Option Valuation: basic concepts Value boundaries Simple arbitrage relationships Intuition for the role of volatility S. Mann, 2010 S. Mann, 2010

Call Option Valuation "Boundaries" Value Option value must be within this region Intrinsic Value - Value of Immediate exercise: S - K K S (asset price) Define: C[S(0),T;K] =Value of American call option with strike K, expiration date T, and current underlying asset value S(0) Result proof 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)

European Call lower bound (asset pays no dividend) “Pure time value”: K - B(0,T)K Option Value Option value must be within this region Intrinsic value: S - K KB(0,T) K S (asset price) Define: c[S(0),T;K] =Value of European call (can be exercised only at expiration) value at expiration Position cost now S(T) < K S(T) >K long call + T-bill c[S(0),T;K] + KB(0,T) K S(T) long stock S(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)

Example: Lower bound on European Call “Pure time value”: 50 - 48.91 = $1.09 Option Value Option value must be within this region Intrinsic value: 55 - 50 48.91 50 55 =S(0) S (asset price) Example: S(0) =$55. K=$50. T= 3 months. 3-month simple rate=4.0%. B(0,3) = 1/(1+.04(3/12)) = 0.99. KB(0,3) = 49.50. Lower bound is S(0) - KB(0,T) = 55 – 49.50 = $5.50. What if C55 = $5.25? Value at expiration Position cash flow now S(T) <= $50 S(T) > $50 buy call - $ 5.25 0 S(T) - $50 buy bill paying K - 49.50 50 50 short stock + 55.00 -S(T) -S(T) Total + $0.25 50 - S(T) >= 0 0

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 T1 > T2: C(S(0),T1;K) >= C(S(0),T2;K)

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

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

Volatility Value : Call option Example: Equally Likely "States of World" "State of World" Expected Position Bad Avg Good Value Stock A 24 30 36 30 Stock B 0 30 60 30 Calls w/ strike=30: Call on A: 0 0 6 2 Call on B: 0 0 30 10

Discrete-time lognormal evolution: S. Mann, 2010

Put Option Valuation "Boundaries" K Option value must be within this region Option Value Intrinsic Value - Value of Immediate exercise: K - S K S (asset price) Define: P[S(0),T;K] =Value of American put option with strike K, expiration date T, and current underlying asset value S(0) Result proof 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)

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

American puts and early exercise KB(0,T) Option Value Option value must be within this region Negative “Pure time value”: KB(0,T) - K Intrinsic value: K - S KB(0,T) K S(0) 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 (1+ 0.095(6/12)) = $25.14 > $25