Financial Risk Manager (FRM) Valuation and Risk Models Flashcards

24 sample cards from the Valuation and Risk Models deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. State the Black-Scholes-Merton formula for a European call option.

   $C = S_0 N(d_1) - K e^{-rT} N(d_2)$, where $d_1 = \frac{\ln(S_0/K) + (r + \sigma^{2}/2)T}{\sigma\sqrt{T}}$ and $d_2 = d_1 - \sigma\sqrt{T}$.

2. List the key assumptions of the Black-Scholes-Merton model.

   The underlying follows geometric Brownian motion with constant volatility and drift; no dividends (in the basic form); constant risk-free rate; no transaction costs or taxes; continuous trading; no arbitrage; and lognormally distributed prices (normally distributed returns).

3. What is the put-call parity relationship for European options?

   $C + K e^{-rT} = P + S_0$ (no dividends). Rearranged: $C - P = S_0 - K e^{-rT}$. It links the prices of a European call and put with the same strike and maturity.

4. How is a European put priced under Black-Scholes-Merton?

   $P = K e^{-rT} N(-d_2) - S_0 N(-d_1)$, using the same $d_1$ and $d_2$ as the call. It can also be obtained from put-call parity.

5. Describe how a one-step binomial tree prices an option using risk-neutral valuation.

   The option value is the discounted expected payoff under risk-neutral probabilities: $f = e^{-rT}[p f_u + (1-p) f_d]$, where $p = \frac{e^{rT} - d}{u - d}$ is the risk-neutral up-probability, and $u$, $d$ are the up/down factors.

6. In a Cox-Ross-Rubinstein binomial tree, how are the up and down factors set?

   $u = e^{\sigma\sqrt{\Delta t}}$ and $d = e^{-\sigma\sqrt{\Delta t}} = 1/u$, where $\sigma$ is volatility and $\Delta t$ is the length of each time step. This matches the variance of returns.

7. Why are binomial trees preferred over Black-Scholes for American options?

   Binomial trees allow checking the early-exercise condition at each node (taking the max of the intrinsic value and the continuation value), so they can price American-style early exercise, which the closed-form Black-Scholes cannot handle.

8. Define delta ($\Delta$) of an option and its value for a Black-Scholes call.

   Delta is the sensitivity of the option price to the underlying price: $\Delta = \frac{\partial f}{\partial S}$. For a non-dividend BSM call, $\Delta = N(d_1)$, ranging from 0 to 1; for a put, $\Delta = N(d_1) - 1$.

9. Define gamma ($\Gamma$) and explain its significance.

   Gamma is the rate of change of delta with respect to the underlying: $\Gamma = \frac{\partial^{2} f}{\partial S^{2}}$. It measures convexity of the option value; high gamma means delta changes rapidly, so a delta hedge must be rebalanced frequently. Gamma is largest for at-the-money, near-expiry options.

10. Define theta, vega, and rho.

    Theta ($\Theta = \partial f/\partial t$) is time decay—sensitivity to passage of time. Vega ($\partial f/\partial\sigma$) is sensitivity to volatility. Rho ($\partial f/\partial r$) is sensitivity to the risk-free interest rate.

11. What does it mean for a portfolio to be delta-neutral, and why is it not sufficient alone?

    Delta-neutral means the net delta is zero, so it is immune to small first-order moves in the underlying. It is insufficient because it ignores gamma: for larger moves the delta changes, so the hedge breaks down unless the portfolio is also gamma-hedged.

12. What is the volatility smile (and skew)?

    A plot of Black-Scholes implied volatility against strike price for options of the same maturity. A 'smile' shows higher implied vol for deep ITM and OTM strikes; equity options typically show a 'skew/smirk' with higher implied vol for low strikes (out-of-the-money puts), reflecting crash fear and fat left tails.

13. What does the existence of a volatility smile imply about the Black-Scholes assumptions?

    It implies the true distribution of returns is not lognormal with constant volatility—markets price in fat tails and skewness. If BSM held exactly, implied volatility would be constant (flat) across all strikes.

14. What is the volatility surface?

    A three-dimensional representation of implied volatility as a function of both strike (or moneyness) and time to maturity. Traders interpolate across this surface to price and risk-manage options consistently with quoted market prices.

15. How is default probability commonly measured, and distinguish PD from cumulative/marginal default probability.

    Default probability (PD) is the likelihood a borrower defaults over a horizon. It can be derived from credit spreads, ratings/historical transition data, or structural models (e.g. Merton). Marginal PD is the chance of default in a given period given survival to its start; cumulative PD is the probability of defaulting by some horizon.

16. Write the formula for Expected Loss (EL) on a credit exposure.

    $\text{EL} = \text{PD} \times \text{LGD} \times \text{EAD}$, where PD is probability of default, LGD is loss given default ($=1-\text{recovery rate}$), and EAD is exposure at default.

17. Distinguish Expected Loss from Unexpected Loss in credit risk.

    Expected Loss is the average anticipated loss, covered by loan-loss provisions/pricing. Unexpected Loss is the volatility (standard deviation) of credit losses around the mean—the potential for losses to exceed EL—and is covered by economic/regulatory capital.

18. How does the Merton structural model define default?

    Equity is modeled as a call option on the firm's assets with strike equal to the face value of debt $D$. Default occurs at maturity if asset value $V_T < D$; the default probability is $N(-d_2)$, and distance to default measures how many standard deviations assets are above the default point.

19. Compare external and internal credit ratings.

    External ratings are issued by agencies (Moody's, S&P, Fitch) using public scales and are 'through-the-cycle' (stable across the business cycle). Internal ratings are a bank's own assessments, often 'point-in-time' (reflecting current conditions), used for internal capital and lending decisions.

20. What is a rating transition (migration) matrix?

    A table giving the probabilities that an issuer rated in a given category migrates to each other rating category (including default) over a fixed horizon, typically one year. Rows are starting ratings, columns are ending ratings, and each row sums to 100%.

21. What key empirical properties are observed in rating transition matrices?

    The largest probabilities lie on the diagonal (ratings are sticky/stable); default probabilities rise sharply as credit quality falls; transition probabilities are higher for adjacent ratings; and migrations exhibit some momentum/autocorrelation (a downgrade makes further downgrade more likely).

22. Distinguish 'through-the-cycle' from 'point-in-time' rating philosophies.

    Through-the-cycle ratings assess default risk over a full economic cycle and are stable, avoiding frequent changes due to short-term conditions. Point-in-time ratings reflect current economic conditions and a near-term horizon, so they are more volatile and migrate more with the cycle.

23. How can default probability be inferred from a bond's credit spread?

    Approximately, credit spread $\approx \text{PD} \times \text{LGD}$ (spread $\approx$ PD $\times (1-\text{recovery})$). Solving gives an implied (risk-neutral) PD $\approx \frac{\text{spread}}{1-\text{recovery rate}}$. Risk-neutral PDs from spreads typically exceed real-world historical PDs.

24. What is recovery rate and how does it relate to LGD?

    Recovery rate is the fraction of exposure recovered after default. Loss given default is its complement: $\text{LGD} = 1 - \text{recovery rate}$. Recovery depends on seniority and collateral, with senior secured debt recovering more than subordinated/unsecured debt.

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