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- 1. Tommaso Gabbriellini Siena, 20 Maggio 2011
- 2. Very basic recap The Black&Scholes model assumes a market in which the tradable assets are: - A risky asset, whose evolution is driven by a geometric brownian motion - the money market account, whose evolution is deterministic
- 3. Valuing a derivative contract A derivative can be perfectly replicated by means of a self-financing dynamic portfolio whose value exactly matches all of the derivative flows in every state of the world. This approach shows that the values of the derivative (and of the portofolio) solves the following PDE where the terminal condition at T is the derivative’s payoff.
- 4. There exists a very important result: the Feynman-Kac theorem. It mathematically states the equivalence between the solution of this PDE and an expectation value. If f(t0,S(t0)) solves the B&S PDE, then it is also solution of i.e. it’s the expected value of the discounted payoff in a probability measure where the evolution of the asset is This probability measure is the Risk Neutral Measure
- 5. Since there exist such an equivalence, we can compute option prices by means of two numerical methods PDE: finite difference (explicit, implicit, crank-nicholson) suitable for optimal exercise derivatives Quadrature methods Integration Monte Carlo Methods suitable for path dependent options
- 6. Let’s recall the European Call/Put option: It’s a derivative contract in which the holder of the option has the right to buy/sell the asset at expiry at a fixed price (the strike). The price at time t can be computed as where the expectation is taken in the risk neutral probability.
- 7. The american version of the put option gives the holder the right to exercise it any time before the expiration date. Will there be cases in which it is convenient to “early” exercise the option? Yes. Here’s a case. Imagine you bought an american put and at t1 the stock drops to zero, with no chance to ever going back to a strictly positive value (like in the Black&Scholes model)
- 8. The holder, at t1, wonders if it is worth exercising the option. K = 10 Which is the optimal strategy? I have to compare the values of the possibilities 1. The option is exercised at t1, the holder gets K t1 T 2. The option is exercised later, suppose at maturity, the value is It’s convenient to exercise at t1!
- 9. What about american call option? Will there be cases in which it is convenient to “early” exercise the option? Well, it depends on dividends. Imagine you bought an american call and at t1 the stock goes so high that the probability to finish out of the money at expiry is negligible (S >> K)
- 10. No dividends The holder, at t1, wonders if it is worth exercising the option. Which is the optimal strategy? I have to compare the values of the possibilities 1. The option is exercised at t1, the holder gets S(t1) - K 2. The option is exercised later, suppose at maturity, the value is approximately (remember the assumption) K = 10 t1 T It’s better to wait!
- 11. With dividends With dividends things are different. As in the previous example, but now the stock pays a dividend yield q: The holder, at t1, wonders if it is worth exercising the option. Which is the optimal strategy? I have to compare the values of the possibilities 1. The option is exercised at t1, the holder gets S(t1) - K K = 10 t1 T 2. The option is exercised later, suppose at maturity, the value is approximately (remember the assumption) It might be better to exercise
- 12. The bermudan option is similar to an american option, except that it can be early exercised once only on a specific set of dates In the graph Put at strike K, maturity 6 years, and each year you can choose whether to exercise or wait. t1 t2 t3 t4 t5 T
- 13. Let’s consider a simple example: a put option which can be exercised early only once. t1 T
- 14. Can we price this product by means of a Monte Carlo? Yes, let’s see how. Let’s implement a MC which actually simulates, besides the evolution of the market, what an investor holding this option would do (clearly an investor who lives in the risk neutral world). In the following example we will assume the following data S(t) = 100, K = 100, r = 5%, = 20%, t1 = 1y, T = 2y
- 15. 1. We simulate that 1y has passed, computing the new value of the asset and the new value of the money market account 2. At this point, I (the investor) could exercise. How do I know if it’s convenient? In case of exercise I know exactly the payoff I’m getting. In case I continue, I know that it is the same of having a European put option.
- 16. In mathematical terms I have the payoff in t1 is Where P(t1, T, S(t1), K) is the price of a put (which I can compute analytically!) In the jargon of american products, P is referred to the continuation value, i.e. the value of holding the option instead of early exercising it. So the premium of the option is the average of this discounted payoff calculated in each iteration of the monte carlo procedure
- 17. I could have priced this product because I have an analytical pricing formula for the put. What if I didn’t have it? Brute force solution: for each realization of S(t1) I run another Monte Carlo to price the put. This method (called Nested Monte Carlo) is very time consuming. For this very simple case it’s time of execution grows with N2… which becomes prohibitive when you deal with more than one exercise date!
- 18. A finer solution For each realization of S(t1) I go on with the following step simulating S(T) t0 1 2 3 4 5 6 7 8 9 10 t1 T 100 94.08641 68.09733 100 87.59017 102.2035 100 131.1194 121.3294 100 112.1032 98.53462 100 81.33602 98.15437 100 212.0479 206.7438 100 118.9995 110.1571 100 77.46154 56.85677 100 164.9462 160.4879 100 91.20603 71.07494 For each path compute at time t1 the discounted payoff given the value S(T) i.e. t0 1 2 3 4 5 6 7 8 9 10 t1 T disc .payoff 100 94.08641 68.09733 30.34675427 100 87.59017 102.2035 0 100 131.1194 121.3294 0 100 112.1032 98.53462 1.393911351 100 81.33602 98.15437 1.755616844 100 212.0479 206.7438 0 100 118.9995 110.1571 0 100 77.46154 56.85677 41.03910966 100 164.9462 160.4879 0 100 91.20603 71.07494 27.51436479
- 19. Plot the discounted payoff Pi versus Si(t1) (as an example, by means of the scatter plot in excel) t1 disc .payoff 94.08641 5.62518072 87.59017 11.80459474 131.1194 0 112.1032 0 81.33602 17.75372602 212.0479 0 118.9995 0 77.46154 21.43924169 164.9462 0 91.20603 8.365080751
- 20. On this plot, add the analytical price of the put as a function of Si(t1) t1 disc .payoff put 94.08641 5.62518072 8.06533295 87.59017 11.80459474 11.65149926 131.1194 0 0.511529034 112.1032 0 2.383906432 81.33602 17.75372602 15.95968568 212.0479 0 0.00019481 118.9995 0 1.399417905 77.46154 21.43924169 19.0155111 164.9462 0 0.022265243 91.20603 8.365080751 9.542377625
- 21. The analytical price of the put is a curve which kinds of interpolate the cloud of monte carlo points. Observation. Today the price can be computed by means of an average on all discounted payoff (i.e. the barycentre of the cloud made of discounted payoffs) Maybe…. The future value of an option can be seen as the problem of finding the curve that best fits the cloud of dicounted payoffs (up to the date of interest)???
- 22. Below there’s a curve found by means of a linear regression on a polynomial of 4° order.
- 23. We now have a pricing formula for the put to be used in my MC: The formula is obviously fast: the cost of this algorithm is performing the best fit Please note that I could have used any form for my curve (non only a polynomial). This method has the advantage that it can be solved as a linear regression, which is fast.
- 24. Let’s consider now a generic bermudan option Here’s the Longstaff-Schwarz algoritm 1. Generate the MC trajectories of the underlying up to maturity 100 72.31062 81.05736 96.04066 90.91403 68.04453 66.75914 2. Compute the payoff at maturity and discount it to the previous exercise date 100 72.31062 81.05736 96.04066 90.91403 68.04453 31.61969 3. Regress the last column as a function of the previous one, compute the continuation value for each path and calculate what you would get from exercise. Continuation value Exercise 100 72.31062 81.05736 96.04066 90.91403 20.1 31.95547 31.61969
- 25. 4. Compare those two numbers. In this particular path the payoff in case of exercise is greater than the continuation value. Exercise it and go to next step and discount the payoff. Continuation value Exercise 100 72.31062 81.05736 96.04066 90.91403 20.1 30.39698 5. As in step 3, compute the continuation value and the payoff in case of exercise Continuation value Exercise 100 72.31062 81.05736 96.04066 14.5 9.085973 30.39698 6. Now the continuation value is greater. Don’t exercise: the payoff value is replaced with the discounted adjacent number (more on this in next slide) Continuation value Exercise Continuation value Exercise 14.5 9.085973 100 72.31062 81.05736 96.04066 100 72.31062 81.05736 14.5 96.04066 28.9145 30.39698 30.39698
- 26. Theoretically we should have done this Continuation value Exercise 100 72.31062 81.05736 96.04066 Continuation value Exercise 100 72.31062 81.05736 96.04066 14.5 9.085973 30.39698 14.5 This is correct, but it is generally less accurate because the continuation value provided by the interpolating function is accurate only in a region close to the exercise boundary. That’s why it is used the previous step. 7. Ok, iterate till you get the price!
- 27. Recall that pricing a derivative means solving a backward partial differential equation i.e. starting from the payoff, and proceeding backward in time, you compute at each time and for each value of S the option value. Did I say option value? Well, I could have said continuation value Therefore I can naturally price american/bermudan products
- 28. Longstaff-Schwarz method is thus a way to introduce backward evaluation in a Monte Carlo approach (which is naturally forward looking)

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