Dynamic Hedging and PDE Valuation

Dynamic Hedging and PDE Valuation Dynamic Hedging and PDE Valuation 1/ 36

Introduction Asset prices are modeled as following di usion processes, permitting the possibility of continuous trading. This environment can allow a market with an underlying asset, a contingent claim, and the risk-free asset to be dynamically complete. We illustrate the Black-Scholes-Merton portfolio hedging argument that results in a partial di erential equation for a contingent claim s price. Examples are The Black-Scholes (1973) option pricing model. The Vasicek (1977) equilibrium term structure model. The Merton (1973b) stochastic interest rate option pricing model. Dynamic Hedging and PDE Valuation 2/ 36

Portfolio Dynamics in Continuous Time The insight of Black and Scholes (1973) and Merton (1973) is that when assets follow di usion processes, an option s payo can be replicated by continuous trading in its underlying asset and a risk-free asset. Consider an investor who can trade in any n di erent assets whose prices follow di usion processes. De ne S i (t) as the price per share of asset i at date t, where i = 1; :::; n. The instantaneous rate of return on the i th asset is ds i (t) = S i (t) = i dt + i dz i (1) with expected return and variance i and 2 i. Let F (t) be the net cash out ow per unit time from the portfolio at date t. Dynamic Hedging and PDE Valuation 3/ 36

Portfolio Dynamics in Continuous Time cont d First consider the analogous discrete-time dynamics where each discrete period is of length h. Let w i (t) be the number of shares held by the investor in asset i from date t to t + h. The date t portfolio value is denoted as H (t) and equals the prior period s holdings at date t prices: nx H (t) = w i (t h)s i (t) (2) The net cash out ow over the period is F (t) h which must equal the net sales of assets: nx F (t) h = [w i (t) w i (t h)] S i (t) (3) Dynamic Hedging and PDE Valuation 4/ 36

Portfolio Dynamics in Continuous Time cont d To derive the limits of equations (2) and (3) as of date t and as h! 0, convert backward di erences, such as w i (t) w i (t h), to forward di erences. Update one period: F (t + h) h = = nx [w i (t + h) w i (t)] S i (t + h) nx [w i (t + h) w i (t)] [S i (t + h) S i (t)] + nx [w i (t + h) w i (t)] S i (t) (4) and Dynamic Hedging and PDE Valuation 5/ 36

Portfolio Dynamics in Continuous Time cont d H (t + h) = nx w i (t) S i (t + h) (5) Taking the limits of (4) and (5) as h! 0: F (t) dt = nx dw i (t) ds i (t) + nx dw i (t) S i (t) (6) and H (t) = nx w i (t) S i (t) (7) Dynamic Hedging and PDE Valuation 6/ 36

Portfolio Dynamics in Continuous Time cont d Applying Itô s lemma to (7), the dynamics of the portfolio s value are dh (t) = nx w i (t) ds i (t) + nx dw i (t)s i (t) + Substituting (6) into (8), we obtain nx dw i (t) ds i (t) (8) dh (t) = nx w i (t) ds i (t) F (t) dt (9) Dynamic Hedging and PDE Valuation 7/ 36

Portfolio Dynamics in Continuous Time cont d Thus, the value changes by capital gains income less net cash out ows. Substitute ds i (t) in (1) into (9), dh (t) = = nx w i (t) ds i (t) F (t) dt (10) nx w i (t) [ i S i dt + i S i dz i ] F (t) dt De ne the proportion of H (t) invested in asset i as! i (t) = w i (t)s i (t)=h (t), then (10) becomes Dynamic Hedging and PDE Valuation 8/ 36

Portfolio Dynamics in Continuous Time cont d dh (t) = nx! i (t) H (t) [ i dt + i dz i ] F (t) dt (11) Collecting terms in dt, " nx # dh (t) =! i (t) H (t) i F (t) dt + nx! i (t) H (t) i dz i (12) Note from (7) that P n! i (t) = 1. Adding a riskfree asset that P pays r(t), so that its portfolio proportion is 1 n! i (t), we obtain Dynamic Hedging and PDE Valuation 9/ 36

Portfolio Dynamics in Continuous Time cont d dh (t) = " nx #! i (t) ( i r) H (t) + rh (t) F (t) dt + nx! i (t) H (t) i dz i (13) which is a continuous-time formulation of wealth dynamics. Having derived the dynamics of an arbitrary portfolio, we now consider the Black-Scholes dynamic hedge portfolio that replicates contingent claims. Dynamic Hedging and PDE Valuation 10/ 36

Black-Scholes Model Assumptions Let S(t) be the date t price per share of a stock that follows the di usion process ds = S dt + S dz (14) with time-varying but constant. Let r be the constant rate of return on a risk-free investment B(t): db = rbdt (15) Next, let there be an European call option written on the stock whose date t value is c(s; t). Its maturity value at date T is c(s(t ); T ) = max[ 0; S(T ) X ] (16) Dynamic Hedging and PDE Valuation 11/ 36

Black-Scholes Model Assume that c (S; t) is twice-di erentiable in S and once- in t. Itô s lemma states that the option s value follows the process @c @c dc = S + @S @t + 1 @ 2 c 2 @S 2 2 S 2 dt + @c S dz (17) @S Consider a self- nancing (F (t) = 0 8t), zero net investment portfolio that is short one unit of the call option and hedged with the stock and risk-free asset. Zero net investment implies that the amount invested in the risk-free asset must be B (t) = c (t) w (t) S (t) where w(t) is the number of shares of stock. Thus, the hedge portfolio H(t) has instantaneous return dh (t) = dc(t)+w (t) ds (t)+[c (t) w (t) S (t)] rdt (18) Dynamic Hedging and PDE Valuation 12/ 36

Black-Scholes Model Substituting (14) and (17) into (18), we obtain @c @c dh (t) = S + @S @t + 1 @ 2 c 2 @S 2 2 S 2 dt @c S dz @S +w (t) (S dt + S dz) + [c (t) w (t) S (t)] rdt (19) Set w (t) = @c=@s to hedge the return on the option. Then, @c @c dh (t) = S + @S @t + 1 @ 2 c 2 @S 2 2 S 2 @c dt S dz @S + @c @c (S dt + S dz) + c (t) @S @S S (t) rdt @c 1 = @t 2 2 S 2 @2 c @c + rc (t) rs (t) dt(20) @S 2 @S Dynamic Hedging and PDE Valuation 13/ 36

Black-Scholes Model The portfolio is hedged, so dh(t) is riskfree and must equal r. It is also costless, so H(0) = 0 and dh (0) = rh (0) dt = r 0dt = 0 (21) so H (t) = 0 8t and dh (t) = 0 8t. This implies @c @t + 1 2 2 S 2 @2 c @S 2 + r S @c @S r c = 0 (22) This partial di erential equation has boundary condition c(s(t ); T ) = max[ 0; S(T ) X ] (23) Dynamic Hedging and PDE Valuation 14/ 36

Black-Scholes Formula (1973) The solution to (22) subject to (23) is where c(s(t); t) = S(t) N(d 1 ) X e r (T t) N(d 2 ) (24) d 1 = ln (S(t)=X ) + r + 1 2 2 (T t) p T t d 2 = d 1 p T t (25) and N() is the standard normal distribution function. Similar to the binomial model, (24) does not depend on, but only on S(t) and. Dynamic Hedging and PDE Valuation 15/ 36

Black-Scholes Formula (1973) cont d From put-call parity, the value of a European put is p(s(t); t) = c(s(t); t) + X e r (T t) S(t) (26) = X e r (T t) N( d 2 ) S(t)N( d 1 ) Taking the partial derivatives of (24) and (26) gives the hedge ratios @c @S = N (d 1) (27) @p @S = N ( d 1) (28) which implies 0 < @c=@s < 1 and 1 < @p=@s < 0. Dynamic Hedging and PDE Valuation 16/ 36

Vasicek (1977) Model When the prices of default-free bonds depend on continuous-time stochastic processes, continuous trading and the no-arbitrage condition place restrictions on their prices. We now consider the Vasicek (1977) one-factor term structure model where uncertainty is determined by the yield on the shortest-maturity bond, r(t). De ne P (t; ) as the date t price of a bond that makes a single payment of $1 at date T = t +. The rate of return on the bond is dp(t;) P(t;) and P(t; 0) = $1. The instantaneous-maturity yield, r (t), is de ned as dp (t; ) lim r (t) dt (29)!0 P (t; ) Dynamic Hedging and PDE Valuation 17/ 36

Process for r(t) r (t) is assumed to follow the Ornstein-Uhlenbeck process: dr(t) = [r r (t)] dt + r dz r (30) where, r, and r are positive constants. For r (0) = r = 0:05, = 0:3, and r = 0:02, a typical path is Dynamic Hedging and PDE Valuation 18/ 36

Price Process Assume that bond prices of all maturities depend on only a single source of uncertainty r (t); P(r(t); (t)) where T t. Using Itô s lemma, dp (r; ) = @P @P dr + @r @t dt + 1 @ 2 P 2 @r 2 (dr)2 (31) = P r (r r) + P t + 1 2 P rr 2 r dt + Pr r dz r = p (r; ) P (r; ) dt p () P (r; ) dz r where subscripts on P denote partial derivatives and h P r (r r )+P t + 1 2 Prr 2 r i Pr r p (r; ) P(r ;) and p () P(r ;). Now make a portfolio containing one bond of maturity 1 and p( 1 )P(r ; 1 ) p( 2 )P(r ; 2 ) units of a bond with maturity 2. Dynamic Hedging and PDE Valuation 19/ 36

Hedge Portfolio Since both bond values depend on dz r, the portfolio is hedged if we continually readjust the amount of the 2 -maturity bond to equal p( 1 )P(r ; 1 ) p( 2 )P(r ; 2 ) as r (t) changes. The value of this hedge portfolio, H (t), is H (t) = P (r; 1 ) = P (r; 1 ) p ( 1 ) P (r; 1 ) p ( 2 ) P (r; 2 ) P (r; 2) (32) p ( 1 ) 1 p ( 2 ) and the hedge portfolio s instantaneous return is dh (t) = dp (r; 1 ) p ( 1 ) P (r; 1 ) p ( 2 ) P (r; 2 ) dp (r; 2) (33) Substituting for dp(r; i ) i = 1; 2 from (31): Dynamic Hedging and PDE Valuation 20/ 36

Hedge Portfolio cont d dh (t) = p (r; 1 ) P (r; 1 ) dt p ( 1 ) P (r; 1 ) dz r p ( 1 ) p ( 2 ) P (r; 1) p (r; 2 ) dt + p ( 1 ) P (r; 1 ) dz r = p (r; 1 ) P (r; 1 ) dt p ( 1 ) p ( 2 ) P (r; 1) p (r; 2 ) dt Since the portfolio return is riskless, its rate of return must equal the instantaneous riskless interest rate, r (t): p ( 1 ) dh (t) = p (r; 1 ) p ( 2 ) p (r; 2 ) P (r; 1 ) dt (34) p ( 1 ) = r (t) H (t) dt = r (t) 1 P (r; 1 ) dt p ( 2 ) Dynamic Hedging and PDE Valuation 21/ 36

Bond Risk Premium The second line is from our de nition of H(t) in (32). Equating the two, we get the equality of bond Sharpe ratios: p (r; 1 ) p ( 1 ) r (t) = p (r; 2 ) r (t) p ( 2 ) (35) Condition (35) requires all bonds to have a uniform market price of interest rate risk, as all risk is represented by dz r. Cox, Ingersoll and Ross (1985a,b) derive this price of risk from general equilibrium, but for now we simply assume it is a constant q: p (r; ) r (t) = q (36) p () or p (r; ) = r (t) + q p () (37) Dynamic Hedging and PDE Valuation 22/ 36

Bond Risk Premium cont d Substituting p (r; ) and p () from (33) into (37): P r (r r) + P t + 1 2 P rr 2 r = rp q r P r (38) which can be rewritten as 2 r 2 P rr + (r + q r r) P r rp + P t = 0 (39) Since d = dt, so that P t @P (39) can be rewritten as @t = @P @ P, equation 2 r 2 P rr + [ (r r) + q r ] P r rp P = 0 (40) subject to the boundary condition that at = 0, P (r; 0) = 1. Dynamic Hedging and PDE Valuation 23/ 36

Bond Risk Premium cont d Equation (40) has a solution of the form P (r (t) ; ) = A () e B()r (t) (41) Substituting back into (40) gives ordinary di erential equations for A and B with boundary conditions A ( = 0) = 1 and B ( = 0) = 0 and solutions: B () 1 e A () exp " (B () ) r + q r (42) 1 2 # r 2 r B () 2 2 2 4 (43) Dynamic Hedging and PDE Valuation 24/ 36

Characteristics of Bond Prices Using equation (41) in our de nition of p, we see that p () r P r P = r B () = r 1 e (44) which is an increasing and concave function of. Equation (37), p (r; ) = r (t) + q p (), implies that a bond s expected rate of return increases (decreases) with its time until maturity if the market price of risk, q, is positive (negative). Dynamic Hedging and PDE Valuation 25/ 36

Characteristics of Bond Prices cont d A bond s continuously compounded yield, Y (r (t) ; ), equals Y (r (t) ; ) = = where Y 1 r + q r 1 ln [P (r (t) ; )] 1 B () ln [A ()] + r (t) (45) = Y 1 + [r (t) Y 1 ] B () 1 2 r 2. 2 + 2 r B () 2 4 Note that lim!1 Y (r (t) ; ) = Y 1. Hence, the yield curve, which is the graph of Y (r (t) ; ) as a function of, equals r (t) at = 0 and asymptotes to Y 1 for large. Dynamic Hedging and PDE Valuation 26/ 36

Bond Yield Slopes When r (t) Y 1 = r + q r 4 2 monotonically increasing. 2 r When Y 1 < r (t) < Y 4 2 1 + 2 r curve has a humped shape. 2 r 2 2 3 2 r 4 2, the yield curve is = r + q r, the yield A monotonically downward sloping, or inverted, yield curve occurs when r + q r r (t). Since the yield curve is normally upward sloping, this suggests that r < r + q r 3 2 r, or q > 3r 4 2 4, i.e., a positive market price of bond risk. Dynamic Hedging and PDE Valuation 27/ 36

Option Pricing with Random Interest Rates We now value contingent claims ( rst example) assuming stochastic interest rates (second example) to derive the Merton (1973b) option pricing model (third example). De ne the price of a risk-free bond that pays $X at as P (t; ) X, so the option s value is c (S (t) ; P (t; ) ; t). From Vasicek (1977), this bond s process is dp (t; ) = p (t; ) P (t; ) dt + p () P (t; ) dz p (46) where from equation (31) de ne dz p dz r and assume a bond-stock correlation of dz p dz = dt. Applying Itô s lemma: @c @c dc = S + @S @P pp + @c @t + 1 @ 2 c 2 @S 2 2 S 2 + 1 @ 2 c 2 @P 2 2 pp 2 + @2 c @S@P psp dt + @c @c S dz + @S @P pp dz p (47) Dynamic Hedging and PDE Valuation 28/ 36

Option Pricing with Random Interest Rates cont d = c cdt + @c @c S dz + @S @P pp dz p where c c is de ned as the bracketed terms in (47). Our hedge portfolio is a unit short position in the option, a purchase of w s (t) units of the underlying stock, and a purchase of w p (t) units of the -maturity bond. A zero-net-investment restriction implies c (t) w s (t) S (t) w p (t) P (t; ) = 0 (48) The hedge portfolio s return can then be written as dh (t) = dc(t) + w s (t) ds (t) + w p (t) dp (t; ) (49) Dynamic Hedging and PDE Valuation 29/ 36

Hedge Portfolio with Random Interest Rates = c c + w s (t) S + w p (t) p P dt @c + @S S + w s (t) S dz @c + @P pp + w p (t) p P dz p = w s (t) ( c ) S + w p (t) p c P dt @c + w s (t) Sdz @S + w p (t) @c @P p P dz p Dynamic Hedging and PDE Valuation 30/ 36

Hedge Portfolio with Random Interest Rates cont d If w s (t) and w p (t) are chosen to make the portfolio s return riskless, then from (49) they must equal: w s (t) = @c @S (50) w p (t) = @c (51) @P But from the zero-net-investment condition (48), this can only be possible if it happens to be the case that c = w s (t) S + w p (t) P = S @c @S + P @c @P (52) Dynamic Hedging and PDE Valuation 31/ 36

Hedge Portfolio Dynamics By Euler s theorem, condition (52) holds if the option price is a homogeneous of degree 1 function of S and P. That is, c (ks (t) ; kp (t; ) ; t) = kc (S (t) ; P (t; ) ; t). If so, then no-arbitrage implies dh(t) = 0: w s (t) ( c ) S + w p (t) p c P = 0 (53) or @c @S ( c ) S + @c @P p c P = 0 (54) which, using (52), can be rewritten as @c @c S + @S @P pp c c = 0 (55) Dynamic Hedging and PDE Valuation 32/ 36

Hedge Portfolio Dynamics cont d Substituting for c c from (47), we obtain @c @t 1 @ 2 c 2 @S 2 2 S 2 1 @ 2 c 2 @P 2 2 pp 2 which, since T 1 2 t, can also be written as @ 2 c @S 2 2 S 2 + @2 c @P 2 2 pp 2 + 2 @2 c @S@P psp @ 2 c @S@P psp = 0 (56) The boundary conditions are c (S (T ) ; P (T ; 0) ; T ) = c (S (T ) ; 1; T ) = max [S (T ) X ; 0] where P (t = T ; = 0) = 1. The Merton (1973) solution is @c @ = 0 (57) Dynamic Hedging and PDE Valuation 33/ 36

Merton PDE Solution where where c (S (t) ; P (t; ) ; ) = S(t) N(h 1 ) P (t; ) XN(h 2 ) (58) v 2 = h 1 = ln h 2 = h 1 v Z 0 S(t) P(t;)X 2 + p (y) 2 v + 1 2 v 2 (59) 2 p (y) dy (60) This is the Black-Scholes equation with v 2 replacing 2. v 2 S(t) is the total variance of P(t;)X from date t to date T, an interval of periods. Dynamic Hedging and PDE Valuation 34/ 36

Merton PDE Solution cont d If the bond s volatility is assumed to be that of the Vasicek model, p (y) = r (1 e y ), then (60) is R v 2 = 2 + 2 r 2 1 2e y + e 2y 2 r 1 e y dy 0 = 2 + 2 r 3 + 1 e 2 2 2 1 e 2 r 2 1 e (61) Finally, note that the solution is homogeneous of degree 1 in S (t) and P (t; ), which veri es condition (52). Dynamic Hedging and PDE Valuation 35/ 36

Summary When an underlying asset follows a di usion and trade is can occur continuously, a portfolio can be created that fully hedges the risk of a contingent claim. In the absence of arbitrage, this hedge portfolio s return must equal the riskless rate, which implies an equilibrium partial di erential equation for the contingent claim s value. This Black-Scholes-Merton hedging argument can derive values of options and determine a term structure of default-free interest rates. Dynamic Hedging and PDE Valuation 36/ 36