Option Pricing Models

This sub‑topic covers the major option pricing models used for equity derivatives in India. Understanding these models helps you value European‑type options and answer calculation‑based questions in the NISM Series VIII exam. You will learn the assumptions, key formulas, and practical application of the Black‑Scholes‑Merton and Binomial models.

Learning Objectives

1Identify the assumptions behind each pricing model.
2Apply the Black‑Scholes‑Merton formula to compute a European call price.
3Explain the step‑by‑step construction of a binomial price tree.
4Recognise exam traps related to volatility and time‑to‑maturity.

Why Option Pricing Models Matter

Option pricing models translate market expectations of price movement into a theoretical premium that a buyer should pay. In the Indian equity derivatives market, the Securities and Exchange Board of India (SEBI) requires brokers to quote fair values based on recognised models, making these calculations a core competency for distributors.

For the NISM exam, you will be asked to identify the correct model for a given scenario, compute the option premium using the provided data, and interpret the impact of changing inputs such as volatility or time to expiry. The exam frequently tests the ability to spot the underlying assumptions – for example, whether the option is European (exercisable only at expiry) or American (exercisable any time).

Typical traps include confusing the risk‑free rate with the cost‑of‑carry, or using the wrong time unit (months vs years). Remember that all NISM calculations assume a continuous‑compounding framework unless explicitly stated otherwise.

Model choice directly affects the premium; using BSM for an American option can lead to a lower price than the market expects.
Accurate input of volatility (σ) is crucial because it is the most sensitive variable in the formula.

ℹ️Common Exam Mistake – Volatility Units

Students often enter volatility as a percentage (e.g., 20) instead of a decimal (0.20). The BSM formula requires σ in decimal form, so always divide the quoted percent by 100 before substituting.

Black‑Scholes‑Merton (BSM) Model

The BSM model provides a closed‑form solution for European call and put options on non‑dividend‑paying stocks. It assumes a frictionless market, constant risk‑free rate, constant volatility, and that the underlying price follows a log‑normal distribution.

Key variables are the spot price (S), strike price (K), time to maturity (T in years), risk‑free rate (r, continuously compounded), and volatility (σ). The model also incorporates the cumulative standard normal distribution function N(·), which captures the probability that the option finishes in‑the‑money.

In the NISM exam, you may be given all inputs and asked to compute the call premium, or you may need to select the appropriate model for an American‑style option (answer: BSM is not suitable). Understanding the assumptions helps you quickly eliminate incorrect choices.

∑Formula: Black‑Scholes‑Merton Call Option Price

C = S \times N(d_1) - K \times e^{-rT} \times N(d_2)

Where:

C= Theoretical price of a European call option in rupees

S= Current spot price of the underlying equity in rupees

K= Strike price of the option in rupees

r= Continuously compounded risk‑free interest rate (decimal per annum)

T= Time to expiry expressed in years

σ= Annualised volatility of the underlying (decimal)

N(d)= Cumulative standard normal distribution evaluated at d

Worked Example

Given S = 10,000, K = 9,800, r = 0.06, T = 0.5, σ = 0.20: Step 1: Compute d1 = [ln(10000/9800) + (0.06 + 0.20^2/2)×0.5] / (0.20×√0.5) = 0.426 Step 2: Compute d2 = d1 - 0.20×√0.5 = 0.285 Step 3: N(d1) ≈ 0.666, N(d2) ≈ 0.610 (standard normal table) Step 4: e^{-rT} = e^{-0.06×0.5} = 0.97045 Step 5: C = 10,000×0.666 – 9,800×0.97045×0.610 = 6,664 – 5,800 = 864 Verification: 10,000×0.666 – 9,800×0.97045×0.610 = 864.

∑Formula: Intermediate Variables d₁ and d₂

d_{1}=\frac{\ln\left(\frac{S}{K}\right)+(r+\frac{\sigma^{2}}{2})T}{\sigma\sqrt{T}}\quad\text{and}\quad d_{2}=d_{1}-\sigma\sqrt{T}

Where:

d_{1}= First intermediate variable in BSM

d_{2}= Second intermediate variable, equals d1 minus volatility term

S= Spot price

K= Strike price

r= Risk‑free rate (decimal)

σ= Volatility (decimal)

T= Time to expiry in years

Worked Example

Using the same inputs as the call price example: Step 1: ln(S/K) = ln(10000/9800) = 0.0202 Step 2: (r + σ²/2)T = (0.06 + 0.04/2)×0.5 = 0.04 Step 3: Numerator = 0.0202 + 0.04 = 0.0602 Step 4: σ√T = 0.20×√0.5 = 0.1414 Step 5: d1 = 0.0602 / 0.1414 = 0.426, d2 = 0.426 – 0.1414 = 0.285 Verification: d1 = 0.426 and d2 = 0.285.

⚠️Exam Tip – Time Unit Consistency

If the question gives time in months, convert to years (e.g., 6 months = 0.5 years) before using the BSM formula. Forgetting this conversion leads to a significant pricing error.

Binomial Option Pricing Model

The binomial model builds a discrete‑time price tree where the underlying can move up or down at each step. It is flexible enough to price American options because early‑exercise decisions can be evaluated at every node.

Key inputs are the same as BSM (S, K, r, σ, T) plus the number of steps (n). The up‑move factor u = e^{σ\sqrt{Δt}} and down‑move factor d = 1/u, where Δt = T/n. The risk‑neutral probability of an up‑move is p = (e^{rΔt} - d) / (u - d). After constructing the tree, you work backwards, discounting expected option values at each node.

In the NISM exam, you may be asked to compute the option price for a two‑step tree or to identify why the binomial model is preferred for American options. Remember that increasing n improves accuracy but also increases computation – the exam usually limits n to a small number for hand calculations.

Comparison of BSM and Binomial Models

Model	Key Assumptions	Best Suited For
Black‑Scholes‑Merton	Continuous time, constant σ and r, European‑type, no dividends	European call/put on non‑dividend stocks
Binomial	Discrete steps, can incorporate varying σ, dividends, early exercise	American options, dividend‑paying stocks, scenarios requiring flexibility

Effect of Volatility on European Call Premium (BSM)

Example: NISM‑Style Scenario: Pricing a NIFTY Call

Scenario

Rohit, a retail investor, wants to buy a NIFTY call option with a strike of 18,000. The current NIFTY spot is 18,250, the SEBI‑approved risk‑free rate is 6% p.a., time to expiry is 3 months, and the implied volatility is quoted at 22% p.a. Compute the theoretical premium using the Black‑Scholes‑Merton model.

Solution

Convert inputs: T = 3/12 = 0.25 years, σ = 0.22, r = 0.06. Compute d1 = [ln(18250/18000) + (0.06 + 0.22²/2)×0.25] / (0.22×√0.25) = 0.338. Compute d2 = d1 - 0.22×√0.25 = 0.338 - 0.11 = 0.228. Using standard normal tables, N(d1) ≈ 0.632, N(d2) ≈ 0.590. Discount factor e^{-rT} = e^{-0.06×0.25} = 0.9851. Call premium = 18,250×0.632 – 18,000×0.9851×0.590 = 11,532 – 10,462 = 1,070 rupees (≈₹1,070).

Conclusion

Rohit should expect to pay roughly ₹1,070 for the call. The calculation demonstrates how volatility and time to expiry directly influence the premium, a frequent exam focus.

Practical Use in the Indian Market

Broker‑dealers in India quote option premiums based on the BSM or binomial model, depending on the contract type. SEBI’s “Derivatives Market Regulations” require that the quoted price be the higher of the model price and the market‑driven price to protect investors from under‑pricing.

Implied volatility is back‑solved from market prices using the BSM formula. In practice, traders use software to iterate until the model price matches the observed market price. For the exam, you only need to know the conceptual steps – set the market price equal to the BSM formula and solve for σ.

Remember that for dividend‑paying stocks, the BSM formula is adjusted by subtracting the present value of expected dividends from the spot price (S – PV(D)). The NISM syllabus mentions this adjustment, so be prepared for a quick calculation if a dividend is specified.

⭐Exam Takeaways

BSM provides a closed‑form price for European options; it assumes constant volatility, risk‑free rate, and no dividends.
Use the BSM call formula C = S·N(d₁) – K·e^{-rT}·N(d₂) with d₁ and d₂ defined explicitly; ensure σ and r are in decimal form.
Convert time to years and volatility to decimal before substitution – a common source of errors.
The binomial model can handle American options and dividend adjustments; it uses up/down factors u = e^{σ√Δt}, d = 1/u, and risk‑neutral probability p.
Higher volatility always raises the premium of a call option; the chart illustrates this monotonic relationship.
For dividend‑paying stocks, subtract the present value of expected dividends from the spot price in the BSM formula.
SEBI requires quoted prices to be at least the model‑derived price, reinforcing the practical relevance of accurate calculations.
When the exam asks for the model choice, match the option style (European → BSM, American → Binomial) and note any dividend considerations.

Practice Questions

8 questions on Option Pricing Models

Which of the following is an explicit assumption of the Black‑Scholes‑Merton model for equity options?

What is the formula for the intermediate variable d₁ in the BSM model?

Using S=10,000 ₹, K=9,800 ₹, r=0.06, T=0.5 yr and σ=0.20, what is the value of d₂?

For which type of option is the binomial pricing model preferred over the Black‑Scholes‑Merton model?

Rohit wants to buy a NIFTY call with strike 18,000 ₹, spot 18,250 ₹, r=6% p.a., T=3 months and σ=22% p.a. What is the theoretical premium using the BSM model?

According to the chart on volatility effect, what happens to a European call premium when volatility rises from 20% to 30%?

In the binomial option pricing model, how is the up‑move factor u defined?

If a question provides time to expiry as 6 months, what conversion must be performed before using the BSM formula?

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