MFE230Q - Stochastic Calculus, Solutions to Problem Set 1 Q3) Stock Price: t=0 S_u S S_d 100 65 t=1 125
R u d
1.05 1.25 0.65
a) What is the price of a call with strike K = $105? K Call Price: t=0 C_u C C_d ? 0 C = (qC_u + (1-q)C_d)/R = $ 12.70 t=1 20 105 Since no arbitrage, risk neutral probabilities exist: q 1-q 0.67 0.33
b) Suppose option trades at $5. How can you lock in arbitrage profits? Its price is too low. Buy it and sell replicating portfolio. Replicating portfolio is ΔS + B, where Δ = (C_u - C_d)/((u - d)S) = 0.33 B = -(dC_u - uC_d)/(R(u-d)) = $ (20.63) Selling this yields ΔS + B = Buying call Arbitrage Profit $ 12.70 ($5) $7.70
c) Find arbitrage-free price of a put with strike $105 K = 105 Put Price; K = 105 t=0 P_u P P_d ? 40 t=1 0 i) Price using replication: Replicating portfolio is ΔS + B Δ = (P_d - P_u)/((d - u)S) = B = (1/R)(dP_u-uP_d)/(d-u) = Replicating portfolio, ΔS + B = ii) Price using risk-neutral valuation P = (qP_u + (1-q)P_d)/R = $ 12.70 -0.6667 $ 79.37 $ 12.70
ii) Price using Put-Call Parity: C - P = S - K/R P = C - S + K/R = $ 12.70
Q4) t=0 Stock Price dynamics: 115 R= u= d= 1.05 1.15 0.75 q 1-q 0.75 0.25 100 75 56.25 t=0 Call option payoffs, K = 105 ? ? ? 0 a) find the dynamic, self-financing portfolio that replicates the call payoff Work recursively from the end of the tree (t = 2) to the start (t = 0). In up state at t = 1, the replicating portfolio V_u must replicate the following payoff t=1 t=2 V_uu 27.25 V_u ? Δ_u = (C_uu - C_ud)/((u - d)Su) = 0.5924 V_ud 0 B_u = -(dC_uu - uC_ud)/(R(u-d)) = $ (48.66) V_u = Δ_uS_u + B_u = $ 19.46 It is trivial to see that Δ_d = 0 and B_d = 0, so V_d = 0. At t = 0, we then face the payoffs t=0 t=1 V_u 19.46 Δ = (V_u - V_d)/((u - d)S) = V ? B = -(dV_u - uV_d)/(R(u-d)) = V_d 0 V = ΔS + B = 0 t=1 t=2 27.25 86.25 t=1 t=2 132.25
0.4865 $ (34.75) $ 13.90
This portfolio is self-financing since there is no in- or outflows between t = 0 and maturity at t = 2. Since the portfolio replicates the call option payoffs, it must have the same value as the call option at all times.I.e, C_u = V_u, C_d = V_d and C = V. b) Value the call using risk-neutral valuation. C_u = (qC_uu + (1-q)C_ud)/R = C = (qC_u + (1-q)C_d)/R = $ $ 19.46 13.90 C_d = (qC_du + (1-q)C_dd)/R = 0.00
This verifies that C_u = V_u, C_d = V_d and C = V.
c) Find the arbitrage-free price of a put option with strike $105 using three techniques. Put price: t=0 t=1 ? ? ? 48.75 18.75 t=2 0
i) Find the put price using replication. Again work recursively through the tree. Δ_u = (P_ud - P_uu)/((d - u)S_u) = B_u = (1/R)(dP_uu-uP_ud)/(d-u) = Δ_d = (P_dd - P_du)/((d - u)S_d) = B_d = (1/R)(dP_du-uP_dd)/(d-u) = Δ = (P_d - P_u)/((d - u)S) = B = (1/R)(dP_u-uP_d)/(d-u) = -0.4076 $ 51.34 -1.0000 $ 100.00 -0.5134 $ 60.48 V_u = Δ_uS_u + B_u = $ 4.46
V_d = Δ_dS_d + B_d =
$ 25.00
V = ΔS + B =
$
9.14
ii) Find the value of the put using risk-neutral valuation P_u = (qP_uu + (1-q)P_ud)/R = P_d = (qP_du + (1-q)P_dd)/R = $ $ 4.46 25.00 P = (qP_u + (1-q)P_d)/R = $ 9.14
iii) Find the value using the put-call parity Note, this is a static strategy. We only need to use instruments at t = 0 P = C - S + K/R^2 = $ 9.14
d) Suppose the put trades at $7 at date 0. How would you lock in arbitrage profits. The put is too cheap. Buy it and sell the replicating portfolio. I would use the put call parity. Selling the replicating portfolio yields C - S + K/R^2 = 13.90 - 100 +105/(1.05^2) = $ 9.14 Buying the put cost $7. Arbitrage profits = $9.14 - $7 = $2.14 I have locked in the arbitrage with a self-financing portfolio, i.e.there are no in- or outflows between initiation and maturity. The same will be the case with a dynamic replicating strategy. Therefore, I do not need to do anything more for the arbitrage to work. In particular, future mispricing of the derivative relative to the stock does not affect the time = 0 arbitrage. (Of course, there seems to be a new arbitrage opportunity at t = 1, but you can deal with that separately)… e) Find the arbitrage-free value of a claim that pays max(S_t - S_2), for t = 0, 1, 2. Payoff diagram: V_uu V_u V_ud V V_du V_d V_dd Using risk-netrual valuation: V_u = (qV_uu + (1-q)V_ud)/R = $ 6.85 V = (qV_u + (1-q)V_d)/R = $ 9.71 t=0 t=1 ? 28.75 ? 13.75 ? 43.75 t=2 0
V_d = (qV_du + (1-q)V_dd)/R =
$
20.24