Financial Risk Management

Transcription

1 Extensions of the Black-Scholes model University of Oulu - Department of Finance Spring 018

2 Dividend payment Extensions of the Black-Scholes model ds = rsdt + σsdz ÊS = S 0 e r S 0 he risk-neutral price process provides a prediction of the level of the asset price at the maturity. S 0 0 Dividend payment 0 ÊS = S 0 e r ÊS In a case of a dividend-paying asset, the price falls below the predicted level.

3 Dividend-adjustment in the initial price S 0 S 0 e q 0 Dividend payment D d ds = rsdt + σsdz ÊS = S 0 e r q he return of the asset must be adjusted downwards by the amount of the dividend yield q. If we replace the observed current price S 0 with a dividend-adjusted price S 0 e q, the unadjusted return r results the expected dividend-adjusted terminal price: S 0 e q e r = S 0 e r q he Black-Scholes model assumes the unadjusted return r and thus we need to substitute the dividend-adjusted price for the observed current price everywhere in the model.

4 Dividend-adjustment in the initial price he Black-Scholes model of a non-dividend-paying asset: Asset price lns0 /X + r + σ c 0 = S 0 σ Xe r lns0 /X + r σ σ he Black-Scholes model of a dividend-paying asset, where the dividend-adjusted price S 0 e q is is substituted for S 0 : Dividend-adjusted asset price {}}{{}}{{}}{ c 0 = S 0 e q lns0 e q /X + r + σ σ Xe r lns0 e q /X + r σ σ = S 0 e q lns0 /X + r q + σ σ Xe r lns0 /X + r q σ σ.

5 Dividend-adjustment in the price process ds = r qsdt + σsdz ÊS = S 0 e r q S 0 0 An alternative approach is to use the observed current price S 0, but replace the price process with a dividend-adjusted price process: ds = r qsdt + σsdz. In the Black-Scholes model we need to adjust the risk-free rate r downwards by the amount of the dividend yield q, wherever it appears, either explicitly of implicitly, as the risk-neutral expected return of the underlying asset.

6 Dividend-adjustment in the price process he Black-Scholes model of a non-dividend-paying asset: Risk-free rate as an expected return c 0 = S 0 e r e r lns0 /X + r + σ σ Xe r lns0 /X + r σ σ Risk-free rate as a discount rate

7 Dividend-adjustment in the price process he Black-Scholes model of a non-dividend-paying asset: Risk-free rate as an expected return c 0 = S 0 e r e r lns0 /X + r + σ σ Xe r lns0 /X + r σ σ he Black-Scholes model of a dividend-paying asset, where the dividend-adjusted return r q is is substituted for r: Dividend-adjusted expected return c 0 = S 0 e r q e r lns0 /X + r q + σ σ Xe r lns0 /X + r q σ σ Risk-free rate as a discount rate

8 Dividend-adjustment in the price process he Black-Scholes model of a non-dividend-paying asset: Risk-free rate as an expected return c 0 = S 0 e r e r lns0 /X + r + σ σ Xe r lns0 /X + r σ σ he Black-Scholes model of a dividend-paying asset, where the dividend-adjusted return r q is is substituted for r: Dividend-adjusted expected return c 0 = S 0 e r q e r lns0 /X + r q + σ σ Xe r lns0 /X + r q σ σ = S 0 e q lns0 /X + r q + σ σ Xe r lns0 /X + r q σ σ.

9 Black-Scholes models for a dividend-paying asset he Black-Scholes model of a European call option on a dividend-paying asset: c t = S t e q t lnst /X + r q+ σ t σ Xe r t lnst /X + r q σ t t σ. t he Black-Scholes model of a European put option on a dividend-paying asset: p t = Xe r t lnst /X + r q σ t σ S 0 e q t lnst /X + r q+ σ t t σ. t

10 Currency exchange rate process ds = r r f Sdt + σsdz ÊS = S 0 e r r f t S 0 0 Let us consider a currency exchange rate S in terms of the price of one unit of foreign currency in domestic currency. If the domestic risk-free interest rate r is higher than the foreign risk-free interest rate r f, the drift r r f is positive and predicts the weakening of the domestic currency. If the foreign risk-free interest rate r f is higher than the domestic risk-free interest rate r, the drift r r f is negative and predicts the strengthening of the domestic currency. A higher risk-free interest rate of a currency is a compensation for the expected weakening of the relative purchasing power of the currency.

11 Currency exchange rate process ds = r r f Sdt + σsdz ÊS = S 0 e r r f t S 0 0 he rates r and r f are risk-free rates, and thus the process of an exchange rate is a risk-free process as such. here is a clear analogy between the risk-neutral processes of a dividend-paying asset and an exchange rate: ds = r qsdt + σsdz, ds = r r f Sdt + σsdz. When pricing an option on currency exchange rate the Black-Scholes model of a dividend-paying asset is applied, the current exchange rate S 0 is substituted for the current asset price S 0, the foreign currency risk-free rate r f is substituted for the dividend yield q.

12 Adjustment in the currency exchange rate process he Black-Scholes model of a dividend-paying asset: Dividend-adjusted expected return c 0 = S 0 e r q e r lns0 /X + r q + σ σ Xe r lns0 /X + r q σ σ Risk-free rate as a discount rate

13 Adjustment in the currency exchange rate process he Black-Scholes model of a dividend-paying asset: Dividend-adjusted expected return c 0 = S 0 e r q e r lns0 /X + r q + σ σ Xe r lns0 /X + r q σ σ he Black-Scholes model of a currency exchange rate, where an adjusted return r r f is is substituted for r: Adjusted expected return c 0 = S 0 e r r f e r lns0 /X + r r f + σ σ Xe r lns0 /X + r r f σ σ Risk-free rate as a discount rate

14 Adjustment in the currency exchange rate process he Black-Scholes model of a dividend-paying asset: Dividend-adjusted expected return c 0 = S 0 e r q e r lns0 /X + r q + σ σ Xe r lns0 /X + r q σ σ he Black-Scholes model of a currency exchange rate, where an adjusted return r r f is is substituted for r: Adjusted expected return c 0 = S 0 e r r f e r lns0 /X + r r f + σ σ Xe r lns0 /X + r r f σ σ = S 0 e r f lns0 /X + r r f + σ σ Xe r lns0 /X + r r f σ σ

15 Garman-Kohlhagen models for currency options he Garman-Kohlhagen model of a European call option on a currency exchange rate: c t = S t e r f t lnst /X + r r f + σ t σ Xe r t lnst /X + r r f σ t t σ. t he Garman-Kohlhagen model of a European put option on a currency exchange rate: p t = Xe r t lnst /X + r r f σ t σ S 0 e r f t lnst /X + r r f + σ t t σ. t

16 Futures price process F 0 df = µ rfdt + σfdz EF = ES = S 0 e µ S 0 ds = µsdt + σsdz 0 EF = F 0 e µ r = S 0 e r e µ r = S 0 e µ F 0 = S 0 e r Futures price F equals to the underlying asset price S at the maturity of the contract. If the drift rate of the underlying asset price is µ, the drift rate of the futures price must be µ r. r t F F = Se df = t F = er S t = rse r t F S = 0 F F µs + S t + 1 F S σ S dt + F S σsdz = µ r Fdt + σfdz.

17 Risk-neutral futures price process F 0 df = r rfdt + σfdz ÊF = ÊS = S 0 e r S 0 ds = rsdt + σsdz 0 ÊF = F 0 e r r = S 0 e r e r r = S 0 e r F 0 = S 0 e r In the risk-neutral process of a futures price, the drift rate of the futures price is r r. here is a clear analogy between the risk-neutral processes of a dividend-paying asset and a futures price: ds = r qsdt + σsdz, df = r rfdt + σfdz. When pricing an option on a futures contract the Black-Scholes model of a dividend-paying asset is applied, the current futures price F 0 is substituted for the current asset price S 0, the risk-free rate r is substituted for the dividend yield q.

18 Adjustment in the futures price process he Black-Scholes model of a dividend-paying asset: Dividend-adjusted expected return c 0 = S 0 e r q e r lns0 /X + r q + σ σ Xe r lns0 /X + r q σ σ Risk-free rate as a discount rate

19 Adjustment in the futures price process he Black-Scholes model of a dividend-paying asset: Dividend-adjusted expected return c 0 = S 0 e r q e r lns0 /X + r q + σ σ Xe r lns0 /X + r q σ σ he Black-Scholes model of a futures contract, where the adjusted return r r is is substituted for r: Adjusted expected return c 0 = F 0 e r r e r lnf0 /X + r r + σ σ Xe r lnf0 /X + r r σ σ Risk-free rate as a discount rate

20 Adjustment in the futures price process he Black-Scholes model of a dividend-paying asset: Dividend-adjusted expected return c 0 = S 0 e r q e r lns0 /X + r q + σ σ Xe r lns0 /X + r q σ σ he Black-Scholes model of a futures contract, where the adjusted return r r is is substituted for r: Adjusted expected return c 0 = F 0 e r r e r lnf0 /X + r r + σ σ Xe r lnf0 /X + r r σ σ = = e [F r lnf0 /X + σ 0 lnf0 σ /X σ X ] σ.

21 Black-76 models for futures options he Black-76 model of a European call option on futures: c t = e [F r t lnf0 /X + σ t lnft t σ /X σ ] t X t σ. t he Black-76 model of a European put option on futures: p t = e [X r t lnft /X σ σ t t F t ] t σ. t lnft /X + σ

22 Bond options Extensions of the Black-Scholes model A bond price B at time t = can be assumed to follow lognormal distribution ˆpB = ln B 1 ˆm { ˆm = Êln B e s s B π s = Stdln B However, a bond price cannot be assumed to follow a geometric Brownian motion, and thus the pricing of the bond options is based on the following models: Call option s value under an assumption of lognormally distributed bond price: ] c t = e [ÊB r t ˆm ln X + s ˆm ln X X. s s Put option s value under an assumption of lognormally distributed bond price: p t = e r t [X ˆm ln X s ÊB ] ˆm ln X + s. s he parameters ˆm and s of the model are typically expressed in such a way that the model appears in the same form as the Black-76 model for futures options.

23 Bond options Extensions of the Black-Scholes model he risk-neutral expected bond price ÊB is equal to the forward price F t = F t, of the bond, and thus we are able to express the model of a call option as ] c t = e [F r t ˆm ln X + s ˆm ln X t X. s s ext we solve the risk-neutral expected logarithmic bond price ˆm at time t = in terms of the current forward price F t of the bond and the standard deviation s of the logarithmic bond price at time t = : F t = ÊB = e ˆm+s / ˆm = ln F t s he model appears now as ln c t = e [F r t Ft s ln X + s ln Ft s t X ln X ] s s = e [F r t lnft /X + s lnft /X s ] t X. s s

24 Bond options Extensions of the Black-Scholes model Finally, we express the standard deviation s of the logarithmic bond price at time t = in terms of a function of the volatility rate σ and the time-to-maturity t of the option: s = σ t. he model appears now in the same form as the Black-76 model for futures options: Forward price of the bond Volatility rate of the bond c t = e [F r t lnft /X + σ t lnft t σ /X σ ] t X t σ. t σ = s t In a case of a futures option the model is Futures price Volatility of the underlying asset c t = e [F r t lnft /X + σ t lnft t σ /X σ ] t X t σ. t

25 Forward price of a bond e30 e30 e30 B = 8.50 B 0 = e Let us consider a e1000 bond which pays e30 seminannual coupons. he next payment is to take place three months from now, and thereafter every six months until the maturity of the bond, say eight years and three months from now. he current quoted cash price of the bond is B 0 = e Sometimes in the US market, for instance, bond price is quoted in terms of a clean price, where the interest accrued since the beginning of the current coupon period is cleaned off from the current cash price of the bond. In our example, one half of the next e30 coupon is already accrued since the beginning of the current coupon period, he current clean price of the bond would be e e15 = e894.85, and should be adjusted back to a cash price.

26 Forward price of a bond e30 e30 e30 B = 8.50 B 0 = e e30 e30.7%.9% = 1.00 B 0 = e Let us consider an option on the bond, with the time-to-maturity of one year. he current bond price contains two such coupon payments which are to take place before the maturity of the option, and must be cleaned off from the current price before the determination of the one-year forward price of the bond. he risk-neutral valuation approach requires us to calculate the present values of the coupon payments in terms of the corresponding continuously compounded risk-free rates. he coupon-cleaned current price of the bond is B 0 = e e30e e30e = e

27 Forward price of a bond e30 e30 e30 B = 8.50 B 0 = e e30 e30.7%.9% = % B 0 = e F 0 = e he one-year forward price of the bond is obtained by the coupon-cleaned current price of the bond and the one-year continuously compounded risk-free rate. he one-year forward price of the bond is F 0 = e850.70e = e

28 Bond option Extensions of the Black-Scholes model % = 1.0 r = 3.0% F 0 = e X = e900 σ = s t Regarding the pricing of the option: We know the time to maturity of the option: = 1.0, We know the corresponding risk-free rate : r = 3.0%, We know the corresponding forward price of the bond: F 0 = e876.60, he strike price of the option is specified in the contract: X = e900, What remains to be solved is the volatility rate σ of the forward price. As the amount of accrued interest is e15 at the maturity of the option, the same strike price could be given in terms of a clean price of e900 e15 = e885, and should be adjusted back to a cash price.

29 Bond price volatility We are interest in the standard deviation s of the logarithmic bond price ln B at the maturity of the option. he change db in the price B of the bond can be approximated by the duration D and the change dy of the forward yield y of the bond at the maturity of the option: db = D dy B. Correspondingly, the relative change in the bond price appears as db B = D dy. At the same time we know that d ln B = db B. he combining of the results shows that the change in the logarithmic bond price can be approximated by d ln B = D dy. In terms of the relative change in the forward yield of the bond it appears as d ln B = D y dy y.

30 Bond price volatility he variance of the logarithmic bond price can be expressed as Var dy ln B = Var d ln B = D y Var = D y dy Var. y y Correspondingly, the standard deviation of the logarithmic bond price is s = Std ln B = D y Std dy y. he standard deviation of the relative change in the forward yield is considered to be proportional to the square root of time, and can be expressed in terms of the forward yield volatility σ y : dy Std = σ y t. y ow we are able to rewrite the standard deviation of the logarithmic bond price as s = D y σ y t. Finally, the volatility σ of the forward bond price or the volatility rate of a bond is defined as σ = D y σ y.

31 Bond price volatility he above expression is in line with our definition of the volatility rate: σ = s = D y σy t = D y σ y. t t he forward yield y of the bond is calculated on the basis of the forward price F of the bond, and the remaining payments between the time and the maturity of the bond: = 30e y e y e y 7.5 y = he duration D of the bond at time is then calculated on the basis of the forward price F of the bond, the forward yield y of the bond, and the remaining payments between the time and the maturity of the bond: D = e e e = he forward yield volatility σ y may be estimated by an appropriate interest rate model, or a possibly available quoted rate may be used. Here we assume the forward rate volatility of σ y = 0.0.

32 Bond option Extensions of the Black-Scholes model % = 1.0 r = 3.0% F 0 = e X = e900 σ = s t = D y σ y = 9.7% D = 5.77 y = 8.4% σ y = 0% Regarding the pricing of the option: We know the time to maturity of the option: = 1.0, We know the corresponding risk-free rate : r = 3.0%, We know the corresponding forward price of the bond: F 0 = e876.60, he strike price of the option is specified in the contract: X = e900, We know the duration of the bond at the maturity of the option: D = 5.77, We know the forward yield of the bond at the maturity of the option: y = 8.4%, We know the forward yield volatility: y = 0%, We are able to calculate the volatility rate of the bond price at the maturity of the option: σ = 9.7%.

33 Value of a call option % = 1.0 r = 3.0% F 0 = e X = e900 σ = s t = D y σ y = 9.7% D = 5.77 y = 8.4% σ y = 0% Regarding the pricing of the option: We know the time to maturity of the option: = 1.0, We know the corresponding risk-free rate : r = 3.0%, We know the corresponding forward price of the bond: F 0 = e876.60, he strike price of the option is specified in the contract: X = e900, We know the volatility rate of the bond price at the maturity of the option: σ = 9.7%. he value of a call option is c 0 = e [F r lnf0 /X + σ 0 lnf0 σ /X σ X ] σ = e3.8.

34 Value of a put option % = 1.0 r = 3.0% F 0 = e X = e900 σ = s t = D y σ y = 9.7% D = 5.77 y = 8.4% σ y = 0% Regarding the pricing of the option: We know the time to maturity of the option: = 1.0, We know the corresponding risk-free rate : r = 3.0%, We know the corresponding forward price of the bond: F 0 = e876.60, he strike price of the option is specified in the contract: X = e900, We know the volatility rate of the bond price at the maturity of the option: σ = 9.7%. he value of a put option is c 0 = e [X r lnf 0/X σ σ F 0 lnf 0/X + σ ] σ = e45.99.