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Derivative Securities (Fach) / Calculations (Lektion)
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- Calculate with Black-Scholes Formula d1= ln(S0/X)+(r+0.5σ2)•T /σ • √T d2= d1 - σ • √T N(d1) N(-d2) = 1- N(d2) C0 = S0 • N(d1) - X • e -r•T • N(d2) P0 = C0 + X • e -r•T - S0
- High implied volatility reasons expectations rise demand increases high-priced premium r=0 l no other option
- What happens if the Stock increases by 1$ in a Collar Buy a share, Buy a Put, Sell Call Put - N(d1) = 0.6 => 0.6-1 Call - N(d2) = 0.35 If it increases by 1 => Δ Long Stock: 1 Short Call: - N(d1) Long Put: N(d1) - 1 Large S => N(d1,2) turn to 1 and the Delta becomes 0 Small S => N(d1,2) turn to1 and the Delta becomes 0
- How could you exploit the conviction that the stock will move out of the price range? Buy a straddle p0 + c0 = necessary for profit
- Maximum profit if you sell a straddle p0+c0 = most range the price can move
- Market price for a zero-coupon bond with a face value of 119 and 1 month to maturity X/(1+r)T = p0 - c0 + S0
- Show that the range of S is 50 while P ist 30 Hedge ratio = p1-p2/s1-s2 Hedge ratio = 3/5 => portfolio of 3 shares and 5 puts - riskless portfolio = Present Value Put value for the replicating portfolio - P= PV - 3•S0 / 5
- Cost of a Protective Put Lock in a minimum value of 107 How much would it cost - replicating portfolio Cost = cost of a share + cost of a put
- Calculate the value of a Stock 1. Tree 2. Hedge ratio caluclation => Δ = OU - Od / Su - Sd 3 Portfolio 4. Guaranteed value
- How much of the market portfolio can we buy or sell to hedge? Beta-Confers How much of the market portfolio can we buy or sell to hedge? increases in 1% on market portfolio, increases of ß in shares 1. increase = ß • S0 • nr. of shares (total) 2. incrase in options = Δ • .... • total
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- Hedge ratio of a straddle hedge ratio of call + hedge ratio of put
- Collar Collar X = Buy put, buy stock, sell call Long put => N(d1)-1 Long call => N(d1)
- hedge ratio of a put -P/S > Buy put PV(X) = S*2+5*P
- Hedge ratio of call P/S => Sell Call PV(X) = S*2-5*P
- Swap contracts price FV = 1-B0(h) / ∑B0(h) => 1/1(y • h/360) at time= t => FSold • ∑B0(h) new+ B0(h) new annualized => FV • 360/m
- Small cap payments (new rate/old rate - 1) • Notional Principle
- Payment of swap positions floating: dealer => NP • Lo(h) • 180/360 fixed: company => NP • Lo(h) • 180/365 net position:
- If the stock price increases by one Δ • Notional principal • ß
- Covered call position Share of a stock + sell a call naked option => writing without offsetting stock position Payoff stock l ST ST Paypff Call l -0 -(ST-X) ST X
- Protective Put Position Buy a Put + Buy a stock => lock in a stock price Payoff stock l ST ST Paypff Call l -0 -(ST-X) ST ST
- Straddle position Buy a call + Buy a Put => Cost= P+C High movement: Buy the straddle No movement: Sell the straddke Strap= 2 C + 1 P Strip = 2 P + 1 C
- Spread Position 2 or more calls with different expiration dates
- Offsetting a position He would offset his position. He would reenter the swap market and offer a June Forward contract. He ha then both Long and short position on the futures contract. From the point of view of the clearing house, he no longer holds a position
- Contago contango: S0 > fo backwardation So < fo
- Second way to determine the Spot price for the T-Bill 1. Spot rate for compounding h+m Bill ) 1/Bo = 1.0045 2. bo (h+m) * 1.0045 Thus, futures prices should be sport price of the 120 sday Bill compounded by 1.0045
- Overpriced OVERPRICED < 0 => SHORT CONTRACT
- IMM IMM = 25 Dollar increase for each Basic point. Libor = 100 - IMM
- Eurodollar face value = 1000000 time = 90 days 1 Mio * (1-Libor * 90/360)
- When interest rate decrases the price of the futures rise
- Semi-annual coupon Coupon-rate / 2 !
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- Hedge ratio for puts Change in numerator Δ = p- - p+ / S+ - S+ Always go LONG 10000 calls / puts, and Δ*10000 stocks
- Calculate price for the 8-percent coupon bond at t=2 0.08*(...)+1.08*(...)