Financial Economics II Project Yajun Xiao Semester II, 2016-2017 Form into groups of your choice with 3 or 4 students to a group. If you are having problems getting into a group contact me at [email protected]. Forward and Futures Pricing • Assume a stock price follows geometric Brownian motion (GBM) dS = µSdt +sSdz: (1) The initial stock price is S0 = 100 and the instantaneous drift and volatility are µ = 7% and s = 25%. The continuously compounded risk-free is r = 2%. They all measured annually. • Determine the price of a forward contract F that obligates the holder of the contract to purchase the stock in one year at the forward price F. • Conduct a Monte Carlo study to examine the equivalence between forward and futures contracts. Assume marking-to-market occurs weekly (i.e. the margin account associated with the futures contract is changed each week to reflect any profits or losses incurred by the futures holder). Use the Monte Carlo study to illustrate that the profit and loss distributions of a future contract and a forward contract are equivalent when interest rates are constant. • Use a simple model of your choice to allow for a stochastic interest rate that is correlated with the returns of the stock. Outline by the Monte Carlo study how forward and futures contracts differ when there is a non-zero correlation between the interest rate and the stock return. Black-Scholes Option Pricing • Use the same stock price model in (1). Consider options with a maturity of 1 year and a strike price of 100. 1• Determine the price of a European call and put option on this stock using the Black-Scholes model. • Determine the delta, gamma and vega for the European call and put option and interpret the results. • Describe how an investor would delta hedge the writing of (i) the call option, (ii) the put option and (iii) the writing of both the call and put option (i.e. the investor has written a straddle) and illustrate how these hedges perform for a 1%, 5% and 10% change in the underlying stock price. • Describe how an investor would simultaneously delta and vega hedge these positions. To do this you will need to include an additional “hedging option” in your portfolio (with a different strike price or maturity) and you can assume the hedging option can also be priced using the Black-Scholes model. • Furthermore illustrate the performance of the delta-vega hedge for changes in the underlying stock price and the volatility parameter and compare the resilience of the delta-vega hedge relative to the delta hedge when both the stock price and volatility change. Numerical Methods for Option Pricing • Use the same stock price model in (1).and consider the European call option, European put option, and American put option with maturity of 1 year and strike price of 100 • Determine the prices of three options using a 52 step binomial tree (one time step equals one week). • Determine the prices of two European options using Monte Carlo simulation with 52 time steps (one step equals one week) and 1000 simulations (more simulations can be used if using a programming language). • In the case of European options, take prices in the Black-Scholes formula as the exact ones. Tell which numerical method performs better and explain why. Hedging Options in Discrete Time • Use the same stock price model in (1) and consider an European put option with maturity of 1 year and strike price of 100. Consider an investor who constructs a replicating portfolio to replicate the European put option. This replicating portfolio has a set up cost equal to the option price where the set up cost plus additional money obtained by shorting delta units of the stock is spent on purchasing Nb units of the risk-free bond with a face value of K. Assume the investor rebalances the portfolio every time period Dt with the objective of replicating the option. 2• Illustrate the concept of the replicating portfolio on the European put option using the binomial tree as your model and explain how the replication is perfect in the binomial model (i.e. there is no difference between the option value and the replicating portfolio). • Next conduct a Monte Carlo study of the replicating portfolio by simulating the stock price on a weekly basis from the options inception to its expiry assuming at each point in time that the option satisfies the Black-Scholes equation. In the Monte Carlo study the replication will no longer be perfect because the portfolio is rebalanced discretely rather than continuously (unlike the binomial model where discrete rebalancing still works as the stock price can only jump to two possible future states at each point in time). Estimate the average error between the replicating portfolio and the “true” Black-Scholes option price over the life of the option for each simulation. Give a plot that shows the average error in each simulation. • Repeat the Monte Carlo hedging exercise but augment the GBM stock price process with another feature of your choice that more realistically models financial assets. For example you could allow volatility to vary stochastically where the expectation of the volatility is equal to 25% or you could build discontinuous jumps into the model using a Poisson process. However assume the replicating portfolio investor still uses the Black-Scholes formula to price the option and to calculate the options delta at each point over the life of the option. • Illustrate the difference between the replication performance when using GBM as the data generating process for the stock price and when using an “augmented” GMB model of your choice for the data generating process of the stock price. Project Instructions 1. Write up a project report on the work carried out. The submitted project should take the form of a scientific report (4000 - 6000 words) with an introduction, a section on the methods/models used, a results section and a conclusion. The inclusion of tables and graphs in your report is recommended but remember to label both axes and provide a graph title. 2. By all means use external resources for your report but you should paraphrase your research and findings in your own words and you should cite your references if any. 3. Please submit a hard copy of the project on Friday the 28th of April. A drop box will be organised for the week prior to the due date. Those who finish early can hand in their projects during lectures. I will also require the submission of an electronic version of the report to SafeAssign in Blackboard with details to follow. 31 Note on Monte Carlo simulation Monte Carlo simulation to value option is the application of law of large number. The stock price follows GBM hence the risk-neutral stock price at time t +Dt can be expressed as a function of the stock price at time t via the following equation lnSt+Dt =lnSDt +r − 1 2s2Dt +sDz =lnSDt +r − 1 2s2Dt +sepDt; where e ∼ N(0;1). Calculate a payoff for each simulation i, denote each payoff by h(Si) for i = f1;2;:::;1000g. Calculate the present value of each payoff and denote it by pi = PV(h(Si)) = e−rTh(Si) for i = f1;2;:::;1000g. The Monte Carlo estimate of the option price, ˆ p, is given by the average of the discounted payoffs i.e. p ˆ = 1 1000 1000 ∑i=1 pi Let s p denote the standard deviation of the discounted payoffs fp1; p2;:::; p1000g that are i.i.d. The standard deviation of the estimated option price, sp ˆ, is given by sp ˆ = s p p1000 where sp ˆ is the average deviation of the estimated option price about its true value. Note that the Monte Carlo estimate uses the average of the payoffs, not the payoff of the average stock price. 2 Note on the Replicating Portfolio The replicating portfolio (RP) is denoted by P and the put option by p(S;K;r;T;s) where S is the stock price, K is the strike price, r is the interest rate, T is time-to-expiry and s is the instantaneous volatility of the stock price process. The RP can be expressed at time 0 as P(S0;t = 0) = p(S0;K;r;T;s) = Ke−rTN(−d2) − S0N(−d1) = Ke−rTNb − S0Ns, i.e. the replicating portfolio has a long position of Nb = N(−d2) units of the risk-free bond that pays K at time T and has a short position in Ns = N(−d1) units of the stock whose current price is S0. Suppose the RP is set up to replicate the option at time tk and one week passes without rebalancing the RP. Then at time tk+1 we have P(Sk+1;tk+1) = Ke−r(T−tk+1)Nb(tk)− Sk+1Ns(tk) i.e. the number of units of the bond and stock held in the RP have not been updated since last week although the bond price has moved one week closer to maturity and the stock price is now 4Sk+1 rather than Sk. Hence the RP is no longer equal to the current option price since: P(Sk+1;tk+1) = Ke−r(T−tk+1)Nb(tk)− Sk+1Ns(tk) 6= Ke−r(T−tk+1)N (−d2(tk+1))− Sk+1N (−d1(tk+1)) = Ke−r(T−tk+1)Nb(tk+1)− Sk+1Ns(tk+1) = p(Sk+1;K;r;T −tk+1;s): The difference between the RP and the option price at time tk+1 is known as the hedging or replicating error e(tk+1). To put the RP back on course and make it replicate the option once again at time tk+1 the number of units of the bond and stock held must be updated to Nb(tk+1) and Ns(tk+1) respectively. Hence in the Monte Carlo study of the RP in the Black-Scholes model the RP will not perfectly replicate the option at each point in time tk+1 = (k + 1)Dt for k = 0;1;:::;n− 1 where n = D Tt . The cost of putting the RP back on track again at time tk+1 is equal to the hedging error e(tk+1). 5