Insurance against Market Crashes Hongzhong Zhang a Tim Leung a Olympia Hadjiliadis b a Columbia University b The City University of New York June 29, 2012 H. Zhang, T. Leung, O. Hadjiliadis (Columbia Insurance University, against Market The City Crashes University of New York) June 29, 2012 1 / 26
Outline 1 Motivation 2 Mathematical Formalism 3 Insurance claims Drawdown insurance Cancellable drawdown insurance Drawdown insurance contingent on drawups 4 Reference H. Zhang, T. Leung, O. Hadjiliadis (Columbia Insurance University, against Market The City Crashes University of New York) June 29, 2012 2 / 26
Motivation Market Turbulence 1600 1400 S&P500, 2007 Now Historical High SP500 Historical Low 1200 1000 800 600 01 Jan 2007 19 Dec 2007 05 Dec 2008 22 Nov 2009 How to insure? How much is the insurance? H. Zhang, T. Leung, O. Hadjiliadis (Columbia Insurance University, against Market The City Crashes University of New York) June 29, 2012 3 / 26
Mathematical Formalism Setup Filtered probability space (Ω, F, {F t } t 0, Q) with filtration F = {F t } t 0 Price/value process: {S t } t 0 : ds t S t = rdt + σdw t Log price/value process: {X t } t 0 where X t = log S t and x = X 0 ) Drawdown process: D t = X t X t, where X t = x (sup s [0,t] X s Drawup process: U t = X t X t, where X t = x ( ) inf s [0,t] X s Reference period s low & high: x x < x + k First hitting times of the drawdown/drawup process: τ D (k) = inf{t 0 D t k} τ U (k) = inf{t 0 U t k} A market crash is modeled as τ D (k)! H. Zhang, T. Leung, O. Hadjiliadis (Columbia Insurance University, against Market The City Crashes University of New York) June 29, 2012 4 / 26
Mathematical Formalism A snapshot of log S&P index 7.25 7.2 Initial drawdown y Log price 7.15 7.1 7.05 7 01 Jul 2011 01 Aug 2011 01 Sep 2011 01 Oct 2011 01 Nov 2011 Figure: July of 2011 is the reference period. x = 7.21 and x = 7.16. Initial drawdown y = D 0 = 0.05. The large drawdown in August is due to the downgrade of US debt by S&P. H. Zhang, T. Leung, O. Hadjiliadis (Columbia Insurance University, against Market The City Crashes University of New York) June 29, 2012 5 / 26
Insurance claims against a market crash Let r 0 be the risk-free interest rate, Q the risk-neutral measure Drawdown insurance: time-0 value is (seen from the protection buyer) { τd V 0 (p) = E Q (k) T } pe rt dt + αe rτd(k) I {τd (k) T } 0 Ways to terminate a drawdown insurance when necessary Callable drawdown insurance: the time-0 value is (seen from the protection buyer, τ is the cancelation time) V0 c (p) = sup E { Q 0 τ<t τd (k) τ 0 pe rt dt+αe rτ D(k) I {τd (k) τ} ce rτ I {τ<τd (k)} Drawdown insurance contingent on drawups: time-0 value is (seen from the protection buyer) { τd (k) τ V0 U (p) = E U (k) T } Q pe rt dt + αe rτd(k) I {τd (k) τ U (k) T } 0 H. Zhang, T. Leung, O. Hadjiliadis (Columbia Insurance University, against Market The City Crashes University of New York) June 29, 2012 6 / 26 }
Drawdown insurance Fair evaluation The premium is the rate P such that the time-0 value of a insurance is zero: v 0 (P ) = 0 Value calculation: { τd v 0 (p) =E Q (k) T } pe rt dt + αe rτd(k) I {τd (k) T } 0 {( ) p =E Q r + αi {τ D (k) T } e r(τ D(k) T ) p } r If perpetual T =, then v 0 (p) = p r ( α + p r ) ξ(d 0 ) := f (D 0, p) where ξ(y) = E Q {e rτ D(k) D 0 = y} H. Zhang, T. Leung, O. Hadjiliadis (Columbia Insurance University, against Market The City Crashes University of New York) June 29, 2012 7 / 26
Drawdown insurance The conditional Laplace transform ξ(y) For µ = r 1 2 σ2, 0 y 1, y 2 < k, Ξ r µ,σ = 2r σ 2 + µ2 σ 4 The quantity ξ(y) = E Q {e r(τ D(k)) D 0 = y} satisfies functional equation: ξ(y 2 ) = e µ σ 2 (y 2 k) sinh(ξr µ,σ(y 2 y 1 )) sinh(ξ r µ,σ(k y 1 )) + e µ σ 2 (y 2 y 1 ) sinh(ξr µ,σ(k y 2 )) sinh(ξ r µ,σ(k y 1 )) ξ(y 1) Equivalently, Λ(y 2 ) λ(y 1 ) = e µk σ 2 sinh(ξ r µ,σ(y 2 y 1 )) sinh(ξ r µ,σ(k y 1 )) sinh(ξ r µ,σ(k y 2 )) ξ(0) is calculated by H. Taylor 1975. H. Zhang, T. Leung, O. Hadjiliadis (Columbia Insurance University, against Market The City Crashes University of New York) June 29, 2012 8 / 26
Drawdown insurance More properties of ξ(y) ξ(y) is increasing over [0, k]: continuity of path and Markov property Neumann condition at 0: ξ (0) = 0 ODE: Feymann-Kac 1 2 σ2 ξ (y) µξ = rξ(y) ξ(y) is strictly convex, i.e., ξ (y) > 0 for all y (0, k) H. Zhang, T. Leung, O. Hadjiliadis (Columbia Insurance University, against Market The City Crashes University of New York) June 29, 2012 9 / 26
Cancellable drawdown insurance Pricing a cancellable drawdown insurance Callable drawdown insurance: recall that the time-0 value is (seen from the protection buyer, τ is the cancellation time, c is the cancellation fee) V c 0 (p) = sup E { Q 0 τ<t τd (k) τ 0 pe rt dt+αe rτ D(k) I {τd (k) τ} ce rτ I {τ<τd (k)} To find the fair premium p, we need to first solve the above optimal stopping problem to find the value function V0 c (p), and then solve for P in V0 c (P ) = 0 } H. Zhang, T. Leung, O. Hadjiliadis (Columbia Insurance University, against Market The City Crashes University of New York) June 29, 2012 10 / 26
Cancellable drawdown insurance Premium of cancellation To avoid unnecessary complications, we consider perpetual insurances, i.e., T = Notice that, for any cancellation time τ < τ D (k), = + τd (k) τ 0 τd (k) 0 τd (k) pe rt dt + αe rτ D(k) I {τd (k) τ} ce rt I {τ<τd (k)} pe rt dt + αe rτ D(k) pe rt dt ce rτ I {τ<τd (k)} αe rτd(k) I {τ<τd (k)} τ D (k) τ } {{ } Extra premium from cancellation Let V 0 (p) be the time-0 value of a perpetual drawdown insurance, then necessarily, V 0 (p) V c 0 (p). H. Zhang, T. Leung, O. Hadjiliadis (Columbia Insurance University, against Market The City Crashes University of New York) June 29, 2012 11 / 26
Cancellable drawdown insurance The cancellable drawdown insurance as an American call type contract Recall that: V 0 (p) = f (D 0, p), where f (y) := p r ( α + p r ) ξ(y) The value function of the cancellable drawdown insurance can also be computed: V0 c (p) = V 0(p) + sup E Q {e rτ (f (D τ ) c)}, S = {τ 0 τ < τ D (k)} τ S Since ξ( ) is increasing, f ( ) is decreasing. To avoid trivial optiomal cancellation strategy (τ ), it is necessary to have f (0) > 0. In other words, r(c + αξ(0)) Cond :p > 0 1 ξ(0) Under condition Cond, we seek the optimal exercise time: sup τ S E Q {e rτ f (D τ )I {τ<τd (k)}}, with f = f c H. Zhang, T. Leung, O. Hadjiliadis (Columbia Insurance University, against Market The City Crashes University of New York) June 29, 2012 12 / 26
Cancellable drawdown insurance Method of solution Conjecture a stopping time of the form τ θ := τ D (θ) τ D(k) S, where τ D (θ) = inf{t 0 D t θ}, 0 < θ < k We seek a θ through smooth pasting y E Q {e rτθ f (Dτ θ)i {τ θ <τ D (k)} D 0 = y} = f (θ) y=θ Let V (θ, y) = E Q {e rτθ f (Dτ θ )I {τ θ <τ D (k)} D 0 = y}, show that {e r(t τ D(k)) V (θ, D t τd (k))} t 0 is the smallest supermartingale dominating { f (D t τd (k))} t 0 Verify the cancellation strategy based on θ is indeed optimal H. Zhang, T. Leung, O. Hadjiliadis (Columbia Insurance University, against Market The City Crashes University of New York) June 29, 2012 13 / 26
Cancellable drawdown insurance Smooth pasting Figure: Model parameters: r = 2%, p = P = 1.5245, σ = 30%, k = 30%, α = 1, c = 0.05 and D 0 = 10%. The intrinsic function f ( ) is shown in red dash line, the optimal extra premium from cancellation is shown in blue solid line. The only point determined by smooth pasting is θ 5% H. Zhang, T. Leung, O. Hadjiliadis (Columbia Insurance University, against Market The City Crashes University of New York) June 29, 2012 14 / 26
Cancellable drawdown insurance Theorem Under the proposed model, there exists a unique solution θ (0, θ 0 ) to equation y E Q {e rτθ f (Dτ θ)i {τ θ <τ D (k)} D 0 = y} = f (θ). y=θ Moreover, for any θ (θ, k), E Q {e rτθ f (Dτ θ )I {τ θ <τ D (k)} D 0 = θ} > f (θ) Here θ 0 (0, k) is the unique root to equation f (θ) = 0. Mean value theorem implies existence Uniqueness: We use properties of Λ( ) and representation f (θ) = (α + p r )(ξ(θ 0) ξ(θ)) to prove it. The last result in the theorem asserts that {e r(t τ D(k)) V (θ, D t τd (k))} t 0 is the smallest supermartingale dominating {f (D t τd (k))} t 0 H. Zhang, T. Leung, O. Hadjiliadis (Columbia Insurance University, against Market The City Crashes University of New York) June 29, 2012 15 / 26
Cancellable drawdown insurance Determine the fair premium implicitly If f (0) 0, the fair premium is obtained from V 0 (P ) = 0 If f (0) > 0, the fair premium is obtained from V 0 (P ) + E Q {e rτθ f (D τ θ )I {τ θ <τ D (k)} } = 0 Notice that in this case, θ = θ (P ) depends on P. To see the dependence of P on the size of drawdown k, we plot the value function V0 c on a grid of (k, p), and then find the zero contour. H. Zhang, T. Leung, O. Hadjiliadis (Columbia Insurance University, against Market The City Crashes University of New York) June 29, 2012 16 / 26
Cancellable drawdown insurance The value function and the fair premium p vs. k (B-S model) Figure: The fair premium of the cancelable drawdown insurance decreases with respect to the drawdown strike level k. Model parameters: r = 2%, σ = 30%, α = 1, c = 0.05 and D 0 = 10%. H. Zhang, T. Leung, O. Hadjiliadis (Columbia Insurance University, against Market The City Crashes University of New York) June 29, 2012 17 / 26
Cancellable drawdown insurance Cancellable vs. non-cancellable 0.4 0.35 0.3 Callable drawdown swaps with various fees c=0.01 c=0.03 c=0.07 Non callable The fair premium p * 0.25 0.2 0.15 0.1 0.05 0 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 k Figure: Model parameters: r = 1%, σ = 15%, α = 1, and D 0 = 10%. The fair premium p (c = ) for the non-callable drawdown insurance is shown in red. It is seen that the fair premium P (c) is decreasing in the cancellation fee c. H. Zhang, T. Leung, O. Hadjiliadis (Columbia Insurance University, against Market The City Crashes University of New York) June 29, 2012 18 / 26
Drawdown insurance contingent on drawups The fair premium of drawdown insurances with contingency Recall that V U 0 (p) = ( αi {τd (k) τ U (k) T } + p r For drawdown insurance contingent on drawups: ) E Q {e r(τ D(k) τ U (k) T ) } p r P = rαe Q {e rτ D(k) I {τd (k) τ U (k) T }} 1 E Q {e r(τ D(k) τ U (k) T ) } Examine the dependence of P on interest rate, volatility, maturity and other model parameters. H. Zhang, T. Leung, O. Hadjiliadis (Columbia Insurance University, against Market The City Crashes University of New York) June 29, 2012 19 / 26
Drawdown insurance contingent on drawups Finite time-horizon Using Zhang&Hadjiliadis 10, the following probability can be obtained for drifted Brownian motion X Q{τ D (k) τ U (k) T }, Q{τ D (k) τ U (k) T } Explicit computation of the fair premium P = rα T ( 0 e rt t Q{τ D(k) τ U (k) t} ) dt 1 T ( 0 e rt t Q{τ D(k) τ U (k) t} ) dt H. Zhang, T. Leung, O. Hadjiliadis (Columbia Insurance University, against Market The City Crashes University of New York) June 29, 2012 20 / 26
Drawdown insurance contingent on drawups Large-time and infinite time-horizons For a large time-horizon T, it is known that P (T ) P ( ), as T where P ( ) is the fair premium for perpetual insurance Using Zhang&Hadjiliadis 09, the following Laplace transform can be obtained for a general regular linear diffusion X E Q {e rτ D(k) I {τd (k)<τ U (k)}}, E Q {e r(τ D(k) τ U (k)) } Explicit computation of the fair premium for perpetual drawdown insurance P = P ( ) = rαe Q {e rτ D(k) I {τd (k)<τ U (k)}} 1 E Q {e r(τ D(k) τ U (k)) } H. Zhang, T. Leung, O. Hadjiliadis (Columbia Insurance University, against Market The City Crashes University of New York) June 29, 2012 21 / 26
Drawdown insurance contingent on drawups The fair premium P vs. the size of drawdown k 0.25 0.2 Drawdown swap, various maturity T T=5 T=10 T=20 T= Fair premium p * 0.15 0.1 0.05 0 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 k Figure: Model parameters: r = 1%, σ = 15%, α = 1. The fair premium P ( ) is shown in red. It is seen that P (k, T ) is increasing in T and decreasing in k. H. Zhang, T. Leung, O. Hadjiliadis (Columbia Insurance University, against Market The City Crashes University of New York) June 29, 2012 22 / 26
Drawdown insurance contingent on drawups The fair premium p vs. interest rate r Fair premium p * 0.12 0.1 0.08 0.06 0.04 Drawdown swap, various maturity T T=5 T=10 T=20 T= 0.02 0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Interest rate r Figure: Model parameters: σ = 15%, k = 50%, α = 1. The fair premium P ( ) is shown in red. It is seen that, the fair premium P (r) is eventually decreasing in r. H. Zhang, T. Leung, O. Hadjiliadis (Columbia Insurance University, against Market The City Crashes University of New York) June 29, 2012 23 / 26
Drawdown insurance contingent on drawups The fair premium P vs. volatility σ 0.45 0.4 0.35 T=5 T=10 T=20 T= Drawdown swap, various maturity T Fair premium p * 0.3 0.25 0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 Volatility σ Figure: Model parameters: r = 1%, k = 50%, α = 1. The fair premium P ( ) is shown in red. Like most derivatives, the fair premium P (σ, T ) is increasing both in σ and in T. H. Zhang, T. Leung, O. Hadjiliadis (Columbia Insurance University, against Market The City Crashes University of New York) June 29, 2012 24 / 26
Thank You! H. Zhang, T. Leung, O. Hadjiliadis (Columbia Insurance University, against Market The City Crashes University of New York) June 29, 2012 25 / 26
Reference Carr P., Zhang H., and Hadjiliadis O. Maximum drawdown insurance, IJTAF 2011. Lehoczky J.P. Formulas for Stopped Diffusion Processes with Stopping Times Based on the Maximum, AP 1977. Peskir G., and Shiryaev A. Optimal Stopping and Free-Boundary Problems, 2006. Zhang H., and Hadjiliadis O. Formulas for Laplace transform of stopping times based on drawdowns and drawups, preprint, 2010. Zhang H., and Hadjiliadis O. Drawdowns and rallies in a finite time-horizon, MCAP 2010. Zhang H., and Hadjiliadis O. Drawdowns and the speed of market crash, MCAP 2011. Zhang H., Leung T., and Hadjiliadis O. Insurance contracts for drawdown protection, Submitted 2012. H. Zhang, T. Leung, O. Hadjiliadis (Columbia Insurance University, against Market The City Crashes University of New York) June 29, 2012 26 / 26