814NOTE 5

Risk ManagementNotes # 5Derivatives and Risk Management •Risk management means modifying the shape of the probability distribution.•Hedging (some risk exposure) is an example of risk management.•Recall: in modern portfolio theory e.g. mean-variance analysis, the focus is on the expected value and the dispersion, but not the shape (e.g. tails, skewness, etc.) of the distribution.Derivatives and Risk Management •Consider the protective put strategy:S T < X ST> X––––––––Stock+ST S TPut X –ST 0––––––––X STDerivatives and Risk Management•Manage the risk of large losses:–the downside risk, i.e. realized value below X, is gone.Derivatives and Risk Management •Can do same thing with calls and T-bills:S T < X ST> X––––––––T-bills (face value X) +X XCall0ST –X–––––––––X ST•This equivalence is (no-arbitrage) proof of Put-Call Parity. Since payoffs are the same, portfolios must cost the same (Law of One Price).S 0+ p = c + PV(X)p = c + PV(X) –SFinancial Engineering •Financial Engineering means constructing desired payoffs from given payoffs.•Consider put-call parity: Other than a no-arbitrage statement about values, the equation also tells you how to “engineer” a synthetic security.+ p –c•Example: PV(X) = S–If you buy stock, buy put, and write call, you get a synthetic bill.Intuition: if you take away both up and downside exposure, youhave a risk-free position.Financial Engineering •This can be viewed as a "building block" approach to financial engineering.–one building block gives upside potential (calls)–another for downside exposure (puts)–another for both (stock)–another for neither (bills).•You can combine various building blocks to create other positions.Financial Engineering•So the result is a riskless position, i.e., a synthetic bill:Payoff at time TS T < X ST> XBuy stock ST S TBuy put X −ST 0Write call0−(ST –X)–––––––––X X•The synthetic bill provides the same payoff as the actual bill regardless of the ultimate stock price.More building blocks:synthetic forwards•We saw in the building block example above that you can view a combination of long put and short call as a full hedge for a stock.–buying the put eliminates the downside, whileselling the call eliminates the upside.•Another way to fully hedge a stock is to arrange now to sell it at a future date for a price agreed to today, i.e., to sell a forward or futures contract on the stock.synthetic forwards•This suggests that the long put/short call strategy and the forward contract are substitutes.•In fact, a long put/short call portfolio is a synthetic forward contract:•Using the forward:PV(X) = S + forward contract to sell stock for $X Synthetic bill = stock + hedge•Using the options:PV(X) = S + P –Csynthetic forwards •Therefore, P −C must be a synthetic forward contract to sell stock for $X. Let’s check:S < X S > XLong put X –S0Short call0–(S –X)X –S X –S•This is the payoff to a short forward position with forward price X. No matter what happens to the stock price, you deliver the stock and receive $X.synthetic forwards•We usually think of a forward contract as a zero-cost"side bet" on the stock. The forward price is F* = S ×e rT.•In contrast, the "cost" of the synthetic forward is P –C, which is not necessarily zero, and the forward price is, in this example, X, which can be chosen arbitrarily.synthetic forwards•This synthetic strategy can be thought of as creating an "off-market" forward contract, that is, a forward with arbitrary forward price X and cost to enter the short side of P –C.•When X > F*, P −C is positive. You would be willing to pay to enter the synthetic forward contract to deliver a stock at price X greater than F*. Conversely, when X < F*, P −C is negative.synthetic forwards•What value of X (i.e., what forward price) would be consistent with the synthetic forward contract having zero cost?•Again we start from put-call parity:PV(X) = S + P –Cwhich implies that if you choose X to make P –C = 0 then S = PV(X) = e−rT X ⇒X = S ×e rT•Therefore, the synthetic forward contract will indeed have zero cost if the forward price is S ×e rT, which of course is the usual parity result.synthetic forwards•As a digression, people often wonder whether at the money calls and puts should have equal value. •This analysis shows that the answer is "yes" only if we define at the money "correctly," i.e., not that S = X, but more appropriately, that S = PV(X).•In contrast, if S = X, then C > P.•What about more complicated positions?–E.g., suppose you want to engineer a portfolio with this payoff structure.The moral here: you can engineer very complex payoffs, with custom tailored risk patterns, using derivatives.Bull Spread Using Calls K1K2Profit ST c(K2,T)-c(K1,T)Long call Short callBull Spread Using Puts K1K2Profit STp(K2,T)-p(K1,T)Short put Long putBear Spread Using CallsK1K2ProfitSTLong callShort callBear Spread Using PutsK1K2ProfitSTShort putLong putCalendar and Diagonal Spread •Calendar –constructed with calls with common strike price but different expiration dates.•Depending on the strike price can be neutral, bullish or bearish.•Diagonal –constructed with calls with different strike prices and different expiration dates.A Straddle CombinationLong call K ProfitSTKLong put KStrip & StrapProfitK ST ProfitK STStrip Strap Long 2 puts K Long 2 calls KA Strangle CombinationProfitLong call K2K1K2STLong put K1Strangle or Straddle?Equity Linked CDs•1999 CD issued by First Union National Bank and Charles Schwab•The 5.5-year CD promises to repay initial invested amount and 70% of the gain in S&P 500 index–Assume $1,000 invested when S&P 500 = 1,000) -1000, 0]–Final payoff =1,000 + .7 max[1000 x (1 + rS&P= 1,000 + .7 max(S-1000, 0)T•So instead of repaying principal plus interest, the firm has implicitly issued its customers a $1,000 face value zero coupon bond and .7 call options.Equity Linked CDs•Suppose that at the time the CD was issued, the risk-free rate was 6%, the dividend yield on the S&P was d = 1.5%, and the volatility of the S&P index was 25%.•Questions for the security designer:1.What is the profit on this offering?2.How should the bank set the “participation rate” to provide a givenprofit target?3.How will the participation rate vary with factors like marketvolatility?4.How should the bank hedge and/or fund an obligation that dependson market returns?5.Variations on security design: e.g., how can we set the participationrate if the bank pays some interest on the CD?Equity Linked CDs•How much have the customers paid for this package? They would have earned interest of1000 ×e0.06 ×5.5−1000which has present value today:e−0.06 ×5.5 ×[1000 ×e0.06 ×5.5−1000] =1000 −1000 ×e−0.06 ×5.5= 281.08So the bank’s profit is[1000 −1000 ×e−0.06 ×5.5 ] −.7 ×Call valueCall value(1000, 1000, 0.06, 0.25, 5.5, 0.015)=?Equity Linked CDs•Call value(1000, 1000, .06, .25, 5.5, .015)=?•We can estimate the profit using an option value equation such as Black-Scholes.•In practice, we might not even need the B/S equation. We simply obtain the value of an at-the-money 5 ½ year call option from an option dealer.Equity Linked CDs•The bank can break even offering a participation rate, p, that makes the PV of profits zero:[1000 −1000 ×e−0.06 ×5.5] −p×Call value (1000, 1000, .06, .25, 5.5, .015) = 0p= PV(foregone interest)/Call valuep= PV interest as % of call value•Given r, σ, T, d[dividend yield], we can solve for the break-even value of p. Any participation rate below this value offers aprofit.Equity Linked CDs•How would the bank hedge and fund such a product?–It can engineer a synthetic ELCD: buy one zero coupon bond with face value 1000 and .7 calls•How will participation rate depend on factors like market volatility?•There are more complicated designs, for example, capped index-linked CDs, and so on.•For example, what if the bank paid some below market interest on the CD? E.g., what if the CD pays a guaranteed minimum of 1% interest?。