF3 augments the market factor with size (SMB) and value (HML) factors, capturing more dimensions of systematic risk that the CAPM omits.

Fama and French sort stocks first by size (market equity) and then by book-to-market (value) to capture cross-sectional variation along two key dimensions simultaneously. This double sorting ensures that the size factor (SMB) isolates the performance of small minus big firms and the value factor (HML) isolates the performance of high book-to-market minus low book-to-market firms, reducing noise that might appear if each dimension were sorted independently.

In FF3, the market loading (b) is estimated alongside SMB and HML exposures. In the CAPM, there is only one factor— the market. Because the presence of SMB and HML can absorb some of the variance originally attributed to the market, the estimated market loading (b) in the three-factor setup can shift from the single-factor beta in the CAPM.

If the FF3 model is truly correct, then the CAPM is missing the size and value risk premiums. Thus, a portfolio’s expected alpha under the CAPM would systematically deviate from zero. Specifically, if the portfolio has non-zero exposure to SMB or HML, its CAPM alpha would not be zero on average, indicating mispricing in the single-factor framework.

Under the CAPM, the factor contribution is 1.42 * 0.42% = 0.60%, implying an alpha of 0.27% − 0.60% = −0.33%. Under the FF3, the factor contribution is (1.03 * 0.42%) + (1.42 * 0.28%) + (−0.30 * 0.48%) = 0.43% + 0.40% − 0.14% = 0.69%, implying an alpha of 0.27% − 0.69% = −0.42%. So the CAPM attributes 0.60% per month to market exposure, while the FF3 attributes 0.69% to all three factors, leaving slightly different alphas in each model.

You would regress the strategy’s excess returns on the three FF3 factors and check if the intercept (alpha) is statistically different from zero. You would also examine the strategy’s Sharpe ratio. If the alpha persists out of sample and remains statistically significant after transaction costs, then there is evidence supporting the existence of genuine alpha. Otherwise, it may be a product of data mining or insufficient risk adjustment.

An American option can be exercised at any time up to and including its expiration date, whereas a European option can only be exercised at maturity. The additional flexibility of American options can make them more valuable or at least as valuable as European options on the same underlying.

For a call, it is max(S − X, 0), representing the immediate exercise value if S > X. For a put, it is max(X − S, 0). It is the payoff if you were to exercise the option right now, ignoring any remaining time value.

Generally, a European call option’s value decreases as the strike price increases. A higher strike means the option finishes in-the-money less frequently, so its expected payoff and thus its present value shrink.

By put–call parity, a forward contract to buy the underlying at maturity can be replicated by a call minus a put plus a risk-free bond (or net stock position). The forward price F = S0 − (C0 − P0) after adjusting for the present value of the strike. This ensures no-arbitrage pricing across these instruments.

From put–call parity, P0 = C0 + X / Rf − S0. Rearranging, you can form a put by buying a call, shorting the underlying, and investing the strike amount in a risk-free bond. At expiration, this position mimics the payoff of a put.

Put–call parity implies C0 − P0 = S0 − X/ (1 + rf)^T. If the equality does not hold, you can construct a long–short position with the underlying, the call, and the put, plus borrowing or lending at the risk-free rate, to lock in a riskless profit. In this example, you would check whether 4.9255 − 4.9255 = 131 − 134/(1.02^0.75) to see if there is a mismatch.

They are identical only under very special conditions where the forward price of the underlying equals the strike. In that case, the cost of carry plus the spot price aligns perfectly so that the call and put have the same present value. More practically, it happens when S0 = X / (1+rf)^T, ignoring dividends.

Actual probabilities reflect real-world likelihoods of future states, while risk-neutral probabilities incorporate market prices of risk. Under no-arbitrage and market completeness, there exists a unique risk-neutral measure that discounts expected payoffs by the risk-free rate. These risk-neutral probabilities reweight outcomes so that every asset’s discounted expected payoff equals its current price.

Typically you choose u and d to reflect the expected volatility or range of possible stock price movements over a period. For example, one might set u = e^(σ√Δt) and d = 1 over u (1 / u), or use another parameterization that matches implied volatilities. The key is that u > Rf > d to avoid arbitrage in the binomial framework.

You would specify the up and down moves for Apple’s stock price over each period, choose a suitable risk-free rate, and then compute the option payoffs at each node. By working backward through the tree with risk-neutral probabilities, you find the option’s current fair value. Altering the strike or maturity just changes the nodes at which exercise might be beneficial (for American options) or the payoff at maturity (for European ones).

The payoff looks like a V shape, combining upside from a long call with downside from a long put. You gain if the stock price moves significantly in either direction away from the strike, but lose if it stays near the strike. Traders use this when they expect volatility to increase but are uncertain about the direction of price movement.

(i) Buy a call: You gain if the stock rises above the strike, and your downside is limited to the call premium. (ii) Sell a put: You profit if the stock stays above the strike, collecting the premium, but face the risk of having to buy the stock at the strike if Apple declines below it. The call buyer has lower risk but pays a premium; the put seller collects premium but faces higher downside risk if the stock drops.

A call spread strategy could fit. You might buy a call with a strike at 130 and simultaneously sell a call with a strike at 150. The net premium is lower than buying a single call, and it monetizes your view that Apple will likely rise to around 150 but not beyond. Your upside is capped at 150, and your net cost is reduced.