Option Pricing: basic principles S. Mann, 2009 Value boundaries Simple arbitrage relationships Intuition for the role of volatility

Call Option Valuation "Boundaries" Option Value Define: C[S(0),T;K] =Value of American call option with strike K, expiration date T, and current underlying asset value S(0) Resultproof 1) C[0,T; K] = 0 (trivial) 2) C[S(0),T;K] >= max(0, S(0) -K) (limited liability) 3) C[S(0),T;K] <= S(0) (trivial) Intrinsic Value - Value of Immediate exercise: S - K KS (asset price) 0 Option value must be within this region

European Call lower bound (asset pays no dividend) Option Value Define: c[S(0),T;K] =Value of European call (can be exercised only at expiration) value at expiration Positioncost nowS(T) K A) long call + T-bill c[S(0),T;K] + KB(0,T)KS(T) B) long stockS(0)S(T)S(T) position A dominates, so c[S(0),T;K] + KB(0,T) >= S(0) thus 4)c[S(0),T;K] >= Max(0, S(0) - KB(0,T) Intrinsic value: S - K KB(0,T) KS (asset price) 0 Option value must be within this region “Pure time value”: K - B(0,T)K

Example: Lower bound on European Call Option Value Example: S(0) =$55. K=$50. T= 3 months. 3-month simple rate=4.0%. B(0,3) = 1/(1+.04(3/12)) = KB(0,3) = Lower bound is S(0) - KB(0,T) = 55 – = $5.50. What if C 55 = $5.25? Value at expiration Positioncash flow nowS(T) $50 buy call- $ S(T) - $50 buy bill paying K short stock S(T) -S(T) Total+ $ S(T) >= 00 Intrinsic value: =S(0) S (asset price) 0 Option value must be within this region “Pure time value”: = $1.09

American and European calls on assets without dividends 5) American call is worth at least as much as European Call C[S(0),T;K] >= c[S(0),T;K](proof trivial) 6) American call on asset without dividends will not be exercised early. C[S(0),T;K] = c[S(0),T;K] proof: C[S(0),T;K] >= c[S(0),T;K] >= S(0) - KB(0,T) so C[S(0),T;K] >= S(0) - KB(0,T) >= S(0) - K and C[S(0),T;K] >= S(0) - K Call is: worth more alive than dead Early exercise forfeits time value 7) longer maturity cannot have negative value: for T 1 > T 2: C(S(0),T 1 ;K) >= C(S(0),T 2 ;K)

Call Option Value Option Value 0 Intrinsic Value: max (0, S-K) lower bound No-arbitrage boundary: C >= max (0, S - PV(K)) 0 KS

Volatility Value : Call option Call payoff KS(T) (asset value) Low volatility asset High volatility asset

Volatility Value : Call option Example: Equally Likely "States of World" "State of World" Expected Position Bad Avg Good Value Stock A Stock B Calls w/ strike=30: Call on A: Call on B:

Discrete-time lognormal evolution:

Put Option Valuation "Boundaries" Option Value Define: P[S(0),T;K] =Value of American put option with strike K, expiration date T, and current underlying asset value S(0) Resultproof 8) P[0,T; K] = K (trivial) 9) P[S(0),T;K] >= max(0, K - S(0)) (limited liability) 10) P[S(0),T;K] <= K (trivial) Intrinsic Value - Value of Immediate exercise: K - S KS (asset price) 0 Option value must be within this region K

European Put lower bound (asset pays no dividend) Option Value Define: p[S(0),T;K] =Value of European put (can be exercised only at expiration) value at expiration positioncost nowS(T) K A) long put + stockp[S(0),T;K] + S(0)KS(T) B) long T-billKB(0,T)KK position A dominates, sop[S(0),T;K] + S(0) >= KB(0,T) thus 11)p[S(0),T;K] >= max (0, KB(0,T)- S(0)) Intrinsic value : K - S KB(0,T) KS(0) 0 Option value must be within this region Negative “Pure time value”: KB(0,T) - K KB(0,T)

American puts and early exercise Option Value Define: P[S(0),T;K] =Value of American put (can be exercised at any time) 12) P[S(0),T;K] >= p[S(0),T;K] (proof trivial) However, it may be optimal to exercise a put prior to expiration (time value of money), hence American put price is not equal to European put price. Example: K=$25, S(0) = $1, six-month simple rate is 9.5%. Immediate exercise provides $24 ( (6/12)) = $25.14 > $25 Intrinsic value : K - S KB(0,T) KS(0) 0 Option value must be within this region Negative “Pure time value”: KB(0,T) - K KB(0,T)