Stochastic Calculus, Week 9. Applications of risk-neutral valuation. Dividends

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1 Dividends Stochastic Calculus, Week 9 Applications of risk-neutral valuation The standard Black-Scholes analysis does not allow for dividend payments. We may introduce them in two alternative ways: Dividend is paid continuously, with a dividend yield δ; Outline Dividend is paid at discrete time points. 1. Dividends 2. Foreign exchange 3. Quantos 4. Market price of risk Continuous dividends Suppose that the dividends are paid continuously at the rate δ, and that all dividends are reinvested in the stock. Then the value S t of a portfolio that starts with one stock evolves as d S t = a t ds t + δa t S t dt, where a t = S t /S t, the number of shares at time t. Thus, d S t = S t (µdt + σdw t )+δ S t dt = (µ + δ) S t dt + σ S t dw t. This implies that S t = e δt S t, and hence a t = e δt. 1

2 A strategy (φ t,ψ t ) in terms of (S t,b t ) may be written as a corresponding strategy ( φ t,ψ t ) in terms of ( S t,b t ), with φ t = e δt φ t. The self-financing restriction now is (with V t = φ t St + ψ t B t ) dv t = φ t d S t + ψ t db t = φ t ds t + φ t δs t dt + ψ t db t. The essential point is that, whereas the replicating portfolio is in terms of S t, the derivative is in terms of S t. Note that the measure Q which makes Z t = Bt 1 S t a martingale, does not make Bt 1 S t a martingale. We find This yields, via risk-neutral valuation: The forward price of S t : setting e r(t t) E Q [(S T F t ) F t ]=0and solving for F t gives F t = e (r δ)(t t) S t. The price of a call option struck at K. Following the same steps as in the standard Black-Scholes model, we find V t = e r(t t) E Q [(S T K) + F t ] = e δ(t t) S t Φ( d 1 ) e r(t t) KΦ( d 2 ) = e {F r(t t) t Φ( d 1 ) KΦ( d } 2 ) d Z t = (µ + δ r) Z t dt + σ Z t dw t = σ Z t d W t, where W t = W t + γt, with γ = µ + δ r, which is a σ Brownian motion under the measure Q defined by dq/dp = exp( γw T 1 2 γ2 T ). Hence ds t = (r δ)s t dt + σs t d W t, = S 0 exp ([r δ 12 σ2 ]t + σ W ) t S t with d 1,2 = log(s t/k)+[(r δ) ± 1 2 σ2 ](T t) σ T t = log(f t/k) ± 1 2 σ2 (T t) σ. T t 2

3 Discrete dividends When dividends are paid at discrete time points T 1,...,T n, then the stock goes ex-dividend, which means its price falls instantaneously by the amount of the dividend. When these dividends are immediately reinvested, the value S t of that strategy of course does not display these discontinuities; i.e., we simply may assume d S t = µ S t dt + σ S t dw t. (Note: µ here should be compared with µ + δ in the continuous dividend model). When the dividend payments are δs t, we obtain S t =(1 δ) n[t] St, where n[t] is the number of dividend payments made by time t. This implies F t =(1 δ) n[t ] n[t] e r(t t) S t, Foreign exchange Let C t denote the exchange rate, in US dollar per pound sterling. We ll assume a geometric Brownian motion for C t : dc t = µc t dt + σc t dw t. Next, consider a US cash bond B t = e rt and a UK cash bond D t = e ut ; i.e., the interest rates r and u may be different. From the perspective of the US investor, there are two assets: the domestic cash bond with price B t, and the foreign cash bond with price S t = C t D t. Note that the latter is a risky asset; its SDE is ds t = C t dd t + D t dc t = (µ + u)s t dt + σs t dw t = rs t dt + σs t d W t, where W t = W t + γt, γ = µ + u r. This again defines σ Q, the risk-neutral measure. and the value of a call option remains the same in terms of F t. 3

4 Notice that under this measure, E Q [C T F t ] = e ut E Q [S T F t ] = e ut e r(t t) S t = e (r u)(t t) C t, which yields the uncovered interest rate parity: (r u)(t t) = log E Q[C T F t ], C t where the right-hand side is the conditionally expected continuous depreciation. The forward exchange rate (for delivery at time T ) F t should solve e r(t t) E Q [(C T F t ) F t ]=0, ( and since C T = C t exp [r u 1 2 σ2 ](T t)+σ[ W T W ) t ], this will yield F t = e (r u)(t t) C t, which gives the covered interest rate parity: Again, the value of a call option on the exchange rate struck at K is the same as before, when expressed in terms of F t. Change of numeraire The entire analysis could be repeated from the perspective of the UK investor, who has the choice between a sterling cash bond D t and the sterling value of a dollar cash bond, S t = Ct 1 B t. The discounted price then is Dt 1 Ct 1 B t = Z 1 t, where Z t = Bt 1 S t. The martingale measure is not the same as before: a measure which makes Z t a martingale does not make Zt 1 a martingale. However, the prices and hedge ratios are the same, regardless of the choice of the measure. Similarly, in the standard Black-Scholes model we may also work with a measure Q which makes St 1 B t a martingale. The important thing is to make relative prices martingales the choice of the numeraire is not important. (r u)(t t) = log F t C t. 4

5 Quantos Quantos are derivatives which have a payoff in another currency than the underlying asset, using a fixed, prespecified exchange rate C (e.g., one dollar per pound sterling). For example, when S t is a sterling stock price and K is a corrresponding strike price, then a quanto call has the dollar payoff C(S T K) +. In order to price such a derivative, one has to set up a joint process for (S t,c t ), which is a vector diffusion with two independent Brownian motions (W 1 (t),w 2 (t)): ds t = µdt + σ 1 dw 1 (t), S t dc t C t = νdt + σ 2 dw 2 (t) = νdt + ρσ 2 dw 1 (t)+ 1 ρ 2 σ 2 dw 2 (t), where W2 (t) = ρw 1 (t) + 1 ρ 2 W 2 (t) is a standard Brownian motion, which has correlation ρ with W 1 (t), i.e., E P [W 1 (t)w2 (t)] = ρt. The equivalent martingale measure now should turn both dollar assets Bt 1 C t S t and Bt 1 C t D t into martingale, which now involves a vector γ =(γ 1,γ 2 ), with dq dp = exp ( γ W T 1 2 γ γt ), where W T = (W 1 (T ),W 2 (T )). The actual definition of γ follows from deriving the SDE for the discounted dollar assets and setting the drifts to zero. Note that CSt is not a tradable dollar asset; hence its discounted value need not be a martingale. In fact its drift is (u ρσ 1 σ 2 ) CS t dt, which in general does not equal r CS t dt. It can be derived that the dollar forward price on CS t will be F Qt = C exp( ρσ 1 σ 2 [T t])f t, where F t is the sterling forward price. The quanto call value then is the usual, with F t replaced by F Qt, K replaced by CK, and σ replaced by σ 1. 5

6 Market price of risk The fundamental theorem of asset pricing states that the absence of arbitrage opportunities is equivalent to the existence of a measure Q under which all asset prices relative to some numeraire are martingales. The equivalent martingale measure is unique if markets are complete, i.e., if any claim is replicable. Note that only tradable asset need to be martingales under Q; the previous examples all had a payoff in terms of a nontradable asset, which was an explicit function of another tradable. The existence of Q implies a common market price of risk γ t, which determines the change of measure via the CMG theorem. For example, if two tradable asset prices S 1 (t) and S 2 (t) are driven by the same Brownian motion W t : then ds 1 (t) = µ 1t S 1 (t)dt + σ 1t S 1 (t)dw t, ds 2 (t) = µ 2t S 2 (t)dt + σ 2t S 2 (t)dw t, γ t = µ 1t r t σ 1t = µ 2t r t σ 2t, where r t is the risk-free interest rate. In models with more than one driving Brownian motion, there is a vector of market prices of risk, one for each source of risk (i.e., each Brownian motion). Exercises 1. Consider a bivariate geometric Brownian motion of the form ds 1 (t) = S 1 (t) {µ 1 dt + σ 11 dw 1 (t)+σ 12 dw 2 (t)}, ds 2 (t) = S 2 (t) {µ 2 dt + σ 21 dw 1 (t)+σ 22 dw 2 (t)}, where W 1 (t) and W 2 (t) are independent Brownian motions, and µ i and σ ij are constants, i, j = 1, 2. Find the vector γ of market prices of risks, and check that e rt S 1 (t) and e rt S 2 (t) are both martingales under the measure Q defined by this γ. 2. Suppose that S 1 and S 2 are geometric Brownian motions, driven by the same Brownian motion W t. Show that if both are tradable asset prices, but with a different market price of risk, then an arbitrage opportunity exists. 6