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Black-Scholes Model for European Call Option

Topic: Options ValuationChapter: Chapter 1
C=S0N(d1)−Ke−rTN(d2)C = S_0 N(d_1) - K e^{-rT} N(d_2)
This formula calculates the theoretical price of a European call option. S0S_0 is the current stock price, KK is the strike price, TT is the time to expiration (in years), rr is the risk-free interest rate (annualized, continuous compounding), N(x)N(x) is the cumulative standard normal distribution function, and d1d_1 and d2d_2 are parameters calculated using specific formulas.

Black-Scholes Model for European Put Option

Topic: Options ValuationChapter: Chapter 1
P=Ke−rTN(−d2)−S0N(−d1)P = K e^{-rT} N(-d_2) - S_0 N(-d_1)
This formula calculates the theoretical price of a European put option. Alternatively, it can be derived from the Black-Scholes call option formula using the Put-Call Parity relationship. All variables are as defined for the call option.

d1 parameter (Black-Scholes Model)

Topic: Options ValuationChapter: Chapter 1
d1=ln⁡(S0/K)+(r+σ2/2)TσTd_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2)T}{\sigma \sqrt{T}}
d1d_1 is a key parameter in the Black-Scholes model, representing the expected return of the stock adjusted for volatility. σ\sigma is the volatility of the stock's returns (annualized standard deviation).

d2 parameter (Black-Scholes Model)

Topic: Options ValuationChapter: Chapter 1
d2=d1−σTd_2 = d_1 - \sigma \sqrt{T}
d2d_2 is another key parameter in the Black-Scholes model, related to the probability of the option expiring in the money. It is derived directly from d1d_1.

Put-Call Parity for European Options

Topic: Options ValuationChapter: Chapter 1
C+Ke−rT=P+S0C + K e^{-rT} = P + S_0
This fundamental no-arbitrage relationship links the prices of a European call option (C), a European put option (P), the underlying stock price (S_0), the strike price (K), the risk-free rate (r), and the time to expiration (T). It holds true for options on non-dividend paying stocks.

Payoff for a Long Call Option

Topic: Options ValuationChapter: Chapter 1
Max(0,ST−K)Max(0, S_T - K)
This formula calculates the profit or loss from holding a long call option at expiration. STS_T is the stock price at expiration, and KK is the strike price. The payoff is positive if ST>KS_T > K and zero otherwise.

Payoff for a Long Put Option

Topic: Options ValuationChapter: Chapter 1
Max(0,K−ST)Max(0, K - S_T)
This formula calculates the profit or loss from holding a long put option at expiration. STS_T is the stock price at expiration, and KK is the strike price. The payoff is positive if K>STK > S_T and zero otherwise.

Delta of a European Call Option

Topic: Options ValuationChapter: Chapter 1
ΔC=N(d1)\Delta_C = N(d_1)
Delta measures the sensitivity of the option price to a change in the underlying asset's price. For a call option, it ranges from 0 to 1. N(d1)N(d_1) is the cumulative standard normal distribution function of d1d_1.

Delta of a European Put Option

Topic: Options ValuationChapter: Chapter 1
ΔP=N(d1)−1\Delta_P = N(d_1) - 1
Delta for a put option measures the sensitivity of its price to a change in the underlying asset's price. For a put option, it ranges from -1 to 0.

Black-Scholes-Merton Model for Call Option Valuation

Topic: Mergers and AcquisitionsChapter: Chapter 1
C=S0N(d1)−Ke−rTN(d2)d1=fracln(S0/K)+(r+sigma2/2)TsigmasqrtTd2=d1−sigmasqrtTC = S_0 N(d_1) - K e^{-rT} N(d_2)\\ d_1 = \\frac{\\ln(S_0/K) + (r + \\sigma^2/2)T}{\\sigma \\sqrt{T}}\\ d_2 = d_1 - \\sigma \\sqrt{T}
This model is used to estimate the theoretical price of European-style call options. In M&A, it can be applied to value embedded options in deal structures, contingent consideration (earn-outs) that behave like options, or warrants issued as part of the acquisition. S0S_0 is the current stock price, KK is the strike price, TT is the time to expiration (in years), rr is the risk-free interest rate (annualized), σ\sigma is the volatility of the stock's returns, and N(x)N(x) is the cumulative standard normal distribution function.

Delta of a Call Option (Hedging Ratio)

Topic: Mergers and AcquisitionsChapter: Chapter 1
Deltac=N(d1)\\Delta_c = N(d_1)
Delta measures the sensitivity of an option's price to a $1 change in the underlying asset's price. For a call option, Δc\Delta_c is given by N(d1)N(d_1) from the Black-Scholes-Merton model. It represents the number of shares of the underlying asset needed to hedge one option, or the proportion of the underlying asset that should be held to replicate the option's payoff. This is crucial for delta hedging strategies to manage risk exposures arising from options in M&A.

Forward Rate Agreement (FRA) Settlement Amount

Topic: Mergers and AcquisitionsChapter: Chapter 1
Settlement=fracNotionaltimes(RM−RF)times(Days/360)1+RMtimes(Days/360)Settlement = \\frac{Notional \\times (R_M - R_F) \\times (Days/360)}{1 + R_M \\times (Days/360)}
This formula calculates the cash settlement amount for a Forward Rate Agreement (FRA) at its expiration. FRAs are used to hedge against future interest rate movements, which is particularly relevant for M&A financing where future borrowing costs are uncertain. NotionalNotional is the principal amount, RMR_M is the market reference rate (e.g., LIBOR) at settlement, RFR_F is the agreed-upon FRA rate, and DaysDays is the number of days in the interest period. The settlement is typically discounted back to the start of the interest period.

Value at Risk (VaR) - Variance-Covariance Method (Single Asset)

Topic: Mergers and AcquisitionsChapter: Chapter 1
VaR=PtimesZtimessigmaVaR = P \\times Z \\times \\sigma
Value at Risk (VaR) quantifies the potential loss in value of a portfolio or asset over a defined period for a given confidence level. The variance-covariance method assumes asset returns are normally distributed. PP is the portfolio or asset value, ZZ is the Z-score corresponding to the desired confidence level (e.g., 1.645 for 95% one-tailed, 2.33 for 99% one-tailed), and σ\sigma is the standard deviation of the asset's returns over the specified period. For portfolios, the standard deviation σ\sigma would be the portfolio's standard deviation, incorporating asset weights and correlations.

Forward Rate Calculation (Interest Rate Parity basis)

Topic: Foreign Exchange Risk ManagementChapter: Chapter 1
F=S×(1+rd×t)(1+rf×t)F = S \times \frac{(1 + r_d \times t)}{(1 + r_f \times t)}
This formula calculates the theoretical forward exchange rate (F) based on the current spot rate (S), the domestic interest rate (r_d), the foreign interest rate (r_f), and the time period (t) in years. It assumes that there are no arbitrage opportunities and is a cornerstone of covered interest arbitrage. 't' should be expressed as a fraction of a year (e.g., 3 months = 0.25).

Purchasing Power Parity (PPP) - Relative Form

Topic: Foreign Exchange Risk ManagementChapter: Chapter 1
S1=S0×(1+Id)(1+If)S_1 = S_0 \times \frac{(1 + I_d)}{(1 + I_f)}
This formula estimates the future spot exchange rate (S_1) based on the current spot rate (S_0) and the inflation rate differentials between two countries. I_d represents the domestic inflation rate, and I_f represents the foreign inflation rate. It suggests that exchange rates adjust to equalize the purchasing power of different currencies over time.

Cross Rate Calculation

Topic: Foreign Exchange Risk ManagementChapter: Chapter 1
AC=AB×BC\frac{A}{C} = \frac{A}{B} \times \frac{B}{C}
This formula is used to derive an exchange rate between two currencies (A and C) when their direct exchange rate is not available, but their exchange rates against a common third currency (B) are known. For example, if you have USD/EUR and EUR/GBP, you can calculate USD/GBP.

Option Premium Components (Call Option)

Topic: Foreign Exchange Risk ManagementChapter: Chapter 1
Option Premium=Intrinsic Value+Time ValueIntrinsic Value (Call)=Max(0,Spot Price−Strike Price)\text{Option Premium} = \text{Intrinsic Value} + \text{Time Value}\\ \text{Intrinsic Value (Call)} = \text{Max}(0, \text{Spot Price} - \text{Strike Price})
The premium paid for an option consists of two parts: Intrinsic Value and Time Value. Intrinsic value is the immediate profit if the option were exercised today. For a call option, it's the maximum of zero or the difference between the current spot price and the strike price. For a put option, it would be Max(0, Strike Price - Spot Price). Time value is the portion of the premium that reflects the probability of the option becoming more profitable before expiration.

Hedging Effectiveness (Simplified)

Topic: Foreign Exchange Risk ManagementChapter: Chapter 1
Net Gain/Loss (after hedging)=Unhedged Outcome−Hedged Outcome\text{Net Gain/Loss (after hedging)} = \text{Unhedged Outcome} - \text{Hedged Outcome}
This is a conceptual formula to evaluate the financial impact of a hedging strategy. It involves comparing the financial outcome (e.g., revenue, cost) if no hedging was undertaken versus the outcome achieved after implementing a hedge. A positive value indicates the hedge was beneficial in mitigating adverse movements, while a negative value might suggest the hedge was costly or unnecessary given market movements.
Chapter 1 Formulas — CA Final | Kartavvya