Determinants of Option Premium
This sub‑topic explains what determines the premium of a commodity option. Understanding each determinant helps you calculate and interpret option prices, a frequent requirement in the NISM Series XVI exam. The content links theory to the Black‑Scholes model and practical Indian market scenarios.
Learning Objectives
- 1Define option premium and its components.
- 2Identify and explain each determinant of option premium.
- 3Apply the Black‑Scholes formula for European commodity options.
- 4Analyse how changes in market variables affect call and put premiums.
Option Premium Overview
An option premium is the price that the buyer pays to the seller for the right, but not the obligation, to buy (call) or sell (put) the underlying commodity at a predetermined strike price.
The premium consists of two parts: intrinsic value, which reflects any immediate payoff, and time value, which compensates the writer for the risk of future price movements.
In the NISM exam, candidates are often asked to decompose a quoted premium, identify which component dominates, or predict how a premium will change when a market variable shifts.
Intrinsic Value
Intrinsic value represents the amount by which an option is already in‑the‑money. For a call, it is the excess of the spot price (S) over the strike price (K); for a put, it is the excess of K over S.
The formal definition is:
Call Intrinsic Value = max(S − K, 0)
Put Intrinsic Value = max(K − S, 0). If the result is zero, the option is out‑of‑the‑money and its premium consists entirely of time value.
Exam tip: Never forget the max function. A common mistake is to subtract without applying the max, which leads to negative intrinsic values – something the syllabus explicitly prohibits.
Students sometimes treat a deep‑out‑of‑the‑money option as having zero premium. Remember, the premium can never be less than the intrinsic value; it may be equal only when time value is zero.
Time Value
Time value is the portion of the premium that exceeds intrinsic value. It reflects the probability that the option will become more valuable before expiry, and therefore compensates the writer for the uncertainty.
Key drivers of time value include time to expiry, volatility of the underlying, the risk‑free interest rate, and any expected dividend or convenience yield on the commodity.
For the exam, you may be asked to compute time value as: Time Value = Premium − Intrinsic Value. If the result is negative, the answer is wrong – time value is never negative.
A negative time value indicates a calculation error. The premium always equals or exceeds intrinsic value.
Determinants of Option Premium
The premium is shaped by six primary determinants: the current spot price of the commodity, the strike price (or moneyness), the time remaining to expiry, the volatility of the underlying, the prevailing risk‑free rate, and any dividend or convenience yield associated with the commodity.
Each determinant influences call and put premiums differently. For example, a rise in the spot price raises call premiums but lowers put premiums, while higher volatility lifts both because the chance of large moves increases.
In NISM questions, you may be presented with a scenario where one variable changes and you must identify the direction of the premium’s movement for both call and put options.
- Spot price (S) – higher S ↑ call premium, ↓ put premium.
- Strike price (K) – deeper‑in‑the‑money (lower K for calls) ↑ premium.
- Time to expiry (T) – longer T ↑ time value, thus higher premium.
- Volatility (σ) – higher σ ↑ both call and put premiums.
- Risk‑free rate (r) – higher r ↑ call premium, ↓ put premium.
- Dividend/Convenience Yield (q) – higher q ↓ call premium, ↑ put premium.
Effect of Key Determinants on Call and Put Premiums
| Determinant | Effect on Call Premium | Effect on Put Premium |
|---|---|---|
| Spot price (S) ↑ | Premium ↑ | Premium ↓ |
| Strike price (K) ↑ | Premium ↓ | Premium ↑ |
| Time to expiry (T) ↑ | Premium ↑ | Premium ↑ |
| Volatility (σ) ↑ | Premium ↑ | Premium ↑ |
| Risk‑free rate (r) ↑ | Premium ↑ | Premium ↓ |
| Dividend/Convenience Yield (q) ↑ | Premium ↓ | Premium ↑ |
Impact of Underlying Price & Time to Expiry
The relationship between the spot price and the option premium is captured by the Greek Delta (Δ). For a call, Δ is positive (0 to 1) and measures how much the premium changes for a one‑rupee move in the underlying. For a put, Δ is negative (‑1 to 0).
Time to expiry influences the premium through Theta (Θ), the time‑decay factor. As expiry approaches, Θ becomes more negative, eroding time value faster. This effect is strongest for at‑the‑money options.
Exam relevance: A typical NISM question may give a change in days to expiry and ask whether the premium will increase or decrease, and by which Greek the effect is explained.
Impact of Volatility & Interest Rate
Volatility is measured by the standard deviation of the underlying’s returns and is represented by the Greek Vega (ν). Higher σ raises the probability of large price swings, thereby increasing both call and put premiums.
The risk‑free interest rate enters the pricing model via Rho (ρ). An increase in r raises the present value of the strike price paid in the future, which benefits call writers (premium ↑) and hurts put writers (premium ↓).
Common exam mistake: Confusing the direction of Rho’s impact on calls versus puts. Remember that a higher r makes calls more valuable and puts less valuable.
Black‑Scholes Pricing Formula
Where:
C= European call premium (rupees)P= European put premium (rupees)S= Current spot price of the commodity (rupees)K= Strike price (rupees)r= Risk‑free interest rate (annual, decimal)q= Continuous dividend or convenience yield (annual, decimal)T= Time to expiry (years)σ= Volatility of the underlying (annual, decimal)N(·)= Cumulative standard normal distribution functiond_1= Intermediate variable in Black‑Scholesd_2= Intermediate variable in Black‑ScholesWorked Example
Given S = 100, K = 100, r = 0.05, q = 0, σ = 0.20, T = 0.5 years: Step 1: d1 = (ln(100/100) + (0.05 + 0.20^2/2)*0.5) / (0.20*√0.5) = (0 + 0.035) / 0.1414 = 0.2475 Step 2: d2 = d1 - 0.20*√0.5 = 0.2475 - 0.1414 = 0.1061 Step 3: N(d1) ≈ 0.598, N(d2) ≈ 0.542 Step 4: C = 100 × 0.598 – 100 × e^{-0.05×0.5} × 0.542 = 59.8 – 100 × 0.9753 × 0.542 = 59.8 – 52.9 = 6.9 Verification: (100 × 0.598) – (100 × e^{-0.025} × 0.542) = 6.9
Scenario
Rohit, an Indian distributor, wants to buy a European call option on crude oil futures. The spot price of crude is ₹4,500 per barrel, the strike price is ₹4,600, time to expiry is 90 days (0.246 years), the annual risk‑free rate is 6%, the annual volatility is 25%, and the convenience yield is 2%. Compute the approximate call premium using the Black‑Scholes model.
Solution
Convert all rates to decimals: r = 0.06, q = 0.02, σ = 0.25, T = 0.246. Compute d1 = [ln(4500/4600) + (0.06‑0.02 + 0.25^2/2)×0.246] / (0.25√0.246). ln(4500/4600)=‑0.0216. σ^2/2 = 0.03125. (r‑q+σ^2/2)=0.06‑0.02+0.03125=0.07125. Multiply by T: 0.07125×0.246=0.0175. Numerator =‑0.0216+0.0175=‑0.0041. √T = √0.246 = 0.496. Denominator = 0.25×0.496=0.124. d1 =‑0.0041/0.124 =‑0.033. d2 = d1‑σ√T =‑0.033‑0.124 =‑0.157. N(d1)≈0.487, N(d2)≈0.438. Compute e^{-qT}=e^{-0.02×0.246}=e^{-0.00492}=0.9951. e^{-rT}=e^{-0.06×0.246}=e^{-0.01476}=0.9854. Call premium C = 4500×0.9951×0.487 – 4600×0.9854×0.438 = 2185.5 – 1985.6 ≈ 199.9 rupees.
Conclusion
The premium of roughly ₹200 reflects the out‑of‑the‑money position (spot < strike) and the moderate time left. In the exam, selecting the correct sign for ln(S/K) and applying the convenience yield are common pitfalls.
Effect of Volatility on a Call Premium (S=100, K=100, T=0.5 yr, r=5%)
⭐Exam Takeaways
- Option premium = Intrinsic Value + Time Value; never forget the max() function for intrinsic value.
- Higher spot price raises call premium but lowers put premium; the opposite holds for a higher strike price.
- Longer time to expiry and higher volatility increase the time value component for both calls and puts.
- Risk‑free rate benefits calls (premium ↑) and hurts puts (premium ↓); dividend or convenience yield has the reverse effect.
- Black‑Scholes provides the standard formula for European commodity options; ensure correct input of r, q, σ, and T.
Practice Questions
8 questions on Determinants of Option Premium
What are the two components that together make up an option premium?
How is the intrinsic value of a call option calculated?
According to the determinants of option premium, what is the effect of a higher volatility on call and put premiums?
If the spot price of a commodity rises while all other determinants remain unchanged, how will the premiums of a call and a put option respectively be affected?
In the Black‑Scholes example where S=100, K=100, r=0.05, σ=0.20 and T=0.5 years, what is the sign of the calculated d₁?
Using the crude‑oil call scenario (S=₹4,500, K=₹4,600, r=6%, q=2%, σ=25%, T=0.246 yr), what is the approximate call premium obtained from the Black‑Scholes calculation?
When the spot price rises, time to expiry stays the same, volatility is unchanged, and the risk‑free rate increases, what is the overall impact on a European call premium?
In the Black‑Scholes call pricing formula C = S e^{-qT} N(d₁) - K e^{-rT} N(d₂), which variable directly influences the present value of the strike price component?