Advanced Financial Management · Derivatives Analysis and Valuation

Black-Scholes Option Valuation for CA Final AFM

Updated 5 October 2026 · Fact-checked

Black-Scholes prices a European option from the spot price, strike, risk-free rate, volatility and time. Calculate d1 and d2, read N(d1) and N(d2) from the normal table, then compute C = S·N(d1) − X·e^(−rt)·N(d2). Get the put from its formula or put-call parity. Greeks measure sensitivity to each input.

Understand Black-Scholes Option Valuation

An option's value depends on five inputs: the spot price (S), the strike price (X), the time to expiry (t), the risk-free rate (r) and the volatility of the share (σ). The Black-Scholes model combines them into one price for a European option, which can be exercised only at expiry.

Think of the call value as two parts. S·N(d1) is the present value of the share you expect to receive if the option is exercised. X·e^(−rt)·N(d2) is the present value of the strike you expect to pay. N(d2) is the risk-neutral probability that the call ends in the money. The call value is the first part minus the second.

N(d) is the cumulative standard normal probability. It is the area under the bell curve to the left of d. Your exam gives a table, or it gives the N values directly. For a negative d, use N(−d) = 1 − N(d).

The model assumes the share price is lognormal with constant volatility, the risk-free rate is constant, there are no transaction costs or taxes, and trading is continuous. It assumes no dividends unless you adjust for them. Here r is the continuously compounded rate, which is why e^(−rt) is used for discounting.

The Greeks tell you how the option value moves. Delta is the change in option value for a ₹1 change in the share price. Gamma is the change in delta for a ₹1 change in the share price. Theta is the change per unit of time passing. Vega is the change per unit change in volatility. Rho is the change per unit change in interest rate. Traders use them to hedge, and the exam tests them in short numerical or theory parts.

Key rules to remember

d1: d1 = [ln(S ÷ X) + (r + σ² ÷ 2) × t] ÷ (σ × √t); S = spot, X = strike, r = continuously compounded risk-free rate, σ = annual volatility, t = years to expiry. ln is the natural log.
d2: d2 = d1 − σ × √t; Compute d1 first, then subtract σ√t.
Call value: C = S × N(d1) − X × e^(−rt) × N(d2); European call, no dividends.
Put value: P = X × e^(−rt) × N(−d2) − S × N(−d1); N(−d) = 1 − N(d). Check with put-call parity.
Put-call parity: C + X × e^(−rt) = P + S; For European options on a share with no dividends, same strike and expiry. Use it for a quick put value.
Dividend adjustment (discrete): Use S* = S − PV of dividends expected before expiry in place of S; Discount each dividend at the risk-free rate from its date.
Dividend adjustment (continuous yield q): Use S* = S × e^(−qt) in place of S; Use this only if the question gives a continuous dividend yield.
Delta: Call delta = N(d1); Put delta = N(d1) − 1; Call delta is between 0 and 1. Put delta is between −1 and 0.
Gamma: Gamma = N′(d1) ÷ (S × σ × √t), where N′(d1) = e^(−d1²÷2) ÷ √(2π); Same for a call and a put.
Vega: Vega = S × √t × N′(d1); Per 1.00 (100%) change in σ. Divide by 100 for a 1% change. Same for call and put.
Rho (call): Rho = X × t × e^(−rt) × N(d2); For a put: −X × t × e^(−rt) × N(−d2).
Approximate price change: ΔC ≈ Delta × ΔS + ½ × Gamma × (ΔS)²; Delta alone is a first approximation. Gamma corrects for curvature.

How to solve Black-Scholes Option Valuation questions

Follow the same order every time. Most marks are for correct d1, d2, the table reading and the final formula.

1List the inputs: S, X, r, σ, t. Convert t to years and make sure σ and r are annual and in decimals.
2Adjust S for dividends if the question gives them: subtract the PV of dividends, or multiply by e^(−qt).
3Compute ln(S ÷ X), then (r + σ²÷2) × t, then σ√t. Show each part.
4Calculate d1 and then d2 = d1 − σ√t.
5Read N(d1) and N(d2) from the table. For a put, use N(−d1) = 1 − N(d1) and N(−d2) = 1 − N(d2).
6Calculate e^(−rt) and the present value of the strike, X × e^(−rt).
7Apply the call or put formula. If the put is asked after the call, you can use put-call parity and cross-check.
8Compute any Greek asked using the d1 you already have, and state the interpretation in one line.

Quickest way: Call first, put by parity, Greeks from d1

When to use it: Use this when the question asks for both call and put values, or a call value plus Greeks, and time is short.

Compute d1 and d2 once and write them clearly.
Calculate the call value only from C = S·N(d1) − PV(X)·N(d2).
Get the put as P = C − S + PV(X). This avoids two more table readings.
If S = X, ln(S ÷ X) = 0 and d1 = (r + σ²÷2) × t ÷ (σ√t), which saves effort.
Use N(d1) as call delta and N(d1) − 1 as put delta. Do not recompute.
If the question asks only for delta, compute d1 and read N(d1). Skip d2.

Common mistakes in Black-Scholes Option Valuation

Using a percentage like 10 instead of 0.10 for r and σ
The question gives rates as percentages and students plug them in directly.
Fix: Convert r and σ to decimals before starting. Write them as 0.10 and 0.20 in your working.
Forgetting to compute the discount factor and using X instead of X × e^(−rt)
Students remember the S·N(d1) part and treat the strike as a plain number.
Fix: Write PV(X) = X × e^(−rt) as a separate line before applying the formula.
Reading N(−d) from the table as if it were N(d)
Tables often show only positive d, so negative values cause confusion.
Fix: Use N(−d) = 1 − N(d). Remember the two values always add up to 1.
Ignoring dividends or deducting them wrongly
Students forget that the share drops by the dividend and keep S unchanged, or subtract the undiscounted dividend.
Fix: Subtract the present value of dividends falling before expiry from S, then use S* throughout, including in ln(S* ÷ X).
Using the call delta for a put, or giving a positive put delta
Students memorise N(d1) and apply it to both options.
Fix: Put delta = N(d1) − 1, always negative. Check that call delta minus put delta equals 1.
Applying the model to an American option or dividends without comment
Students treat the model as universal.
Fix: State that Black-Scholes values European options. For early exercise, mention that the binomial model is more suitable.

Worked examples

Example 1

The share of Meru Ltd trades at ₹100. A European call and a European put on it each have a strike of ₹100 and expire in one year. The risk-free rate is 10% (continuously compounded), volatility is 20% a year, and no dividend is expected. Given N(0.6) = 0.7257, N(0.4) = 0.6554, e^(−0.10) = 0.9048. Find the call value and the put value.

Show the solution

Inputs: S = 100, X = 100, r = 0.10, σ = 0.20, t = 1.
ln(S ÷ X) = ln(1) = 0.
d1 = [0 + (0.10 + 0.20² ÷ 2) × 1] ÷ (0.20 × 1) = (0.10 + 0.02) ÷ 0.20 = 0.60.
d2 = 0.60 − 0.20 = 0.40.
N(d1) = 0.7257 and N(d2) = 0.6554.
PV of strike = 100 × 0.9048 = ₹90.48.
Call: C = 100 × 0.7257 − 90.48 × 0.6554 = 72.57 − 59.30 = ₹13.27.
Put by parity: P = C − S + PV(X) = 13.27 − 100 + 90.48 = ₹3.75.
Check with the direct formula: N(−0.4) = 1 − 0.6554 = 0.3446 and N(−0.6) = 1 − 0.7257 = 0.2743. P = 90.48 × 0.3446 − 100 × 0.2743 = 31.18 − 27.43 = ₹3.75.

Answer: Call value ≈ ₹13.27 and put value ≈ ₹3.75.

Example 2

Using the data of Meru Ltd from the previous question (S = ₹100, X = ₹100, r = 10%, σ = 20%, t = 1 year, d1 = 0.60, N(d1) = 0.7257), a trader has written 1,000 call options. Find the call delta, put delta, gamma and vega. How many shares should the trader hold to be delta-neutral? Given N′(0.6) = 0.3332.

Show the solution

Call delta = N(d1) = 0.7257.
Put delta = N(d1) − 1 = 0.7257 − 1 = −0.2743.
Gamma = N′(d1) ÷ (S × σ × √t) = 0.3332 ÷ (100 × 0.20 × 1) = 0.3332 ÷ 20 = 0.0167 (rounded).
Vega = S × √t × N′(d1) = 100 × 1 × 0.3332 = 33.32 per 1.00 change in volatility, which is about ₹0.33 for a 1 percentage point rise in volatility.
The trader is short 1,000 calls, so the position delta is −1,000 × 0.7257 = −725.7.
To offset it, buy 725.7 shares, so about 726 shares.
Interpretation: if the share price rises by ₹2, each call gains about 0.7257 × 2 + ½ × 0.0167 × 2² = 1.45 + 0.03 = ₹1.48, so the hedge must be rebalanced because delta changes.

Answer: Call delta 0.7257, put delta −0.2743, gamma ≈ 0.0167, vega ≈ 33.32 (₹0.33 per 1% change in volatility). Buy about 726 shares to be delta-neutral.

Exam tips

Show d1, d2, the table values, and the discount factor on separate lines. Marks are given for each step even if the final figure differs slightly.
Use the N(d) values and e^(−rt) supplied in the question exactly as given. Small rounding differences are accepted.
If the question gives dividends, adjust S first and say so in a line. This is the most common hidden step.
For Greeks questions, write the one-line meaning with the number, for example delta 0.73 means the call moves about ₹0.73 for a ₹1 move in the share.
In theory parts on binomial versus Black-Scholes, give two or three clear points: discrete versus continuous time, handling of early exercise, and the assumptions.

Practice questions from Derivatives Analysis and Valuation

Black-Scholes Option Valuation in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Black-Scholes Option Valuation: frequently asked questions

How do I calculate d1 and d2 in Black-Scholes?

First find ln(S ÷ X). Add (r + σ²÷2) × t to it and divide the total by σ√t to get d1. Then subtract σ√t from d1 to get d2. Use decimals for r and σ, and t in years.

How do I find the Black-Scholes put value?

Use P = X·e^(−rt)·N(−d2) − S·N(−d1), where N(−d) = 1 − N(d). Alternatively, compute the call first and use put-call parity: P = C − S + X·e^(−rt). Both give the same answer for a European option.

What is the difference between the binomial and Black-Scholes models?

The binomial model values an option over a number of discrete time steps with up and down moves of the share price. Black-Scholes assumes continuous trading and gives a closed-form price for European options. The binomial model can handle early exercise, and as the steps increase its value approaches the Black-Scholes value.

How do I treat dividends in Black-Scholes?

If dividends are paid before expiry, subtract their present value from the spot price and use that adjusted price in all the formulas. If a continuous dividend yield is given, multiply S by e^(−qt). Do not change the strike.

What do delta, gamma, theta and vega mean?

Delta is the change in option value for a ₹1 change in the share price. Gamma is the change in delta for that same move. Theta is the change in value as time passes, and vega is the change for a change in volatility.