Module 8 · Derivatives

Pricing and Valuation of Options

EN: Payoffs at expiry, intrinsic and time value, and the six pricing factors.
VN: Payoff khi đáo hạn, giá trị nội tại và thời gian, sáu yếu tố định giá.

1. Option Payoffs at Expiry Core

About: Call payoff = max(S−K,0). Put payoff = max(K−S,0). Profit subtracts the premium paid. Hockey-stick payoff diagrams characterize all options.Tóm tắt: Call = max(S−K,0). Put = max(K−S,0). Profit trừ premium. Đồ thị 'gậy hockey'.

Hockey-stick payoffs at expiry. Buyer's loss capped at premium paid.

\[ \text{Call payoff: } \max(S_T - K,\,0) \] \[ \text{Put payoff: } \max(K - S_T,\,0) \]

Profit (after premium)

\[ \text{Long call profit} = \max(S_T - K, 0) - c_0 \] \[ \text{Long put profit} = \max(K - S_T, 0) - p_0 \]

Practice problem

ST = $55, K = $50. Compute call and put payoffs at expiry.

Show solution

Call = max(55 − 50, 0) = 5

Put = max(50 − 55, 0) = 0

Call payoff $5; Put payoff $0

2. Intrinsic Value & Time Value Core

\[ \text{Intrinsic (call)} = \max(S - K,\,0), \quad \text{Intrinsic (put)} = \max(K - S,\,0) \] \[ \text{Option price} = \text{Intrinsic} + \text{Time value} \]

Moneyness

ITM Positive intrinsic value (call: S > K, put: S < K).
ATM S ≈ K.
OTM Zero intrinsic value.

3. Six Factors Affecting Option Value Core

About: S, K, σ, r, T, dividends. σ helps both calls and puts. Dividends hurt calls, help puts. Memorize these directional effects for the exam.Tóm tắt: 6 yếu tố: S, K, σ, r, T, div. σ giúp cả call & put. Div hại call giúp put.

Effects on call (c) and put (p)

S₀ ↑ c ↑, p ↓
K ↑ c ↓, p ↑
σ ↑ Both c ↑ and p ↑ (volatility helps both)
r ↑ c ↑, p ↓
T ↑ Generally both ↑ (American); for European it's usually true
CF on under. ↑ c ↓, p ↑ (dividends reduce stock value at expiration)

4. Option Price Bounds (No Arbitrage) Core

About: No-arb bounds keep options from being absurdly cheap/expensive. Lower bound for call: max(0, S − PV(K)). Tight when ITM.Tóm tắt: Cận no-arbitrage. Lower bound call: max(0, S − PV(K)). Chặt khi ITM.

\[ \max(0,\,S_0 - K(1+r)^{-T}) \le c_0 \le S_0 \] \[ \max(0,\,K(1+r)^{-T} - S_0) \le p_0 \le K(1+r)^{-T} \]

Practice problem

S0 = $50, K = $45, r = 5%, T = 1 yr. Compute lower bound for European call.

Show solution

PV(K) = 45/1.05 ≈ 42.86

Lower bound = max(0, 50 − 42.86)

Lower bound ≈ $7.14

Practice problem Practice

Practice problem

S = $50, K = $45, call premium = $7. Compute intrinsic and time value.

Show solution

Intrinsic = max(50 − 45, 0) = 5

Time value = premium − intrinsic = 7 − 5

Intrinsic = $5; Time value = $2