Stochastic Calculus, Week 8 Intuition via Binomial Approximation n Equivalent Martingale Measures Let Wt (ω),t∈ [0,T] be defined by √ Xi T Outline W n(ω)= δt Y (ω),t= iδt, δt= , t j n j=1 1. Intuition via binomial approximation where ω =(ω1,...,ωn) represents the combined outcome of n consecutive binary draws, where each draw may be u 2. Cameron-Martin-Girsanov Theorem (up) or d (down); further, 3. Application to derivative pricing: ( 1 if ωi = u, equivalent martingale measures/risk neutral measures, Yi(ω)= −1 if ωi = d. martingale representation theorem 4. Obtaining the Black-Scholes formula once more Let Ω be the set of all possible ω’s, let F be the appropri- ate σ-field, and let P and Q be two probability measures on (Ω, F), such that the draws are independent under both P and Q, and 1 P(ω = u)=p = , i 2 1 √ Q(ω = u)=q = 1 − γ δt , i 2 for some constant γ. 1 n d ˜ n d ˜ As n →∞, Wt (ω) −→ Wt(ω) (convergence in distribu- Then Wt (ω) −→ Wt(ω), where tion), where Brownian motion with drift γ under P, ˜ Standard Brownian motion under P, Wt(ω)= W (ω)= t Standard Brownian motion under Q. Brownian motion with drift − γ under Q. In summary, we can write down the following table with The measures P and Q are equivalent, which means that drifts: for any even A ∈F,wehaveP(A) > 0 if and only if PQ Q(A) > 0. This requires that γ is such that 0 <q<1. W 0 −γ If q would be either 0 or 1, the two measures would not be t W˜ γ 0 equivalent (why?). t Thus, by changing the measure from P to Q, we change the drift in Wt from 0 to −γ. n We can also change the drift by adding γt to Wt (ω). Thus, define ˜ n n Wt (ω)=Wt (ω)+γt √ Xi T = δt Y˜ (ω),t= iδt, δt= , j n j=1 ( √ ˜ 1+γ δt if ωi = u, Yi(ω)= √ −1+γ δt if ωi = d. 2 Cameron-Martin-Girsanov Theorem It may be best thought of as the ratio of densities of a the process under Q and P, respectively. This is seen as follows. Suppose that Wt(ω) is a P-Brownian motion, let γt be Suppose that γt = γ (constant), so that adapted to this Brownian motion, and consider the Itoˆ pro- ( N(0,T) with density fP(w) under P, cess WT ∼ N(−γT,T) with density fQ(w) under Q. ˜ ˜ dWt = γtdt + dWt, W0 =0, Z t Then ˜ Z ∞ Wt = γsds + Wt. 0 EQ(WT )= wfQ(w)dw Z−∞ ∞ f (w) Question: does a measure Q, equivalent to P exist, such that = w Q f (w)dw f (w) P ˜ −∞ P Wt is a Q-Brownian motion? fQ(WT ) = EP WT . fP(WT ) Answer: (C-M-G theorem): Yes: Q is characterized by Z Z Therefore, dQ T 1 T = exp − γ dW − γ2dt . t t t 1 1 2 dP 0 2 0 √ exp − (WT + γT) dQ fQ(WT ) 2πT 2T = = dP f (W ) 1 1 dQ/dP P T √ exp − W 2 The derivative is a so-called Radon-Nikodym deriva- 2T T 2πT tive. In general this is a random variable, such that for any 1 1 = exp − (W + γT)2 + W 2 X (Ω, F) 2T T 2T T random variable on : 1 dQ 2 E (X)=E X . = exp −γWT − γ T . Q P dP 2 Z 1 Z 1 1 2 = exp − γtdWt − γt dt . 0 2 0 3 Note that ζT = dQ/dP is a random variable. We can create Application to derivative pricing a martingale ζt from that as usual: ζt = EP(ζT |Ft), which S is the Radon-Nikodym process. This satisfies: Consider a stock price t generated by a geometric Brown- ian motion under P: −1 EQ(Xt|Fs)=ζs EP (ζtXt|Fs) . dSt = µStdt + σStdWt, The C-M-G theorem allows us to change the drift of an Itoˆ with Wt a P-Brownian motion. process by changing the probability measure. Let Wt be a 1 2 Remark: the solution is St = S0 exp([µ − 2σ ]t + σWt). P-Brownian motion, and Xt be an Itoˆ process with drift µt Baxter and Rennie start from St = S0 exp(µt + σWt),so and volatility σt under P. Then 1 2 their µ corresponds to our µ − 2σ . dXt = µtdt + σtdWt rt = ν dt +(µ − ν )dt + σ dW Consider also a cash bond with price Bt = e , so that t t t t t µt − νt = νtdt + σt dt + dWt dBt = rBtdt. σt ˜ = νtdt + σtdWt, −1 −rt R Define the discounted stock price Zt = Bt St = e St,so ˜ t ˜ where Wt = γ ds+Wt, with γ =(µ −νt)/σt. Thus Wt 0 s t t that, by Ito’sˆ lemma, is a Brownian motion under the measure Q defined by the −rt −rt dZt = e dSt + Stde C-M-G theorem, and hence Xt is an Itoˆ process with drift νt = e−rt[µS dt + σS dW ] − re−rtS dt and volatility σt under Q. Note that we can only change t t t t the drift, not the volatility. If Xt would have a different =(µ − r)Ztdt + σZtdWt. volatility under P and Q, then these two measures are no longer equivalent. 4 Finally, let X be some claim, determined at time T . For a Thus, under Q, the stock price has an average (continu- + European call option, X =[ST − K] . We wish to find the ously compounded) growth rate of r, the risk-free rate. no-arbitrage value of this claim at time t<T. Hence under this measure there is no compensation for risk. Therefore Q is also called the risk-neutral measure. We proceed in the following steps: −1 2. Define Et = EQ(BT X|Ft), which is a martingale with respect to Ft under Q (just like Zt). 1. Find the measure Q under which Zt is a martingale: (µ − r) 3. The martingale representation states that because both dZt = σZt dt + dWt σ Et and Zt are Q-martingales with respect to Ft, ˜ = σZtdWt, dE = φ dZ , R t t t ˜ t where Wt = 0 γsds + Wt, with γt =(µ − r)/σ. This for some Ft-adapted φt. is the market price of risk: the excess expected return Define ψt = Et − φtZt, and the strategy of holding φt required as compensation for one additional unit of stan- stocks and ψt bonds, so that the value of the portfolio is dard deviation. Thus Zt is a martingale under Q, defined from the C-M- Vt = φtSt + ψtBt γ G theorem for this choice of t. Hence we often refer to = φtSt + EtBt − φtZtBt Q as the equivalent martingale measure. = BtEt. The martingale property for Zt under Q implies: −1 Since ET = BT X, this implies that VT = X, i.e., the E (S |F )=ertE (Z |F ) Q t s Q t s strategy is replicating. rt = e Zs r(t−s) = e Ss. 5 What remains to be shown is that it is also self-financing. Obtaining the Black-Scholes formula once more This follows from Vt = BtEt,so + For a European call option, X =[ST − K] . Defining the dV = B dE + E dB t t t t t time to expiration τ = T − t, this means that its value is = φ BtdZt +(ψ + φ Zt)dBt t t t −rτ + Vt = e EQ [ST − K] |Ft . = φtd(BtZt)+ψtdBt ˜ dSt = rStdt + σStdWt = φ dSt + ψ dBt. Because ,wehave t t 1 2 ˜ ˜ ST = St exp [r − 2σ ]τ + σ[WT − Wt] Thus the strategy is self-financing and replicates the √ = S exp [r − 1σ2]τ − σ τY , claim; that means that, by no-arbitrage, Vt must equal t 2 the value of the claim at time t. ˜ ˜ √ where Y = −(WT − Wt)/ τ ∼ N(0, 1) under Q. The option expires exactly at the money if rt In other words, the value of the claim is Vt = BtEt = e Et, √ S = S exp [r − 1σ2]τ − σ τY = K, so T t 2 −r(T −t) Vt = e EQ(X|Ft) which can be rewritten as ln(S /K)+(r − 1σ2)τ This is the discounted expected payoff, with the expectation Y = t √ 2 = d . σ τ 2 taken under the risk-neutral measure. This is a very gen- This implies that eral expression for the value of a claim, that will also hold ( for, e.g., interest rate derivatives (which are not driven by + ST − K if Y ≤ d2, [ST − K] = a geometric Brownian motion). It is the central result that 0 if Y>d2. underlies risk-neutral valuation. 6 Thus Exercises Z d2 √ −rτ 1 2 Vt = e [St exp [r − 2σ ]τ − σ τy − K]φ(y)dy −∞ 1. Suppose that Xt(ω) is an Itoˆ process with volatility σt = Z d2 1 √ σXt under P, whereas it is an Itoˆ process with volatility √ 1 2 1 2 = St exp −σ τy − 2σ τ − 2y dy 2π σt = σ under Q. Argue that P and Q are not equivalent. −∞ Z d2 −e−rτ K φ(y)dy 2. Show that −∞ Z t Z t 1 2 Z ζt := EP(ζT |Ft) = exp − γsdWs − γ ds . d2 1 √ 2 s = S √ exp −1 y + σ τ 2 dy − e−rτ KΦ(d ) 0 0 t 2π 2 2 −∞ 3. Use risk-neutral valuation to obtain the value of a claim Z ( d1 1 √ 1 2 −rτ 1 if ST ≥ K = St exp −2v dv − e KΦ(d2) X = .