I have difficulty with those 3 questions. please help.

Question # 1 (10 Points): Consider the following market price information about bonds. Fill up the four shaded cells for 2 points each. Fill up the other cells for the sake of completeness and if the entire set of cells is correct, you earn 2 more points. ____________________________________________________________ _________ Question # 2 (5 Points): Say, at the end of 2013, BX Gold Inc. entered into a commitment to sell 1.4 million ounce of gold @$1,100/ounce at the end of 2015 and to sell a second lot of 1.8 million ounces @$1,175/ounce at the end of 2016. As of the end of 2014, say the spot price of gold is $1,300/ounce, and the continuously compounded riskfree rate is 6% (ignore other carrying costs). Calculate the end of 2014 total value of BX Gold's forward sale commitments. Question # 3 (10 Points): Consider the following information about the market and about MoC Inc's Swap Portfolio comprising of two existing swap positions. Calculate the current market value of MoC Inc's Swap Portfolio. [Need to imply some missing information] DERIVATIVES FORMULA SHEET Futures Hedging Full Long Hedge: Full Short Hedge: All-In-price (Cost) =S2 - (F2 - F1) = S2+ (F1 - F2) = F1 + Basis2 All-In-Price (Proceeds) =S2+ (F1 - F2) = F1 + Basis2 h* = S/F Measure of Hedging Effectiveness: *2 Optimal Number of Futures Contracts: N* = h* QA/QF , Tailing the Hedge N* = h* VA/VF Minimum Variance Hedge Ratio: Optimal Number of Index Futures Contracts: To Reduce Positive Beta to Zero: N* = VA/VF To Reduce Positive Beta to *: N* = ( -*) VA/VF To Increase Positive Beta to *: N* = (* -) VA/VF Forward and Futures Prices Investment Asset F0: Forward/Futures Price Provides: [Arbitrage-Free] No Income: S0 erT PV of Known Income (I) and Storage Costs U: (S0 I+U) erT Known Yield q and Storage Cost Proportion u: S0e(rq+u)T Consumption Asset with Convenience Yield y : S0e(rq+uy)T Foreign Currency: q= rf, Foreign Risk-Free Rate _________________________________________________________________ Value of Position in a T-Maturity Forward Contract with Delivery Price K: Long: + (F0 K) erT, Short: + (K F0) erT Systematic Risk and Futures Price: F0 = E(ST) e(rk)T, where k is required return on investment in asset. Binomial Model Units of Underlying Asset (S) to be traded for Hedging a Position in one unit of the Derivative (f): = (fu - fd) / (Su - Sd) = (fu - fd) / (S0u- S0d) Risk Neutral Probability of UP: p= (a-d)/(u-d), where a= e(rq)t, For Futures Options, a=1.0. t To match a given volatility rate of per annum: u= e+ t, d=1/u = e Black-Scholes-Merton Model Lognormal Distribution Model: S/S (t, 2t) ln(St+t) ln(St) [( 0.52)t, 2t], ln(St+t) [ln(St)+( 0.52) t, 2t] ln(ST) ln(S0) [( 0.52)T, 2T], ln(ST) [ln(S0)+( 0.52)T, 2T] ST = S0 exp(xT), x=(1/T) ln(ST/ S0), x [( 0.52), 2/T] E(ST)= S0 exp(T), Variance(ST)= S02 exp(2T)[exp(2T)-1] Estimating Volatility from Historical Data: ui =ln (Si / Si-1 ), s= [ {1/(n-1)} ui2 {1(n-1)}(ui)2] Estimate of =s / Derivatives Formula Sheet and Probability Table, Mo Chaudhury, McGill 1 Derivatives Formula Sheet and Probability Table, Mo Chaudhury, McGill 2 Price Dynamics and No Arbitrage: 2 S 2 2 S 2 t S z S S t S z , t S S S Portfolio: 1 Derivative and f/S underlying asset Value of the Portfolio: = f + S (f/S) Change in the Value of the Portfolio: = f + S (f/S) =r t Black-Scholes No-Arbitrage Differential Equation: 2 (r q) S 2S 2 r t S S2 Option Valuation Formula For Lognormal Distribution C = S0 exp(qT) N(d1 ) K exp(rT) N(d2) ] d1 = [ln (S0/K) + (rq)T + 0.5 2T]/ T, or d1 = ln [S0 exp(qT) /K exp(rT)] /T + 0.5 T, d2 = d1 T P = S0 exp(qT) [N(d1) 1] K exp(rT) [N(d2) 1] , or P = K exp(rT) [1N(d2)] S0 exp(qT) [1N(d1)] EUROPEAN OPTION PUT CALL PARITY C P = S0 exp(qT) K exp(rT) Warrant or Employee Stock Option Value: [N/(N+M)] * C, where N: Number of Shares Outstanding Prior to Exercise, M: Number of Shares Issued due to Exercise Options on Indices, Foreign Currencies and Futures r: Risk-Free Interest Rate q=Dividend Yield: Stock or Index Option, q=Foreign Risk-Free Rate rf: Foreign Currency Option, q= r: Futures Option Black-Scholes No-Arbitrage Differential Equation with q: 2 (r q) S 2S 2 r t S S2 Risk-Neutral Process for the Asset: dS = (rq)S dt + S dz Expected Asset Price at Maturity: S0 exp[(rq)T] Derivatives Formula Sheet and Probability Table, Mo Chaudhury, McGill 2 Derivatives Formula Sheet and Probability Table, Mo Chaudhury, McGill 3 THE GREEKS N(d1) = f(d1) = [1/(2)]exp(0.5 d12) =0.398942 exp(0.5 d12) Call Delta = exp(qT)N(d1 ) Call Gamma = exp(qT) f(d1 )/ S0 T Call Theta (w.r. to Time) = [f(d1) S0 exp( qT)/(2T) ] +[qS0N(d1)exp( qT)] [r K exp( rT) N(d2) ] Call Rho (w.r. to r) = K T exp(rT) N(d2) Call Phi (w.r. to q) = T exp(qT) S0 N(d1 ) Call Vega = S0 exp(qT) f(d1 )T ---------------------------------------------------------------------------------------------------------Put Delta = exp(-qT)[N(d1 ) 1] Put Gamma = exp(qT) f(d1 )/ S0 T Put Theta (w.r. to Time) = [f(d1) S0 exp(qT)/(2T) ] [q S0 exp(qT) {1N(d1)}] +[r K exp(rT) {1N(d2)} ] Put Rho (w.r. to r)= K T exp(rT) [1N(d2)] Put Phi (w.r. to q) = T exp(qT) S0 [1N(d1)] Put Vega = S0 exp(qT) f(d1 )T --------------------------------------------------------------------------------------------------------Financial Forward Long Position Value Delta = exp(qT) Financial Futures Long Position Value Delta = exp[(rq)T] Delta-Hedge with Futures: HA: Required Position in Asset for Delta Hedging HF: Alternative Required Position in Futures for Delta Hedging HF= exp[(rq)T] HA --------------------------------------------------------------------------------------------------------No-Arbitrage Relationship between Delta ( =f/S0), Theta (=f/t), and Gamma (=2f/S02): + r S0 + 0.5 2 S02 = r f For Delta-Neutral Portfolio: Change in Portfolio Value= t + 0.5 (S)2 --------------------------------------------------------------------------------------------------------- Two Equations, Two Unknowns: ni: Signed # of Instrument i (e.g., +1.2 means Long 1.2 Units, -1.2 means SHORT 1.2 units) xi: Greek x (e.g., Delta) of Instrument i yi: Greek y (e.g., Gamma) of Instrument i X0: Target Value for Greek x for the positions in the two instruments taken together Y0: Target Value for Greek y for the positions in the two instruments taken together Equation 1: n1x1 + n2x2= X0 Equation 2: n1y1 + n2y2= Y0 Solution: n1 = (X0y2 - x2Y0)/ (x1y2 - x2y1) n2 = (Y0 - n1y1)/y2 Derivatives Formula Sheet and Probability Table, Mo Chaudhury, McGill 3 Derivatives Formula Sheet and Probability Table, Mo Chaudhury, McGill 4 FORMULA SHEET FOR EXOTICS Short Range Forward: Long Put (K1) + Short Call (K2), K1 K, 0 otherwise Digital (Binary) Cash or Nothing Put Option: Q if ST K, 0 otherwise Digital (Binary) Asset or Nothing Put Option: ST if ST