(Step‑by‑step practical examples for the key formulas and relations used in option valuation, using plain numbers and intuitive explanations)
Option valuation estimates the fair value of an option by modeling expected payoffs under risk‑neutral probabilities and discounting at the risk‑free rate, with price sensitivities captured by Greeks and implied volatility extracted from market prices for calibration and risk management.
Core idea
Option valuation uses risk‑neutral pricing to discount expected payoffs at the risk‑free rate, with model calibration anchored to implied volatility from market prices to ensure consistency between theoretical values and observed quotes.
Key models
- Black–Scholes–Merton (BSM): Closed‑form solution for European calls and puts on non‑dividend or constant‑dividend assets, assuming lognormal returns and constant volatility; efficient for benchmarks but cannot capture early exercise or volatility smiles without adjustments.
- Binomial/Lattice: Discrete‑time tree with up/down moves, risk‑neutral probabilities, and backward induction; handles American early exercise, discrete dividends, and path‑dependent features, converging to BSM as steps increase.
- Monte Carlo: Simulation of many price paths to estimate the expected discounted payoff; flexible for exotics and path dependence, with specialized methods required for early‑exercise features.
Black–Scholes formulas
For a non‑dividend European call and put, with spot SSS, strike KKK, risk‑free rate rrr, volatility σ\sigmaσ, and time TTT:
d1=ln(S/K)+(r+σ2/2)TσTd_1=\frac{\ln(S/K)+(r+\sigma^2/2)T}{\sigma\sqrt{T}}d1=σTln(S/K)+(r+σ2/2)T and d2=d1−σTd_2=d_1-\sigma\sqrt{T}d2=d1−σT.
Call: C=S N(d1)−Ke−rTN(d2)C=S\,N(d_1)-K e^{-rT} N(d_2)C=SN(d1)−Ke−rTN(d2).
Put: P=Ke−rTN(−d2)−S N(−d1)P=K e^{-rT}N(-d_2)-S\,N(-d_1)P=Ke−rTN(−d2)−SN(−d1).
These formulas reflect risk‑neutral pricing where expected growth equals the risk‑free rate, with standard extensions for continuous dividends or foreign interest rates in FX options.
Binomial method essentials
Construct a tree with up factor uuu, down factor ddd, and risk‑neutral probability ppp; compute terminal payoffs and back‑propagate discounted values node by node.
For American options, at each node take the maximum of intrinsic value and continuation value to capture optimal early exercise.
As the number of steps grows large, binomial prices for European claims converge to BSM values, offering a cross‑check between methods.
Greeks and risk Delta:
Delta: Sensitivity to underlying price; for European calls near at‑the‑money, delta approximates the probability of finishing in‑the‑money and sets the hedge ratio for small spot moves.
Gamma: Sensitivity of delta to spot; high gamma indicates curvature and a need for more frequent re‑hedging, typically largest near at‑the‑money and at short maturities.
Theta: Time decay of option value; generally negative for long options and accelerates close to expiration as extrinsic value erodes.
Vega: Sensitivity to implied volatility; largest near at‑the‑money and increases with time to maturity, central to volatility risk management.
Rho: Sensitivity to interest rates; more material for longer‑dated options and underlyings with strong rate linkages.
Practical risk control uses delta‑gamma‑vega‑theta monitoring and scenario testing for combined shocks to spot, vol, and rates across the book.
Implied volatility, smile, and surface:
Implied volatility is the σ\sigmaσ that aligns a model price to the observed market price, creating a common quoting convention and a calibration anchor for valuation and risk.
Smiles and skews: Equity options often show higher IV for out‑of‑the‑money puts than for calls, reflecting downside risk aversion and crash‑risk premia that contradict constant‑vol assumptions.
Term structure and surfaces: IV varies by strike and maturity; a smooth 3D volatility surface across expiries and moneyness guides quoting, hedging, and stress testing.
Practical calibration and usage
Derive ATM implied vols from liquid options, then interpolate across strikes and maturities to build a smooth surface for consistent valuation and Greeks.
Use lattices for American options and discrete dividend schedules, and BSM (with dividend yields) for European‑style contracts where appropriate.
For exotics and path‑dependent payoffs, use Monte Carlo with variance reduction and techniques like Longstaff–Schwartz for early exercise.
Model limitations and extensions
Constant‑vol and lognormal assumptions miss fat‑tails, skew, and volatility dynamics; models such as local volatility, stochastic volatility, and jump‑diffusions better fit smiles and term structures.
Continuous‑time hedging is infeasible; discrete re‑hedging leaves residual gamma and vega risks that widen P/L distributions during volatile regimes.
Indian market notes
For Indian single‑stock and index options, exercise styles and discrete dividends influence model choice; lattices handle early exercise and dividend timing, while BSM with dividend yields is a practical approximation for European‑style series.
Trading and risk management are commonly expressed in volatility terms, with surface construction and maintenance central to pricing discipline and supervision.
Quick formulas and relations
Put‑call parity (European, no dividends): C−P=S−Ke−rTC-P=S-K e^{-rT}C−P=S−Ke−rT, enabling arbitrage checks and derivation of missing prices given counterparts and forward relations.
Delta in BSM: Call delta =N(d1)=N(d_1)=N(d1); put delta =N(d1)−1=N(d_1)-1=N(d1)−1, guiding hedge sizing and rebalancing.
Vega profile: Vega peaks near at‑the‑money and increases with time to expiry, helping identify which options dominate volatility P/L.
Workflow for valuation and risk
Ingest option chain data, compute per‑strike implied vols, and smooth to a stable surface; recompute Greeks on the surface for consistency.
Value European options with BSM on the surface; value American/dividend‑heavy names via binomial; validate marks with parity and cross‑model checks.
Hedge to delta/gamma thresholds, monitor vega buckets and theta decay, and stress test under vol‑of‑vol and skew shifts.
Here are simple, step‑by‑step practical examples for the key formulas and relations used in option valuation, using plain numbers and intuitive explanations. Black–Scholes call price Example: Spot S = 100, Strike K = 110, Time T = 0.25 years (3 months), Risk‑free rate r = 5% (0.05), Volatility σ = 20% (0.20).
Steps:
Compute
d1 = [ln(S/K) + (r + σ²/2)T] / (σ√T). Using ln(100/110) ≈ −0.0953, σ²/2 = 0.02, so numerator ≈ −0.0953 + (0.05 + 0.02)×0.25 = −0.0953 + 0.0175 = −0.0778. Denominator = 0.20×√0.25 = 0.10. Thus d1 ≈ −0.778.
d2 = d1 − σ√T = −0.778 − 0.10 = −0.878.
Look up N(d1) and N(d2) from a normal table or calculator: N(−0.778) ≈ 0.218, N(−0.878) ≈ 0.190.
Discounted strike = K e^{−rT} = 110 × e^{−0.05×0.25} ≈ 110 × 0.9876 ≈ 108.64.
Call value C = S×N(d1) − Discounted strike×N(d2) ≈ 100×0.218 − 108.64×0.190 ≈ 21.8 − 20.64 ≈ 1.16.
I nterpretation: With only 3 months left and the strike above spot, the model gives a low call value; most of the price is time value, and it’s sensitive to volatility and time. Black–Scholes put price
Use the same inputs: S = 100, K = 110, T = 0.25, r = 5%, σ = 20%. Either price directly with P = K e^{−rT} N(−d2) − S N(−d1), or use put‑call parity. Direct: N(−d2) = N(0.878) ≈ 0.810, N(−d1) = N(0.778) ≈ 0.782. P ≈ 108.64×0.810 − 100×0.782 ≈ 88.00 − 78.20 ≈ 9.80.
Parity check: C − P = S − K e^{−rT} ⇒ 1.16 − P = 100 − 108.64 ⇒ 1.16 − P = −8.64 ⇒ P ≈ 9.80.
Interpretation: The put is valuable because the strike is above spot; parity keeps call and put prices consistent. Put‑call parity intuition
Formula (European, no dividends): C − P = S − K e^{−rT}.
Quick use: If three items are known, the fourth is implied. For example, if S = 200, K = 190, T = 0.5, r = 6%, and C = 18, then Discounted strike = 190×e^{−0.06×0.5} ≈ 190×0.9704 ≈ 184.38. Thus P = C − (S − Discounted strike) = 18 − (200 − 184.38) = 18 − 15.62 = 2.38.
Interpretation: This prevents arbitrage by linking calls, puts, and forwards embedded via S and discounted K.
Delta (call and put) : Using the first example with d1 ≈ −0.778: Call delta = N(d1) ≈ 0.218. That means if the stock rises by 1, the call price rises by about 0.22 (for small moves).Put delta = N(d1) − 1 ≈ −0.782. If the stock rises by 1, the put price falls by about 0.78 (for small moves).
Hedge intuition: To be roughly delta‑neutral when long 1 call, short about 0.22 shares; for 100 options (each on 1 share), short about 22 shares. Gamma (curvature)
Conceptual example: Suppose gamma = 0.015 for the call at the current spot. A 1‑point rise in S increases delta by about 0.015; a 1‑point fall reduces delta by about 0.015.
Practical effect: If S jumps from 100 to 102, delta might move from 0.22 to roughly 0.25, so a static hedge becomes misaligned and needs rebalancing. Theta (time decay) Example intuition: If the call’s theta is −0.03 per day, then, all else equal, the option loses about 0.03 in value overnight due to the passage of time. Short‑dated, at‑the‑money options often see larger daily theta in absolute terms near expiration, as time value collapses faster. Vega (volatility sensitivity)
Example: If vega is 0.12, a 1 percentage point rise in implied volatility (say 20% to 21%) increases the option’s value by about 0.12, holding spot and time fixed.
Practical takeaway: Longer‑dated, near‑ATM options usually have higher vega and thus larger P/L swings when implied volatility shifts. Rho (rate sensitivity)
Example: If rho for the call is 0.20, then a 1 percentage point rise in interest rates (e.g., 5% to 6%) raises the call’s value by about 0.20, other inputs unchanged. Longer maturities have larger rho; for puts, rho is typically negative, so higher rates reduce put values. Binomial lattice: simple two‑step example
Setup: S = 100, up factor u = 1.10, down factor d = 0.90, risk‑free r = 5%, T = 1 year, two steps of 0.5 years each, European call with K = 100. Risk‑neutral probability per step p = (e^{rΔt} − d) / (u − d) = (e^{0.05×0.5} − 0.90) / (1.10 − 0.90) ≈ (1.0253 − 0.90)/0.20 ≈ 0.6265.
Tree prices: After two steps, S_uu = 100×1.10×1.10 = 121; S_ud = S_du = 99; S_dd = 81.
Payoffs: max(121−100,0)=21; max(99−100,0)=0; max(81−100,0)=0.
Backward induction: At t = 0.5, the node after one up move has continuation value = e^{−rΔt}[p×21 + (1−p)×0] ≈ e^{−0.025}[0.6265×21] ≈ 0.9753×13.16 ≈ 12.84.
The node after one down move leads to zero payoff either way, so its value is 0. At t = 0, price = e^{−rΔt}[p×12.84 + (1−p)×0] ≈ 0.9753×(0.6265×12.84) ≈ 0.9753×8.04 ≈ 7.85.
Interpretation: The binomial model recovers a reasonable European call price using simple up/down logic and discounting; more steps make it more accurate and allow American‑style checks. Implied volatility from a market price Suppose the 3‑month call from the first example trades in the market at 2.00 instead of the 1.16 theoretical value using σ = 20%.
Task: Find the σ that, when plugged into the Black–Scholes formula, yields price = 2.00.Process: Use a calculator or solver to adjust σ until the model price equals 2.00; the resulting σ is the implied volatility.
Interpretation: Implied volatility translates market prices into the volatility level the market is “implying,” creating a common language for quoting and hedging.
Using put‑call parity to detect issues Suppose observed quotes are C = 12 and P = 10 with S = 150, K = 140, T = 0.5, r = 4%.Discounted K = 140×e^{−0.04×0.5} ≈ 140×0.9802 ≈ 137.23.
Parity says C − P should be S − discounted K = 150 − 137.23 = 12.77.Observed C − P = 12 − 10 = 2.The gap (2 vs 12.77) suggests mispricing or stale quotes; in frictionless markets, this would imply an arbitrage opportunity. In reality, costs, constraints, and spreads may explain the discrepancy.
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