CONTINUOUS TIME PRICING AND TRADING: A REVIEW, WITH SOME EXTRA PIECES THE SOURCE OF A PRICE IS ALWAYS A TRADING STRATEGY SPECIAL CASES WHERE TRADING STRATEGY IS INDEPENDENT OF PROBABILITY MEASURE COMPLETENESS, RISK NEUTRAL PRICING including quantile hedging (in the lect 7 case) SUPERREPLICATION American options unknown probability measure known probability measure, incomplete market MEAN-VARIANCE, UTILITY BASED PRICING such as the paper by Schweizer unknown probability measure known probability measure, incomplete market Winter 25 1 Per A. Mykland
THE SIMPLEST CASE Problem: A contract pays the owner 1 share of stock S at the first time t, t T that the share price S t exceeds $ X. If the share price does not exceed X at any time t =,..., T, the contract pays one share of stock S at time T. Find the price of this contract. Solution: To satisfy this contract, you need to buy one share of stock at time zero THIS IS A TRADING STRATEGY, BUT DOES NOT REQUIRE STOCHASTIC CALCULUS IF SUCH A STRATEGY IS AVAILABLE, USE IT! OTHER EXAMPLES prices of forward contracts put call parity for European options Winter 25 2 Per A. Mykland
IN MOST CASES: REDUCE PROBLEM BY NUMERAIRE INVARIANCE S t = discounted price η = discounted payoff at T η can be exactly financed if and only if η = c + T θ t d S t ( ) (Lect 9 last quarter, p. 9-1) c is initial (discounted) price θ t is the delta (independent of numeraire) This is a self financing strategy (SFS) Same principle for multiple securities This does not depend on risk free or actual measure IF YOU FIND SUCH A STRATEGY W/OUT GOING THROUGH THE USUAL MACHINERY: USE IT! Winter 25 3 Per A. Mykland
CLASSICAL EXAMPLE: THE VOLATILITY SWAP d S t = S t µ t dt + S t σ t dw t Ito s formula: log S T = log S + T Or: ( T σ2 t dt = 2 log S T + log S + T 1 S t d S t 1 2 T 1 d S ) t S t σ2 t dt Read directly from this that discounted payoff T σ2 t dt can be replicated by: initial capital: 2 log S + by owning an option with payoff 2 log S T Dynamic hedge: θ t = 2 S t Note: we have not told you which probability distribution, P or P, we are using Winter 25 4 Per A. Mykland
VOLATILITY SWAP, CONTINUED If you wish to replicate actual (not discounted payoff) T σ2 t dt: Suppose discounting by zero coupon bond Λ t, Λ T = 1 The discounted payoff is 1 Λ T σ2 t dt Therefore: same strategy as on previous page, multiplied by 1 Λ WHICH MEANS REPLICATION BY: initial capital: 2 Λ log S + by owning an option with payoff 2 Λ log S T Dynamic hedge: θ t = 2 Λ S t IMPORTANT: THIS IS A HEDGE FOR THE CUMULATIVE VOLATIL- ITY OF THE DISCOUNTED SECURITY (This is the same as the cum. vol. for the original security if r is constant) Winter 25 5 Per A. Mykland
THE USUAL MACHINERY IN MORE COMPLEX CASES, CANNOT READ HEDGE DIRECTLY SIMPLEST APPROACH: COMPLETENESS FROM GE- OMETRIC BROWNIAN MOTION system: ds t = µ t S t dt + σs t dw t and r = constant payoff: η = function of the path of S IN THIS CASE, THE ALGORITHM IS... Define P to be such that ds t = rs t dt + σs t dw t Compute C t = E ( η F t ) The delta is θ t = d[ C, S] t d[ S, S] t This works (gives SFS) because of the martingale representation theorem (p. 11-12 in Lect 9 of last quarter) YOU NEED TO KNOW HOW TO DO THIS FOR MORE EXERCISE, COMPUTE THE ANALYTIC EXPRESSIONS GIVEN IN HULL S BOOK Winter 25 6 Per A. Mykland
A MORE COMPLEX COMPLETE (?) CASE: THE HESTON MODEL ds t = µ t S t dt + v 1/2 t S t dw t dv t = a(b v t )dt + cv 1/2 t db t with d[w, B] t = ρdt and constant interest rate r System is generated by two Brownian motions. possibilities: Two if only S is traded: market is incomplete, need to use methods to this case if one derivative is traded, may be able to complete market with this derivative (need as many securities as you have Brownian motions) But a problem is as follows: under the risk neutral measure ds t = rs t dt + v 1/2 t S t dwt dv t =???dt + cv 1/2 t dbt The market may not be complete under P Winter 25 7 Per A. Mykland
BACK TO THE DRAWING BOARD... Suppose the model is valid under P ds t = rs t dt + v 1/2 t S t dwt dv t = a(b v t )dt + cv 1/2 t db t Suppose for simplicity that ρ = (otherwise numerical solution only). European call payoff η K = (S T K) +. If B( S t, σ 2 (T t)) is Black-Scholes price (for constant σ 2 ), then, since ρ = : E ( η K (v u ) u T, (S u ) u t ) = B( S t, T t v u du) and so the discounted price for payoff η K is C t K = E ( η K F t ) = E (B( S t, = f K ( S t, v t, T t) T t v u du) F t ) which can be calculated if one knows a, b, c Winter 25 8 Per A. Mykland
HOW TO HEDGE IN THIS MODEL Since the system is generated by two Brownian motions under the risk neutral measure: need two securities to hedge, say S t, and Ct X for one strike price X If f K v means f K v ( S t, v t, T t), etc, and since d[ S, v] t = d C K t = f K S d S t + 1 2 f K SSd[ S, S] t + f K v dv t + 1 2 f K vvd[v, v] t f K t dt and the same for C X, so that d C K t d C X t = f K S d S t + f K v dv t + dt-terms = f X S d S t + dv t + dt-terms Express the dv t -term by dv t = 1 d C X t f X S d S t + dt-terms and get d C K t = f K S d S t + f K v ( 1 d C X t f X S d S t ) + dt-terms Winter 25 9 Per A. Mykland
Rearrange: d C K t = ( f K S f K v f X S ) d S t + f K v d C X t + dt-terms However, C K t, S t and C X t are all (local) martingales, so dt-terms = Final self financing strategy: d C K t = ( f K S f K v f X S ) d S t + f K v d C X t in other words: to hedge payoff η K, hold ( ) fs K units of stock S t f K v f X S f K v units of the option with payoff η X Initial starting capital: f( S, v, T ) dollars THE ONLY THING THAT REMAINS IS TO CALCU- LATE THE FUNCTION f Winter 25 1 Per A. Mykland
ANOTHER CASE OF HEDGING IN ADDITIONAL SECURITIES The superreplication in HW 6 THE THREE INVARIANCES SELF FINANCING STRATEGIES ARE INVARIANT UNDER: Change of numeraire Change of measure (so long as absolutely continuous) Change of time Winter 25 11 Per A. Mykland
CHANGE OF TIME Function f : [, T ] [, T ] is a time change if f is increasing f() = and f(t ) = T f ( 1) (t) is a stopping time, for each t Securities on original time scale S t, C t are connected by C t = C + Securities on new time scale t θ u d S u S new t = S f(t) and Cnew t = C f(t) satisfy where C new t = C + θ new t t = θ f(t) θ new u d S new u Winter 25 12 Per A. Mykland
This is because the stochastic integral is a limit of sums: Set u i = f(v i ): C new t C = f(t) u i <f(t) θ u d S u θ ui ( S ui+1 S ui ) = v i <t θ f(vi )( S f(vi+1 ) S f(vi )) = v i <t θ new v i ( S new v i+1 S new v i ) t θ new u d S new u Winter 25 13 Per A. Mykland
IMPORTANT CONSEQUENCE OF INVARIANCE TO TIME CHANGE Suppose that the discounted payoff η satisfies: η = g( S) where g is invariant to time: if s new t = s f(t) for any deterministic time change, then g(s new ) = g(s) Example: European payoffs, barrier and lookback payoffs written on the discounted process, or future THEN: if you know that T σ2 t dt = Ξ, you can price option as if σ 2 t = Ξ/T = constant Technically: time change on the form: f ( 1) (t) = The time changed security is t σ 2 udu d S new t = Ξ T S new t dw new (see p. 4-6 in Lect 5 (part 2)) Winter 25 14 Per A. Mykland
CONVEXITY AND OPTIMAL STOPPING JENSENS INEQUALITY g is a convex function, M t is a martingale, τ 1, τ 2 are stopping times, with τ 1 τ 2 T E(g(M τ2 ) F τ1 ) g(m τ1 ) SEVERAL CONSEQUENCES, SUCH AS APPLICATION 1: If the interest rate is zero, it is never optimal to exercise any convex payoff early. Application to superhedging in Lecture 5. APPLICATION 2: The American option inequality (Lect 5 in Fall (p. 14)) Assume also g(s), g() =, and that exp{ t r udu}s t is a martingale. Then exp{ t r udu}g(s t ) is a submartingale. In particular, it is never optimal to exercise an American call early (when there is no dividend) Winter 25 15 Per A. Mykland