Financial Strategy

Module 69 — Options Theory & Derivatives Pricing

Option Greeks, put-call parity, binomial tree pricing, the volatility surface, exotic options, interest rate derivatives — the complete options theory toolkit for quantitative finance practitioners.

Learning Objectives

  • Calculate and interpret all major option Greeks
  • Apply put-call parity as an arbitrage check
  • Price options using the binomial tree model
  • Understand and construct the volatility surface
  • Price interest rate derivatives: caps, floors, and swaptions

1. Option Fundamentals Review

The Four Basic Options

TypeRightExercise
Call (Long)Right to BUY at strike KProfitable when S_T > K
Put (Long)Right to SELL at strike KProfitable when S_T < K
Call (Short)Obligation to SELL if exercisedLoses when S_T > K
Put (Short)Obligation to BUY if exercisedLoses when S_T < K

European option: Can only be exercised at expiry T. American option: Can be exercised at any time before T. Always worth ≥ European equivalent.

2. The Option Greeks

Greeks measure the sensitivity of option value to various inputs. Every options trader needs to live and breathe Greeks.

Delta (Δ) — Price Sensitivity

Delta = ∂V/∂S: Change in option value per unit change in stock price.

Call delta: N(d₁) ∈ [0, 1]
Put delta: N(d₁) − 1 ∈ [−1, 0]

At-the-money (ATM): Call delta ≈ 0.50; Put delta ≈ −0.50 Deep in-the-money call: Delta → 1 (moves like the stock) Deep out-of-the-money call: Delta → 0 (barely moves with the stock)

Delta hedging: A delta-neutral portfolio has total delta = 0. If you own 100 call options (each with delta 0.5), you have delta = +50. Short 50 shares to make the portfolio delta-neutral.

Gamma (Γ) — Delta Sensitivity

Gamma = ∂²V/∂S² = ∂Δ/∂S: Rate of change of delta as stock price changes.

Always positive for long options (calls and puts).

Practical meaning:

  • High gamma: delta changes rapidly when stock moves — requires frequent rebalancing
  • ATM options have highest gamma (delta changes fastest near the strike)
  • OTM and ITM options have lower gamma

Gamma-delta hedging: A position that is both delta-neutral and gamma-neutral is insensitive to small AND larger price moves.

Theta (Θ) — Time Decay

Theta = ∂V/∂t: Change in option value with passage of time (per day).

Almost always negative for long options — options lose value as time passes (all else equal). The closer to expiry, the faster the decay for ATM options.

Time decay accelerates near expiry for ATM options. This creates the "gamma vs theta" trade-off:

  • Long options: positive gamma (good for big moves), negative theta (paying time decay)
  • Short options: negative gamma (hurt by big moves), positive theta (collecting time decay)

Vega (ν) — Volatility Sensitivity

Vega = ∂V/∂σ: Change in option value per 1% change in implied volatility.

Always positive for long options — higher volatility makes options more valuable.

Vega ≈ S × √T × N'(d₁) / 100    (per 1% change in vol)

Vega is largest for ATM options and for longer-dated options.

Practical implication: An options portfolio with large positive vega profits when volatility rises (long volatility). Large negative vega profits when volatility falls (short volatility).

Rho (ρ) — Interest Rate Sensitivity

Rho = ∂V/∂r: Change in option value per 1% change in risk-free rate.

Calls: Positive rho (higher rates make calls worth more — higher future stock price expectation) Puts: Negative rho

Rho matters more for long-dated options and during periods of rate volatility (e.g., Pakistan's SBP rate cycles).

3. Put-Call Parity

The Relationship

For European options on a non-dividend-paying stock:

C + PV(K) = P + S₀

Where:
C = call price
P = put price
PV(K) = K × e^(-rT) = present value of strike
S₀ = current stock price

Intuition: Two portfolios with the same payoff must have the same value:

  • Portfolio A: Call + PV(K) → at expiry: gets max(S_T, K)
  • Portfolio B: Put + S₀ → at expiry: gets max(S_T, K)

Using Put-Call Parity

Arbitrage check: If put-call parity is violated in the market, there is an arbitrage. Example:

S₀ = 100, K = 100, T = 1yr, r = 5%, Call = PKR 10

PCP implies: P = C + K×e^(-rT) − S₀ = 10 + 100×e^(-0.05) − 100 = 10 + 95.12 − 100 = PKR 5.12

If market put is trading at PKR 7: arbitrage (buy call, sell put, short stock, invest PV(K))

Implied forward price: PCP implies F = S₀ × e^(rT) = (C − P) × e^(rT) + K.

4. Binomial Tree Pricing

One-Period Binomial Tree

The binomial tree model prices options without needing the Black-Scholes PDE. It is conceptually clear and extends easily to American options and exotic payoffs.

Setup (one period):
  Current stock price: S = 100
  Up move: uS = 110 (u = 1.1)
  Down move: dS = 90 (d = 0.9)
  Risk-free rate: r = 5% (for the period)
  
Risk-neutral probability of up move:
  p* = (e^r − d) / (u − d) = (1.05 − 0.9) / (1.1 − 0.9) = 0.75

Option payoffs:
  Call (K=100): Cu = max(110 − 100, 0) = 10; Cd = max(90 − 100, 0) = 0

Price:
  C = e^(-r) × [p* × Cu + (1−p*) × Cd]
    = (1/1.05) × [0.75 × 10 + 0.25 × 0]
    = (1/1.05) × 7.5
    = PKR 7.14

Multi-Period Binomial Tree

For T periods, build the tree forward (computing all nodes) and then work backward from the terminal payoffs. The Black-Scholes formula is the limit of the binomial tree as the number of periods approaches infinity.

American option pricing: At each node, compare the option's continuation value (from binomial backward induction) with the immediate exercise value. Take the maximum. This is why American options are priced on trees (Black-Scholes gives no closed form for American puts).

5. The Volatility Surface

What the Volatility Surface Is

Black-Scholes assumes constant volatility σ. In reality, implied volatility (the σ that makes the Black-Scholes price equal to the market price) varies across:

  • Strike (K): The "volatility smile" or "volatility skew"
  • Maturity (T): The "term structure of volatility"

Together, these form the volatility surface: a 3D plot of implied volatility vs strike and maturity.

The Volatility Smile and Skew

Equity markets — volatility skew (or "smirk"):

  • OTM puts (low strikes) have higher implied volatility than ATM options
  • OTM calls (high strikes) have lower implied volatility than ATM options
  • Reflects demand for downside protection (OTM puts are expensive)
Example: BIQAI Group equity options
  Strike   80    90    100   110   120
  Implied  35%   30%   25%   24%   23%
  
Skew: lower strikes have higher implied vol → reflecting crash risk premium

FX markets — symmetric smile: Both OTM calls and OTM puts have higher implied vol than ATM. Reflects two-way tail risk in exchange rates.

Term Structure of Volatility

Short-dated options often have higher implied vol than longer-dated ones (when near-term uncertainty is elevated — e.g., around a major economic announcement). The opposite holds when long-term uncertainty dominates.

6. Interest Rate Derivatives

Caps and Floors

A cap is a portfolio of call options on interest rates — it caps the maximum interest rate a borrower pays. Each sub-option (a "caplet") pays:

max(L_T − K_cap, 0) × Notional × Δt
Where L_T = LIBOR/KIBOR at reset date, K_cap = cap strike rate, Δt = accrual period

A floor is a portfolio of put options on interest rates — it floors the minimum rate an investor receives:

Floorlet payoff: max(K_floor − L_T, 0) × Notional × Δt

CFO application: BIQAI Group has a PKR 5bn floating rate TFC at KIBOR + 200bps. If KIBOR rises to 18%, interest expense becomes punishing. Buy a cap at KIBOR = 15% to limit upside rate risk.

Swaptions

A swaption is an option to enter into an interest rate swap at a future date.

Payer swaption: Right to PAY fixed rate in a swap (analogous to a call on rates — used to hedge against rising rates) Receiver swaption: Right to RECEIVE fixed rate (analogous to a put on rates)

Black's Model for Caps and Swaptions

Black's model (a modification of Black-Scholes) is the market standard for pricing caps and swaptions, treating forward rates as the underlying:

Caplet price = P(0,T) × [F × N(d₁) − K_cap × N(d₂)]

Where F = forward KIBOR rate, P(0,T) = discount factor, and d₁, d₂ are analogous to Black-Scholes

Self-Assessment

  1. FERROQUANT Capital holds a long position in 1,000 European call options on BIQAI Group equity (each covering 1 share). Current price: PKR 150, strike: PKR 155, T = 3 months, σ = 28%, r = 12%.

    (a) Use Black-Scholes to calculate the call price and delta. (b) How many shares does FERROQUANT need to short to delta-hedge this position? (c) The next day, BIQAI Group rises to PKR 158. Recalculate delta. Why has it changed, and what must FERROQUANT do to rebalance? (d) What is the gamma of this position? Why does positive gamma mean you always benefit from large moves?

  2. FERROQUANT identifies what appears to be a put-call parity violation:

    • BIQAI Group: S₀ = PKR 150
    • Strike K = PKR 150, T = 6 months, r = 12% annualized
    • Market call price: PKR 12.50
    • Market put price: PKR 16.00

    (a) What does put-call parity imply the put should be worth? (b) Is there an arbitrage? If so, describe exactly which trades you would execute today and at expiry. (c) Why do put-call parity violations in Pakistan markets persist longer than in developed markets?

  3. The FERROQUANT options desk observes this implied volatility surface for BIQAI Group equity:

    • 1-month options: ATM vol = 32%, OTM put (80% moneyness) vol = 45%
    • 6-month options: ATM vol = 28%, OTM put vol = 36%
    • 1-year options: ATM vol = 25%, OTM put vol = 30%

    Interpret: (a) the volatility skew in 1-month options vs 1-year options, (b) the term structure (1-month vs 1-year ATM), (c) one trading strategy that would profit if you believe the 1-month skew is too steep, and (d) one trading strategy if you believe the term structure will flatten.