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BFC5915 Options, futures and risk management

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1 | P a g eBFC5951 Workshop 6 SolutionsBFC5915 Options, futures and risk managementWorkshop 6Suggested solutionsQuestion 1From the information give, we can quickly draw the one-step Binomial tree. In part (b), we aretold that the call option has a $79 strike price. This allows us to write the call option payoffs ateach “node” on the tree (either $21 if price goes up or $2 if price goes down).a) We use a multiplicative Binomial tree to model stock-price movements. That is, theup/down movements over each branch of the tree are proportional to the starting stockprice.0.9090811.111190100= == =u dTherefore, stock price will either go up by 11.11% or down by 10%. Note: u and d arecalculated using an estimate of the underlying asset’s volatility (σ). We calculate u andTime 0 Time 1/12S0 = $90C = ?ST = $100C = $21ST = $81C = $ 22 | P a g eBFC5951 Workshop 6 Solutionsthen get d = 1/u. The numbers in this question satisfy this approach in that d = 1/u. Evenif d ≠ 1/u (which will happen in some questions), our Binomial method still works.As an aside, if we know u, we can calculate the volatility driving this stock (althoughwe don’t actually need σ in this question). Just take the formula given in lectures for“u” and re-arrange it as follows:σ δσ δσδ= == = =lnln ln1.11110.3651 /12tu eu tutThe volatility of the underlying stock is 36.5% pa.b) The replication approach constructs a portfolio of ∆ shares and $B in the bank, with ∆and B carefully chosen such that the portfolio value one step ahead (at the end of thebranch) equals the option payoff.( )( ) ( )(1.1111 1.1111 0.902)exp0(0.90 .08 21 1 /12) 78.481.00100 8121 2= –– ×× – ×=––==– –=– –∆ =r td uu du du d euC dCBS SC CδThus, we must purchase 1.00 share and borrow $78.48. Let’s do a quick check to ensurethat this strategy does indeed replicate the call option payoffs. One month afterestablishing the replicating portfolio, we still own ∆ = 1 share and we have to repay theloan of $78.48 plus interest (this adds up to $79). Calculate the value of the portfolio inup and down states:• If share price goes up to $100: (1 × $100) – (78.48 exp(0.08 × 1/12))= $21.00• If share price goes down to $81: (1 × $81) – (78.48 exp(0.08 × 1/12)) = $2.00Clearly, buying 1.00 share and borrowing $78.48 produces a payoff in one months’time which is identical to the call option payoff. It follows that, to prevent arbitrageopportunities, the call option must be valued at the construction cost of thisreplicating portfolio. To establish this replicating portfolio today, we bought one shareand borrowed $78.48:Construction cost = (1.00 × $90) – 78.48 = $11.52.3 | P a g eBFC5951 Workshop 6 SolutionsThe fair price for this call option must be $11.52. If it trades at any other price, thatwould present an arbitrage opportunity.To summarise, the replication approach involves establishing a strategy that generatespayoffs exactly the same as the call option. To replicate the call option, we simplybought ∆ = 1 share and borrowed B = $78.48.1 We proved that this strategy replicatesthe call option payoffs. As such, the cost to establish this replicating portfolio must beidentical to the cost to purchase the call option.c) The method in part (b) is a little slow since we must calculate both ∆ and B. Analternative is to construct a portfolio that contains ∆ stocks and one short option whichprovides a certain payoff in one months’ time (the delta-hedging method). That is, nomatter what the share price ends up being, the portfolio will have aconstant/fixed/guaranteed/certain value. Thus, there is no risk involved in such aportfolio. It follows that the current value of this portfolio must be the certain futurepayoff discounted to present value at riskless rate. Finally, we back out the option price.∆ will again be 1.00 (delta is the same under replication and delta-hedging approaches).So all we need to do to delta hedge is buy ∆ = 1 share and short one call option. Let’sprove that the value of our portfolio is constant, regardless of the stock price:• If share price goes up to $100: Vu = (1.00 × $100) – 21 = $79• If share price goes down to $81: Vd = (1.00 × $81) – 2 = $79.Therefore, irrespective of which direction share price moves, our portfolio is perfectlyhedged – it is certain to be worth $79 one month later. How much would anyone beprepared to pay for a strategy that pays off $79 for certain in one months’ time? Surely,it is just the future cashflow ($79) discounted to present value at the riskless rate:( )$78.48$79exp 0.08 1/120 1== – ×=-r tV V e δThe establishment/construction cost of the portfolio must be $78.48 today. If we knowthat, then we can work backwards to figure out what the call option must be worthtoday.( )$11.52.1.00 90 78.480 0 0== × –C = ∆S -V1 Warning: delta will not always equal one. It depends on the moneyness of the option (how much the option in the money orout of the money).4 | P a g eBFC5951 Workshop 6 SolutionsNo surprise that the call option price under delta-hedging approach ($11.52) is exactlythe same as under the replication approach.d) Last, let’s use the risk-neutral valuation approach. Here we need to calculate the riskneutral probabilities of up and down movements in the tree. Using the formula,
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