Algorithmic Trading: Option Hedging PIMS Summer School

Transcription

1 Algorithmic Trading: Option Hedging PIMS Summer School Sebastian Jaimungal, U. Toronto Álvaro Cartea,U. Oxford many thanks to José Penalva,(U. Carlos III) Luhui Gan (U. Toronto) Ryan Donnelly (Swiss Finance Institute, EPFL) Damir Kinzebulatov (U. Laval) Jason Ricci (Morgan Stanley) July, 06 (c) S. Jaimungal, 06 Algo Trading July, 06 /

2 Motivation We consider the problem that an agent has written an option and would like to hedge it by trading the underlying asset. We provide a framework for replicating option payoffs using limit and market orders while incorporating transaction cost and price impact. (c) S. Jaimungal, 06 Algo Trading July, 06 /

3 Classical option pricing theory An option payos G(S T ) at a future time T, where S T is the price of the underlying asset. Classical option pricing theory tells us that the value of the option is its expected payoff under a risk-neutral measure Q: V (t, S) = E Q [G(S T ) S t = S]. Hedge position is to hold (t, S t ) units of the underlying asset: (t, S t ) = S V (t, S t ). (c) S. Jaimungal, 06 Algo Trading July, 06 /

4 What s missing here? Bid-ask spread Discrete nature of prices Limit and market orders impact (c) S. Jaimungal, 06 Algo Trading July, 06 /

5 The agent s problem The agent maximizes utility of terminal wealth with the option obligation [ { H(t, x, q, s) = sup E t,x,q,s exp γ (X T +S T Q T G(S T, Q T ))} ], (τ,l ± t ) A G(, ) is the option payoff, e.g., for N calls... G(S T, Q T ) = N (S T K) + + l(q T, N) I ST K + l(q T, 0) I ST <K, where l(q, q ) is the cost over midprice to go from q to q shares (c) S. Jaimungal, 06 Algo Trading July, 06 5 /

6 dynamics Bid-ask spread is always (one tick). The asset s midprice S = (S t ) t 0 with S t = [ (Z t + Zt ) + ], Z ± t are independent Poisson processes with rate θ representing shuffling in the LOB (c) S. Jaimungal, 06 Algo Trading July, 06 6 /

7 Sample price path S t ask bid Figure: Sample path for the underlying asset price. (c) S. Jaimungal, 06 Algo Trading July, 06 7 /

8 The agent s inventory The agent: has inventory is Qt {q,..., q} can execute buy (sell) market orders (M 0± t ) at the best ask (bid) Mid-price jumps up(down) by with probability β can post buy (sell) limit orders (l ± t ) at best bid (ask) dq t = LO fills {}}{ l t dl t l + t dl+ t 0+ + dmt dmt 0. }{{} MO executions (c) S. Jaimungal, 06 Algo Trading July, 06 8 /

9 The agent s inventory Other market participants send MOs according to Poisson processes with rate λ If posted, agent s LO is filled with probability ρ midprice jumps up (down) by with probability α where, S t = S + t M ± t S t ± = Z t ± + i=0 S t Mt 0± ξ i ± + i=0 ξ 0± i, ξ + i, ξ i are iid Bernoulli r.v. prob = α ξ 0± i are iid Bernoulli r.v. prob = β(m 0± t M 0± t ) (c) S. Jaimungal, 06 Algo Trading July, 06 9 /

10 Cash process The agent s cash process is cost of limit buy profit of limit sell {}}{{}}{ dx t = (S t ) l t dl t + (S t + ) l+ t dl + t + (S t Υ) dm 0 t }{{} profit of market sell (S t + Υ) dm 0+ t } {{ } cost of market buy. (c) S. Jaimungal, 06 Algo Trading July, 06 0 /

11 The QVI The QVI associated with the value function is { max ( t + L Z )H(t, x, q, s) { + max λe[h(t, x ζl (s ), q + ζl, s ξ ) H(t, x, s, q)] } l {0,,..., q q} + max { λe[h(t, x + ζl + (s + ), q + ζl+, s + ξ ) H(t, x, s, q)] } l + {0,,...,q q} max m + {,..., q q} max m {,...,q q} { E[H(t, x m + (s + Υ), q + m +, s + ζ m+ ) H(t, x, q, s)] } ; { E[H(t, x + m (s Υ), q m +, s ζ m ) H(t, x, q, s)] } } = 0, where, L Z H(t, x, q, s) = θ [H(t, x, q, s ) + H(t, x, q, s ) H(t, x, q, s)] and ζ, ξ, ζ m are independent Bernoulli r.v.s with success prob α, ρ and β(m) The solution admits the ansatz H(t, x, q, s) = exp{ γ(x + q s + h(t, q, s))} (c) S. Jaimungal, 06 Algo Trading July, 06 /

12 Indifference The indifference price f(t, s) is the compensation that makes the agent indifferent between. Receiving the compensation and delivering the option payoff at maturity.. Not delivering the option payoff. (In this case, the agent can trade to maximize her utility) Hence, also introduce the value function [ H 0(t, x, q, s) = sup (τ,l ± t ) A E t,x,q,s { exp γ (X T + S T Q T )} ], i.e., optimally trade without the option obligation. Then, f(t, s) is such that or equivalently H(t, x + f(t, s), 0, s) = H 0 (t, x, 0, s) f(t, s) = h 0 (t, 0, s) h(t, 0, s) (c) S. Jaimungal, 06 Algo Trading July, 06 /

13 Buy LO q= Sell LO q= MO q= (c) S. Jaimungal, 06 Algo Trading July, 06 /

14 Buy LO q= Sell LO q= MO q= (c) S. Jaimungal, 06 Algo Trading July, 06 /

15 Buy LO q= Sell LO q=.8 MO q= (c) S. Jaimungal, 06 Algo Trading July, 06 5 /

16 Buy LO q= Sell LO q= MO q=.5.5 (c) S. Jaimungal, 06 Algo Trading July, 06 6 /

17 Buy LO q= Sell LO q= MO q= 0 - (c) S. Jaimungal, 06 Algo Trading July, 06 7 /

18 Buy LO q=5 Sell LO q=5 MO q= (c) S. Jaimungal, 06 Algo Trading July, 06 8 /

19 Buy LO q= Sell LO q=6 MO q= (c) S. Jaimungal, 06 Algo Trading July, 06 9 /

20 Buy LO q=7 Sell LO q=7 MO q= (c) S. Jaimungal, 06 Algo Trading July, 06 0 /

21 Buy LO q=8.8 Sell LO q=8 MO q= (c) S. Jaimungal, 06 Algo Trading July, 06 /

22 Buy LO q= Sell LO q=9 MO q= (c) S. Jaimungal, 06 Algo Trading July, 06 /

23 Buy LO q= Sell LO q= MO q= (c) S. Jaimungal, 06 Algo Trading July, 06 /

24 .5.5 Sample Path Inventory (c) S. Jaimungal, 06 Algo Trading July, 06 /

25 .9 Sample Path Inventory (c) S. Jaimungal, 06 Algo Trading July, 06 5 /

26 Sample Path Inventory (c) S. Jaimungal, 06 Algo Trading July, 06 6 /

27 .. Sample Path Inventory (c) S. Jaimungal, 06 Algo Trading July, 06 7 /

28 .. Sample Path Inventory (c) S. Jaimungal, 06 Algo Trading July, 06 8 /

29 Terminal payoff 0 0 Payoff 80 0 Payoff (a) γ = (b) γ = 0. Figure: Terminal payoff. (c) S. Jaimungal, 06 Algo Trading July, 06 9 /

30 Number of orders executed 50 γ = 0. γ = γ = 0. γ = 0. Frequency Frequency Numer of market orders (a) Numer of limit orders (b) Figure: Histogram of number of market/limit orders. (c) S. Jaimungal, 06 Algo Trading July, 06 0 /

31 50 Bachelier price Indifference price 0.6 Option price 50 Volatility Spot price (a) Indifference price Spot (b) Implied volatility (c) S. Jaimungal, 06 Algo Trading July, 06 /

32 Thank you! (c) S. Jaimungal, 06 Algo Trading July, 06 /